The Seventh International Conference on Vibration Problems ICOVP 2005: 05-09 September 2005, Istanbul, Turkey (Springer Proceedings in Physics)

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springer proceedings in physics 90 Computer Simulation Studies in Condensed-Matter Physics XV Editors: D.P. Landau, S.P. Lewis, and H.-B. Schüttler 91 The Dense Interstellar Medium in Galaxies Editors: S. Pfalzner, C. Kramer, C. Straubmeier, and A. Heithausen 92 Beyond the Standard Model 2003 Editor: H.V. Klapdor-Kleingrothaus 93 ISSMGE Experimental Studies Editor: T. Schanz 94 ISSMGE Numerical and Theoretical Approaches Editor: T. Schanz 95 Computer Simulation Studies in Condensed-Matter Physics XVI Editors: D.P. Landau, S.P. Lewis, and H.-B. Schüttler 96 Electromagnetics in a Complex World Editors: I.M. Pinto, V. Galdi, and L.B. Felsen 97 Fields, Networks, Computational Methods and Systems in Modern Electrodynamics A Tribute to Leopold B. Felsen Editors: P. Russer and M. Mongiardo 98 Particle Physics and the Universe Proceedings of the 9th Adriatic Meeting, Sept. 2003, Dubrovnik Editors: J. Trampetić and J. Wess 99 Cosmic Explosions On the 10th Anniversary of SN1993J (IAU Colloquium 192) Editors: J.M. Marcaide and K.W. Weiler 100 Lasers in the Conservation of Artworks LACONA V Proceedings, Osnabrück, Germany, Sept. 15–18, 2003 Editors: K. Dickmann, C. Fotakis, and J.F. Asmus

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101 Progress in Turbulence Editors: J. Peinke, A. Kittel, S. Barth, and M. Oberlack

Volumes 64–89 are listed at the end of the book.

The Seventh International Conference on

Vibration Problems ICOVP 2005 05-09 September 2005, İstanbul, Turkey

Edited by

Esin İnan Işık University, İstanbul, Turkey and

Ahmet Kır ış İstanbul Technical University, İstanbul, Turkey

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-5400-9 (HB) 978-1-4020-5400-6 (HB) 1-4020-5401-7 (e-book) 978-1-4020-5401-3 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

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All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

ICOVP-2005

CONTENTS

Preface

xv

Adanur S, Soyluk K, Dumano˘glu A A, Bayraktar A (RC) Asynchronous and antisynchronous effects of ground motion on the stochastic response of suspension bridges

1

Akkas¸ N (Special Talk) European union framework programme 7 building the Europe of knowledge

7

Akköse M, Adanur S, Bayraktar A, Dumano˘glu A A (RC) Dynamic response of rock-fill dams to asynchronous ground motion

9

Akso˘gan O, Choo B S, Bikçe M, Emsen E, Res¸ato˘glu R (RC) A comparative study on the dynamic analysis of multi-bay stiffened coupled shear walls with semi-rigid connections

15

˘ O, Sofiyev A H, Sofiyev A (RC) Aksogan Free vibrations of cross-ply laminated non-homogeneous composite truncated conical shells

21

Akyüz U, Ertepınar A (RC) Symmetric and asymmetric vibrations of cylindrical shells

27

¨ Aydın E (RC) Aldemir U, An active control algorithm to prevent the pounding of adjacent structures ¨ Güney D (RC) Aldemir U, Vibration control of non-linear buildings under seismic loads

33 39

¨ (RC) Aslanyan A G, Dronka J, Mishuris G S, Selsil O Vibrations of damaged 1D-3D multi-structures

45

¨ (RC) As¸çı N, Uysal H, Uzman U Sizing of a spherical shell as variable thickness under dynamic loads

51

v ii

viii

CONTENTS

Aydo˘gdu M, Tas¸kın V (RC) Vibration analysis of simply supported functionally graded beams

57

Baltacı A, Sarıkanat M, Yıldız H (RC) Damping effects of rubber layer in laminated composite circular plate during forced vibration

63

Banerjee M M (GL) A critical study on the application of constant deflection contour method to nonlinear vibration of plates of arbitrary shapes

69

Banks S P, Salamcı M U (RC) Nonlinear wave equations and boundary control using visco elastic dampers

79

¨ C (RC) Birlik G, Sezgin O Effect of vibrations on transportation system

85

Bonifasi-Lista C, Cherkaev E (RC), (presented by Cherkaev A) Identification of bone microstructure from effective complex modulus

91

Boyacı H (RC) Beam vibrations with non-ideal boundary conditions

97

Chakrabarti B K, Chatterjee A (GL) A two-fractal overlap model of earthquakes

103

Cherkaev A, Cherkaev E, Slepyan L (GL) Dynamics of structures with bistable links

111

Çelebi M (GL) Real-time seismic monitoring of the new Cape Girardeau (MO) bridge and recorded earthquake response

123

Demir F, Türkmen M, Tekeli H, C ¸ ırak ˙I (RC) Earthquake response of masonry infilled precast concrete structures

137

Demiray H (GL) Nonlinear waves in fluid-filled elastic tubes: a model to large arteries

143

Do˘gan V (RC) The nonlinear axisymmetric vibrations of circular plates with linearly varying thickness under random excitation

151

Do˘gan V, Kırca M (RC) Dynamic analysis of a helicopter rotor by Dymore program

157

CONTENTS

ix

Ecker H (GL) Parametric stiffness excitation as a means for vibration suppression

163

¨ (RC) Erdem A U A formulation of magnetic-spin and elasto-plastic waves in saturated ferrimagnetic media

175

Erdik M, Apaydın N (GL) Earthquake response of suspension bridges

181

Fedorov V A, Smolskiy S M, Mizirin A V, Shtykov V V, Fomenkov A V, Kaplunov S M (RC, not presented) The microwave sensor of small moving for the active control of vibrations and chaotic oscillations modes

195

Fesenko T N, Foursov V N (RC, not presented) Forced oscillations of tube bundles in liquid cross-flow

205

Filippenko G V, Kouzov D P (RC) The exact and approximate models for the vibrating plate partially submerged into a liquid

213

¨ ¸ a M (RC) Gög˘ u¨ s¸ M T, Tays¸i N , Ozakc A computational tool based on genetic algorithm for determining optimum shapes of vibrating planar and space trusses

219

Güler C, Akbarov S D (RC) On the dynamical stress field in the pre-stretched bilayered strip resting on the rigid foundation

225

Güler K, Celep Z (RC) Dynamic response of a rectangular plate-column system on a tensionless elastic foundation

231

˙Itik M, Salamcı M U, Ulker ¨ F D (RC) Vibration suppression of an elastic beam via sliding mode control

237

Ivanova J, Bontcheva N, Pastrone F, Bonadies M (RC) Thermoelastic stress analysis for linear elastic bodies

243

x

CONTENTS

Kan C, Urey H, S¸enocak E (RC) Discrete and continuous mathematical models for torsional vibration of micromechanical scanners

249

Kaplunov S M, Makhutov N A, Solonin V I, Shariy N V (RC, not presented) Physical modeling of stationary and impulsive processes for large255 scaled constructions of fluid elastic systems Karmakar B, Biswas P, Kahali R, Karanji S (RC) Thermal stresses and nonlinear thermal deformation analysis of shallow shell panel

265

Karmakar B, Karanji S B, Kahali R, Biswas P (RC) Nonlinear thermal vibrations of a circular plate under elevated temperature

271

¨ ¨ (RC) Kaya M O, Ozdemir O Flexural-torsional coupled vibration analysis of a thin-walled closed section composite Timoshenko beam by using the differential transform method

279

Kırıs¸ A, ˙Inan E (RC) Estimation of microstretch elastic moduli by the use of vibrational data ¨ Parlak I˙ B (RC) Korfalı O, Vibrations of a circular membrane subjected to a pulse

285 291

Kumbasar N (RC) A method of discrete time integration using Betti’s reciprocal theorem

297

Mehta A (GL) Bridges in vibrated granular media

305

Michelitsch T M, Askes H, Wang J, Levin V M (RC) Solutions for dynamic variants of Eshelby’s inclusion problem

317

Mondal S (RC) Note on large deflections of clamped elliptic plates under uniform load

323

Movchan A B, Haq S, Movchan N V (GL) Localised defect modes and a macro-cell analysis for dynamic lattice structures with defects

327

CONTENTS

xi

Mukherjee A, Samadhiya R (RC) Acoustic wave propagation through discretely graded materials

337

Nordmann R (GL) Use of mechatronic components in rotating machinery

345

Okrouhl´ık M (GL) Computational limits of FE transient analysis

357

Oskouei A V, Akso˘gan O (RC) The effect of non-linear behavior of concrete on the seismic response of concrete gravity dams

371

¨ ¨ Onbas¸lı U ¨ (RC) Ozdemir Z G, Aslan O, Calculation of microwave plasma oscillations in high temperature superconductors

377

¨ ¨ ˙Inan E (RC) Ozer A O, One-dimensional wave propagation problem in a nonlocal finite medium with finite difference method

383

¨ Ozer M, Alıs¸veris¸çi F (RC) Dynamic response analysis of rocking rigid blocks subjected to halfsine pulse type base excitations

389

¨ Ozeren M S, Postacıo˘glu N, Zora B (RC) A new spectral algorithm for 3-D wave field in deep water

395

◦

Pe˘sek L, Pust L (RC) Experimental and numerical assessment of vibro-acoustic behavior of rubber-damped railway wheels

403

Raamachandran J (GL) Charge simulation method applied to vibration problems

409

Singh S, Patel B P, Sharma A, Shukla K K, Nath Y (GL) Nonlinear stability and dynamics of laminated composite plates and shells ˘ Skliba J, Siv˘ca k M, Skarolek A (RC)

415

On the stability of a vibroisolation system with more degrees of freedom

429

xii

CONTENTS

Sofiyev A H, Deniz A (RC) The stability and vibration of conical shells composed of SI3 N4 and SUS304 under axial compressive load Sofiyev A, Schnack E (RC) Dynamic buckling of elasto-plastic cylindrical shells under axial load ¨ (RC) Sönmez U Dynamic buckling analysis of imperfect elastica

437

443 449

˘ Svoboda R, Skliba J, Mat˘ejec R (RC) Specification of flow conditions in the mathematical model of hydraulic damper

455

Tanrıöver H, S¸enocak E (RC) Nonlinear transient analysis of rectangular composite plates

463

Tas¸çı F, Emiro˘glu ˙I, Akbarov S D (RC) On the “resonance” values of the dynamical stress in the system comprises two-axially pre-stretched layer and half-space

469

¨ Tays¸i N, Gög˘ u¨ s¸ M T, Ozakc ¸ a M (RC) Optimization of vibrating arches based on genetic algorithm

475

Teymür M (RC) Propagation of long extensional nonlinear waves in a hyper-elastic layer

481

Tondl A, Nabergoj R, Ecker H (RC) Quenching of self-excited vibrations in a system with two unstable vibration modes ¨ (RC) Topal U, Uzman U

487

Free vibration analysis of laminated plates using first-order shear deformation theory

493

¨ (RC) Turhan O The generalized Bolotin method as an alternative tool for complete dynamic stability analysis of parametrically excited systems: application examples

499

¨ Bulut G (RC) Turhan O, Coupling effects between shaft-torsion and blade-bending vibrations in rotor-blade systems

505

CONTENTS

xiii

Yahnio˘glu N, Akbarov S D (RC) Forced vibration of the pre-stretched simply supported strip containing two neighbouring circular holes

511

Yavuz M, Ergüven M E (RC) Free vibration of curved layered composite beams

519

Yeliseyev V V, Zinovieva T V (RC) Vibrations of beam constructions submerged into a liquid

525

Yıldırım V (RC) Vibration behavior of composite beams with rectangular sections considering the different shear correction factors

531

Yüksel H M, Türkmen H S (RC) Air blast-induced vibration of a laminated spherical shell

537

Author Index

543

List of Participants

547

The names of the authors who actually delivered the lectures, talk, or communications at the conference are underlined. Abbreviations: GL = general lectures, RC = research communications

PREFACE

The Seventh International Conference on Vibration Problems (ICOVP-2005) took place in S¸ile Campus of Is¸ık University, ˙Istanbul, Turkey, between the dates 59 September 2005. First ICOVP was held during October 27-30, 1990 at A.C. College, Jalpaiguri under the co-Chairmanship of two scientists, namely, Professor M. M. Banerjee from the host Institution and Professor P. Biswas from the sister organization, A. C. College of Commerce, in the name of “International Conference on Vibration Problems of Mathematics and Physics”. The title of the Conference was changed to the present one during the third conference. The Conferences of these series are: 1. ICOVP-1990, 20-23 October-1990, A.C. College, Jalpaiguri- India 2. ICOVP-1993, 4-7 November 1993, A.C. College, Jalpaiguri- India 3. ICOVP-1996, 27-29 November 1996, University of North Bengal, India 4. ICOVP-1999, 27-30 November 1999, Jadavpur University, West Bangal, India 5. ICOVP-2001, 8-10 October 2001, (IMASH), Moscow, Russia 6. ICOVP-2003, 8-12 September 2003, Tech. Univ. of Liberec, Czech Republic 7. ICOVP-2005, 5-9 September 2005, Is¸ık University, S¸ile, ˙Istanbul, Turkey The General Lecturers of ICOVP-2005 have been personally invited by the International Scientific Committee, which this time comprised the following members: ˘ ¨ (Turkey), Orhan AKSOGAN Nuri AKKAS¸ (Turkey), Yal¸cın AKOZ (Turkey), Fikret BALTA (Turkey), M. M. BANERJEE (India), Victor BIRMAN (USA), ˙ Paritosh BISWAS (India), Bikas K. CHAKRABARTI (India), Hilmi DEMIRAY ¨ ¨ (Turkey), Ali Unal ERDEM (Turkey), Ragıp ERDOL (Turkey), Aybar ERTEPINAR (Turkey), K. V. FROLOV (Rusia), Avadis HACINLIYAN (Turkey), xv

xvi

PREFACE

¨ ˙ (Canada), Esin INAN ˙ Richard B. HETNARSKI (USA), Koncay HUSEY IN (Chair˘ person, Turkey), S. M. KAPLOUNOV (Russia), Faruk KARADOGAN (Turkey), Nahit KUMBASAR (Turkey), J. MAZUMDAR (India), Yal¸cın MENGI˙ (Turkey), Nikita MOROZOV (Russia), Natasha MOVCHAN (UK), Abhijit MUKHERJEE (India), Jiri NAPRSTEK, (Czech Republic), J. RAAMACHANDRAN (India), ˘ G. A. ROGERSON (UK), J. SKLIBA (Czech Republic), T. R. TAUCHERT ( USA), ¨ Mevlut TEYMUR (Turkey), A. TONDL (Czech Republic), S¸enol UTKU (USA), F. VERHUST (The Nederlands), H. I. WEBER (Brazil), Vebil YILDIRIM (Turkey). The General Lecturers (GL) who kindly accepted our invitaton and delivered a lecture in ICOVP-2005 are: Professor M. M. BANERJEE (Asanson, India) Professor B. CHAKRABARTI (Saha Institute, Calcutta, India) Professor A. CHERKAEV (Utah University, Salt Lake City, USA) Professor M. C ¸ ELEBI˙ (USGS, Menlo Park, CA, USA) ˙ Professor H. DEMIRAY (Is¸ık University, ˙Istanbul, Turkey) Professor H. ECKER (Institute for Machine Dynamics, Vienna, Austria) ˙ (Bogazii University, ˙Istanbul, Turkey) Professor M. ERDIK Professor A. MEHTA (National Centre for Basic Sciences, Calcutta, India) Professor A. B. MOVCHAN (University of Liverpool, Liverpool, UK) Professor Y. NATH (NIT, New Delhi, India) Professor R. NORDMANN (Technical University, Darmstadt, Germany) Professor M. OKROUHLIC (Academy of Science, Dolejskova, Czech Republic) Professor J. RAAMACHANDRAN (IIT, Madras, India) The Chairperson, Esin ˙Inan (Is¸ık University) has been supported by the Local Committee including her colleagues O. Akso˘gan (Çukurova University) and H. Demiray (Is¸ık University). As with the earlier Conferences of the ICOVP series, the purpose of ICOVP-2005 was to bring together scientists with different backgrounds, actively working on vibration problems of engineering both in theoretical and applied fields. The main objective did not lie, however, in reporting specific results as such, but rather in joining different languages, questions and methods developed in the respective disciplines and to stimulate thus a broad interdisciplinary research. Judging from the lively discussions, the friendly, unofficial and warm atmosphere, both inside and outside Conference rooms, this goal was achieved.

PREFACE

xvii

The following broad fields have been chosen by the International Scientific Committee to be of special importance for the ICOVP-2005: TOPIC 1. Mathematical models of vibration problems in continuum mechanics, TOPIC 2. Vibration problems in non-classical continuum models and wave mechanics, TOPIC 3. Vibrations due to solid / liquid phase interaction, TOPIC 4. Vibration problems in structural dynamics, damage mechanics and composite materials, TOPIC 5. Analysis of the linear / non-linear and deterministic / stochastic vibrations phenomena, TOPIC 6. Vibrations of transport systems, TOPIC 7. Computational methods in vibration problems and wave mechanics, TOPIC 8. Vibration problems in earthquake engineering, TOPIC 9. Vibration of granular materials, TOPIC 10. Active vibration control and vibration control in space structures, TOPIC 11. Vibration problems associated with nuclear power reactors. Other topics concerned with vibration problems, in general, were open as well, but it was understood that the bulk of presentations were within the above fields. All of the lecturers were carefully “nominated” by the International Scientific Committee, so as to illustrate the newest trends, ideas and the results. Altogether there were 81 active participants from 11 different countries, who presented 13 “general lectures (GL)” 1 Special Talk and 59 “research communications (RC)”, of which 49 were oral presentations and 10 were posters. Each GL was 50 minutes long, including questions and discussion, while each oral RC was 20 minutes long, including questions and discussion. The posters were exhibited throughout the 5 days of the Conference. There was ample time and opportunity for private discussions. Many private scientific meetings and interactions took place. Hopefully these will lead to new collaborations and other research developments in the coming years. It is indicated in the Table of Contents (TOC) the character of each presentation, i.e., GL or RC. In the case of more than one author, the name of the presenting author is underlined in the TOC. At the end of this volume, there appears a list of all the presenting participants, along with addresses, both postal and e-mail. An Author Index (Al) is also included to the end of this volume; all the manuscript authors appear in it along with the page number(s) where their article(s) begins. The presenting author is always indicated by underlining their names. Authors not actually participating in or present at the symposium are marked in the Author

xviii

PREFACE

lndex (AI) by an asterisk. All lectures delivered at the Conference are recorded in this volume with the full text. It is real pleasure to express our sincere gratitude to the people and organizations for their contributions, help and support for this Conference. In the first place, I have to mention here that it would not be possible to organize this symposium without the support of the Is¸ık University. The full support, understanding and encouragement of Prof. Ersin Kalaycıo˘glu, the Rector of the Is¸ık University, were made the life easy for us. Secondly the support of the Faculty of Arts and Sciences was immeasurable and the staff of the Dean’s office and my secretary, Ms. Filiz ¨ Ozsobacı were really worked very hard. I am very grateful to all of them. On the other hand, I would like to express my deep gratitude to the members of The International Scientific Committee for their valuable and vitalizing ideas, comments, suggestions, and criticism on the scientific program of the Conference. It is with great pleasure and gratitude that we acknowledge the support of The ¨ ˙ITAK- which made Technical and Scientific Research Council of Turkey -TUB possible to publish this Proceeding and also to give a small contribution to some of the General Lecturers who could not get any financial support from their own Institutes and to cover some part of their travel and local expenses of the participants coming from India, Russia and the former Soviet Union Countries. Finally, we would like to send our cordial thanks to all lecturers for their excellent presentations and careful preparation of the manuscripts. We are looking forward to come together at 8th ICOVP conference, which will tentatively take place in India in 2007. Esin ˙Inan, Chairperson Arıköy, ˙Istanbul, January 2006

ASYNCHRONOUS AND ANTISYNCHRONOUS EFFECTS OF GROUND MOTION ON THE STOCHASTIC RESPONSE OF SUSPENSION BRIDGES Süleyman Adanur1 , Kurtulus¸ Soyluk2 , A. Aydın Dumano˘glu3 and Alemdar Bayraktar1 1 Deparment of Civil Engineering, KT , Trabzon, Turkey Ü 2 Deparment of Civil Engineering, Gazi University, Ankara, Turkey 3 Grand National Assembly of Turkey, Ankara, Turkey

Abstract. In this paper, the stochastic dynamic responses of a suspension bridge subjected to asynchronous and antisynchronous ground motions are performed. Asynchronous and antisynchronous dynamic analyses are carried out for various wave velocities of the traveling earthquake ground motion. As an example Bosporus Suspension Bridge, built in Istanbul, is chosen. Filtered white noise (FWN) ground motion model modified by Clough and Penzien is used as ground motion model. The intensity parameter for the FWN model is obtained by equating the variance of this model to the variance of the two thirds of the S16E component of Pacoima Dam record of 1971 San Fernando earthquake, applied in the vertical direction. Mean of maximum displacements and internal forces obtained from the considered analyses are compared with each other. Key words: suspension bridge, stochastic response, asynchronous ground motion, antisynchronous ground motion, filtered white noise (FWN) ground motion model

1. Introduction Under the effect of dynamic loading one of the uncertainties in the structural analysis arises from the dynamic loading itself to which the structure is subjected. Because dynamic effects like earthquake motions are random there is a need to a process taking into account the uncertainty of the dynamic loading in the analysis. The analysis due to random loading is defined as the stochastic analysis. In long-span structures, like suspension bridges, dynamic ground motion will arrive to the support points at different times. At this time period, the content as well as the phase of the motion is likely to be changed depending on the distance between support points and the local soil conditions. If an earthquake ground motion applied to one support propagates with finite velocity is in phase with the motions applied at the other supports, the analysis that takes into account this variation is defined as asynchronous dynamic analysis. In the opposite case, when 1 .

E. Inan and A. Kırıs¸ (eds.), Vibration Problems ICOVP 2005, 1–6. c 2007 Springer.

2

˘ ADANUR, SOYLUK, DUMANOGLU, BAYRAKTAR

a ground motion applied to one support propagates with finite velocity is out of phase relative to the motions applied at the other supports, this analysis is defined as antisynchronous dynamic analysis. Since the wave passage effect of ground motions between the supports of lifeline structures has drawn the attention of researchers, stochastic and deterministic analyses of various structural configurations have been analysed. The objective of this paper is to determine the stochastic response of a suspension bridge, which has not been analysed comprehensively subjected to asynchronous and antisynchronous ground motions together. For this purpose, the stochastic responses of suspension bridges subjected to antisynchronous ground motion as well as asynchronous ground motion are investigated. Mean of maximum displacements and internal forces obtained from the considered analyses are compared with each other. 2. Random vibration theory In the random vibration theory, the variance of the total response component is expressed as (Harichandran et al., 1996) σ2z = σ2zs + σ2zd + 2Cov(z s , zd )

(1)

where, σ2zs and σ2zd are the pseudo-static and dynamic variances, respectively and Cov(z s , zd ) is the covariance between the pseudo-static and dynamic response components. In the stochastic analysis depending on the peak response and standard deviation of the total response, the mean of maximum value (µ) can be written as µ = pσz

(2)

where p is a peak factor and σz is the standard deviation of the total response (Der Kiureghian and Neuenhofer, 1991). 3. Ground motion model for random vibration response The cross-power spectral density function of the accelerations at the support points l and m is expressed as (Hawwari, 1992), S v¨ gl v¨ gm (ω) = γlm (ω)S v¨ g (ω)

(3)

where γlm (ω) is the coherency function describing the variability of the ground acceleration processes for support degrees of freedom l and m as a function of

STOCHASTIC RESPONSE OF SUSPENSION BRIDGES

3

frequency ω, and S v¨ g (ω) is auto-power spectral density function of the ground acceleration. Spatial variability of the ground motion is characterised with the coherency function in frequency domain. This function is dimensionless and complex valued. Recently Der Kiureghian (Der Kiureghian, 1996) proposed a general composite model of the spatial seismic coherency function in the following form γlm (ω) = γlm (ω)i γlm (ω)w γlm (ω) s = γlm (ω)i exp[i(θlm (ω)w + θlm (ω) s )]

(4)

where γlm (ω)i , γlm (ω)w and γlm (ω) s characterise the incoherence, the wave-passage and the site-response effects, respectively. In this study all the spatial effects are disregarded except the wave-passage effect and the coherency function is written as γlm (ω) = γlm (ω)w = exp[i(θlm (ω)w )] (5) L dlm θlm (ω) = −τω = − ω (6) V where τ is the arrival time of the ground motion between support points l and m, V L is the projection of d on the ground is the apparent propagation velocity and dlm lm surface along the direction of propagation of seismic waves. The auto-power spectral density function of the ground acceleration characterizing the earthquake process is assumed to be of the following form of FWN ground motion model modified by Clough and Penzien (Clough and Penzien, 1993)

S v¨ g (ω) = S 0 [

ω4l + 4ξl2 ω2l ω2 (ω2l − ω2 )2 + 4ξl2 ω2l ω

][ 2

ω4 ] (ω2f − ω2 )2 + 4ξ2f ω2f ω2

(7)

where S 0 is the amplitude of the white-noise bedrock acceleration, ωl , ξl and ω f , ξ f are the resonant frequency and damping ratio of the first and second filters, respectively. In this study firm soil type proposed by Der Kiureghian and Neuenhofer (Der Kiureghian and Neuenhofer, 1991) is used. The filter parameters for this soil type are ωl = 15.0 rad/s, ξl = 0.6, ω f = 1.5 rad/s, ξ f = 0.6. The amplitude of the white-noise bedrock acceleration (S 0 ) is obtained for this soil type by equating the variance of the ground acceleration to the variance of S16E component of Pacoima Dam acceleration records of 1971 San Fernando earthquake. The calculated value of the intensity parameter for the firm soil type is S 0 (firm)=0.021338 m2 /s3 . 4. Example In this study, as an example the Bosporus Suspension Bridge (Brown and Parsons, 1975) built in Turkey and connects Europe to Asia in Istanbul is selected. The

4


bridge has flexible steel towers of 165m high, inclined hangers and a steel boxdeck of 1074m main span, with side spans of 231m and 255m on the European and Asian sides, respectively, supported on piers. The horizontal distance between the cables is 28m and the roadway is 21m wide, accommodating three lanes each way. The roadway at the mid-span of the bridge is approximately 64m above the see level. Two-dimensional finite element model of Bosporus Suspension Bridge with 202 nodal points, 199 beam elements and 118 truss elements is considered for the analyses. While the deck, towers and cables are represented by beam elements, the hungers are represented by truss elements. The selected finite element model of the bridge is represented by 475 degrees of freedom. As apparent wave velocities of the ground motion, V=1000 m/s, 2000 m/s and infinite wave velocities are used. It is assumed that the vertical ground motion is propagating from the European side to the Asian side. The analyses are obtained for 2.5 percent damping ratio and for the first 15 modes. The stiffening effects of the cables caused by the dead load are also accounted for in the analyses. Mean of maximum total response values are calculated for the asynchronous and antisynchronous ground motions. Mean of maximum total response values of the deck and tower are shown in Fig. 1, respectively. It is shown that the response values calculated for the asynchronous ground motion are generally larger than the values calculated for the antisynchronous ground motion and the values increase with decreasing wave velocities in the analyses for the asynchronous and antisynchronous ground motions. It is also shown that the infinite wave velocity case in the analysis of the antisynchronous ground motion produces nil displacement value at the middle of the deck because of the out of phase motion of the applied forces at the supports. 5. Conclusions In this study, the stochastic responses of a suspension bridge are calculated considering the antisynchronous ground motion as well as the asynchronous ground motion. The response values are obtained and compared with each other. The results at the deck and tower obtained from the asynchronous ground motion case are generally larger than those of the response values obtained from the antisynchronous ground motion case. Because of the pseudo-static effects, the response values increase with decreasing wave velocities for both ground motions.

5

STOCHASTIC RESPONSE OF SUSPENSION BRIDGES

Asynchronous

V=infinite

V=infinite

V=infinite

V=2000 m/s

V=2000 m/s

V=2000 m/s

V=2000 m/s

V=1000 m/s

V=1000 m/s

V=1000 m/s

V=1000 m/s

200

Tower height (m)

Displacement (cm)

0 -25 -50 -75

150 100 50 0

-100 -600 -450 -300 -150

0

150

300

450

0

600

2

(a) Deck vertical displacement

6

8

(a) Tower horizontal displacement

200

25000

Tower height (m)

Bending moment (kNm)

4

Displacement (cm)

Bridge span (m)

20000 15000 10000 5000 0 -600 -450 -300 -150

150 100 50 0

0

150

300

450

0

600

25000

50000

75000

100000

Bending moment (kNm)

Bridge span (m)

(b) Deck bending moment

(b) Tower bending moment

200

Tower height (m)

800

Shear force (kN)

Antisynchronous

Asynchronous

Antisynchronous

V=infinite

600 400 200 0 -600 -450 -300 -150

150 100 50 0

0

150

300

450

600

0

500

1000

Bridge span (m)

Shear force (kN)

(c) Deck shear force

(c) Tower shear force

Figure 1.

1500

Mean of maximum total response values of the deck and tower

2000

6


References Brown W. C., Parsons M. F. (1975) Bosporus Bridge, Part I, History of design, Proc. Instn Civ. Engrs, Part 1 58 505-532. Clough R. W., Penzien J. (1993) Dynamics of Structures, Second Edition, McGraw Hill, Inc., Singapore. Der Kiureghian A., Neuenhofer A. (1991) A response spectrum method for multiple-support seismic excitations. Report No. UCB/EERC-91/08, Berkeley (CA), Earthquake Engineering Research Center, College of Engineering, University of California. Der Kiureghian A. (1996) A coherency model for spatially varying ground motions, Earthquake Engineering and Structural Dynamics 25 99-111. Harichandran R. S., Hawwari A., Sweiden B. N. (1996) Response of long-span bridges to spatially varying ground motion, Journal of Structural Engineering 122 476-484. Hawwari A. R. (1992) Suspension bridge response to spatially varying ground motion, Ph.D. Thesis, Michigan State University, Michigan.

EUROPEAN UNION FRAMEWORK PROGRAMME 7 BUILDING THE EUROPE OF KNOWLEDGE Nuri Akkas¸ Department of Engineering Sciences, METU, 06531 Ankara, Turkey

1. Introduction In March 2000, the Lisbon European Council set the goal of becoming by 2010 “the most competitive and dynamic knowledge-based economy in the world, capable of sustainable economic growth with more and better jobs and greater social cohesion”. This was called the Lisbon Strategy. The project of creating a European Research Area (ERA) was endorsed as a central element of the Lisbon Strategy to achieve this goal. However, EU still invests too little in R & D. In 2003, top ˜ 500 private R & D spenders in EU decreased their R & D investment by 2.0%. Top 500 private R & D spenders outside EU increased their R & D investment ˜ by 3.9%. Overall R &D investments are as follows: EU: 1.96%; US: 2.59%; S. Korea: 2.91%; Japan: 3.12%. ERA is implemented through so-called Framework Programmes (FP). FP7 is proposed on the basis of a doubling of funds and the duration is 7 years (2007-13). FP7 will fund R& D projects of immediate industrial relevance & needs of industry. Projects will include both public research institutions and private companies (PPP). FOUR MAJOR COMPONENTS OF EUROPEAN RESEARCH IN FP7: I. Cooperation: COLLABORATIVE RESEARCH COMPONENT Support transnational cooperation in 9 themes: 1. Health 2. Food, agriculture and biotechnology 3. Information and communication technologies 4. Nanosciences, Nanotechnologies, Materials and new Production Technologies 5. Energy 6. Environment and Climate Change 7. Transport and Aeronautics 8. Socio-economic sciences and the humanities 9. Space and Security Research

7 E. Inan and A. Kırıs¸ (eds.), Vibration Problems ICOVP 2005, 7– 8 . c 2007 Springer. .

8

AKKAS¸

II. Ideas: FRONTIER RESEARCH COMPONENT III. People: HUMAN POTENTIAL COMPONENT IV. Capacities: RESEARCH CAPACITY COMPONENT FOUR TOOLS TO IMPLEMENT EU SUPPORT IN THE ABOVE 9 THEMES: 1) Collaborative projects and networks. 2) Joint Technology Initiatives (JTI). 3) Coordination of national research programs. 4) International Cooperation (INCO): - Open all activities carried out in 9 themes to researchers and institutions from all 3rd countries. - Implemented on the basis of cost-shared funding. Bilateral agreements (US, Canada, Japan, Russia, Brazil, China, India, etc). - Targeting developing countries in fields of their particular needs (health, agriculture, energy, etc). - Specific actions aiming at reinforcing research capacities of Candidate Countries to the EU. The following questions will be answered: What are Technology Platforms? What are their Rationale, Characteristics, Approach, Aims, Criteria for Establishment, and Expected Results? What are the existing TPs? What are Joint Technology Initiatives (JTI)? JTI topics are proposed by Commission for a limited number of key technologies, considering industrial impact, feedback to public, national support, added value of European coordination. What are the areas already identified by Commission?

DYNAMIC RESPONSE OF ROCK-FILL DAMS TO ASYNCHRONOUS GROUND MOTION Mehmet Akköse1 , Süleyman Adanur1 , Alemdar Bayraktar1 and A. Aydın Dumano˘glu2 1 Department of Civil Engineering, KT Ü, 61080, Trabzon, Turkey 2 Grand National Assembly of Turkey, 06543, Ankara, Turkey

Abstract. In this study, dynamic response of rock-fill dams to asynchronous ground motion is investigated. The Keban Dam, constructed in Elazı˘g, Turkey, is chosen as a numerical example. In the asynchronous dynamic analysis, wave velocities of 500 m/s, 1000 m/s, 2000 m/s and infinite are used for the travelling ground motion. Stresses are obtained for the wave velocities, and compared with each other. It is observed that the propagation velocity of the ground motion greatly influences the response of the rock-fill dam. Key words: rock-fill dams, asynchronous ground motion

1. Introduction In the classical dynamic analysis, it is assumed that the ground motion is uniform and has an infinite wave velocity so that same ground motion affects all support points of structure at the same time. In reality, however, the ground motion has finite wave velocity, so it will arrive to support points at different times (SanchezSesma, 1987) and the effect of finite wave velocity results in various arrival time to support points. Because structures such as dams, nuclear power stations, suspension bridges and cable-stayed bridges are hundreds of meters long, ground motions will arrive from one support to the other in a few seconds. If the acceleration records are applied with different arrival times, the displacement components occurring with the movement of the support points will not be the same at every point on the structure. Because of the finite velocity of ground motion, the support points will move relatively towards each other. In addition to the dynamic displacements, quasi-static displacements will take place on the structure due to this movement (Clough and Penzien, 1993). When the ground motion is considered to be travelling with finite velocity, the equation of motion, therefore, has to be written in terms of total displacements that have quasi-static and dynamic components. In the classical dynamic analysis, no extra inertial forces will come from the first component of the displacement. So the first component is subtracted from the 9 .

E. Inan and A. Kırıs¸ (eds.), Vibration Problems ICOVP 2005, 9 –14. c 2007 Springer.

10

¨ ˘ AKKOSE, ADANUR, BAYRAKTAR, DUMANOGLU

total displacements and only the dynamic displacements are considered. Analysis that includes the travelling effect of the ground motion is called the asynchronous dynamic analysis (Dumanoglu and Severn, 1984; Bayraktar et al., 1998; Soyluk and Dumanoglu, 2000). 2. Formulation of asynchronous dynamic analysis The formulation of asynchronous dynamic analysis is given in the previous studies in detail (Dumanoglu and Severn, 1984; Bayraktar et al., 1998). Therefore, in this study, equation of motion of the system to asynchronous ground motion is shortly presented. The mentioned equation of motion is given by M u¨ + C u˙ + Ku = F

(1)

where M, C and K are mass, damping and stiffness matrices, respectively. u¨ , u˙ , and u are total acceleration, velocity, and displacement vectors, respectively. F is external load vector. In the asynchronous dynamic analysis, total displacement consists of the sum of quasi-static and dynamic components, and can be expressed as u(t) = r j u jg (τ j , t) + Φi Yi (τ j , t) (2) j

i

where r j is the jth displacement shape function due to unit displacement assigned to ground degree of freedom; u jg is the jth ground displacement at the support points; τ j is the arrival time of the jth ground motion at a specific support point; Φi is the modal vector for mode i and Yi is the modal amplitude for mode i. 3. Numerical example In this study, the Keban dam constructed in Elazı˘g, Turkey is chosen as a numerical example to investigate the dynamic response of a rock-fill dam to asynchronous ground motion. The finite element mesh of the dam is shown in Fig. 1. The Keban Dam is 163m high from river bed. The crest has a maximum length of 1097m. The main purpose of the dam is to regulate river flow and supply energy. In the finite element mesh of the dam, there are 326 nodes and 286 quadrilateral elements. The structure is treated as a plane strain problem. The interaction of the rock-fill dams with the foundation rock and the reservoir has generally neglected (Priscu et al., 1985). In this study, the interaction with the reservoir is ignored, but not the foundation rock. Materials in the dam can be grouped in three main categories: compacted rock-fill placed at various lifts, the impervious clay core flanked by transition filters and a concrete core at the bottom of the dam (Akay and Gülkan, 1975). The properties of these materials are as follows. For the compacted rock-fill, elasticity modulus E = 1.632x1010 N/m2 , mass density ρ = 2120.29

DYNAMIC RESPONSE OF ROCK-FILL DAMS

Figure 1.

11

Finite element mesh of Keban Dam

kg/m3 , and Poisson’s ratio ν = 0.36. For the impervious clay core, elasticity modulus E = 1.015x1010 N/m2 , mass density ρ = 2089.70 kg/m3 , and Poisson’s ratio ν = 0.45. For the concrete core, elasticity modulus E = 2x1010 N/m2 , mass density ρ = 2446.48 kg/m3 , and Poisson’s ratio ν = 0.15. Also, the foundation rock is taken into account in the study. Its elasticity modulus, mass density and Poisson’s ratio are taken as 1.379x1010 N/m2 , 2689.09 kg/m3 , and 0.24, respectively. The materials used are assumed to be linearly elastic, homogenous and isotropic. The program MULSAP (Dumanoglu and Severn, 1984) is employed in the response calculations. The E-W component of the Erzincan Earthquake, March 13, 1992, Erzincan, Turkey is chosen as ground motion and given in Fig. 2. The component considered is applied in the upstream-downstream direction. In the asynchronous dynamic analysis of the Keban Dam, wave velocities of 500 m/s, 1000 m/s, 2000 m/s and infinite are used for travelling ground motion. Infinite velocity case corresponds to classical dynamic analysis.

Figure 2.

The E-W component of the Erzincan Earthquake, March 13, 1992, Erzincan, Turkey


12

500 m/s

1000 m/s

2000 m/s

Infinite

10000

Vyy (kN/m2)

8000 6000 4000 2000 0 0

100

200

300

400

500

Lateral Distance (m)

500 m/s

1000 m/s

2000 m/s

Infinite

Vzz (kN/m2)

3000

2000

1000

0 0

100

200

300

400

500


500 m/s

1000 m/s

2000 m/s

Infinite

Vyz (kN/m2)

4000 3000 2000 1000 0 0

100

200

300

400

500


Figure 3. Horizontal, vertical and shear stresses at I-I section for wave velocities of 500 m/s, 1000 m/s, 2000 m/s and infinite of asynchronous ground motion

The absolute maximum horizontal, vertical and shear stresses at section I-I shown in Fig. 1 are presented in Fig. 3. The stresses are given at the centroid of the elements. As expected, all the stress components generally increase with decreasing velocity of the earthquake waves. This situation is clearly seen in horizontal stresses. But, this situation couldn’t be said fully for vertical and shear stresses. It is thought that this arose from horizontal quasi-static displacements caused by asynchronous horizontal ground motion. In order to investigate the variation of the frequency content of the stresses, time-histories of only horizontal stresses at element A shown in Fig. 1 were plotted in Fig. 4 for wave velocities of 500

13

DYNAMIC RESPONSE OF ROCK-FILL DAMS Velocity of 500 m/s 6000

Vyy (kN/m2)

4000 2000 0 -2000 -4000 -6000 0

3

6

9

12

15

18

21

12

15

18

21

12

15

18

21

12

15

18

21

Time (s) Velocity of 1000 m/s 6000

Vyy (kN/m2)

4000 2000 0 -2000 -4000 -6000 0

3

6

9

Time (s) Velocity of 2000 m/s 6000

Vyy (kN/m2)

4000 2000 0 -2000 -4000 -6000 0

3

6

9

Time (s) Velocity of infinite 6000

Vyy (kN/m2)

4000 2000 0 -2000 -4000 -6000 0

3

6

9

Time (s)

Figure 4. The time-histories of horizontal stresses at the element A for wave velocities of 500 m/s, 1000 m/s, 2000 m/s and infinite of asynchronous ground motion

14


m/s, 1000 m/s, 2000 m/s and infinite of asynchronous ground motion. It is seen from Fig. 4 that both amplitude and frequency contents of the time-histories of the stresses are considerably affected by decreasing velocity of the earthquake waves. 4. Conclusions In this study, dynamic response of a rock-fill dam to asynchronous ground motion is investigated by using the finite element method. In the asynchronous dynamic analysis, wave velocities of 500 m/s, 1000 m/s, 2000 m/s and infinite are used for the travelling ground motion. As decreasing velocity of the earthquake waves, all the stress components generally increase. It is also understood from the timehistories of the horizontal stresses that both amplitude and frequency contents of the time-histories of the stresses are considerably affected by decreasing velocity of the earthquake waves. References Akay H. U., Gülkan P. (1975) Earthquake Analysis of Keban Dam, Fifth European Conference on Earthquake Engineering, Istanbul, Turkey 1-3 Paper No:40. Bayraktar A., Dumano˘glu A. A., Calayir Y. (1998) Asynchronous Dynamic Analysis of DamReservoir-Foundation Systems by The Lagrangian Approach, Computers and Structures 58 925-935. Clough R. W., Penzien J. (1993) Dynamics of Structures, Second Edition, McGraw-Hill Book Company, Singapore. Dumanoglu A. A., Severn R. T. (1984) Dynamic Response of Dams and Other Structures to Differential Ground Motions, Proc. Instn. Civ. Engrs., Part 2 77 333-352. Priscu R., Popovici A., Stematiu D., Stere C. (1985) Earthquake Engineering for Large Dams, Second Edition, Editura Academiei, Bucureti. Sanchez-Sesma F. J. (1987) Site Effects on Strong Ground Motion, Soil Dynamics and Earthquake Engineering 6 124-132. Soyluk K., Dumanoglu A. A. (2000) Comparison of Asynchronous and Stochastic Dynamic Responses of A Cable-Stayed Bridge, Engineering Structures 22 435-445.

A COMPARATIVE STUDY ON THE DYNAMIC ANALYSIS OF MULTI-BAY STIFFENED COUPLED SHEAR WALLS WITH SEMI-RIGID CONNECTIONS O. Akso˘gan1 , B.S. Choo2 , M. Bikçe3 , E. Emsen1 and R. Res¸ato˘glu1 1 Department of Civil Engineering, C ¸ ukurova University, 01330 Adana, Turkey 2 School of the Built Environment Faculty of Engineering & Computing, Napier University, Edinburgh, EH10 5DT, UK 3 Department of Civil Engineering, University of Mustafa Kemal, 31024 Hatay, Turkey

Abstract. The present work considers dynamic analysis of multi-bay shear walls having any number of stiffening beams resting on rigid foundations. Key words: coupled shear wall, dynamic analysis, Newmark method

1. Introduction To carry out a quick predesign of coupled shear walls, an elegant method called Continuous Connection Method (CCM) has been widely used. In this method, the connecting beams are assumed to have the same properties and spacing along the entire height of the wall (or in a particular section of the wall). The discrete system of connecting beams is replaced by continuous laminae of equivalent stiffness (Rosman, 1964). For applying the method to the dynamic analysis of coupled shear walls, the structure is considered as a discrete system of lumped masses with their lateral displacements as the only degrees of freedom. Then, to find the stiffness matrix, a unit loading is applied for each freedom and the deflection forms are found by the CCM (Akso˘gan et al., 2003). Substituting this stiffness matrix and the previously obtained mass matrix in the free vibration equation, the natural frequencies and the corresponding mode vectors are obtained. The forced vibration analysis under time dependent loading has been carried out using the mode superposition technique and the Newmark method (Clough and Penzien, 1993).

15 .


˘ ˘ AKSOGAN, CHOO, B˙IKÇE, EMSEN, RES¸ATOGLU

16 2. Analysis

In the coupled shear wall seen in Figure 1, the bending stiffness and the shear forces in the discrete connecting beams are replaced by equivalent continuous functions. The flexibility matrix is found by applying a unit force in the horizontal direction at the height of each lumped mass, one at a time. The horizontal displacements found from each unit loading case constitute a column of the flexibility matrix, the inverse of which yields the stiffness matrix. The main assumption of the CCM renders the horizontal disFigure 1. A multi-bay shear wall with stiffening placements equal at the ends of beams a connecting beam which is a natural result of the well accepted rigid diaphragm assumption. The relations among the shear force functions, q j−1,i and q j,i , and the corresponding contributions to the axial forces in the piers, Q j−1,i and Q j,i , can be written from the vertical force equilibrium (Figure 2). Consequently,

Figure 2.

The vertical forces on an isolated region of the shear wall

dQ j,i = −q j,i (1) dx where q j−1,i and q j,i are the shear force functions and the unknown functions Q j−1,i and Q j,i , each, are the sums of the shear forces starting from the top in a span between two neighboring piers. Defining Macaulay’s brackets by Q0,i = Qm+1,i = q0,i = qm+1,i = 0

17

MULTI-BAY STIFFENED COUPLED SHEAR WALLS

n f or x > x < x − x >n = x − x and < x − x >0 = 1 < x − x >n = 0 and < x − x >0 = 0 f or x ≤ x

      

(2)

for a unit force at height H p , the moment-curvature relation, yields d2 yi 1 =< H − x > − Q j,i L j j = 1, 2, ..., m, i = 1, 2, ..., n p dx2 j=1 m

EIi

(3)

For the compatibility of vertical displacements at the midpoints of the connecting beams (similarly for stiffening beams):  xk  1   (Q − Q )dx j,k j−1,k  A j,k  hi a3j hi a2j x dyi  k+1 L j dx − 2Ccbi q j,i − 12EIc j,i q j,i − E1  xk   1 + A j+1,k (Q j,k − Q j+1,k )dx   xk+1  x 1   (Q − Q )dx j,i j−1,i A j,i  n  xi+1  − δ = 0 j = 1, 2, . . . , m, i = 1, 2, . . . , n  − E1 x 0   k=i+1  + A1 (Q j,i − Q j+1,i )dx  j+1,i

xi+1

(4) The terms of the compatibility equation (4) are the relative vertical displacements of the two sides of the midpoints due, respectively, to the bending of the piers, the relative rotation of the beams with respect to the piers, the bending of the connecting beams due to the shear forces, the axial deformations of the piers in section i, the axial deformations of the parts of the piers between section i and the foundation and the relative vertical displacements in the foundation. Differentiating (4) with respect to x, using equations (1, 3) for j = 1, 2, ..., m, i = 1, 2, ..., n, k = 1, 2, ..., m

γ2j,i Qj,i γ2j,i

=

m×1

− α2jk

EIi Lj

α2jk = 1 + α2jk = 1 −

m×m

hi a2j 2Ccbi

+

Ii L2j

1 A j,i

Q j,i L j

hi a3j 12EIc j,i

+

Ii L j L j−1 A j,i

1 A j+1,i

,

m×1

= − < H p − x >1 , xi+1 ≤ x ≤ xi ,

α2jk

= 1−

( j = k),

( j = k + 1),

Ii L j L j+1 A j+1,i

(5)

( j = k − 1),

α2jk = 1( j < k − 1),

α2jk = 1( j > k + 1)

(6)

˘ ˘ AKSOGAN, CHOO, B˙IKÇE, EMSEN, RES¸ATOGLU

18

The coupled set of differential equations (5) is solved by the matrix orthogonalization method (Meirovitch, 1980), using a variable transformation, Q j,i L j = R j,i

j = 1, 2, ..., m, i = 1, 2, ..., n

 0 . 0   R   −α2 −α2 . −α2    1,i   12 1m  2 γ2,i   R   −α11 2 2 . −α2  −α 21 22 2m  L2 . 0   2,i  +   . . . . .   .   .  2 2 2   R 2 −α −α . −α γm,i mm m1 m2 m,i 0 . Lm R B  γ2  1,i  L1   0   .  0

(7)

     R1,i   − < H p − x >1   R   − < H − x >1  p   2,i  =    .   .      Rm,i − < H p − x >1 R

Me

A

(8) where A and B are m × m and R and Me are m × 1 dimensional matrices. The homogeneous part of this matrix equation, which is an eigenvalue problem in the following form A R + B R = 0

(9)

is solved, the eigenvectors yielding the transformation matrix T . Since the coefficient matrices A and B are constant, (8) can be diagonalized, for a new variable W, using R = T W, R = T W , A T W + B T W = Me

(10)

Multiplying both sides of the last equation in (10) by the transpose of T , ˜e A˜ W + B˜ W = M

(11)

where A˜ and B˜ are diagonal vectors and (11) is uncoupled. The solution of (11) is ( j = 1, 2, ..., m, i = 1, 2, ..., n)      B˜   B˜  1 j j,i  j j,i  x + D j,i S inh  x + < H p − x >1 W j,i = C j,iCosh  A˜ j j,i A˜ j j,i B˜ j j,i

(12)

The integration constants in (12) are found from vertical force equilibrium and compatibility at the ends (including the level of the unit force). Finding Q j,i from (7), (10) and (12) and plugging in (3), integration with respect to x yields (i = 1, 2, ..., n)   # #  m   1   Q j,i L j  dx dx + Hi x + Gi (13) yi =  < H p − x >1 − EIi j=1

MULTI-BAY STIFFENED COUPLED SHEAR WALLS

19

The constants in (13) are determined from the conditions for the displacements and slopes on the boundaries. Using the unit horizontal displacement expressions, the flexibility, and thereof, the stiffness matrices are found. Substituting the mass and stiffness matrices in the free vibration equation, the natural frequencies, and therefrom, the pertinent mode shapes can be found. The equation of motion has been written as M Y¨ + C Y¨ + K Y = P(t)

(14)

and solved numerically by using the Newmark method (Clough and Penzien, 1993). 3. Numerical results To verify the present method a coupled shear wall with three rows of openings (Figure 3), has been solved under the effect of a dynamic loading P(t) of 1000 kN at the top for 4 seconds. The maximum top displacements of the shear wall were determined, for the undamped and damped (5% damping ratio) cases by the present method and by SAP2000 (Wilson, 1997) and Figure 3. Shear wall with three rows of openings the results were compared (Table 1). The variation of the top displacement for the undamped and damped cases is presented in Figure 4. The example structure has been also solved considering a varying beam-wall connection stiffness against rotation and the results are compared in Figure 5. 4. Conclusions Since the present method proves to be accurate enough, it should be preferred for two reasons. Firstly, the data preparation is much easier than those of the equivalent frame and finite element methods. Besides, it renders possible quick trials of many different cases of a structure for optimization purposes. Secondly, the computation time of the present method is much less than those of the other two. Hence, the present method can be used effectively for predesign or dimensioning purposes.

20

˘ ˘ AKSOGAN, CHOO, B˙IKÇE, EMSEN, RES¸ATOGLU TABLE I.

The comparison of the maximum top displacements

Damping ratio

Present study CCM (m)

Frame method

SAP2000 (m) % Finite element diff. method

% diff.

0%

0.1828

0.1832

0.21

0.1841

0.69

5%

0.1638

0.1643

0.33

0.1651

0.81

Figure 4. Variation of top displacement in undamped and damped cases

Figure 5. Variation of maximum top displacement with connection stiffness

References Akso˘gan O., Arslan H. M., Choo B. S. (2003) Forced Vibration Analysis of Stiffened Coupled Shear Walls Using Continuous Connection Method, Eng. Struct. 25 499-506. Clough R. W., Penzien J. (1993) Dynamics of Structures, McGraw-Hill Inc., USA. Meirovitch L. (1980) Computer Methods in Structural Dynamics, Sijthoff and Norrdhoff, Netherlands. Rosman R. (1964) Approximate Analysis of Shear Walls Subject to Lateral Loads, Journal of the American Concrete Institute 61 717-732. Wilson E. L. (1997) SAP2000 Integrated Finite Element Analysis and Design of Structures 1-2, C&S Inc., USA.

FREE VIBRATIONS OF CROSS-PLY LAMINATED NON-HOMOGENEOUS COMPOSITE TRUNCATED CONICAL SHELLS ˘ 1 , Abdullah H. Sofiyev2 and Ali Sofiyev3 Orhan Aksogan 1 Department of Civil Engineering, C ¸ ukurova University, Adana, Turkey 2 Technical Sciences Department of Kazakh Branch of Teachers Institute, Azerbaijan 3 Department of Civil Engineering, SDÜ, Isparta, Turkey

Abstract. In this study, the free vibration of cross-ply laminated non-homogeneous orthotropic truncated conical shells is studied. At first, the basic relations have been obtained for cross-ply laminated orthotropic truncated conical shells, the Young’s moduli and density of which vary piecewise continuously in the thickness direction. Applying Galerkin method to the foregoing equations, the frequency of vibration is obtained. Finally, the effect of non-homogeneity, the number and ordering of layers on the frequency is found for different mode numbers, and the results are presented in tables and compared with other works. Key words: free vibration, laminated composite, conical shell

1. Introduction Laminated structural elements composed of non-homogeneous materials with different elastic properties are frequently used in contemporary engineering applications. These materials have properties that vary as a function of position in the body. Non-homogeneous materials can frequently be found in nature as well as in man-made structures. However, typically non-homogeneous materials seem to be those with elastic constants varying continuously in different spatial directions. Continuous non-homogeneity is a direct generalization of homogeneity in theory; besides, material non-homogeneity becomes essential and must sufficiently be considered in a number of practical situations (Lomakin, 1976). In recent years, noteworthy studies have been carried out on the stress-strain relations and vibrations of membranes and shells made of non-homogeneous materials (Massalas et al., 1981 and Gutierrez et al., 1998). Leissa (1973) made a comprehensive study on the vibrations of plates and gave all previous studies in summary. Most of the recent works on the vibration 21 .


˘ AKSOGAN, SOF˙IYEV, SOF˙IYEV

22

Figure 1.

Geometry and the cross-section of a conical shell with N layers

problems of cylindrical shells made of composite materials cover the vibration problems of composites (Lam and Loy, 1995, and Liew et al., 2002). In the vibration problems, the coverage of non-homogeneity of constituent materials renders the problem a bit more complicated. Moreover, in the case of laminated shells the difficulties increase and vibration characteristics need to be determined by a dependable computation method. In recent years, some studies have been carried out on the vibrations of cylindrical shells (Akso˘gan and Sofiyev, 2000). The free vibration problem of laminated truncated conical shells, made of non-homogeneous cross-ply layers, has not been studied yet. The aim of the present work is to study the vibration of an orthotropic composite conical shell, the Young’s moduli and density of which vary piecewise continuously with the coordinate in the thickness direction. 2. Basic equations Consider a circular conical shell as shown in Fig. 1, which is assumed to be thin, laminated and composed of N layers of equal thickness δ of non-homogeneous orthotropic composite materials perfectly bonded together. A set of curvilinear coordinates (ς, θ, S ) is located on the middle surface. R1 and R2 indicate the radii of the cone at its small and large ends, respectively, 2h is the thickness, L is the length and γ is the semi-vertex angle. Furthermore, the ς-axis is always normal to the moving S -axis, lies in the plane generated by the S -axis and the axis of the cone, and points inwards. The θ-axis is in the direction perpendicular to the S − ς plane. The axes of orthotropy are parallel to the curvilinear coordinates S and θ. The Young’s moduli and densities of all layers are defined as continuous functions of the thickness coordinate ς as (Akso˘gan and Sofiyev, 2000): (k+1) (k+1) (k+1) (k+1) (ς) ( ES E , , E , G ¯ , Eθ(k+1) (ς) ¯ , G(k+1) (ς) ¯ = ϕ¯ (k+1) ς) ¯ 1 0S 0θ 0 (k+1) (k+1) (k+1) ( ¯ ¯ (1) (ς) ρ ς) ¯ = ρ0 ϕ¯ 2 ¯ , −1 + kδ ≤ ς¯ ≤ −1 + (k + 1) δ, ¯δ = δ/h, k = 0, 1, 2, ..., (N − 1) (k+1) (k+1) whereE0S and E0θ are the Young’s moduli in the S and θ directions for the is the shear modulus on the plane of the layer k+1, layer k+1, respectively, G(k+1) 0

CROSS-PLY LAMINATED NON-HOMOGENEOUS COMPOSITE

23

is the density of homogeneous orthotropic material of layer k+1, δ = and ρ(k+1) 0 2h/N is the thickness of the layers. Additionally, (ς) ϕ¯ (k+1) ¯ = 1 + µϕα(k+1) (ς) ¯ , α

α = 1, 2

(2)

(ς) ¯ are continuous functions giving the variations of the where ϕ(k+1) α $$ Young’s $$ $$ ≤ 1, and ( ς) ¯ moduli and densities of the layers, satisfying the condition $$ϕ(k+1) α µ is a variation coefficient satisfying 0 ≤ µ < 1. The middle surface ς = 0 is located at a layer interface for even values of N, whereas, for odd values of N the middle surface is located at the center of the middle layer. After lengthy computations, the free vibration and compatibility equations of a non-homogeneous laminated conical shell, in terms of a displacement function w and a stress function Ψ are obtained as follows: 2 4 3 c11 −2c31 +c22 ∂4 Ψ c21 ∂4 Ψ c12 ∂∂SΨ4 + c11 +2cS12 −c22 ∂∂SΨ3 + cotS γ − cS212 ∂∂SΨ2 + cS213 ∂Ψ ∂S + S 4 ∂θ4 + S2 ∂S 2 ∂θ2 1

11 ) ∂ Ψ + 2(c31S−c + 3 ∂S ∂θ2

1

2(c11 −c31 +c21 ) ∂2 Ψ 2(c14 +c32 ) ∂3 w c14 +c23 +2c32 ∂4 w c24 ∂4 w 4 2 − S 4 ∂θ4 − 2 2 ∂θ2 + S S S3 ∂θ ∂S ∂S ∂θ12 1 1 1 1 cot γ ∂2 Ψ 2(c14 +c32 +c24 ) ∂2 w c23 −c14 −2c13 ∂3 w c24 ∂2 w c24 ∂w ∂4 w − − c13 ∂S 4 + + S 2 ∂S 2 − S 3 ∂S + S ∂S 2 S S4 ∂S 3 ∂θ12 1 2 −ρh ˜ ∂∂tw2 = 0, 3

(3)

3 +b11 ) ∂2 Ψ +b12 ∂4 Ψ b11 ∂4 Ψ b11 ∂2 Ψ 21 ) ∂ Ψ + 2b31 +bS 21 − 2(b31S+b + 2(b31 +bS 21 + bS113 ∂Ψ 2 3 4 ∂S − S 2 ∂S 2 S 4 ∂θ14 ∂S 2 ∂θ12 ∂S ∂θ12 ∂θ12 3 4 4 3 −b24 ∂4 w 32 ) ∂ w + b21 +2bS22 −b12 ∂∂SΨ3 + b22 ∂∂SΨ4 − − bS144 ∂∂θw4 + 2b32 −bS 13 + 2(b24S−b 2 3 ∂S 2 ∂θ12 ∂S ∂θ12 1 b 3 4 cot γ ∂2 w −b14 ) ∂2 w 14 + 2(b32 −bS 24 − − bS143 ∂w + b13 −b24S −2b23 ∂∂Sw3 − b23 ∂∂Sw4 = 0 4 ∂S + S 2 + S ∂S 2 ∂θ12

(4) where t is time, ρ, ˜ ai j , bi j and ci j (i, j = 1, 2, 3, 4 and k1 = 0, 1, 2) are the parameters depending on the material properties and the characteristics of the laminated conical shell. 3. The solution of basic equations Consider a cross-ply laminated truncated conical shell with simply supported edge conditions. The solutions for equation (3) take the following form (Akso˘gan and Sofiyev, 2000): S f (t) sin β1 ξ sin β2 ϕ (5) S1 where f(t) is the time dependent amplitude and the following definitions apply: w=

β1 =

mπ n S2 S , ξ0 = ln , ξ = ln , β2 = ξ0 sin γ S1 S1

(6)

24


Function (5) satisfies the following geometrical boundary conditions: w(S 1 , θ) = w(S 2 , θ) = 0;

∂2 w(S 1 , θ) ∂2 w(S 2 , θ) = =0 ∂S 2 ∂S 2

(7)

Substituting Eq. (5) into Eq. (3) F1 = (K1 sin β1 ξ + K2 cos β1 ξ + K3 e−ξ sin β1 ξ + K4 e−ξ cos β1 ξ) f (t) sin β2 ϕ (8) where Ki (i = 1, 2, 3, 4) are the parameters depending on the material properties and the characteristics of the laminated truncated conical shell. Substituting Eqs. (5) and (8) into Eq. (4) and by applying Galerkin’s method, the following equation is obtained: ∂2 f (t) Λ f (t) = 0 + ρh ˜ ∂t2 where the following definition applies: Λ = (J1 Γ1 + J2 Γ2 + J3 Γ3 + J4 Γ4 + Γ8 )/Γ9

(9)

(10)

in which Ji (i = 1, 2, 3, 4) and Γi (i = 1, 2, 3, 4) are the parameters depending on the material properties and the characteristics of the laminated conical shell. The following expression is obtained from Eq. (9) for the dimensionless frequency parameter of free vibration: % Ω = ωR2 (1 − ν2 )ρ/E (11) & where ω = Λ/(ρh). ˜ 4. Numerical computations and results To verify the present analysis, our results are compared with those presented by Lam and Loy (1995) and Liew et al. (2002) for the homogeneous cross-ply laminated cylindrical shells of lamination scheme (0o /90o /0o ). To be able to make a comparison with cross-ply laminated cylindrical shells, γ ≈ 0o must be substituted for the semi vertex angle and R2 ≈ R1 ≈ R must be assumed. The comparisons for the shell with the following material properties (Lam and Loy, 1995 and (k+1) (k+1) (k+1) Liew et al., 2002): E0S = 2.5E0θ , E0θ = 7.6 × 109 N/m2 , νS(k+1) = θ (k+1) (k+1) 3 o = 1643kg/m and γ ≈ 0 are presented in Table 1. 0.26, νθS = 0.104, ρ0 It is obvious that agreement is achieved for all cases. Numerical computations, for cross-ply laminated conical shells have been carried out using expression (11) and the results are presented in Table 2. In this table, the minimum values of the frequency parameter are given for varying values of

CROSS-PLY LAMINATED NON-HOMOGENEOUS COMPOSITE

25

% 2 TABLE I. Comparisons of non-dimensional frequency parameter Ω = ωR (1 − ν12 )ρ/E22 o o o with (0 /90 /0 ) for a simply supported laminated cylindrical shell (m=1, R2 /h=1000, L/R=5) n

Lam and Loy (1995)

Liew et al.(2002)

Present

3 4 5 6

0.0551 0.0338 0.0258 0.0259

0.0551 0.0338 0.0258 0.0259

0.0553 0.0337 0.02556 0.02557

TABLE %II. Variations of non-dimensional frequency parameter 2 )ρ/E22 for different numbers and ordering of layers and different Ω = ωR (1 − ν12 semi-vertex angles γ for a simply supported cross-ply laminated conical shell (m=1, R2 /h=1000, Lsinγ/R2 =0.25) (ς) ¯ = ς,α ¯ = 1, 2) Ω(ϕ(k+1) α N 1 2 2 3 3

Ordering of layers (0◦ )1 (0◦ /90◦ )2 (90◦ /0◦ )2 (0◦ /90◦ /0◦ )3 (90◦ /0◦ /90◦ )3

γ = 30◦ µ=0 0.909(17) 0.869(18) 0.884(19) 0.927(19) 0.870(18)

γ = 30◦ µ = 0.9 0.830(20) 0.822(20) 0.816(20) 0.840(20) 0.809(19)

γ = 45◦ µ=0 0.581(16) 0.573(16) 0.574(17) 0.586(17) 0.583(15)

γ = 45◦ µ = 0.9 0.535(18) 0.539(18) 0.539(17) 0.534(18) 0.543(17)

0.581(16) 0.573(16) 0.574(17) 0.586(17) 0.583(15)

0.694(16) 0.681(15) 0.683(16) 0.701(16) 0.691(15)

0.581(16) 0.573(16) 0.574(17) 0.586(17) 0.583(15)

0.565(17) 0.563(17) 0.563(17) 0.571(17) 0.570(16)

(ς) ¯ = ς¯ 2 ,α = 1, 2) Ω(ϕ(k+1) α 1 2 2 3 3

(0◦ )1 (0◦ /90◦ )2 (90◦ /0◦ )2 (0◦ /90◦ /0◦ )3 (90◦ /0◦ /90◦ )3

0.909(17) 0.869(18) 0.884(19) 0.927(19) 0.870(18)

1.091(16) 1.035(17) 1.057(18) 1.111(18) 1.035(17)

(ς) ¯ = ς¯ 3 ,α = 1, 2) Ω(ϕ(k+1) α 1 2 2 3 3

(00 )1 (00 /900 )2 (900 /00 )2 (00 /900 /00 )3 (900 /00 /900 )3

0.909(17) 0.869(18) 0.884(19) 0.927(19) 0.870(18)

0.882(18) 0.857(19) 0.861(19) 0.899(19) 0.849(18)

Note: The values in parentheses are the wave numbers.

the vertex angle, different circumferential wave numbers, and different numbers and ordering of layers. The Young’s moduli and density variation function of the materials of the layers are assumed to be power functions which, together with the other composite material properties, are given as follows (Liew et al., 2002):


26

(ς) ϕ(k+1) ¯ = ς¯ q (q = 1, 2, 3), α = 1, 2 α

(k+1) (k+1) E0S = 2.5E0θ ,

(k+1) = 7.6 × 109 N/m2 , νS(k+1) = 0.26, E0θ θ

ρ(k+1) = 1643 kg/m3 0

It is easily observed that, as the number and ordering of the layers change, the non-dimensional frequency parameter takes its maximum value at different wave numbers. For the homogeneous case (µ = 0) and for the non-homogeneous case (ς) ¯ = ς¯ q (q = 1, 2, 3) and (µ = 0.9), as the semi-vertex angle γ increases, ϕ(k+1) α the value of the non-dimensional frequency parameter decreases. It has been observed that the effect of the Young’s moduli and density variation on the non-dimensional frequency parameter is highest for being parabolic and lowest for being cubic. 5. Conclusions In this study, the free vibration of cross-ply laminated non-homogeneous orthotropic truncated conical shells is studied. At first, the basic equations have been obtained for cross-ply laminated orthotropic truncated conical shells, the Young’s moduli and density of which vary piecewise continuously in the thickness direction. Applying Galerkin method to the foregoing equations, the frequency parameter of vibration is obtained. Finally, the effect of non-homogeneity, the number and ordering of layers on the frequency is found for different mode numbers, and the results are presented in tables and compared with other works. References Akso˘gan, O., and Sofiyev, A. H. (2000) The Dynamic Stability of a Laminated Non-homogeneous Orthotropic Elastic Cylindrical Shell under a Time Dependent External Pressure, Int. Con. on Modern Practice in Stress and Vib. Anal. Nottingham, UK, 349-360. Gutierrez, R. H., Laura, P. A. A., Bambill, D. V., Jederlinic, V. A., and Hodges, D. H. (1998) Axisymmetric Vibrations of Solid Circular and Annular Membranes with Continuously Varying Density, J. Sound and Vib. 212, 611-622. Lam, K. Y., and Loy, C. T. (1995) Analysis of Rotating Laminated Cylindrical Shells by Different Thin Shell Theories, J. Sound and Vib. 186, 23-35. Leissa, A. W. (1973) Vibration of Shells, NASA SP 288. Liew, K. M, Ng, T. Y., and Zhao, X. (2002) Vibration of Axially Loaded Rotating Cross-ply Laminated Cylindrical Shells via Ritz Method, J. Eng. Mecs. 128, 1001-1007. Lomakin, V. A. (1976) The Elasticity Theory of Non-homogeneous Materials (in Russian), Moscow, Nauka. Massalas, C., Dalamanagas, D., and Tzivanidis, G. (1981) Dynamic Instability of Truncated Conical Shells with Variable Modulus of Elasticity under Periodic Compressive Forces, J. Sound and Vib. 79, 519-528. Volmir, A. S. (1967) The Stability of Deformable Systems, Moscow, Nauka.

SYMMETRIC AND ASYMMETRIC VIBRATIONS OF CYLINDRICAL SHELLS U˘gurhan Akyüz and Aybar Ertepınar Department of Civil Engineering, Earthquake Engineering Research Center, Middle East Technical University, Ankara, Turkey

Abstract. The stability of cylindrical shells of arbitrary wall thickness subjected to uniform radial tensile or compressive dead-load traction is investigated. The material of the shell is assumed to be a polynomial compressible material which is homogeneous, isotropic, and hyperelastic. The governing equations are solved numerically using the multiple shooting method. The loss of stability occurs when the motions cease to be periodic. The effects of several geometric and material properties on the stress and the deformation fields are investigated. Key words: hyperelastic, compressible, cylindrical shell

1. Introduction The earlier research on compressible, hyperelastic solids were mostly limited to determining a suitable model representing the behavior (Ericksen, 1955) - (Blatz and Ko, 1962), and to analyzing deformations and stresses in bodies of different geometries (Levinson and Burgess, 1971) - (Carroll and McCarthy, 1995). The objective of this work is to investigate the stability and free vibrations of cylindrical shells of arbitrary wall thickness subjected to external uniform, tensile or compressive dead load traction. The material of the shell is assumed to be compressible, isotropic, homogenous and hyperelastic. Under radial surface tractions, the shell undergoes first a finite static deformation, the vibrational behavior and the associated stability characteristics of the pre-stressed shell are then investigated. 2. The analysis of cylindrical shells Consider a long, circular cylindrical shell of arbitrary wall thickness. The inner and the outer radii in the undeformed and the deformed configurations are respectively denoted by r1 , r2 , and R1 , R2 . Let the cylindrical coordinates (R, θ, z) be identified with θi in the deformed state B. Then, a mass point P, whose coordinates are (R(r), θ, z) in the deformed state B, is originally at (r, θ, z) in the undeformed 27 .


¨ ERTEPINAR AKYUZ,

28

state B0 . Referring the undeformed and the deformed configurations to the fixed rectangular coordinates xi and Xi , respectively (Figure 3.1), one has x1 = r cos θ,

x2 = r sin θ,

x3 = z

(1)

and X1 = R(r) cos θ,

X2 = R(r) sin θ,

X3 = z

(2)

The non-zero components of the metric tensor of the undeformed state B0 (gi j ) and deformed state B (Gi j ) are g11 , g22 , g33 = R2 , 1/r2 , 1 , G11 , G22 , G33 = 1, 1/R2 , 1 (3) In equation (3) a prime denotes differentiation with respect to r. The three strain invariants I1 , I2 , and I3 of the deformation field are I1 = R2 +

R2 + 1, r2

I2 = R2 +

R2 R2 R2 + 2 , r2 r

I3 =

R2 R2 r2

(4)

For bodies made of homogeneous, isotropic and hyperelastic materials, the components of the stress tensor τi j are given by & ∂ W ij 2 ∂ W ij 2 ∂ W ij τi j = √ g + √ B + 2 I3 G ∂ I3 I3 ∂ I1 I3 ∂ I2

(5)

Bi j = I1 gi j − gir g jsGrs

(6)

where The associated boundary conditions for the shell, which is assumed to be free of tractions on its inner surface and subjected to a uniform tensile or compressive dead load on its exterior surface, are given as 2 r2 11 11 τ (R1 ) = q1 = 0, τ (R2 ) = q2 = ∓q (7) R2 It is now further assumed that the strain energy density function W has the form ' & & 2 ( I2 µ f (I1 − 3) + (1 − f ) − 3 + 2 (1 − 2 f ) I3 − 1 + (2 f + β) I3 − 1 W= 2 I3 (8) which has been proposed by Levinson and Burgess (Levinson and Burgess, 1971) and has been named ‘polynomial compressible material’ by the authors. In equation (8) µ is the shear modulus of the material for vanishingly small strains, f is a material constant whose value lies between zero and unity, and β is expressed as β=

4ν − 1 1 − 2ν

(9)

29

VIBRATIONS OF CYLINDRICAL SHELLS

where ν is the Poisson’s ratio for the material as the deformations become vanishingly small. It is noted that, for highly elastic rubbers and rubber-like materials f =0 while for solid natural and synthetic rubbers f =1. The only non-zero equation of equilibrium for this finitely deformed state is the one in the radial direction which is given by ∂τ11 τ11 − R2 τ22 + =0 ∂R R

(10)

The shell is now exposed to a secondary dynamic displacement field described by w1 = u (R (r) , θ, t)

w2 = Rv (R (r) , θ, t)

w3 = 0

(11)

to investigate the existence of small, free, radial vibrations about the finitely deformed state. The formulation of this state is based on the theory of small deformations superposed on large elastic deformations1 . The incremental metric tensors and the incremental stresses are given by G∗i j = wi, j + w j,i − 2Γrij wr ,

G∗i j = −Gir G jsG∗rs

τ∗i j = gi j Φ∗ + Bi j Ψ∗ + B∗i j Ψ + G∗i j p + Gi j p∗ where

Φ∗ = Ψ∗ =

√2 I3 2 √ I3

2W ∂2 W ∗ I + √2I ∂I∂1 ∂I I∗ ∂I12 1 2 2 3 2 ∂2 W ∗ √2 ∂ W ∗ ∂I1 ∂I2 I1 + I3 ∂I 2 I2

∂2 W

2

+ +

∂2 W

√2 I3 2 √ I3

p∗ = I3 √2I ∂I1 ∂I3 I1∗ + √2I ∂I2 ∂I3 I2∗ + 3 3 B∗i j = gi j grs − gir g js G∗rs

∂2 W ∗ ∂I1 ∂I3 I3 ∂2 W ∗ ∂I2 ∂I3 I3 ∂2 W

− −

√2 I∗ I3 ∂I32 3

Φ ∗ 2I3 I3 Ψ ∗ 2I3 I3

+

p ∗ 2I3 I3

(12) (13)

(14)

and Γrij are the Christoffel symbols of the second kind, and a comma denotes differentiation with respect to the following subscript. The incremental strain invariants I1∗ , I2∗ , and I3∗ are given by 2 2 I1∗ = 2R2 w1 ,R + r22 (w2 ,θ +Rw1 ) , I3∗ = 2 R rR2 w1 ,R + R12 w2 ,θ + R1 w1 (15) 2 2 2 I2∗ = 2R2 + 2Rr2R w1 ,R + r22 + 2Rr2 (w2 ,θ +Rw1 ) The non-zero incremental equations of motion are ∂τ∗11 ∂τ∗21 1 1 ∗11 − R2 τ∗22 + 2w , τ τ11 + w1 ,θθ τ22 + + + w , 1 RR 2 Rθ 2 ∂R ∂θ R R + R12 Rw1 ,R − R2 w2 ,θ −w1 τ11 + R2 τ22 = ρw¨ 1 1

A detailed discussion of the theory has been given in reference (Green and Zerna, 1968)

(16)

¨ ERTEPINAR AKYUZ,

30

∗22 + ∂τ∂θ + R3 τ∗12 + R12 w2 ,RR + R22 w2 − R2 w2 ,R τ11 + w1 ,Rθ − R22 w2 ,θθ + R3 w1 ,θ + R1 w2 ,R − R22 w2 τ22 = Rρ2 w¨ 2 ∂τ∗12 ∂R

(17)

in radial and in circumferential directions respectively. In equations (16) and (17) ρ is the mass density of finitely deformed body and a dot denotes differentiation with respect to time. The current mass density ρ is related to the mass density ρ 0 of the natural state by √ g (18) ρ = √ ρ0 G using the principle of conservation $$ $$ of$$ mass. $$ In equation (18), g and G denote, $ $ $ respectively, the determinants gi j and Gi j $. For this secondary state, the boundary conditions, which are obtained from the requirement that the secondary surface tractions vanish, are τ∗11 − G∗11 τ11 = 0 τ∗12 − G∗12 τ11 = 0

@ R = R1 @ R = R1

& &

R = R2 R = R2

(19)

The solution of equations (16) and (17) may be assumed to be of the form u=

∞ n=0

R∗1n (R) cos (nθ) eiωt ,

v=

∞ n=0

R∗2n (R) sin (nθ) eiωt

(20)

where ω is the frequency of vibrations about the finitely deformed state, n is the circumferential mode number, and R∗1n and R∗2n are unknown functions of R which, in turn, is a function of r. The case n = 0 corresponds to pure radial vibrations about the prestressed state. The case n = 1 includes rigid body motions and is left out of discussion with the understanding that the rigid body motions of the shell are prevented. The higher modes corresponding to n ≥ 2 are considered in what follows. The governing equations of both states are non-dimensionalized by introducing R∗ R∗ r¯ = rr2 , R¯ = rR2 , R¯ ∗1n = r1n2 , R¯ ∗2n = r2n2 , ) (21) ρ0 r22 q τ11 u 11 ¯ =ω τ¯ = µ , q¯ = µ , u¯ = r2 , ω µ For the finitely deformed state, no closed form solution of the highly non-linear system of equations, equation (10), seems possible. To solve these equations numerically, the boundary value problem is converted to an initial value problem by using the multiple shooting method. The solution of finitely deformed state equation of equilibrium together with the corresponding boundary conditions is obtained by using a FORTRAN code called BVPSOL and developed by Deuflhard and Bader (Deuflhard and Bader, 1982). Equations of motion, equations (16) and (17) governing the secondary state are coupled linear ordinary differential equations. Since no closed form solution

VIBRATIONS OF CYLINDRICAL SHELLS

a. f = 0.0 and ν = 0.25 Figure 1.

31

b. f = 1.0 and ν = 0.49

Natural frequencies of vibrations versus shell thickness

is available for the finitely deformed state, a numerical approach must be used to obtain the solution of this state also. The method of complementary functions is used to transform the linear boundary value problem to an initial value problem. 3. Discussion of the results In this study, the vibrational characteristic and the loss of stability of hollow circular cylindrical shells are investigated using the theory of finite elasticity in conjunction with the theory of small deformations superposed on large elastic deformations. For breathing motions (i.e. when n = 0), and asymmetric vibrations (i.e. when n ≥ 2), several examples are worked out numerically to study the effects of material constants f and ν, the outer stretch ratio (λ = R¯ 2 /¯r2 ) and the thickness ratio of the shell (χ = r¯1 /¯r2 ) on the non-dimensionalized natural frequencies of vibrations ω, ¯ and the critical outer stretch ratio, λcr , which is defined as the stretch ratio of the outer surface when the frequency approaches to zero. In particular, the behaviors of the foam rubber and the slightly compressible hyperelastic materials represented by polynomial model are investigated. When the applied dead load traction is equal to zero the shells are unstrained, and the natural frequencies of the vibrations about the natural state of the shell can be calculated. In Figure (1), the frequencies of vibrations about the unstrained shells are given as a function of shell thickness χ for a foam rubber, and a polynomial material with f =1.0 and ν=0.49. It is seen that for the asymmetric vibrations the frequencies increase as the shell becomes thicker while for breathing motion the frequencies increase as the shell becomes thinner. It is also observed that, for the asymmetric vibrations higher harmonics correspond to higher frequencies for a given shell thickness. Figure (2) show the variation of the frequency ω ¯ as a function of the radial stress on the outer surface, for a foam rubber with with f =0.0, ν=0.25, and for a polynomial material with f =0.9, ν=0.49, respectively, having a shell

¨ ERTEPINAR AKYUZ,

32

a. f = 0.0, ν = 0.25 and χ = 0.80 Figure 2.

b. f = 0.9, ν = 0.49 and χ = 0.80

The effect of harmonic mode number n on frequencies versus outer radial stress

thickness χ = 0.80. Four different modes, corresponding to the harmonic numbers n = 0, 2, 3, 4, are shown both in the tension and in the compression region. It is seen that, in the tension region, the lowest frequency of vibrations depends on the harmonic number n, and the state of the initial strain. A similar behavior is also observed for natural frequencies (see Figure 1). In the early stage of the inflation region, n = 2 is found to be the governing mode of vibration, and the shell behaves as a hardening system. But, as the tensile dead load traction increases, the breathing mode, n = 0, becomes the governing mode. In this stage of the inflation process, the shell behaves as a softening system, and as the limiting pressure τ¯ 11 max is reached, the shell fails in radial expansion without bound. When the shell is subjected to an increasing compressive dead load traction, the behavior mentioned above is reversed. The hardening effect is now associated with the breathing mode, while the asymmetric modes, n = 2, 3, 4, have a softening effect on the system. It is also seen that, in the compression region, n = 2 is the governing mode of vibration. References Blatz P. J. and Ko W. L. (1962) Application of Finite Elasticity Theory to the Deformation of Rubbery Materials, Trans. Soc. Rheo. 6, 223. Carroll M. M. and McCarthy M. F. (1995) Conditions on Elastic Strain Energy Function, Zangew Math. Phys. 46, S172. Deuflhard P. and Bader G. (1982) SFB 123, Tech. Rep. 163, University of Heildelberg. Ericksen J. L. (1955) Deformations Possible in Every Compressible, Isotropic, Perfectly Elastic Body, J. Math. Phys. 34, 198. Green A. E. and Zerna W. (1968) Theoretical Elasticity, 2nd Edition, Oxford. Levinson M. and Burgess I. W. (1971) A Comparison of Some Simple Constitutive Equations for Slightly Compressible Rubber-Like Materials, Int. J. Mech. Sci. 13, 563. Mooney M. (1940) A Theory of Large Elastic Deformations, J. Appl. Phys. 11, 582.

AN ACTIVE CONTROL ALGORITHM TO PREVENT THE POUNDING OF ADJACENT STRUCTURES ¨ Unal Aldemir and Ersin Aydın ˙ ˙ Faculty of Civil Engineering, Istanbul Technical University, Istanbul, Turkey

Abstract. Since the adjacent structures can collide due to their out-of-phase vibration, structural pounding is an additional problem except for the other harmful damage that occurs during the earthquake vibrations. In this study, a simple active control algorithm which results in an asymptotically stable system is proposed for adjacent structures. To be able to prevent a possible pounding, proposed control is calculated based on the measured relative displacement of the structures. The structural responses of the example structure subjected to El Centro ground motion are investigated for optimal passive control and proposed control and compared to uncontrolled case. It is shown by numerical results that the proposed active control algorithm can reduce the response of the adjacent structures under the earthquake forces and prevent pounding. Key words: active control, pounding, output state feedback control

1. Introduction Earthquakes cause many harmful effects in the structures. A significant amount of seismic energy can be absorbed with supplemental energy dissipation systems. Different supplemental energy dissipation systems have been proposed to prevent harmful effects of the strong ground motions. In addition to these passive systems, active and semi-active systems have also been employed by (Bakioglu and Aldemir, 2001) and (Aldemir, 2003). Buildings in a crowded city are often built closely to each other because of the limited availability of land. In most cases, these buildings have been built without link elements between each other. The interaction of adjacent buildings has been repeatedly identified as a common cause of damage. The possibility of added active or passive control devices to link adjacent structures has been improved in recent decades. This application takes advantages of the interaction between adjacent structures which is expected not only to prevent the pounding problem of the adjacent structures but also to reduce seismic response of the adjacent structures. Structural pounding has been studied and reported widely in literature (Anagnostopoulos, 1995), (Bertero, 1986), (Athanasiadou et al., 1994). After strong earthquakes the reason of numerous field damages and collapses was pounding. Especially, in the big cities, damage 33 .


34

ALDEM˙IR, AYDIN

and harmful effects have occured because of pounding in the cente! r of cities. In the Mexico City earthquake in 1985 had attributed to pounding a big part of the observed damage (Anagnostopoulos, 1995 ), (Bertero, 1986). In addition strength building may be benefit to weak building provided that pounding will not cause any serious local damage from which failure could be initiated (Bertero, 1986), (Athanasiadou et al., 1994). An innovative control approach was proposed which can be realized by means of installing an interaction element between two parallel structures (Zhu et al., 2001). The possibility of using dissipative links and semi active devices to reduce the response of adjacent buildings under the wind excitation was studied (Klein et al., 1972). The two adjacent buildings were modeled by a couple of uniform shear beams coupled by a single flexible and damped link and obtained the optimal stiffness and damping in the link in case of primary structures under the wind excitation (Gurley et al., 1991). 2.

Formulation of the problem

Without Control The interaction control strategy takes advantage of the configuration and interaction of two different structural systems (Primary-Auxiliary) to reduce the vibration response of primary-auxiliary structure (P-A structure) during the ground motion. In this study, only response of the P-A structures is reduced the simplest form as shown in Fig. 1. The mass of P-structure is larger than the mass of Astructure. This may cause different vibration modes between P and A structures. P and A structures can be simplified to two single-degree-of-freedom-systems without control element subject to ground excitation as shown in Fig. 1. Without control element, equation of motion is given as follows: .. . .. m p x p + c p x p + k p x p = −m p xg (1) .. . .. ma xa + ca xa + ka xa = −ma xg

(2)

where m p , ma are first modal mass of the P and A structures, k p , ka are stiffness coefficients of the P and A structures, c p , ca are damping coefficient of the P and A structures, x p , x..a represent relative displacement of the P and A structures with respect to ground, xg represents horizontal ground acceleration. The new variable z is defined as: z = xa − x p

(3)

where z represents relative displacement between adjacent structures. First and second order differentiation can be written as in following form: . . . .. .. .. xa = z + x p =⇒ xa = z + x p =⇒ xa = z + x p (4)

POUNDING OF ADJACENT STRUCTURES xp

xa

ma

mp

kp

35

ka

cp

ca

x g ( t )

Figure 1.

The simplest mechanical model of adjacent structure without control.

If eq. (4) is used in eq. (2), eq. (2) can be written depending the displacement of primary structures and relative displacement between the adjacent structures as follows: .. .. . . .. (5) ma (x p + z)! + ca (x p + z) + ka (x p + z) = −ma xg Eqs. (1) and (5) are written in matrix form as follows: * .. + *. + * + * + x..p x x −m p .. x = −M −1C . p − M −1 K p + M −1 z z z −ma g

(6)

H In state space form, eq. (6) is written as . .. X = AX + Exg x   p   z  in which X =  .  where,  x.p  z + * + * I2x2 02x1 02x2 E = A= −M −1 K −M −1C M −1 H

(7)

(8)

Optimal Passive Control A passive energy dissipation element is implemented between two structures, as shown in Fig. 2. The optimum parameters of the implemented passive device (Zhu et al., 2001)

kopt

µ(1 − β1 2 ) = kp (1 + µ)2

copt

mpωp = (1 + µ)

µ(1 + β1 2 )2 (1 + µ)(µ + β1 2 )

(9)

ALDEM˙IR, AYDIN

36 xp

xa

Up

ma

mp kop kp

cop ka

cp

ca

x g ( t ) Figure 2.

where β1 =

The simplest mechanical model of adjacent structure optimal passive control ωa ωp

µ=

mp ma

. Then, the passive control force, . U p = −kopt z − copt z

Control force vector U

(10)

+ * + *. + xp x −U p − G2p . p (11) = −G1p U= Up z z * * + + 0 −kopt 0 −copt , G2p = under the effects of U, eq. (6) is where G1p = 0 kopt 0 copt rewritten as *. + * + * .. + .. x x x..p −1 (12) = −M (C + G2p ) . p − M −1 (K + G1p ) p + M −1 Hxg z z z! *

in eq. (6), K and C are replaced by (K + G1p ) and (C + G2p ), respectively. Output State Feedback Control Active force actuators are placed on each structure as shown in Fig. 3. It is assumed that the required control energy to generate the calculated forces is available and time delay and force saturation effects are neglected. u1 and u2 are given as

Using z = xa − x p

. u1 = −g1 x p − g2 x p

(13)

. u2 = −g3 xa − g4 xa

(14)

eq. (14) is . . u2 = −g3 z − −g3 x p − g4 z − g4 x p

(15)

POUNDING OF ADJACENT STRUCTURES u1

u2

xp

xa

ma

mp

kp

37

cp

ka

ca

x g ( t ) Figure 3.

The simplest mechanical model of adjacent structure active control

Then, the control force vector U for this case is * + * + *. + u1 xp x U= − G2a . p (16) = −G1a u2 z z + + * * g1 0 g2 0 , G2a = is positive definite matrices. For where G1a = g3 g3 g4 g4 active case, the stiffness and damping matrices K and C of uncontrolled structure in eq. (6) are replaced by (K + G1a ) and (C + G2a ), respectively. 3.

Numerical examples

Example adjacent structures are chosen as shown in Fig 1. Dynamic parameter of the structures are given as ma = 0.75105 kg ka = 2.087106 N/m ca = 7 N/m c = 6.33104 kg/s G 3.165104 kg/s m p = 1.5105 kg k* p = 1.6710 1a + *p + 1 0 1 0 6 6 and G2a are selected as G1a = 10 G2a = 10 . For three cases, 1 1 1 1 the response of adjacent structures is calculated with time history analysis under El Centro earthquake. It is shown in Fig. 4 uncontrolled relative displacement is excessive and pounding may occur depending on distance between structures. So passive control may decrease the relative displacement, but it may not be sufficient. Proposed active control gives the best performance and prevents pounding.

ALDEM˙IR, AYDIN

Relative Displacement Z (m)

38 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 0 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18

Without Control

2

6

8

10

Output State Feedback Control

12

14

Time (sec)

Figure 4.

4.

4

Passive Control

Relative displacement time history for three cases

Conclusions

If the dynamic characteristics of the adjacent structures are different, pounding may occur because of their out-of-phase vibration according to distance of adjacent structures. In this study, a simple active control algorithm is proposed to prevent pounding for adjacent structures. Output state feedback control is calculated based on the measured relative displacement of the structures. The results of proposed active control method to prevent pounding of adjacent structures show that the control performance of active control is better than the optimal passive control and uncontrolled case. References Aldemir U. (2003) Optimal control of structures with semiactive-tuned mass dampers, Journal of sound and vibration 266 847-874. Anagnostopoulos S. A. (1995) Earthquake induced pounding, State of the art Proceedings of the 10th European Conference on Earthquake Engineering 2 897-905. Athanasiadou C. J., Penelis G. G. and Cappos A. G. (1994) Seismic response of adjacent buildings with similar or different dynamic characteristics, Earthquake spectra 10. Bakioglu M, Aldemir U. (2001) A new numerical algorithm for sub-optimal control of earthquake excited linear structures, Int. Journal of Numer. Meth. Engng. 50 2601-2616. Bertero V. V. (1986) Observations on structural pounding Proceedings of The International Conference on Mexico Earthquakes, ASCE 264-287. Gurley K., Kareem A., Bergman L. A., Johnson E. A. and Klein R. E. (1994) Coupling tall buildings for control of response to wind, Proc. Sixth Int. Conf. on Structural Safety and Reliability (ICOSSAR) AA Balkema Publishers, Rotherdam 1553-1560. Klein R. E., Cusano C. and Stukel J. (1972) Investigation of a method to stabilize wind induced oscillations in large structures, ASME 72 WA/UUT-H. Zhu H., Wen Y. and Iemura H. (2001) A study on interaction control for seismic response of parallel structures, Computers and Structures 79 231-242.

VIBRATION CONTROL OF NON-LINEAR BUILDINGS UNDER SEISMIC LOADS ¨ Unal Aldemir and Deniz Güney ˙ ˙ Faculty of Civil Engineering, Istanbul Technical University, Istanbul, Turkey

Abstract. An optimal control law is derived using a simple integral type quadratic functional. The resulting control scheme is applied to seismically excited non-linear buildings modeled as lumped mass shear frame structures. The non-linearity is reflected by the non-linear stiffness restoring force. Active tendon actuators are considered as control devices. The performance of the proposed control is compared to those of the uncontrolled structure and the passive base isolation. It is shown by numerical simulation results that the proposed control is capable of suppressing the uncontrolled seismic vibrations of the non-linear structures. Key words: non-linear structures, vibration control, optimal control

1. Introduction Passive and active control systems have found increasing applications to alleviate the seismic hazards in civil engineering structures. Passive single and multiple mass dampers have been installed in civil structures to mitigate the wind or earthquake induced vibrations (Chen and Wu, 2001), (Park and Reed, 2001), (Aldemir et al., 2001). The structure having a passive damper or a base isolation system is first constructed, and then it waits for the force to occur. Then, the structure responds to the induced forces without any modification in performance characteristics of the manufactured elements during the time history of the exciting force. Although the passive devices are simple and yet completely applicable in civil structures, they have limitations in achieving desired performance measures primarily due to the highly uncertain nature of the wind and seismic disturbances. Consequently, many active vibration control methodologies for civil structures have been explored in recent years (Aldemir et al.,, 2001), (Yang et al., 1987). In this study, a new simple integral type quadratic functional between the successive control instants is used in deriving the optimal control force. The proposed control is applied to non-linear structures incorporating a full active tendon system. Numerical simulation results indicate that the proposed control can significantly reduce the earthquake- induced vibrations of non-linear structures.

39 .


¨ ALDEM˙IR, GUNEY

40 2. Control algorithm

To give a brief overview of the active control concept, the governing equations of motion for a general multistory shear-beam lumped mass linear building structure are .. . . MX(t) + CX(t) + KX(t) = D1 f (t) + D2 U(t), t ∈ [to , t1 ]; X(to ) = X(to ) = 0 (1) where X(t) = (x1 , ..., xn )T is the n-dimensional response vector denoting the relative displacement of the each story unit with respect to the ground; the superposed dot represents the differentiation with respect to time; M is the (nxn)-dimensional diagonal constant mass matrix with diagonal elements mi = mass of i th story (i = 1, ..., n); C and K are the (nxn)-dimensional viscous damping and the stiffness matrices, respectively; D1 is the (nx1)-dimensional location matrix of excitation and given by DT1 = −(m1 , ..., mn ); D2 is the (nxr)-dimensional location matrix of r controllers; U(t) is the r-dimensional active control force vector and described as UT (t) = (u1 (t), ..., ur (t)) and scalar function f (t) is the one dimensional earthquake acceleration. In the state space, Eq. (1) becomes . (2) Z (t) = AZ(t) + BU(t) + D f (t), t ∈ [to , t1 ]; Z(to ) = 0 A=

*

0nxn −M −1 K

+ * + * + * + I 0 0nxr X ; D = nx1 ; Z = . ; B= −1 −1 −M C M D2 −η X

(3)

such that 0nxm is the (nxm)-dimensional zero matrix; I is the (nxn)-dimensional identity matrix; η = (1, ..., 1)T is the n-dimensional vector. In this study, to determine the active control force, we introduce the integral type quadratic functional # t+h [Z T (s)QZ(s) + UT (s)RU(s)]ds → min (4) It,h (U) = t

with two parameters, t ≥ to and h > 0 where Q is the (2nx2n)-dimensional positive semi-definite symmetric weighting matrix and R is the (rxr)-dimensional positive definite symmetric weighting matrix; h is the time interval between the successive control instants. It is clear that the problem of minimizing the functional (4) under the condition (2) is an optimal control problem. The solution of this problem is given in detail by (Aldemir et al., 2001) as follows;

Z(t) = [I +

−1 T U(t) = −hN−1 0 R B QZ(t)

(5)

h2 h −1 T −1 BN−1 0 R B Q] [y(t − h) + D f (t)] 2 2

(6)

No = I +

h2 −1 T R B QB] 4

(7)

VIBRATION CONTROL UNDER SEISMIC LOADS

41

h y(t − h) = eAh Z(t − h) + ( )eAh [BU(t) + DF(t)] 2

(8)

3. Optimal control of non-linear structures To treat the nonlinear problem of a structural system, Eq. (1) takes the form .. . . MX(t) + FD (t − h) + C∗ (t − h)[X(t) − X(t − h)] + FS (t − h) + K ∗ (t − h) . (9) [X(t) − X(t − h)] = D1 f (t) + D2 U(t) t ∈ [to , t1 ]; X(to ) = X(to ) = 0 . where both FD (t) and FS (t) are general non-linear functions of X(t) and X(t); K ∗ (t − h) and C∗ (t − h) are the influence coefficient matrices of which the elements represent the tangent stiffness and damping at time (t − h). Using the Wilson-θ numerical integration procedure and the previously defined state vector concept . with the assumption of linear viscous damping (FD (t − h) = CX(t − h)), the solution of Eq. (9) can be expressed as follows: Z(t) = y∗ (t − h) + A1 f (t) + A2 U(t)

(10)

in which y∗ (t − h) = A3 Z(t − h) + A4 y∗ (t − h) = A3 Z(t − h) + A4 FS (t − h) + A5 f (t) + A6 U(t − h)

(11)

In Eqs. (9)-(11), A j ( j = 1, ..., 6) are vectors or matrices defined as follows * * + * + + I θ−2 ET 3 T1 T2 A1 = θ−2 ; A2 = θ−2 ; A3 = ; (3/h)T 1 (3/h)T 2 0 I + θ−2 ET 4 A4 = θ−2

*

E E

+ * + * + T5 E T7 E T7 ; A5 = −θ−2 D1 ; A6 = −θ−2 D2 ; (12) T6 E T8 E T8

in which I is an (nxn) identity matrix, θ is Wilson-θ constant and chosen as 1.37, and T 1 = −ED1 ; T 2 = −ED2 ; T 3 = (6/h)M + [3(θ − 1)I − D1 ]C + h(θ2 − 1)K ∗ (t − h); T 4 = (6/h)I + S2 ]C − 3K ∗ (t − h); T 5 = −(3I + S1 ); T 6 = −(6/h)I − S2 ; T 7 = 2I + S1 ; T 8 = (3/h)I + S2 ; E = [(6/(θh)2 M + (3/(θh)C + K ∗ ]−1 ; S1 = [h(1.5θ − 1)C + 0.5h2 (θ2 − θ)K ∗ ]M−1 ; S2 = [3(θ − 1)C + hθ(θ − 1.5)K ∗ ]M−1 Then, using the same procedure, the optimal control force and the corresponding state vectors for

¨ ALDEM˙IR, GUNEY

42

m8

m8

m7

m7

m2

m2

m1

Figure 1.

m1

Structural model a)Active tendon system b) Base isolated system

−1 T the non-linear structure are derived as follows; U(t) = −2N−1 0 R A1 QZ(t) and −1 T −1 ∗ Z(t) = [I + 2A1 No R−1 A1 Q] [y (t − h) + A2 f (t)]

4.

Numerical examples

In order to evaluate the effectiveness and the performance of the proposed control algorithm, simulations for the controlled and uncontrolled response of the investigated non-linear structures are conducted. The seismic input to the example structures is ErzincanNS (Erzincan Earthquake; Dec 26, 1939). An 8-story building shown in Fig. 1.a that exhibits bilinear elastic-plastic behavior is considered and a full active tendon control system in which the control is implemented by means of actuators exerting forces at each floor is used. It is assumed that the columns of the building are massless and the mass of the structure is concentrated at floor levels. Each story has a mass of 155.6 ton, and a linear viscous damping coefficient of 1097.7 kN sec/m. In the study, 2 Cases are considered. In Case 1, the non-linearity for the 8-story structure is reflected by the stiffness restoring force FS [X(t)]. The yield displacement of each floor is 2 cm while the pre- and post-yield stiffness coefficients of each floor are k1 =153333 kN/m and k2 =45333 kN/m, respectively. Uncontrolled hysteresis loops for the shear forces f s1 , ..., f s8 in each floor of the non-linear structure are shown in Fig. 2. Without any control system, it is clear that the deformation of the unprotected structure is excessive and that yielding occurs in the first six floors. Hence, to suppress the excessive deformations, a full active tendon control system is implemented with the weighting matrices. Controlled hysteresis loops for the shear forces are shown in Fig. 3. It indicates that how effective the active tendon system with the proposed control is in reducing the deformations since all interstorey drifts are within the elastic range. This result can be seen obviously also in Fig. 3 in terms of the reduction in displacements. In Case 2, 8-storey structure subjected to Erzincan NS earthquake is isolated by rubber bearings as shown in Fig. 1.b. The superstructure is assumed to be linear elastic with identical story elastic stiffness coefficient of 153333 kN/m while the base isolation material exhibits bilinear elastic-plastic

VIBRATION CONTROL UNDER SEISMIC LOADS

5000

fs (kN)

fs (kN)

5000

0

2

1

0 -5000

-10000 -10

-5

0

-5000

-10000 -10

5

-5000

-6

-4

-2

0

2

fs (kN)

fs (kN)

-2

-1

0

1

2

f s (kN)

-1

0

1

2

0

8

0

Figure 2.

-2

1000

7

f s (kN)

0

-2000 -4000 -3

2

-1.5

-1

-0.5

0

0.5

-1000 -2000 -1

1

-0.5

0

0.5

Relative displacement (cm)

Relative displacement (cm)

Uncontrolled hysteresis loops for the shear forces

2000 fs (kN)

5000

0

0

2

fs 1 (kN)

-2

0

6

5

-3

-2000 -4000 -2

-5000 -2

-1

0

1

-2000 -4000 -2

2

0

1

2

0

4

0

-2000 -4000 -1.5

-1

2000 fs (kN)

2000 fs 3 (kN)

-4

2000

2000

-1

-0.5

0

0.5

1

-2000 -1.5

1.5

-1

-0.5

0

0.5

1

1000 fs (kN)

2000

0

0

6

fs 5 (kN)

5

0

-5000 -6

4

0

-5000 -4

-2000 -1

-0.5

0

0.5

1

-1000 -2000 -1

0

0.5

1

0

8

0

-1000 -0.6

-0.5

500 fs (kN)

1000 fs7 (kN)

0

4

3

0

5000

Figure 3.

-5

5000

fs (kN)

fs (kN)

5000

-10000 -8

43

-0.4

-0.2 0 0.2 Relative displacement (cm)

0.4

-500 -0.3

-0.2 -0.1 0 0.1 Relative displacement (cm)

0.2

Active tendon system controlled hysteresis loops for the shear forces

behavior. Properties of the base isolation system are: elastic stiffness of the base isolators kb1 =158.1 kN/m; post-elastic stiffness of the base isolators kb2 =30.1 kN/m; damping coefficient cb = 18.9 kN sec/m; mass of base isolation mb =160 ton and the yielding deformation (xb )y =10 cm. Uncontrolled time response of base isolation and the hysteresis loop of shear force in the isolators are shown in Fig. 4. It is obvious that excessive deformations occur in the isolators and the isolation system is not within the elastic range. To prevent a possible instability failure due to the excessive base deformations, active tendon system is implemented. Controlled base isolation response and the corresponding hysteresis loop for the base shear force are shown in Fig. 5. It is observed from Fig. 5 that if the active tendon forces are regulated based on the proposed control algorithm, the excessive isolation deformation is significantly reduced so that it works within the elastic range and instability failure can be prevented.

¨ ALDEM˙IR, GUNEY

44

2000 fs (kN)

fs 1 (kN)

5000

0

2

0

-5000 -2

-1

0

1

-2000 -4000 -2

2

-4000 -1.5

-1

-0.5

0

0.5

1

1

2

0

-2000 -1.5

1.5

-1

-0.5

0

0.5

1

1000 fs (kN)

2000 fs 5 (kN)

0

4

0

-2000

0

6

0

-2000 -1

-0.5

0

0.5

-1000 -2000 -1

1

-0.5

0

0.5

1

500 fs (kN)

fs7 (kN)

1000

0

8

0

-1000 -0.6

-0.4

40

30

-0.2 0 0.2 Relative displacement (cm)

-500 -0.3

0.4

-0.2 -0.1 0 0.1 Relative displacement (cm)

25

3

20

2.5

5

4

2

15

3

20

1.5

10

2

x bi (cm )

F s bi (kN)

0

5

F s bi (kN)

1

10

0.2

Interstorey drifts with and without control

Figure 4.

x bi (cm )

-1

2000 fs (kN)

fs 3 (kN)

2000

0.5

0

1

0

0

-5

-0.5

-10

-1

-10 -20

-30 0

2

4

6

time (sec)

Figure 5.

5.

-1

-15

8

10

-20 -40

-2

-1.5

-20

0

20

relative displacement (cm)

40

-2 0

2

4

6

time (sec)

8

10

-3 -2

-1

0

1

2

3

relative displacement (cm)

Uncontrolled/controlled time response of base isolation and the hysteresis loop

Conclusions

A simple integral type quadratic functional is introduced for the purpose of suppressing the seismic vibrations of nonlinear buildings. It is demonstrated by simulation results that the application of the proposed control to non-linear structures can prevent the structural failures and keep the uncontrolled structures within the elastic range. References Aldemir U., Bakioglu M., Akhiev S. S. (2001) Optimal control of linear structures, Earthquake Engineering and Structural Dynamics 30 835-851. Chen G. D., Wu J. N. (2001) Optimal placement of multiple tuned mass dampers for seismic structures, Journal of Structural Engineering ASCE 127 1054-1062. Park J., Reed D. (2001) Analysis of uniformly and linearly distributed mass dampers under harmonic and earthquake excitation, Engineering Structures 23 802-814. Pourzeynali S., Datta T. K. (2002) Control of flutter of suspension bridge deck using TMD, Wind and Structures 5 407-422. Rasouli S. K., Yahyai M. (2002) Control of response of structures with passive and active tuned mass dampers, Structural Design of Tall Buildings 11 1-14. Yang J. N., Akbarpour A., Ghaemmaghami P. (1987) New Optimal Control Algorithms for Structural Control, Journal of Engineering Mechanics ASCE 113 1369-1386.

VIBRATIONS OF DAMAGED 1D-3D MULTI-STRUCTURES ∗ ¨ ur Selsil3 Adolf G. Aslanyan1 , Janusz Dronka2 , Gennady S. Mishuris2 and Ozg¨ 1 Druzhba St. 5, kv. 355, Khimki, Moscow Region, 141400, Russia 2 Rzesz´ ow Univ. of Tech., Dept. of Mathematics, W. Pola 2. 35-959, Rzeszów, Poland 3 The Univ. of Liverpool, Dept. of Mathematical Sciences, L69 7ZL, Liverpool, UK

Abstract. The objective of the paper is to study the vibrations of 1D-3D nondegenerate multistructures, which are partly damaged. Such an analysis is useful to quickly asses the condition of a structure which has been in use over a long period of time. We look at three different positions of damage: top, middle or bottom of one leg. We calculate six eigenfrequencies associated with rigid body motion of the 3D structure and derive simple analytical asymptotic formulae for these eigenfrequencies. Hence, we analyse the effect of weakening of a leg. Accuracy of the asymptotic formulae is also checked with 3D finite element computations. Key words: multi-structures, partial damage, asymptotics, eigenfrequencies

1. Introduction A number of papers on eigenvalue problems posed for 2D-3D and 1D-3D multistructures have been published in recent years (Aslanyan et al., 2002, 2003, 2003). These papers were motivated by the asymptotic study of static and spectral problems associated with 1D-3D multi-structures (Kozlov et al., 1999). More recently the transition between nondegeneracy and degeneracy in 1D-3D multi-structures has been studied (Aslanyan et al., 2005). To our best knowledge, the only study on inhomogeneity in multi-structures is associated with high contrast mass densities between the 3D part and legs of a degenerate 1D-3D multi-structure (see the last chapter of Kozlov et al., 1999). The current problem is an extension to the study of structural damage on the vibrations of 1D-3D multi-structures (Aslanyan et al., 2005). We model the legs to have piecewise homogeneous sections. For the sake of simplicity, we consider only one of the legs to have this property. 2. General considerations for inhomogeneous nondegenerate multi-structures Here, we first formulate an eigenvalue problem of 3D linear elasticity, as described in Kozlov et al., 1999, and discuss the general asymptotics for the skeleton (pile structure) of a 1D-3D nondegenerate inhomogeneous multi-structure. ∗ supported by the EU Marie Curie grant (ToK scheme), contract number MTKD-CT-2004509809

45 .


ASLANYAN, DRONKA, MISHURIS, SELS˙IL

46

2.1. PROBLEM FORMULATION

We consider the following eigenvalue problem: −Lu = ρω2 u, ( j)

x ∈ Ω ∪Kj=1

Πε , [σ(n) (u)] = 0, [u] = 0, x ∈ S int , σ(n) (u) = 0, x ∈ ∂Ωε \ S ε , u = 0, x ∈ S ε , where L = µ∇2 + (λ + µ) grad div is the Lamé operator (λ and µ are the Lamé elastic moduli), u is the displacement vector, σ(n) (u) = σ n is the stress vector, ρ is the mass density, ω is the corresponding eigenfrequency and x = (x1 , x2 , x3 ) are the Cartesian coordinates in R3 . Here Ω is the 3D cap, ( j) Πε , j = 1, 2, . . . , K are the thin cylinders and S ε is the union of their base regions. S int is used for all internal surfaces where the material parameters have discontinuities. These discontinuities are denoted by, for example, [u] for the displacements. We assume that λ, µ and ρ are constants in each subdomain. 2.2. BRIEF DESCRIPTION OF THE GENERAL ASYMPTOTICS

We assume that the 3D body moves like a rigid body to the leading order, i.e. v = α + β × x in Ω, where α and β are constant vectors. We also assume that the displacements above are valid under the effect of body forces given by Ψ = c+d×x, where c and d are constant vectors. The dependency between the vectors v and Ψ can be established by obtaining the resultant force and the resultant moment (which appear in the equations of equilibrium) in terms of the vectors α, β, c and d (see, for example, Aslanyan et al., 2003). It was shown in Aslanyan et al., 2002 and 2003 that the associ spectral parameter

2 ated with the skeleton is found by t ≡ ρν = minv0 { Ω (v, Ψ(v)) dx/ Ω (v, v) dx}, which can be reduced to the following characteristic equation: det(Φ − t Γ) = 0. Here, Φ and Γ are 6 × 6 matrices ; they are symmetric and positive definite, and all the roots of the characteristic equation above coincide with the positive definite matrix Φ Γ−1 . Hence, they are all positive: 0 ≤ t1 ≤ t2 ≤ . . . ≤ t6 . Therefore, it is possible to approximate & the values of the first six eigenfrequencies of the multi-structure Ωε : fk = tk /ρ/(2π), k = 1, 2, . . . , 6. Here and in what follows ρ is associated with the 3D body. 3. A particular example: A 1D-3D multi-structure with six legs Here we study some particular configurations of a 1D-3D multi-structure. In addition, we also consider different spacing between the neighbouring legs. To simplify the finite element modelling, the 3D body is chosen as a parallelepiped of dimensions a1 , a2 , a3 . We assume that the legs are of length L and have b × b cross-sections. They are assumed to be homogeneous with density ρ, isotropic with Poisson’s ratio ν and Young’s modulus E, with the exception that the sixth leg is inhomogeneous, with Young’s modulus is given by: (6) E(y(6) 3 ) = {E 1 , 0 < y3 < L1 ,

E2 , L1 < y(6) 3 < L2 ,

E3 , L2 < y(6) 3 < L}.

(1)

We introduce a dimensionless normalised parameter ε = b/L (the characteristic ratio of the cross-section to the length of a leg) and use the following coordinates of the junction points: a(1) = (a1 /2, 0, −a3 /2 + p3 ), a(2) = (a1 /2, 0, −a3 /2 −

VIBRATIONS OF DAMAGED 1D-3D MULTI-STRUCTURES

47

q3 ), a(3) = (p1 , a2 /2, −a3 /2), a(4) = (−q1 , a2 /2, −a3 /2), a(5) = (0, p2 , 0), a(6) = (0, −q2 , 0), where 0 < pk , qk < ak /2, k = 1, 2, 3. ( j) Using the definitions of the stiffness coefficients Ci , i = 1, 2, 3, 4; j = 1, 2, . . . , 6, and the fact that C3(6) (see, for example, Kozlov et al., 1999) is much larger than the

L other stiffness coefficients, we define τ = ε21EL { 0 (6)dz }−1 . Taking into account C3 (z)

(1), to be able to model different types of damage of the leg, we consider the following cases: E2 = E3 = E, E1 = E∗ ; E1 = E2 = E, E3 = E∗ ; E1 = E3 = E, E2 = E∗ ,

(2)

E −1 where E∗ < E. Hence, we can write τ = [1 + ∆L L ( E∗ − 1)] , where ∆L denotes the length of the damaged part of the rod. The first case in (2) corresponds to the configuration when the bottom part of the rod is damaged, the second to the case when the top part of the rod is damaged and the third to the case when the middle part of the rod is damaged. It is clear that in all cases the length of the damaged part plays an important role. We note that ε is a small parameter connected with the geometry of the 1D-3D multi-structure, whereas τ is a parameter indicating the strength of the damage. It is clear that 0 ≤ τ ≤ 1, where τ = 1 means no damage and τ = 0 means that the weak part of the sixth rod is completely damaged. In fact, the latter one corresponds to a 1D-3D multi-structure with five legs only.

3.1. ASYMPTOTIC FORMULAE

Similar to the derivation in Aslanyan et al., 2002, 2003 and Aslanyan et al., 2005 analytic asymptotic formulae can be derived for certain six eigenfrequencies of the multi-structure. We note that four of these formulae are identical to the ones derived for a homogeneous multi-structure of the same geometry: f1,2

ε ∼ 2π

2EL ε , f3 ∼ ρΩ0 2π

24ELp21

ε , f4 ∼ 2π ρΩ0 (a21 + a22 )

24ELp23 ρΩ0 (a21 + a23 )

, f5,6

ε ∼ 2π

ELz5,6 (τ) , (3) ρΩ0

ε → 0, where Ω0 = a1 a2 a3 . In the above formulae, z5,6 are the roots of the 1 (a22 + a23 ), d1 = p22 + d2 , d0 = equation d2 z2 − (1 + τ)d1 z + d0 τ = 0, where d2 = 12 2 4p2 . Note that asymptotic formulae (3) are valid for all the cases described in (2). We would also like to emphasise that the formulae (3) can be justified when τ ≥ τ0 > 0, where τ0 is some small constant. In the next section, we discuss the range of applicability of these formulae. 3.2. NUMERICAL RESULTS AND DISCUSSION

For the numerical calculations, we consider the following fixed values of the parameters a1 = a2 = 108 cm, a3 = 60 cm, b = 4 cm , L = 200 cm, ∆L = 10 cm, E = 2.1 · 1012 g/(cm sec2 ), ρ = 7.8 g/cm3 , ν = 0.3, and p1 = p2 = q1 = q2 = (2 + 2p) cm, p3 = q3 = (2 + p) cm. Referring to Aslanyan et al., 2005, we note that the skeleton approximation and 3D finite element computations


48 [Hz] f 40 35

f(3) t 30

25

20

15 f(1) r

10

5

0 0

0.05

0.1

0.15

0.2

0.25 τ

Figure 1. Solid lines denote the eigenfrequencies f j , j = 1, . . . , 6 versus the parameter τ for the case when p = 11, obtained by the skeleton approximation, dotted lines denote the analytical asymptotic solution given in (3). The first eight eigenfrequencies obtained by direct 3D finite element calculations are denoted as follows: Star (*), right-sided triangle () and left-sided triangle () corresponds to damage localised in/at the middle, top and bottom of the leg, respectively.

differ in the region where the neighbouring legs are close to the ends of the 3D body, i.e. the parameter p is large, whereas the difference between the skeleton approximation and its asymptotics is negligible. However, when the neighbouring legs are close to each other (p is small), the skeleton approximation and 3D finite element computations almost perfectly match, but the skeleton approximation and its asymptotics differ considerably. This is due to the existence of a second small parameter (the parameter p) appearing in addition to ε, which needs to be taken into account while constructing the asymptotics. Hence, in this short study, we omit the numerical calculations associated with the small values of the parameter p. In Fig. 1, we present numerical computations for the case when p = 11. Here, solid lines denote the eigenfrequencies f j , j = 1, . . . , 6 obtained by the skeleton approximation versus the parameter τ, and dotted lines denote the analytical asymptotic solution given in (3). We show these solid curves only for the damage localised on the top of the leg. This is because these approximations are practically indistinguishable for the damage localised in other parts of the leg, namely in the

VIBRATIONS OF DAMAGED 1D-3D MULTI-STRUCTURES

49

middle and at the bottom. We underline that two of the eigenfrequencies obtained by using the skeleton approximation are very close to each other and their asymptotics are identical (see (3)). Hence, there are only five solid and five dotted curves visible in Fig. 1. It is clear from Fig. 3 that the skeleton approximation and its asymptotics are sufficiently close to each other. This is violated for fr(1) , the eigenfrequency corresponding to the rotation of the 3D body about the x1 −axis. The only other eigenfrequency affected by the damage appears to be ft(3) , the eigenfrequency corresponding to the translation of the 3D body along the x3 −axis. In addition to these, we denote the eigenfrequencies obtained by 3D finite element calculations for three different values of the damage parameter τ, i.e. τ = 0.01, 0.05, 0.25, by stars (*), right () and left () sided triangles. They correspond to damage localised in/at the middle, top and bottom of the leg, respectively. The shift between the symbols on the figure along the horizontal axis is only for presentational purposes. Here, the first eight eigenfrequencies are included since all these are within the range of six eigenfrequencies obtained by using the skeleton approximation and its asymptotics. Clearly, there is almost no difference between the cases when the damage is localised at the top or at the bottom of the leg. In fact, for six eigenfrequencies associated with rigid body motion of the 3D body, the computations indicate that the values of the eigenfrequencies associated with the damage localised in the middle of the leg are also close to the eigenfrequencies obtained for the other two damage configurations discussed above. At this point, we would like to clarify that the remaining two eigenfrequencies plotted in Fig. 1 (indistinguishable on the figure) are not due to rigid body translations and rotations of the 3D body but due to bending type vibrations of the damaged leg. The ends of this leg can be considered as fixed for these vibration modes, since the 3D body remains motionless. Therefore, it is also possible to find these eigenfrequencies analytically. It is evident from a physical point of view that these two eigenfrequencies computed for damage localised at the top or bottom will differ significantly with eigenfrequencies computed for the damage localised in the middle, when the damage parameter τ is small. This is clearly shown in the figure by a double arrow for τ = 0.01. We present numerical computations for the case when p = 19 in Fig. 2. We use the same style of curves and same symbols for finite element computations as in Fig. 1. As expected, the skeleton approximation and its asymptotics are closer to each other, compared with the case when p = 11. On the other hand, the finite element computations for certain eigenfrequencies differ more (see ellipses on the figure). The other effects are as discussed in the case when p = 11.


50 [Hz] f 45 40 35 30 25 20 15 10 5 0 0

0.05

0.1

τ

0.15

0.2

0.25

Figure 2. Solid lines denote the eigenfrequencies f j , j = 1, . . . , 6 versus the parameter τ for the case when p = 19, obtained by the skeleton approximation, dotted lines denote the analytical asymptotic solution given in (3). The first eight eigenfrequencies obtained by direct 3D finite element calculations are denoted as follows: Star (*), right-sided triangle () and left-sided triangle () corresponds to damage localised in/at the middle, top and bottom of the leg, respectively.

References Aslanyan, A. G., Movchan, A. B., Selsil, O. (2002) Vibrations of multi-structures including elastic shells: Asymptotic analysis. Euro. Jnl of Applied Mathematics 11, 109-128. Aslanyan, A. G., Movchan, A. B., Selsil, O. (2003) Estimates for the low eigenfrequencies of a multi-structure including an elastic shell. Euro. Jnl of Applied Mathematics14, 313-342. Aslanyan, A.G., Movchan, A.B., Selsil, O. (2003) Asymptotics for the first six eigenfrequencies of a 1D-3D multi-structure. Proceedings of IUTAM international symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics, Liverpool, UK, 221-228. Aslanyan, A.G., Dronka, J., Mishuris, G.S., Selsil, O. (2005) Vibrations of a weakly nondegenerate 1D-3D multi-structure. Submitted for publication to Archives of Mechanics. Aslanyan, A.G., Dronka, J., Mishuris, G.S., Selsil, O. (2005) The effect of structural damage on the vibrations of 1D-3D multi-structures. Proceedings of IIIrd Syposium on Damage in Materials and Structures, Augustow, Poland, 11-14. Kozlov, V.A., Maz’ya, V.G., Movchan, A.B. (1999) Asymptotic analysis of fields in multistructures, Oxford University Press.

SIZING OF A SPHERICAL SHELL AS VARIABLE THICKNESS UNDER DYNAMIC LOADS ¨ Uzman1 Nurcan As¸çı1 , Habib Uysal2 and Umit 1 G¨ umüs¸hane Engineering Faculty, KTÜ, 29000, Gümüs¸hane, Turkey 2 Engineering Faculty, Atat rk University, 25240, Erzurum, Turkey u¨

Abstract. In this paper, optimization of a spherical shell under various dynamic loads is investigated. The aim of this optimization problem is to minimize the volume of the shell. Design variables are corner thicknesses of each finite element. Constraints are stresses obtained from von Mises stress criterion not to exceed the yield stress in corner nodal points of each finite element at the top and bottom surfaces of shell and thicknesses are restricted not to be less than 2.5 mm. In addition to shell’s own weight, the vertical loads with equal intensity are applied at the nodal points on the upper edge of spherical shell, varying with respect to time function P(t). Time varying load vector is considered three different cases such as step, step after ramp and impulse functions.A program is coded with MapleV for optimization of spherical shell and finite element package program ANSYS is used for structural analysis. Obtained results are presented in graphical and tabular form. Finally, concluded remarks are given. Key words: size optimization, design variables, constraints, dynamic loads

1. Introduction Shell structures are used engineering areas such as water towers, tanks and other liquid containing structures, domes, silos and pressure vessels where optimization procedure have been successfully used. The first research on optimization of structure was done by Zienkiewicz and Campell (1973). They have discussed the problem of finding the optimum shape of two-dimensional structure known as shape optimization. Since then several researches have been contributed in this area (Haftka and Prasad, 1979), (Ding, 1986), (Gates and Accorsi, 193), (Soares et al., 1994), (Pourazady and Fu, 1996) .The growing interest on this subject reflects the importance of the effect of structural shape on structural performance.

51 .

E. Inan and A. Kırıs¸ (eds.), Vibration Problems ICOVP 2005, 51– 56. c 2007 Springer.

52

AS¸ÇI, UYSAL, UZMAN

2. Optimization 2.1. MATHEMATICAL STATEMENT OF AN OPTIMIZATION PROBLEM

Optimization problem is generally defined as the determination of the design variables to minimize F objective function under gi constraints. This problem is stated mathematically as: Find → minF(S ) Constraints → gi (S ) ≥ 0 i=1,2,...,nv S kL ≤ S k ≤ S kU

(1)

k=1,2,...,n

In here F is the objective function, gi ; i=1,2,...,nv are the constraint functions,nv is the number of constraints, S kL and S kU are the lower and the upper limits of the design variable S k , respectively and n is the number of design variables. 2.2. PROBLEM DEFINITION

A spherical shell of radius 10 m and uniform thickness of 25 mm with an apex cut of radius 2.5 m is considered (Asci 2004). Spherical shell made of steel material is used for analysis. It was assumed that the material is isotropic, linear elastic and Mindlin-Reissner shell theory is considered. The Modulus of Elasticity, Poisson ratio, density are taken 2×1011 N/m2 , 0.3 and 7860 kg/m3 , respectively. The finite element mesh is generated using SHELL93 element. Geometry, loading condition and SHELL93 element are given in Fig. 1.

4

z r, T, M)

O

L

3 K7

8

N

SHELL93 P 2

M x

T

Figure 1.

J

y 1

6

M 5

Geometry of spherical shell, loading condition and SHELL93 element

The region (radius 10 m, 0 ≤ θ ≤ 900 and 0 ≤ ϕ ≤ 75.50 ) consisting of intersection of xy, yz and zx planes and a plane with 14.50 angle to z axis is considered. Only, it is supported along lower edge points of the shell at the xy plane as fixed.

SIZING OF A SPHERICAL SHELL

53

2.3. LOADING CONDITION

In this study, in addition to shell’s own weight, the vertical loads with equal intensity are applied at the nodal points on the upper edge of spherical shell, varying with respect to time function P(t) . Time varying load vector is considered three different cases such as step, step after ramp and impulse functions (Fig. 2).

0 ® ¯P

t0 0 d t d tf

P (t )

P (t)

P(t)

P(t )

Step After Ramp Loading

P

0.00

0.50 1.00 Time (s)

Figure 2.

° 0 ° t ®P ° tv °¯ P

t

P (t )

tv

tv d t d t f

P

0.00

Impulse Loading

td0

P(t)

Step Loading

0.50

1.00

Time (s)

0 ° ®P °0 ¯

t0 0 d t d tv tv d t d t f

P

0.00

0.50 1.00 Time (s)

The varying P(t) load vector for three load cases

Vertical loads with equal intensity re applied at the nodal points on the upper edge of spherical shell. For P(t) function, concentrated loads are taken to be 2 × 103 N for step and step after ramp loadings and 10 × 103 N for impulse loading. 2.4. OBJECTIVE FUNCTION, DESIGN VARIABLES AND CONSTRAINT FUNCTIONS

In this study, the objective function is to minimize weight (minimizing volume) of spherical shell. Totally 81 design variables which are thicknesses of corner nodal points of each finite element are considered independently from each other (Fig. 3). S9 SS8 S18 SSS67 SS1617 S27 S15 S S14 S36 S13 S 26 S2425 S35 S10 S11S12 S23 S45 S34 S 22 S S S19 S20 21 S32 33 S S44 S54 43 S31 S53 S42 S28 S29 S30 S41 S52 S63 S40 S62 S S 51 S37 S38 39 S61 S72 S50 S49 S60 S71 S48 S59 S46 S47 S70 S81 S58 S69 S80 S57 S56 S68 S55 S79 S67 S78 S65 S66 S64 S77 S76 S75 S S S1S2S3 4 5

73

Figure 3.

74

Considered design variables for spherical shell

The constraint functions are stresses (σe ) obtained from von Mises stress criterion not to exceed the yield stress(σe ≤ σ0 = 220 MPa) in corner nodal points

54


of each finite element at the top and bottom surfaces of shell and the thicknesses are restricted not to be less than 2.5 mm. σe = [σ2x + σ2y + σ2z − σ x σy − σ x σz − σy σz + 3(τ2xy + τ2xz + τ2yz )](1/2)

(2)

In the optimization process consist of such as description of geometric model, dynamic analyze, mesh generation, sensitivity analysis and mathematical programming method, respectively. The above steps are repeated until convergence is achieved using iterative convergence strategies. 3. Numerical examples At the beginning of the optimization process design variables are chosen as 0.025 m. In this case, spherical shell’s volume is obtained 3.802m3 . In Fig. 4, displacement time curves are illustrated for three different load functions. 2.00 Step Step After Ramp

Displacement (m)

1.50

Impulse

1.00

(Uz)max (Uz)max

0.50

(Uz)max

0.00 0.0

0.1

0.2

0.3

0.4

-0.50

Figure 4.

0.5 0.6 Time (s)

0.7

0.8

0.9

1.0

Displacement time curves for three load cases at the first iteration

According to Fig. 4, maximum displacements are occurred 1.161 m at 0.525 s for step loading, 1.129 m at 0.62 s for step after ramp loading and 0.931 m at 0.37 s for impulse loading at 1 st and 9th design variables at the first iteration. In Fig. 5, displacement time curves for step loading are given. Displacement (m)

1.20

I

1.10 1.00 0.90 0.80

II

IV III

0.70 0.60 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60

Time (s)

Figure 5. Displacement-time curves for several sequential optimization processes for step loading

It can be seen from Fig. 5 the time values for maximum displacement and optimum volumes approach each other after several sequential optimization processes. For step loading, optimization time is considered as 0.525th s. Also, time is obtained 0.44 s, 0.47 s, 0.465 s and maximum displacements are obtained 0.875 m,

55

SIZING OF A SPHERICAL SHELL

0.795 m, and 0.838 m for II, III and IV curves, respectively. For step after ramp loading, optimization time is considered as 0.62nd s . In Fig. 6 time values for maximum displacements are given for step after ramp loading. Displacement (m)

1.20

I

1.10 1.00 II III

0.90 0.80 0.70 0.60

0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70

Time (s)

Figure 6. loading

Displacement time curves for several sequential optimizations for step after ramp

In Table 1, optimum volumes and design variables are given for three load cases at the end of the optimization process. TABLE I. Obtained optimum volumes and design variables for three load cases at the end of the optimization process Design Variables (m)

Iteration Number S tep A f ter Ramp L. 30 45 Opt.

30

S tep L. 45

Impulse L. 30 45 Opt.

Opt.

S 3, S 7 S 4, S 6 S5 S 11 , S 17 S 20 , S 26 S 28 , S 36 S 37 , S 45 S 46 , S 54 S 64 , S 72 S 74 , S 80

.0467 .0481 .0518 .0597 .0711 .0636 .0664 .0577 .0745 .0529

.0468 .0478 .0514 .0592 .0699 .0642 .0663 .0575 .0743 .0569

.0468 .0478 .0513 .0592 .0698 .0642 .0663 .0575 .0743 .0570

.0444 .0467 .0503 .0584 .0712 .0646 .0686 .0606 .0739 .0352

.0450 .0469 .0505 .0581 .0704 .0648 .0685 .0605 .0736 .0386

.0450 .0469 .0505 .0580 .0704 .0649 .0685 .0605 .0736 .0388

.0779 .0789 .0802 .0884 .1125 .0519 .0614 .0369 .0438 .0135

.0767 .0749 .0785 .0858 .1039 .0560 .0696 .0478 .0418 .0088

.0774 .0768 .0806 .0854 .1040 .0565 .0681 .0471 .0416 .0084

Volume(m3 )

1.788

1.781

1.781

1.790

1.787

1.786

1.983

2.004

2.007

Because of loading condition and geometry are symmetric, design thicknesses are also obtained as symmetric. The objective function became stable at the sequential optimization steps and finally changes in the objective function approach zero. In Fig. 7, time values for maximum displacements are given for impulse loading and optimization time is considered as 0.37th s for this loading.

56


Displacement (m)

1.00 I 0.90 II III 0.80 0.70 0.60 0.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

Time (s)

Figure 7. loading

Displacement-time curves for several sequential optimization processes for impulse

4. Conclusions In this study, optimization of spherical shell is investigated under dynamic loading. The stresses are evaluated at the top and bottom surfaces of the spherical shell at each optimization process. Then, von Mises criterion is compared with yield stress in corner nodal points of finite elements. In some points, the thicknesses are increased in case criterion is higher than the yield stress; otherwise they are decreased in these points by program. The changes in the thicknesses of shell affect the rigidity of shell, the natural frequency of shell and the time values for maximum displacement. In other points, the minimum thickness criterion is active. At the end of the optimization process, it is concluded that the stress criterion is active in the near points of the shell edges, and the thickness criterion is active in the far points from the shell edges. So, the optimum shell is resulted as a shell strengthened at the edges. Hence, it can be considered as a spherical shell supported by beams at the edges. References Asci, N. (2004) Optimization of a Spherical Shell with Variable Thickness Under the Various Dynamic Loads, Ph.D. Thesis, Department of Civil Engineering, Karadeniz Technical University, Turkey. Ding Y. (1986) Shape Optimization of Structures: A Literature Survey, Computers and Structures 24 985-1004. Gates, A. A., Accorsi, M. L. (1993) Automatic Shape Optimization of Three Dimensional Shell Structures with Large Shape Changes, Computers and Structures 49 167-178. Haftka, R. T., Prasad, B. (1979) Programs for Analysis and Resizing of Complex Structures, Computers Structures 10 323-330. Mota Soares, C. M., Mota Soares, C. A., Barbosa, J. I. (1994) Sensitivity Analysis and Optimal Design of Thin Shells of Revolution, AIAA Journal 32 1034-1042. Pourazady, M. Fu, Z. (1996) An integrated Approach to Structural Shape Optimization, Computers and Structures 60 279-289. Zienkiewicz, O. C. and Campbell, J. S. (1973) Shape Optimization and Sequential Linear Programming. In Optimum Structural Design, John Wiley, New York.

VIBRATION ANALYSIS OF SIMPLY SUPPORTED FUNCTIONALLY GRADED BEAMS Metin Aydo˘gdu and Vedat Tas¸kın Department of Mechanical Engineering, Trakya University, 22030, Edirne,Turkey

Abstract. In this study, free vibration of simply supported functionally graded (FG) beam was investigated. Young modulus of beam is varying in the thickness direction according to power law and exponential law. Governing equations are found by applying Hamilton’s principle. Navier type solution method was used to obtain frequencies. Different higher order shear deformation theories and classical beam theories were used in the analysis. Results were given for different material properties and different slenderness ratios. Key words: functionally graded beam, vibration, shear deformation theories

1. Introduction Use of structures like beams, plates and shells, which are made from FG materials, is increasing because of smooth variation of material properties. Compared with FG plates and shells, studies for FG beams are relatively less (Sankar, 2001; Vel and Batra, 2002). Different theories can be used to analyse structural members. For laminated composite plates, some higher order shear deformation theories (HSDT) are developed by Reddy, 1984; Soldatos, 1992 and Karama et al., 2003. Generalization of these theories is made by Soldatos and Timarci, 1993. This new theory called Unified Shear Deformation Theory (USDT) has been applied to vibration of composite cylindrical shells by Timarci and Soldatos, 1995. In the present study, free vibration of simply supported FG beams investigated by using Classical Beam Theory (CBT) and USDT. Firstly, governing equation of FG beam for free vibration problem are found by applying Hamilton principle and Navier type solution method is used to solve vibration problem. It is assumed that elasticity modulus is changing in the thickness direction and all other material properties are taken to be constant. Variation of elasticity modulus in the thickness direction is taken in two different forms like power product form and exponential form. Different shear deformation theories are used in USDT when finding frequencies. Frequency results are given for FG beams for different material properties and different length to thickness ratios and some mode shapes are also given for different material properties. 57 .


˘ AYDOGDU, TAS¸KIN

58 2. Analysis

A functionally graded beam of constant thickness of h with cross-section dimensions L and b is considered. It is assumed that the Young modulus of FG beam vary through the thickness of the beam according to exponential and power law form. The exponential law, which is generally used in fracture studies (Kim and Paulino, 2002) is given by Et 1 (−δ(1−2z/h)) E(z) = Et e δ = ln (1) 2 Eb and the power law introduced by Wakashima et al., 1990 is given by

z 1 E(z) = (Et − Eb ) + h 2

k + Eb

(2)

Et and Eb denote values of the elasticity modulus at top and bottom of the beam, respectively, and k is a variable parameter. According to this distribution, bottom surface (z=-h/2) of functionally graded beam is pure metal, the top surface (z=h/2) is pure ceramics, and for different values of k one can obtain different volume fractions of metal. The state of stress in the beam is given by the generalised Hooke’s law as follows: σ x = Q11 ε x ,

τ xz = Q55 γ xz ,

(3)

where Qi j are the transformed stiffness constants in the beam co-ordinate system and defined as: Q11 =

E(z) 1 − v2

Q55 =

E(z) 2(1 + v2 )

(4)

Assuming that the deformations of the beam are in the x-z plane and denoting the displacement components along the x, y and z directions by U, V and W respectively, the following displacement field for the beam is assumed on the basis of the general shear deformable shell theory presented by Soldatos and Timarci, 1993: U(x, z; t) = u(x; t) − zw,x + Φ(z)u1 (x; t), V(x, z; t) = 0, W(x, z; t) = w(x; t),

(5)

Here u and w represent middle surface displacement components along the x and z directions respectively, while u1 is unknown function that represents the effect of transverse shear strain on the beam middle surface, and Φ represents

VIBRATION ANALYSIS OF FUNCTIONALLY GRADED BEAMS

59

the shape function determining the distribution of the transverse shear strain and stress through the thickness. CBT is obtained as a particular case by taking the shape function as zero. Although different shape functions are applicable, only the ones which convert the present theory to the corresponding Parabolic Shear Deformation Beam Theory (PSDBT), First Order Shear Deformation Beam Theory (FSDBT) and Exponential Shear Deformation Beam Theory (ESDBT) are employed in the present study. This is achieved by choosing the shape functions as follows: 2 2 FS DBT : Φ(z) = z, PS DBT : Φ(z) = z(1 − 4z /3h ), ES DBT : Φ(z) = z exp −2(z/h)2

(6)

By substituting the stress-strain relations into the definitions of the force and moment resultants of the theory given in Soldatos and Timarci, 1993 and using the notation given by Soldatos and Sophocleous, 2001, the following constitutive equations are obtained:    c   c  N x   A11 Bc11 Ba11   u,x   M c  =  Bc Dc Da   −w,xx  (7) 11 11     ax   11 Mx Ba11 Da11 Daa u 1,x 11 Qax = Aa55 u1

(8)

The extensional, coupling and bending rigidities appearing in Eq. (7) are, respectively, defined as follows: # h/2 c A11 = Q11 dz, −h/2

Bc11 , Ba11

# =

# Dc11 , Da11 , Daa 11 =

h/2 −h/2

h/2 −h/2

, Q11 z, φ(z) dz,

Q11 z2 , zφ(z), φ2 (z) dz,

(9)

Moreover, the transverse shear rigidity appearing in Eq.(8) is defined according to # h/2 , a A55 = Q55 φ (z) 2 dz. (10) −h/2

It should be pointed out that the extensional Ac11 , coupling Bc11 and bending c D11 rigidities are the ones usually appearing even in the CBTs. Among the additional rigidities in Eq.(7), the one denoted as Ba11 is considered as additional coupling rigidity while the ones denoted as Da11 and Daa 11 are considered as additional bending rigidities.


60

Upon employing the Hamilton’s principle, the three variationally consistent equilibrium equations of the beam are obtained as: c = (ρ u + ρ u − ρ w ) , N x,x 0 01 1 1 ,x ,tt c = (ρ1 u,x + ρ11 u1,x + ρ0 w − ρ2 w,xx ),tt , M x,xx a − Qa = (ρ u + ρ u − ρ w ) , M x,x 01 02 1 11 ,x ,tt x

(11)

Here ,tt denotes time derivatives and the ρ’s are defined as: h

h

#2 ρi =

#2 ρzi dz, (i = 0, 1, 2), ρ jm =

−h 2

ρz j Φmj dz, ( j = 0, 1; m = 1, 2).

(12)

−h 2

and the force and moment components: #h/2 (N xc ,

M xc )

=

#h/2 σ x (1, z)dz, Qax

−h/2

=

τ xz φ

#h/2 (z)dz, M xa

−h/2

=

σ x φ(z)dz. −h/2

(13)

The simply supported boundary conditions are given as follows: N xc = w = M xc = M xa = 0

(14)

Navier-type solutions to Eq.(11) that satisfy the boundary conditions Eq.(14) can be expressed in the form of u = Am cos

mπx mπx mπx sin ωt, L u1 = Bm cos sin ωt, w = Cm sin sin ωt (15) L L L

where m is the half wave number in the x direction and Am , Bm and Cm are undetermined constant coefficients. Inserting Eq.(15) in Eq.(11) leads to following eigen-value problem: (K − λ2 M) {∆} = 0

(16)

here K and M are stiffness matrix, inertia matrix, respectively and ∆ is column vector of unknown coefficients and λ is free vibration frequencies. 3. Results and discussions The constituent material properties of the FG beam were chosen as follows: Al: Em =70GPa, νm = 0.3, Ceramic: Ec =380GPa, νc = 0.3

VIBRATION ANALYSIS OF FUNCTIONALLY GRADED BEAMS

61

and constant density is assumed for FG beam. Frequency parameter is non dimensionalized as ) ωL2 ρm λ= . (17) h Em For given material properties δ = 0.846 was found and different k values can be chosen. Non-dimensional frequency parameters were given for fundamental frequency and fifth frequency in Table 1 and Table 2 for L/h=5 respectively for different theories and for different material distributions. It is seen from tables that since increasing k leads to approach to metal behaviour, frequency parameter is decreasing with increasing k and increasing with increasing L/h ratios. Since deformations are increasing with increasing mode number, difference between CBT and shear deformation theories for fifth mode is greater than the difference for the first mode. TABLE I. Comparison of frequency parameter with different theories for different material distribution (L/h=5, fundamental mode). Theory

k=0

k=0.1

k=1

Exp

k=2

k=10

Metal

PSDBT ESDBT FSDBT CBT

6.5742 6.5849 6.5633 6.8470

6.2483 6.2581 6.2371 6.4999

4.6595 4.6659 4.6528 4.8216

4.2667 4.2725 4.2639 4.4235

4.1037 4.1091 4.1019 4.2517

3.5482 3.5539 3.5639 3.7372

2.8216 2.8262 2.8169 2.9387

TABLE II. Comparison of frequency parameter with different theories for different material distribution (L/h=5, fifth mode). Theory

k=0

k=0.1

k=1

Exp

k=2

k=10

Metal

PSDBT ESDBT FSDBT CBT

92.7810 94.0910 91.1638 128.8669

88.593 89.810 87.019 122.12

67.088 67.916 65.946 86.409

60.5028 61.2693 59.6582 77.9335

58.2308 58.9629 57.4233 74.0286

46.2902 46.9512 46.7166 65.9794

39.821 40.383 39.127 55.309

62


4. Conclusions This study dealt with free vibration of simply supported FG beams. Young modulus of beam assumed varies in the thickness direction according to power law and exponential law. Governing equations are found by applying Hamilton’s principle. Navier type solution method was used to obtain frequencies. Different Higher Order Shear Deformation Theories and Classical Beam Theories were used in the analysis. It is found that CBT gives higher frequencies especially for higher modes. This study can be extended to study different boundary conditions and temperature effects can be included in the analyses.

References Karama, M., Afaq, K. S., Mistou, S. (2003) Mechanical Behaviour of Laminated Composite Beam by the New Multi-Layered Laminated Composite Structures Model with Transverse Shear Stress Continuity, Int. J. of Solids and Struct 40 1525-1546. Kim, J., Paulino, G. H. (2002) Finite Element Evaluation of Mixed Mode Stress Intensity Factors in Functionally Graded Materials, International Journal for Numerical Methods in Engineering 53 1903-1935. Reddy J. N. (1984) A Simple Higher-Order Theory for Laminated Composite Plates, Journal of Applied Mechanics 51 745-752. Sankar B. V. (2001) An Elasticity Solution for Functionally Graded Beams, Composites Science and Technology 61 689-696. Soldatos K. P. (1992) A Transverse Shear Deformation Theory for Homogeneous Monoclinic Plates, Acta Mechanica 94 195-220. Soldatos, K. P., Timarci T. (1993) A Unified Formulation of Laminated Composite, Shear Deformable, Five Degrees of Freedom Cylindrical Shell Theories, Composite Structures 25 165-171. Soldatos, K. P., Sophocles C. (2001) On Shear Deformable Beam Theories: The Frequency and Normal Mode Equations of the Homogeneous Orthotropic Bickford Beam, J. of S. and Vibration 242 215-245. Timarci T., Soldatos, K. P. (1995) Comparative Dynamic Studies for Symmetric Cross-Ply Circular Cylindrical Shells on the Basis of a Unified Shear Deformable Shell Theory, Journal of S. and Vibration 187 609-624. Vel S. S., Batra R. C. (2002) Exact Solution for the Cylindrical Bending Vibration of Functionally Graded Plates, Proceedings of the American Society of Composites, Seventh Technical Conference, Purdue University West Lafayette, Indiana. Wakashima K., Hirano T., Niino M. (1990) Space Applications of Advanced Structral Materials, ESA, SP303:97.

DAMPING EFFECTS OF RUBBER LAYER IN LAMINATED COMPOSITE CIRCULAR PLATE DURING FORCED VIBRATION Aysun Baltacı, Mehmet Sarıkanat and Hasan Yıldız ˙ Mechanical Engineering Department, Ege University, Izmir, Turkey

Abstract. In this study, the effects of the location of the rubber between composite layers which is located to supply damping on the vibration frequency of laminated composite circular plates having different fiber directions are investigated by using the finite element method. The laminated composite circular plate having free outer edge is manufatured as having a central hole. Hybrid composite circular plate is clamped at the inner edge and subjected to transverse impuls load on a location at the outer edge. The best location of the rubber layer in terms of the damping capability are determined by using finite element analysis. Key words: damping, forced vibration, composite circular plate

1. Introduction Damping is a usefull for controlling vibration and movement in the design of the structure. The plate with the constrained damping layer treatment is well known to have high damping capasity and high resistance to resonant noside and vibration. Also, numerous researchers have developed the vibration and damping properties of basic structures, such as beams, rectangular and circular plates with constrained damping layer treatment in laminated composites. Haddad and Feng (2003) developed Quasi-static models by using a forced balance approach, to define the effects of selected microstructural parameters, on the damping and stiffness of a class of poliymeric, discontinuous fibre composite systems. Chen and Levy (1999) discussed the temperature effects on frequency, loss factor and control of a flexible beam with a constrained viscoelastic layer and shape memory alloy layer. In that study, they also discussed the effects of damping layer shear modulus and damping layer height as affected by the temperature. The axisymmetric vibration and damping of rotating annular plates with constrained damping layer treatments are analyzed by Wang and Chen (2004). In the present work, the effects of the location of the rubber between composite layers which is located to supply damping on the vibration frequency of laminated composite circular plates having different fiber directions are investigated by using the finite element method.

63 .


64

BALTACI, SARIKANAT, YILDIZ

2. Transient analysis Transient dynamic analysis (sometimes called time-history analysis) is a technique used to determine the dynamic response of a structure under the action of any general time-dependent loads. You can use this type of analysis to determine the time-varying displacements, strains, stresses, and forces in a structure as it responds to any combination of static, transient, and harmonic loads. The time scale of the loading is such that the inertia or damping effects are considered to be important. If the inertia and damping effects are not important, you might be able to use a static analysis instead. The basic equation of motion solved by a transient dynamic analysis is Mu¨ + Cu˙ + Ku = F(t)

(1)

Where M = mass matrix, C = damping matrix, K = stiffness matrix, u¨ = nodal acceleration vector, u˙ = nodal velocity vector, u = nodal displacement vector, F(t) = load vector. At any given time, t, these equations can be thought of as a set of ”static“ ¨ and damping equilibrium equations that also take into account inertia forces (Mu) ˙ forces (Cu). 3. Results and discussion Figure 1-3 show the effects of the position of the rubber on the time response of the composite circular plate. As seen from figures, when the rubber is located at the middle layer, the shortest damping time is obtained. Figure 4 shows the effect of the position of the rubber on the maximum amplitude of vibration of the composite circular plate having [0/0]2s fiber orientation for different rubber thickness. Figure 5 shows the effect of the position of the rubber on the maximum amplitude of vibration of the composite circular plate for different fiber orientations. Figure 6 and 7 show the effect of the rubber thickness on the maximum amplitude and damping time of composite circular plate, respectively. As seen from figures, the maximum amplitude and damping time desrease as the thickness of the rubber increases. 4. Conclusions The maximum amplitude of the vibration decreases as the thickness of the rubber increases. The damping time decreases as the thickness of the rubber increases. There is no effect of the fiber orientation on the damping time. The maximum amplitude of the vibration decreases as the location of the rubber gets closer to the middle axis of the laminated composite. Amplitude response is symmetric

DAMPING EFFECTS OF RUBBER LAYER

65

Figure 1. The effects of the position of the rubber on the time response of the composite circular plate (rubber is pleased top of the composite plate)

Figure 2. The effects of the position of the rubber on the time response of the composite circular plate (rubber is pleased middle of the composite plate)

66


Figure 3. The effects of the position of the rubber on the time response of the composite circular plate (rubber is pleased botton of the composite plate)

Figure 4. The effect of the position of the rubber on the maximum amplitude of vibration of the composite circular plate having [0/0]2s fiber orientation for different rubber thickness

DAMPING EFFECTS OF RUBBER LAYER

67

Figure 5. The effect of the position of the rubber on the maximum amplitude of vibration of the composite circular plate for different fiber orientations

Figure 6. plate

The effect of the rubber thickness on the maximum amplitude of composite circular

68

Figure 7.


The effect of the rubber thickness on the damping time of composite circular plate

with respect to the middle axis. When the rubber is located as the top or the bottom layer of the composite structure: the structure produces different vibration pattern in response to impulse input, damping time reaches to a maximum value. Difference between damping times decreases when the rubber is located at the other layers. The shortest damping time is obtained when the rubber is located at the middle layer. References Chen Q. and Levy C. (1999) Vibration analysis and control of flexible beam by using smart damping structures, Composites: Part B 30, 395-406. Haddad Y. M. and Feng J. (2003) On the trade-off between damping and stiffness in the design of discontinuous fbre-reinforced composites, Composites: Part B 34, 11-20. Wang, H. J. and Chen L. W. (2002) Vibration and damping analysis of a three-layered composite annular plate with a viscoelastic mid-layer, Composite Structures 58, 563-570. Wang H. J. and Chen L. W. (2004) Axisymmetric vibration and damping analysis of rotating annular plates with constrained damping layer treatments, Journal of Sound and Vibration 271, 25-45.

A CRITICAL STUDY ON THE APPLICATION OF CONSTANT DEFLECTION CONTOUR METHOD TO NONLINEAR VIBRATION OF PLATES OF ARBITRARY SHAPES M. M. Banerjee Retired Reader, Department of Mathematics, A.C.College, Jalpaiguri-735101, W. B., India [202 Nandan Apartment, Hillview (N), S B Gorai Road, Asansol-713 304, W. B., India]

Abstract. The present work attempts to utilize the Constant Deflection Contour Method in the investigation of nonlinear vibration of thin elastic plates. An attempt has also been made to use conformal mapping technique for plates having uncommon or complex boundary. The usual boundary conditions have also been transformed accordingly. The equations for some iso-deflection curves of practical interest have been presented here for further investigations in conjunction with the present theory. Key words: constant deflection method, conformal mapping, vibration of plates

1. Introduction (Leissa, 1969) in his book provides a detailed study of linear problems and numerous references of significant contributions, within the linear framework, may be found. The paucity of literature concerning nonlinear (large amplitude) vibration analysis is, probably, due to the fact that the two basic Von Kármán field equations extended to the dynamic case, involved the deflection and the stress functions in a coupled form. For solution of plate problems having uncommon boundaries is a difficult task and a more complicated one when geometrical nonlinearities are involved. (Nowinski and Ohanabe, 1972) made some reservations on the free hand use of this method. Recently a new idea has been put forward by (Banerjee, 1997) to study the dynamic response of structures of arbitrary shapes based on “Constant Deflection Contour” (CDC) method. (Mazumdar and others, 1975, 1978, 1979, 1997) have previously developed the method. However, the application of this method has been restricted to linear cases only. The present paper aims at extending this method to the nonlinear analysis of plates of arbitrary shapes vibrating at large amplitude. The analysis carried 69 .


70

BANERJEE

out in this paper may readily be applied to other geometrical structures and as a byproduct the static deflection is also obtainable. A combination of the constant deflection contour method and the Galerkin procedure is employed. 2. Some preliminary remarks on the CDC method When the plate vibrates in a normal mode then at any instant tθ , the intersections between the deflected surface and the parallels z = constant, will yield contours which after projection onto z = 0 surface are a set of level curves, u(x, y) = constant, called the “Lines of Equal Deflections” (Banerjee, 1976), which are isoamplitude contours. The boundary of the plate, irrespective of any combination of support, is also a simple curve Cu belonging to the family of lines of equal deflections As defined by (Mazumdar, 1970) the family of nonintersecting curves may be denoted by Cu , for Cu , 0 ≤ u ≤ u , so that Cu (u = 0) is the boundary and Cu coincides with the point(s) at which the maximum u = u is obtained. 2.1. GOVERNING EQUATIONS - A NEW APPROACH

Consider a thin elastic plate, which vibrates with moderately large amplitude in the transverse direction, under the action of a uniform load q. The usual procedure is to consider Karman type field equations extended to the dynamic case are D ∇4 w = h S (φ, w) + q − ρ h wtt

(1)

∇4 φ = −(E/2) S (w, w),

(2)

with usual notations and a suffix is taken as an indication of partial differentiation with respect to the implied variable and the operator S is defined by S (I, J) = I xx Jyy − 2 I xy J xy + Iyy J xx . As an alternative to deriving subsequent equations followed earlier (Banerjee and Rogerson, 2002), has already been treated where the deflection function and the stress function in the separable form (Yamaki, 1961) w(x, y, t)=h W(x, y) F(t),

φ(x, y, t)=h Φ(x, y) F 2 (t)

(3)

where F(t) is an unknown function of time to be determined. Let us now make the following transformation ∂w dw d2 w dw dw d2 w dw =w x = u x , w xx = 2 u2x + u xx , wy = uy , w xy = 2 u x uy + u xy etc. ∂x du du du du du du (4)

71

CONSTANT DEFLECTION CONTOUR METHOD

With transformations exemplified by those shown above, Eqns.(1) and (2) are transformed to 4

5−i

λi ddu5−iW hF(t) i=1 2 d2 Φ dW dW dΦ 3 2 ¨ ¨ + λ + λ = h3 λ5 dduW2 dΦ 6 7 du du du F (t) + q − ρh W F, F = du2 du D

4 i=1

( ' dW d5−i W d2 W dW + λ9 λi 5−i = Eh λ8 2 du du du du

d2 F dt2

(5)

(6)

in which λi (i = 1 − 9) are functions of partial derivatives of u only. Eqns. (5) and (6) can then be transformed into a set of equations (Banerjee and Rogerson, 2002) as hF(t)

4 i=1

ρh2 f16 d4−i W h3 dW dΦ 3 ¨ f15 F (t) + q+ f17 F(t) f1i 4−i + 2D du du D D du

dW dΦ , , . . . = 0 or, and Λ2 Φ, du du

4 i=1

#u Wdu0 = 0, (9) u

2 dW d4−i Φ Eh g15 g1i 4−i + = 0 (10) 2 du du

u

where f1i and g1i are functions of u and only. To avoid the integral appearing in Eqn.(9), (Jones and Mazumdar, 1997) have taken the derivative of it making the equation of order four again, viz., hF(t)

5 i=1

d5−i W ρh2 f26 3 f25 d dW dΦ ¨ = 0 (11) F 3 (t) + q+ f27 W F(t) f2i 5−i + h 2D du du du D D du

Eqns. (9) and (10) or Eqns. (10) and (11) may be utilized to form the basic equations governing the motion of the structure. 2.2. SUPPORTING CONDITIONS

Different types of boundary conditions have also been considered in our previous paper (Banerjee and Rogerson, 2002). Their transformations to u−variables have already been established therein and hence for brevity the readers may be referred to (Chakraborty et al., 1997). 3. The line of equal deflection -u(x, y) It is very important to know the equation to the line of equal deflection but it is not always possible to know the exact eqution of the line of equal deflection.

72

BANERJEE

3.1. SOME KNOWN EQUATIONS FOR LINE OF CONTOUR LINES Shapes of structures

u(x, y)

(a) Elliptic plates

u(x, y) = 1 − (x/a)2 − (y/b)2

(b) Circular plates

u(x, y) = a2 − x2 − y2

(a, b) length of the semi axes Annular plates

u = a2 − b2 on the inner boundary

a, b are the outher and inner radii

u = 0 on the outher boundary

Equilateral Triangular plates (a) Clamped : (b) Simply-supported :

1 x3 − 3xy2 − ax2 − ay2 + 274 a3 2a u(x, y) = x3 − 3xy2 − ax2 − ay2 + 274 a3 × 49 a2 − x2 − y2 u(x, y) =

Clamped right-angled

u(x, y) = a2 (x + y) − xy − 12 (x2 + y2 )

isosceles triangular plate

4 πa2

2

∞ n=0

(−1)n {sinh(Kn y) cos(Kn y)+sinh(K x y) cos(Ky y)} (2n+1)3 sinh(Kn a/2)

Doubly-curved shallow shells Elliptical in plan ; a, b are the semi-major and semi-minor axes of the elliptic base Shallow spherical shell on an equilateral triangular base

u(x, y) = 1 − (x/a)2 − (y/b)2 u(x, y) = x3 − 3xy2 − ax2 − ay2 + × 49 a2 − x2 − y2

4 3 a 27

Shallow spherical shell upon a square base a = the dimension of the base

u(x, y) = (x2 − a2 /4)(y2 − a2 /4)

4. Application of conformal mapping technique (Pombo et al., 1977) presented a very convenient tool to conformably transform the given domain into a simpler one, i.e., onto a unit circle for such complicated boundaries. Yet their problems are confined to linear in character only. The present author has made some honest attempt to extend this technique to plates exhibiting large deflections (Banerjee, 1976) and vibrating at large amplitudes (Laura and Chi, 1965). Introducing the complex coordinates z = x + iy, z = x − iy in conjunction with the concept of the present CDC-Method the basic governing equations (1-2) will take (on transformation to u−variables as performed earlier) further reduce to ( ' 4 d5−i w d dw dφ dw dφ 16D ) + 2l6 + q − ρhwtt li 5−i = −4h l5 ( (12) du du du du du du i=1

where


73

' 2 ( 4 dw 2 d5−i φ d w dw 16 + l6 ( ) li 5−i hF(t) = 4E l5 2 du du du du i=1

(13)

l1 = u2z u2z¯ , l2 = (u2z uz¯z¯ + 4uz uz¯ uz¯z + uzz u2z¯ ), l3 = (2u2z¯z + 2uz uz¯zz¯ + 2uz¯ uzz¯z + uzz uz¯z¯ ), l4 = uzz¯zz¯ , l5 = (u2z uzz − 2uz uz¯ uz¯z + uz¯z¯ u2z ), l6 = (uzz uz¯z¯ − u2z¯z ).

(14)

Now, if there be a functional relation z = f (ς) which maps the given shape of the plate in the z− plane into a unit circle in the ς− plane, one obtains equations (12-13) more or less in the same form. The only difference is that the li ’s are substantially changed to, say li/ dependent on ς, ς, ¯ namely, 16D

' ( 4 d5−i w d dw dφ dw dφ ) + 2l6 li 5−i = −4h l5 ( + q − ρhwtt du du du du du du i=1 ' 2 ( 4 5−i φ d d w dw dw 2 + l6 ( ) li 5−i hF(t) = 4E l5 2 16 du du du du i=1

(15)

(16)

¯ z z¯ and where li/ are functions of derivatives of ς, ς¯ and and z = f (ς), z¯ = f (ς), iθ ς(= re ), ς¯ are complex conjugate in their complex planes, respectively. 5. Method of solution a) For the real plane Considering (9) and (10) as the basic equations with appropriate boundary conditions, it starts with finding the exact or approximate solution for Φ from Eqn. (10). For nonlinear analysis one may have to seek an approximate solution for which the form of the deflected function must be first assumed compatible with the boundary conditions. Next we solve for Φ from equation (10) in conjunction with a Galerkin procedure. With this expression for Φ and previously assumed form of W yield an ordinary time differential equation when the Galerkin procedure is applied again. Let u(x, y) = u be the representative of one of the family of the iso-deflection curves, then for any prescribed boundary conditions the deflection function w(u, t) can be assumed to take the form W=h

n

Ai Γ(u)F(t)

(17)

i=1

and

φ = hΦ(u)F 2 (t)

(18)

74

BANERJEE

Equation (10), in combination with (15) and (16), will yield Φ = Φ(u). Substituting this value of Φ with W as in (17), Eqn. (7) or Eqn.(9) will yield the error function ¨ (19) ε = Λ1 u, F(t), F(t) Since equation (17) is not an exact expression for W, rather an approximation, the associated error function may be minimized using Galerkin method. The appropriate orthogonality condition applied to Eqn. (19) will yield the following “Time Differential Equation” with known constants C1 and C2 , e.g., ¨ + C1 F(t) + C2 F 3 (t) = C3 q F(t)

(20)

the solution of which is well known from which the required investigations can be carried out. A detailed outline and illustrations are cited in (Banerjee and Rogerson, 2002), (Chakraborty and Banerjee, 1999). For brevity interested readers may go through those papers. b) For the ς− plane (complex plane) It is not possible to obtain an exact solution for the transformed equations (12,15) and we have to search for a suitable expression for the mapping function z = f (ς) −

and the deflection function in the form of W(ζ, ζ) compatible with the prescribed boundary conditions. About the mapping function While employing a conformal technique one must be careful that a proper mapping function for the contour is known. Sometimes the mapping function may be expressed in a power series. For example, in the case of a regular polygon, the power series may be expressed in the form (Laura and Luisoni, 1976) z = f (ζ) = a p A s

∞

an ζ 1+ns

(21)

n=0

where A s are known as the mapping function coefficients. (Pombo et al., 1977) has prescribed the values of z = f (ς) and A s for some structures of arbitrary shapes which may be given here for ready reference. For the deflection function a general expression has also been reported in (Pombo et al., 1977) in the form z = f (ς) = a p λ ζ + µζ m+1 (22) −

W(ζ, ζ) =

N n=1

'

( − n

bn 1 − ζ ζ

(23)

75


Shapes

s

As

Triangular plate

3

1.1352

Square plate

4

1.0788

Pentagon

5

1.0516

Hexagon

6

1.0376

Heptagon

7

1.0279

Octagon

8

1.0220

m

Circular plate Circular plate with Two flat sides

2

λ

µ

1.00

0.00

0.9224

0.00

0.99

0.101

25/45

-0.4

Elliptic plate x2

2

y 4/ + 4/ = 1 3 5 Square plate with rounded corner

4

6. Some results carried out in this investigation (a) Using CDC-Method TABLE I. Frequency parameter Ω = ωa2 ν = 0.3, (b/a = 0.5)

%

ρh D

for a clamped annular plate

Finite Strip (Banerjee 1976)

FEM (Laura et al., 1973)

(Leissa 1969)

Present

Ω =17.747

17.85

17.70

17693

76

BANERJEE

The static deflections may be obtained as a byproduct as   5.8489ξ + 2.754ξ3 ..................(Yamaki, 1961)    4 qa  5.861ξ + 2.762ξ3 ‘..................(RCP-rigid-circular-plate, present) =  Eh4    17.655ξ + 44.3237ξ3 ..............(AP-annular plate, present)  (b) Using conformal mapping technique Considered here vibration of a square plate of side 2a for which z = f (ς) = a(1.08ς − 0.108ς5 + 0.045ς9 ) has been considered with k k − − − W ζ, ζ, t = bk ψk (ζ, ζ)F(t) = bk [1 − (ζ, ζ)2 F(t) k=1

k=1

and following the method described earlier, the time differential equation is obtained as .. . . 0.1599 F + 6.4851F + Eh ρa4 F = 1.7033 q ρh2 , ρ = h f (t) which is in excellent agreement with that of (Yamaki, 1961). Moreover, the conformal approach has been successfully made in author’s note (Laura and Chi, 1965) when the large deflection problem was investigated in close agreement with known available results. Also the results are in excellent agreement with those given in (Banerjee and Rogerson, 2002). The results for static deflection of annular plates cannot be compared for non-availability of such studies. However, for the linear case the result can be compared with those given in (Timoshenko and Woinowisky-Krieger, 1959), e.g., Wmax /h = 0.0575

qa4 (Ref.[22]), Eh4

Wmax /h = 0.0566

qa4 (Present) Eh4

Moreover, in the nonlinear static case the results for RCP are even better than those of (Yamaki, 1961), in the sense that the present results are closer to those of (Way, 1934). In conclusion it may be accepted that the application of constant deflection contour method is justified and appears to be simpler than existing methods. The application of polynomial expressions for the deflection and the stress functions


77

in conjunction with the Galerkin procedure appears to produce excellent results. In the case of nonlinear vibration, the results compare very well with those previously obtained (Nowinski and Ohanabe, 1972). Additionally, the load deflection relations coincide very closely with that of (Way, 1934). The present analysis offers potential for analyzing the nonlinear response of complex structures. The negative aspect of this approach is that the use of Conformal mapping technique restricts one in finding the proper mapping function. References Banerjee M. M. (1976) Note on the large deflections of irregular shaped plates by the method of conformal mapping, Trans. ASME, 356-357. Banerjee M. M., Das J. N. (1990) Use of conformal mapping technique on dynamic response of plate structures, Proc. of ICOVP-1, A. C. College, Jalpaiguri, 129-131. Banerjee M. M. (1997) A new approach to the nonlinear vibration analysis of plates and shells, trans. 14t h Intl. Conf. On Struc. Mech. In Reactor Tech., (SMIRT-13), Divn. B, Lyon, France 247. Banerjee M. M., Rogerson G. A. (2002) An application of the constant contour deflection method to non-linear vibration, Archive of Applied Mechanics 72, 279-292. Bucco D., Mazumdar J. (1979) vibration analysis of plates of arbitrary shape- A new approach, J. Sound and Vibration 67, 253-262. Chakraborty S., Roychoudhury B., Banerjee M. M. (1997) Nonlinear vibration of elliptic plates by the application of constant deflection contour method, Maths. and stats. In Eng. & Tech. Education, Ed. A. Chattopadhyay, Narosa Publishing House, New-Delhi., National Seminar on recent trends and advances of Maths. and Stats. In Eng. & Tech. Chakraborty S., Banerjee M. M. (1999) Large amplitude vibration of plates of arbitrary shape with uniform thickness, Proc. 4t h ICOVP, Jadavpur University (INDIA) 85-89. Jones, R., Mazumdar J., Chiang F. P. (1975) Further studies in the application of the method of constant deflection lines to plate bending problems, Intl. J. Eng. Sci. 13, 423-443. Jones R., Mazumdar, J. (1997) Transverse vibration of shallow shells by the method of constant deflection contours, J. Accoust. Soc. Am. 56, 1487-1492. Laura P. A. A., Chi M., (1965) An application of conformal mapping to a three dimensional unsteady heat conduction problem, Aeronautical Quarterly 16, 221-230. Laura P. A. A., Rumanelli E. (1973) Determination of eige values in a class of wave-gudes of doubly connected cross section, JI. Sound and Vibration 26, 395-400. Laura P. A. A., Luisoni L. E. (1976) Discussion on t paper “An analytical and experimental study of vibrating equilateral triangular plates”, Proc. of the Soc. of Experimental stress analysis 33, 279-280. Leissa A. W., Laura P. A. A., Gutierrez R. H. (1967) Transverse vibrations of circular plates having non-uniform edge constraints, JI. Acous. Soc. Am. 66, 180-184. Leissa A. W. (1969) Vibration of plates, NASA SP-160. Mazumdar J. (1970) A method for solving problems of elastic plates of arbitrary shapes, J. Aust. Math. Soc. 11, 95-112. Mazumdar J., Bucco D. (1978) Transverse vibrations of visco-elastic shallow shells, J. Sound and Vibration 57, 323-331. Nowinski J., Ohanabe H. (1972) On certain inconsistencies in Berger equations for large deflections of plastic plates, Intl. J. Mech. Sc. 14, 165-170.

78

BANERJEE

Pombo J. L., Laura P. A. A., Gutierrez R. H., Steinberg D. S. (1977) Analytical and experimental investigaiton of the free vibrations of clamped plates of regular polygonal shape carrying concentrated masses, J. Sound and Vibration 55, 521-532. Timoshenko S. P., Woinowisky-Krieger, (1959) Theory of plates and shells, 2n d Edn. McGraw-Hiil, Newyork. Yamaki N. (1961) Influence of large amplitude on flexural vibrations of elastic plates, ZAMM 41, 501-510. Volmir A. S. (1956) flexible plates and shells, Moscow, 214. Way S. (1934) Bending of circular plates with large deflection, Trans. ASME, 56, 627-636.

NONLINEAR WAVE EQUATIONS AND BOUNDARY CONTROL USING VISCO ELASTIC DAMPERS Stephen P. Banks1 , Metin U. Salamcı2 1 The University of Sheffield, Automatic Control and Systems Eng., Sheffield, UK 2 Gazi University, Mechanical Engineering Department, Ankara, Turkey

Abstract. In this paper we shall study the boundary control and stabilization of nonlinear wave systems which are controlled by viscoelastic dampers on the boundary. This requires the use of fractional derivatives and a recent ‘iteration’ technique for converting nonlinear problems into a sequence of time-varying linear ones. We shall use the theory of evolution operators on Hilbert space and obtain an abstract setting for the problem. Key words: nonlinear systems, wave equations, time varying approximations

1. Introduction In this paper we generalize results of (Mbodje), (Krall, 1989) to the case of the boundary control of nonlinear wave-type systems with a boundary viscoelastic damper of fractional derivative type (used, for example, in high performance aircraft, etc.). Using a recent iteration technique which converts the system into a sequence of linear, time-varying systems we show that the nonlinear controlled system is asymptotically stable. The full proofs of the results, which are fairly technical, will be given in an extended version of the paper. We shall require the exponentially modified versions of fractional derivative and integration of order α, 0 < α < 1, as follows (the basic properties of fractional derivatives and their calculus can be found in (Podlubny, 1999)): # t (t − τ)−α e−η(t−τ) d f (τ) α,η (D f )(t) = dτ (1) Γ(1 − α) dτ 0 and (I

α,η

# f )(t) =

0

t

(t − τ)α−1 e−η(t−τ) f (τ)dτ, Γ(α)

which have the relationship Dα,η f = I 1−α,η D f . The systems we consider are of the form ∂2t u(x, t) = ∂2x u(x, t) − ∂t u(x, t) · f (u, ∂t u, ∂ x u) 79 .


(2)

80

BANKS, SALAMCI

u(0, t) = 0 α,η ∂ x u(1, t) = −k∂t u(1, t), 0 < α < 1, η ≥ 0, k > 0, (the boundary control) u(x, 0) = u0 (x), ∂t u(x, 0) = v0 (x) (3) for some nonlinear, bounded function f (not necessarily positive). This system will be studied as the limit of a sequence of linear, time-varying systems of the form ∂2t u[i] (x, t) u[i] (0, t) ∂ x u[i] (1, t) u[i] (x, 0)

= = = =

∂2x u[i] (x, t) − ∂t u[i] (x, t) · f (u[i−1] , ∂t u[i−1] , ∂ x u[i−1] ) 0 α,η −k∂t u[i] (1, t), 0 < α < 1, η ≥ 0, k > 0, u0 (x), ∂t u[i] (x, 0) = v0 (x)

(4)

with the starting system ∂2t u[1] (x, t) u[1] (0, t) ∂ x u[1] (1, t) u[1] (x, 0)

= = = =

∂2x u[1] (x, t) − ∂t u[1] (x, t) · f (u0 , v0 , u0x ) 0 α,η −k∂t u[1] (1, t), 0 < α < 1, η ≥ 0, k > 0, u0 (x), ∂t u[1] (x, 0) = v0 (x).

(5)

As shown in (Mbodje), the boundary control term is equivalent to the system ∂t ϕ(ξ, t) + ξ2 ϕ(ξ, t) + ηϕ(ξ, t) − ∂t u(1, t)µ(ξ) = 0, ϕ(ξ, 0) = 0 # ∞ k µ(ξ)ϕ(ξ, t)dξ ∂ x u(1, t) = − sin(απ) π −∞

(6) (7)

where, µ(ξ) = |ξ|(2α−1)/2 . Hence we have a sequence of systems of the form ∂2t u[i] (x, t) = ∂2x u[i] (x, t) − ∂t u[i] (x, t) · f (u[i−1] , ∂t u[i−1] , ∂ x u[i−1] ) u[i] (0, t) = 0 ∂t ϕ[i] (ξ, t) + ξ2 ϕ[i] (ξ, t) + ηϕ[i] (ξ, t) − ∂t u[i] (1, t)µ(ξ) = 0, −∞ < ξ < ∞ # ∞ k ∂ x u[i] (1, t) = − sin(απ) µ(ξ)ϕ[i] (ξ, t)dξ π −∞ u[i] (x, 0) = u0 (x), ∂t u[i] (x, 0) = v0 (x), ϕ[i] (ξ, 0) = 0.

(8) (9)

(10)

In order to apply the general theory of evolution operators, we write these systems in abstract form as follows: first introduce the Hilbert space H = W 1 (0, 1) × L2 (0, 1) × L2 (−∞, ∞)

(11)

81

NONLINEAR WAVE EQUATIONS

with the inner product 1 (u, v, ϕ), ( f, g, h)H = 2

#

1

0

# (∂ x u)(∂ x f )dx +

1 0

k v · gdx + sin(απ) π

#

∞

−∞

ϕhdx (12)

and define the unbounded operator A by       u   v     2 A  v  =  ∂ x u     ϕ −(ξ2 + η)ϕ + v(1)µ(ξ) The domain of A is

  (u, v, ϕ) ∈ H : u ∈ W 2 (0, 1), v ∈ W 1 (0, 1)      ∈ L2 (−∞, ∞)  −(ξ2 + η)ϕ + v(1)µ(ξ)

D(A) =  ∞   µ(ξ)ϕ(ξ)dξ = 0 ∂ x u(1) + πk sin(απ)     |ξ|ϕ ∈ L2 (−∞, ∞). −∞

(13)

We can now write the original system in the form d X(t) = AX(t) + F (X(t)) · X(t) dt

(14)

where, X(t) = (u(t), ∂t u(t), ϕ(t))T , X(0) = (u0 , v0 , ϕ0 )T ∈ H and    0 0 0    F (X) =  f (u, ∂t u, ∂ x u) 0 0    0 0 0 In the abstract form, the iteration scheme now takes the form d [i] X (t) = AX [i] (t) + F (X [i−1] (t)) · X [i] (t), i ≥ 2 dt

(15)

with X [i] (0) = (u0 , v0 , ϕ0 )T ∈ H and d [1] X (t) = AX [1] (t) + F (X(0)) · X [1] (t) dt

(16)

with X [1] (0) = (u0 , v0 , ϕ0 )T . As in (Mbodje), the Lumer-Phillips theorem shows that A generates a strongly continuous semigroup, i.e. a set of operators {T (t) : t ≥ 0} such that AX = lim

t→0+

1 (T (t) − I)X t

(17)

for X ∈ D(A). Thus, for X ∈ D(A), we have d (T (t)X) = AT (t)X = T (t)AX, t > 0. dt

(18)

82

BANKS, SALAMCI

Note that ∩n D(An ) is dense in H. Moreover, we shall see that X [i−1] (t) → X(t) (in H), the solution of the nonlinear problem, and each X [i−1] (t) is stable, so that A + F (X [i−1] (t)) generates an evolution operator. 2. Stability of the controlled system Consider the energy measure E1 (t) =

1 2

#

12

0

3 |∂t u(t, x)|2 + |∂ x u(t, x)|2 dx.

(19)

We have #

1

∂t u∂2t u + ∂ x u∂ xt u dx 0 # 1 # 1 = ∂t u∂2t udx − ∂2x u∂t udx 0 0 # ∞ k µ(ξ)ϕ(ξ, t)dξ +∂t u(1, t) sin(απ) π −∞ # 1 = ∂t u(−∂t u f )dx 0 # ∞ k µ(ξ)ϕ(ξ, t)dξ. +∂t u(1, t) sin(απ) π −∞

dE1 (t) = dt

The energy term connected with the diffusion is # ∞ k sin(απ) |ϕ(ξ, t)|2 dξ E2 (t) = 2π −∞ and we have

# ∞ k dE2 (t) = − sin(απ) (ξ2 + η)ϕ2 (ξ, t)dξ dt π −∞ # ∞ k + sin(απ)∂t u(1, t) µ(ξ)ϕ(ξ, t)dξ. π −∞

Hence, for the total energy E = E1 + E2 , we have # 1 # ∞ dE k = − (∂t u)2 f dx − sin(απ) (ξ2 + η)ϕ2 (ξ, t)dξ dt π 0 −∞ # ∞ # 1 k (∂t u)2 f− dx − sin(απ) (ξ2 + η)ϕ2 (ξ, t)dξ ≤ π 0 −∞

(20)

(21)

(22)

(23)

NONLINEAR WAVE EQUATIONS

4 where f− =

0 if f ≥ 0 . Hence dE dt ≤ 0 if f if f < 0 # ∞ # 1 k 2 (∂t u) f− dx ≤ sin(απ) (ξ2 + η)ϕ2 (ξ, t)dξ π 0 −∞

83

(24)

i.e. if the positive part of the system ‘damping’ is dominated by the boundary damping, which will be the case if | f− | is sufficiently small. The following results, for each typical time-varying approximation above are now simple extensions of the theory given in (Rodriguez and Banks, 2003; Banks, 2001; Cimen and Banks, 2004): Theorem 1 Although not exponentially stable, in general, the system ∂2t u[i] (x, t) = ∂2x u[i] (x, t) − ∂t u[i] (x, t) · f (u[i−1] , ∂t u[i−1] , ∂ x u[i−1] ) u[i] (0, t) = 0 ∂t ϕ[i] (ξ, t) + ξ2 ϕ[i] (ξ, t) + ηϕ[i] (ξ, t) − ∂t u[i] (1, t)µ(ξ) = 0, −∞ < ξ < ∞ # ∞ k [i] µ(ξ)ϕ[i] (ξ, t)dξ ∂ x u (1, t) = − sin(απ) π −∞ u[i] (x, 0) = u0 (x), ∂t u[i] (x, 0) = v0 (x), ϕ[i] (ξ, 0) = 0

(25) (26)

(27)

is asymptotically stable, for sufficiently high gain k, in the sense that it converges to the maximal invariant subset of dE(t) = 0, ∀ t ≥ 0}. (28) I = {X ∈ H : dt (This follows from LaSalle’s invariance principle.) Theorem 2 If η > 0 then the solution of each time-varying approximation decays as 1/t, so that K (29) X(t) ≤ t where K depends on sup f and X0 , but not on the iteration argument i, provided the fedback gain k is large enough. The stability of the nonlinear system now follows from the general theory of linear approximation developed in (Rodriguez and Banks, 2003; Banks, 2001; Cimen and Banks, 2004): Theorem 3 The system ∂2t u(x, t) = ∂2x u(x, t) − ∂t u(x, t) · f (u, ∂t u, ∂ x u) u(0, t) = 0

(30)

84

BANKS, SALAMCI

∂t ϕ(ξ, t) + ξ2 ϕ(ξ, t) + ηϕ(ξ, t) − ∂t u(1, t)µ(ξ) = 0, −∞ < ξ < ∞ # ∞ k ∂ x u(1, t) = − sin(απ) µ(ξ)ϕ(ξ, t)dξ π −∞ u(x, 0) = u0 (x), ∂t u(x, 0) = v0 (x), ϕ(ξ, 0) = 0

(31)

(32)

is asymptotically stable, for sufficiently high gain k, in the sense that it converges to the maximal invariant subset of I = {X ∈ H : dE(t) dt = 0, ∀ t ≥ 0}. Moreover we have X(t) ≤ Kt if η > 0. An estimate for the required gain is given by

1 π 0 (∂t u)2 f− dx

∞ . (33) k > sup sin(απ) −∞ (ξ2 + η)ϕ2 (ξ, t)dξ 3. Conclusions In this paper we have proved stabilizability results for nonlinear wave systems with boundary viscoelastic damping. Using an iteration sequence which converts the system into a sequence of linear, time-varying approximations which can be stabilized by a simple extension of the existing theory. The method applies to other types of nonlinearities in the dynamics, such as ∂2t u(x, t) = (1 + ν(u, ∂t u, ∂ x u))∂2x u(x, t) − ∂t u(x, t) · f (u, ∂t u, ∂ x u)

(34)

by using perturbation theory of semigroups. Acknowledgements This work was partially supported by The British Council, Turkey under the science partnership programme. References Banks S. P. (2001) Exact boundary controllability and optimal control for a generalised Korteweg-de Vries equation, J. Nonlinear Anal.- Apps and Methods 47, 5537–5546. Cimen T., Banks S. (2004) Global Optimal Feedback Control for General Nonlinear Systems with Nonquadratic Performance Criteria, Sys. Cont. Letts. pp. 327–346. Krall A. (1989) Asymptotic Stability of the Euler-Bernoulli Beam with Boundary Control, J. Math. Anal. App. 137, 288–295. Mbodje B. (to appear in IMA) Wave Energy Decay under Fractional Derivative Control, Journal of Math. Cont. and Inf. Podlubny I. (1999) Fractional Differential Equations, Maths. in Sci. and Eng., Vol 198, Academic Press. Rodriguez M. T., Banks S. (2003) Linear Approximations to Nonlinear Dynamical Systems with Applications to Stability and Spectral Theory, IMA J. Math. Cont 20, 89–103.

EFFECT OF VIBRATIONS ON TRANSPORTATION SYSTEM ¨ Cem Sezgin2 Gülin Birlik1 and Onder 1 Dept. of Engineering Sciences Middle East Technical University, Ankara, Turkey 2 Health and Counseling Center Middle East Technical University, Ankara, Turkey

Abstract. In overly populated cities people living in suburban areas have to endure long journeys in order to reach their job sites. Whether they go by train, bus or by car they are inevitably exposed to vibrations, of considerable magnitude, in vertical (z) and lateral (x, y) directions. The immediate effect of vibration exposure is the fatigue of ones’ muscles. This is verified by the blood and saliva analysis of the volunteers travelling in a train. Their lactic acid levels were increased by 34% at the end of a 5˜ hr journey. The most affected people by vibration were, without doubt, the train operators and bus drivers. 42% of the suburban train operators had pain complaints at their waists. az( f loor) in the machinist cabin of a suburban train was measured to be, on the average, 0.23 m/s2. Max peak was 1.34 m/s2. The bus and car drivers were exposed to lower vibrations but they were exposed to multiple shocks originating from the non-standardized humps placed on the roads. Peak az(seat) = 0.054 m/s2 (f = 5.25 Hz) (vcar = 30 km/hr) on an asphalt road increased considerably while crossing over a hump. This value was 1.27 m/s2 (f = 4.5 Hz) in case of bus drivers (vbus = 20 km/hr). Studies have been done to provide practical measures for the reduction of the vibrations transmitted to the drivers. The waist belts filled with fluids of different viscosities prepared for this purpose seemed to be promising. The cushions filled with glycerin and gel were observed to be the best alternatives. Key words: vibration, transportation system

1. Introduction It is a well known fact that people who are exposed to vibration may have serious problems not only in their musculo-skeletal systems but also in their cardiovascular and gastrointestinal systems. In this study two groups of people exposed to vibrations are observed both from the point of view of the severity of the vibration exposure and the aftereffects of the vibration. First group was the drivers of the vehicles i.e., the group exposed to vibration in their working environment. Passengers comprised the second group, i.e., the group either exposed to vibrations regularly (but at intermittent intervals) or from time to time. For this purpose acceleration measurements have been done in the suburban train, intercity train, service bus and a car during their routine cruises. Among the drivers of the vehicles train operators were observed to be the ones most severely affected by vibrations transmitted to their bodies. In case of passengers the adverse effects of the humps, crossed over frequently, 85 .


B˙IRL˙IK, SEZG˙IN

86

during the travel was found to be a serious problem which deserves a detailed follow-up study. 2. Acceleration measurements in trains 2.1. MACHINIST CABIN

The measurements were done in machinist cabins of suburban train (ttravel =35 mins), (vtrain =100 km/hr) and Baskent Express (ttravel =5 hrs), (vtrain ≤100 km/hr). Since the speed of the train change continuously during the measurements, only the mean acceleration values are given in Table 1. AEQ and maxP define reTABLE I. direction)

Acceleration Values (m/s2) (Suburban Train) (direction of travel: x AEQ y

Z

x

Max P y

0.195

0.226

0.28

0.92

1.34

0.358

0.197

0.234

0.46

0.90

1.40

0.372

Measurement Point

SUM z

Machinist Cabin Floor (av. of 10 measurements) Machinist Seat (av. of 10 measurements)

spectively the equivalent continuous vibration (acceleration) level and maximum vibration%level, which was reached during the measurement period. SUM values

(SUM= (1.4a x )2 + (1.4ay )2 + a2z ) are calculated using measured a x , ay and az , acceleration values in the x (fore and aft), y (lateral) and z (vertical) directions. Values given in the tables show that the machinists were exposed to peak accelerTABLE II. travel: x )

Acceleration Values (m/s2) (Baskent Express Train) (direction of

y

AEQ z

x

Max P y

z

0.345

0.675

0.46

1.23

2.81

0.843

0.279

0.652

2.66

1.61

3.71

0.952

Measurement Point

SUM

Machinist Cabin Floor (av. of 10 measurements) Machinist Seat (av. of 10 measurements)

EFFECT OF VIBRATIONS ON TRANSPORTATION SYSTEM

87

ations higher than 1 m/s2 both in standing and sitting postures in the lateral and vertical directions. The mean age of the machinists was 40 (youngest: 27, oldest: 50); and they work in this environment on the average 64 hours a week (Minimum: 40 hrs/week, maximum: 75 hrs/week). 75% of the machinists also go to their homes by train. (Home - work: minimum 20 minutes; maximum 120 minutes). 42% of the operators had waist and 17% had back complaints. 42% of them had neck pain. Open windows of the locomotives can be held responsible for the neck pain of the operators. The high accelerations measured on the operator seat displayed clearly the inefficiency of the seat in isolating the vibrations transmitted from the train floor to the operator. Compared to suburban train machinists of the Baskent express were exposed to high levels of vibration (It has to be noted that duration of the exposure is generally > 6 hrs) which is defined in the ISO2631/1 as “undesired” for the health. 2.2. PASSENGER COACHES

Acceleration values (floor SUM = 0.318 m/s2; seat SUM = 0.324 m/s2) in the passenger coach of the suburban train were less than the accelerations measured in the operator cabin. The situation did not change in case of intercity train. Even though, maximum acceleration transmitted, in z direction, to the seat (seat No: 25) of the passenger of Baskent Express (vtrain =98 km/hr) was 0.13 m/s2 (at f = 1.5 Hz) (ay = 0.0508 m/s2 at f = 8.5 Hz). 34% increase in the lactic acid levels (in blood) showed of the 9 volunteers observed at the end of an 5˜ hours of travel indicates the fatigue of the muscles of the passengers during the travel. The decrease of both in hemoglobin and WBC levels, and the increase in ESR levels could be however thought to be as a result of vasodilatation of microcirculation. Saliva analysis did not exhibit meaningful changes in Na+ and K + concentrations. Na+ concentrations increased on the average 4%, and K + decreased 1%, and the rate of K + /Na+ decreased form 1.50 to 1.43. 3. Acceleration measurements in buses Acceleration measurements were done on the floor and at the interfaces between the buttocks and the seat and between the waist (and back) of the driver (or the passenger) and the back rest in x, y and z directions simultaneously, while crossing over a hump (3.1 m wide, 12 cm high) 4. Some measures To reduce the effects of vibrations transmitted to the waist of the driver waist cushion (28.5 cm x 21.5 cm) belts were prepared. The cushions were tied to the waist (cushion part placed at the back of the driver at L3 level) of the bus driver.

88

Figure 1.


Vibrations transmitted to the bus driver (vbus = 20 km/hr) while crossing over the hump

Compared to the cushion filled with gel the cushion filled with glycerin seemed to be more effective in reducing the low frequency vibrations. 5. Acceleration measurements in cars Peak acceleration values measured in an old car (production year: 1973) are tabulated in Table 3. In all the measurements velocity of the car was 30 km/hr. TABLE III.

Peak az values (m/s2)

Seat Location

Floor

Seat

Waist

Driver (f=3.25 Hz) (Birlik and Sezgin, 2001)

0.237

0.503

0.264

Front Seat Passanger (f=3.25 Hz)

0.355

0.660

0.432

EFFECT OF VIBRATIONS ON TRANSPORTATION SYSTEM

89

Figure 2. Vibrations, transmitted to the bus passenger seated at the first row, while crossing over the hump (vbus = 20 km/hr)

Figure 3.

Isolation efficiency of waist cushion belts while crossing over the hump

90


6. Conclusions The analysis of the blood samples taken from the volunteers had clearly displayed the tiredness of the passengers. From the point of view of vehicle drivers this result has to be regarded not as a problem of just tiredness but rather as a risk for health in the long run. Until more secure traffic calming measures are introduced the waist cushion belts may be used in alleviating the adverse effects of vibrations transmitted to the drivers. References Birlik G., Sezgin O. (2001) Speed reducers on the roads, Inter-noise, The Hague, the Netherlands 1073-1077.

IDENTIFICATION OF BONE MICROSTRUCTURE FROM EFFECTIVE COMPLEX MODULUS ∗ Carlos Bonifasi-Lista and Elena Cherkaev University of Utah, Department of Mathematics, Salt Lake City, UT 84112, USA

Abstract. This work deals with the problem of reconstruction of bone structure from measurements of its effective mechanical properties. We propose a novel method of calculation of bone porosity from measured effective complex modulus. Bone is modelled as a medium with a microstructure composed of trabecular bone (elastic component) and bone marrow (viscoelastic component). We model bone as a cylinder subjected to torsion and assume that the effective complex modulus can be measured as a result of experiment. The analytical representation of the effective complex modulus of the two-component composite material is exploited to recover information about porosity of the bone. The microstructural information is contained in the spectral measure in the Stieltjes representation of the effective complex modulus and can be recovered from the measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed and requires regularization. To verify the approach we apply it to analytically and numerically simulated response of a cylinder subjected to torsion assuming that it is filled with a composite material with known (laminated) microstructure. The values of porosity calculated from the effective shear modulus are in good agreement with the model values. Key words: inverse homogenization, bone microstructure, torsion, viscoelasticity, porosity, effective properties

1. Introduction Trabecular bone has a porous structure formed by trabeculae and filled by bone marrow. Various models of trabecular bone have been developed based on homogenization of materials with microstructure. They are used in computations of effective mechanical properties of bone, its static and dynamic responses (Tokarzewski et al., 2001). On the other hand, for practical applications, it is important to be able to estimate parameters of the bone structure from measured effective mechanical properties. Bone porosity is one of such parameters important for osteoporosis monitoring. We propose a novel method of evaluation of bone porosity and structure from measured effective complex modulus based on inverse homogenization approach. Inverse homogenization approach was developed in (Cherkaev, 2001) ∗

Supported by NSF grant DMS-0508901 and The University of Utah Research Foundation seed grant. 91 .


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BONIFASI-LISTA, CHERKAEV

for recovering microstructural information from the effective complex permittivity of a two-component composite material. In the present paper we consider bone as a beam or a cylinder filled with two component composite consisting of an elastic matrix (cancellous bone) and a viscoelastic solid (bone marrow). The cylinder is subjected to torsion; we assume that the effective complex modulus can be measured as a result of torsion experiment. We derive homogenized equations for the torsion of a cylinder filled with material with microstructure. We show that the effective complex modulus admits a Stieltjes analytic representation. The information about the structure of the medium is contained in the spectral measure in the analytic representation, the spectral function can be recovered from the measurements over a range of frequencies. From the computational point of view, the problem of reconstruction of the spectral function is extremely ill-posed and requires regularization. We show computational results of reconstruction of the spectral function and estimation of bone porosity from numerically simulated effective complex modulus of a third-rank composite using a regularized algorithm. 2. Mathematical model We consider bone as a cylinder of finely scaled porous elastic matrix (cancellous bone) filled with a viscoelastic solid (bone marrow) (Tokarzewski et al., 2001). Both materials are assumed to be isotropic. The bottom of the cylinder is fixed and the top is subjected to a force couple. Let ui be the displacement vector field, σ and be the stress and strain respectively. The tensor anijkl , n = 1, 2, represents the material properties which depend on the microstructure, the tensor can take different values corresponding to the elastic and viscoelastic components of the bone. The governing equations and boundary conditions are: σnij, j = 0 on Ωn , n = 1, 2

(1)

where Ωn , n = 1, 2 is the domain occupied by the first or second material. The constitutive relations correspond to isotropic materials with complex Lame moduli λ and µ: (2) σnij = anijkl kln = λn unk,k δi j + µn uni, j + unj,i in Ωn , n = 1, 2 The interface boundary conditions are given by: u1i = u2i ,

σ1i j n j = σ2i j n j on ∂Ω1 ∩ ∂Ω2

(3)

On the side surface of the cylinder: σ1i j n j = 0 on ∂Ω

(4)

IDENTIFICATION OF BONE STRUCTURE

93

3. Homogenization of heterogeneous material in torsion problem We assume that the microstructure is periodic and using two-scale asymptotic expansion method, we follow the homogenization procedure (Sanchez-Palencia, 1980). We introduce a small parameter characterizing the fine scale of variations of microstructure and reformulate the problem: σi j, j = 0 on Ω ,

σi j (x) = ai jkl (x) kl (x)

(5)

The material properties tensor a is -Y periodic in the x scale and is defined as: ai jkl (x) = ai jkl (x/) = ai jkl (y) = χ(y) a1i jkl + (1 − χ(y)) a2i jkl

(6)

where χ(y) is the characteristic function of the domain occupied by the first material: 4 1, if y ∈ Ω1 χ(y) = (7) 0, otherwise Here is the characteristic length of the cell of periodicity Y. We introduce the following two-scale asymptotic expansions for the vector displacement field: u = u(x, y) = u0 (x) + u1 (x, y) + 2 u2 + ...

(8)

The terms u0 and u1 of the expansion of the displacement field u have the form (Muskhelishvili, 1963): u0 (x) = (−βx2 x3 , βx1 x3 , βw0 (x));

u1 (x, y) = (const, const, βw1 (x, y))

where β is the torsional angle, and only the u3 component depends on the microscopic scale y. Notice that u3 only varies in the (x1 , x2 ) (and (y1 , y2 )) plane. For the given displacement field, the only nonzero stresses are σ031 (x, y) and σ032 (x, y). Substituting the asymptotic expansions in (5) we obtain the following local problem: ∂σ0i j −1 → (x, y) = 0 (9) ∂y j Here   ai jlm  ∂u0m ∂u0l ∂u1m ∂u1l   0 0 1 σi j = ai jlm lmx (u ) + lmy (u ) = + + +   2  ∂xl ∂xm ∂yl ∂ym  From the local problem (9), we obtain for i=3: ∇y · a(y)∇y w1 + ∇y · a q = 0 where the vector q is constant over the cell Y: q1 = −x2 +

∂w0 (x) , ∂x1

q2 = x1 +

∂w0 (x) ∂x2

(10)

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BONIFASI-LISTA, CHERKAEV

Solution of (10) can be represented as 1

w (x, y) =

2 k=1

φk (y)

∂ 0 w (x) + (−1)k x j xk ∂xk

(11)

where φk is a Y-periodic solution to the problem ∇y · a(y)∇y φk + ∇y · a(y)∇y yk = 0

(12)

Since we consider the model of bone formed by isotropic materials, the components a3131 , a3232 of the material properties tensor are the same: a3131 (y) = a3232 (y), so that a(y) = µ(y), and µ(y) = χ(y) µ1 + (1 − χ(y)) µ2

(13)

where µi , i = 1, 2 is the complex shear modulus of the bone or the bone marrow. Homogenized complex shear modulus µ∗ is defined now by averaging the solution of the problem over the Y-cell: # ∂φk (y) ∗ µ jk = dy (14) µ(y) δ jk + ∂y j Y 4. Stieltjes analytic representation for the effective shear modulus We derive the Stieltjes analytic representation of the effective shear modulus similar to the analytic integral representation of the effective complex permittivity which was developed for computing bounds for the effective permittivity of a two component composite (Bergman, 1978; Golden and Papanicolaou, 1983). The Stieltjes integral represents a function F(s) as an analytic function off [0, 1]interval in the complex s−plane: # 1 µ∗ 1 dη(z) F(s) = 1 − , s= = (15) µ2 1 − µ1 /µ2 0 s−z To derive this representation, we rewrite the local problem (12) on the cell Y as ∇ · µ(y)∇(φk + yk ) = 0

(16)

and notice that because of Y-periodicity of φk , the average of the corresponding strain field T = ∇(φk + yk ) is given by the unit vector ek in the k−th direction, < T >= ek , where < · > stands for averaging over Y-cell. Using (13), we bring the last problem to the form: ∇ · χ (∇φk + ek ) = s ∆φk

(17)

IDENTIFICATION OF BONE STRUCTURE

95

Applying (−∆)−1 to both sides of this equation, then taking gradient, and introducing the operator Γ as Γ = ∇(−∆)−1 (∇·) , we can express T as a function of Γχ: T = s(sI + Γχ)−1 ek . Using (13), we express F(s) as F(s) = 1 − µT, ek /µ2 = s−1 χ T, ek . The spectral resolution of Γχ with the projection valued measure Q results in the representation, # 1 χ dQ(z) ek , ek −1 F(s) = χ ( sI + Γ χ ) ek , ek = (18) s−z 0 Introducing a function η corresponding to the spectral measure Q, dη(z) = χ dQ(z)ek , ek , we end up with integral representation (15), where the positive measure η is the spectral measure of a self-adjoint operator Γχ . The representation (15) separates information about the microstructure of the composite material, which is contained in the spectral function η from information about the properties of the materials. It is shown in (Cherkaev, 2001) that the spectral function η can be uniquely reconstructed if measurements of the effective property are available on an arc in a complex plane. Because the shear modulus of the viscoelastic component is frequency dependent, measurements of µ∗ in an interval of frequency should provide the required set of data. After the function η is reconstructed, the volume fraction of bone marrow can be calculated as zero moment of the function η (Bergman, 1978; Golden and Papanicolaou, 1983). 5. Computational method and simulation results The problem of reconstruction of the function η is equivalent to inverse potential problem, which is ill-posed problem. Practically this means that small variations in the data or computational noise in a numerical algorithm can lead to arbitrary large variations of the solution. With s = x + iy, real and imaginary parts of F(s) are # 1 # 1 (x − t) dη(t) y dη(t) Re(F(s)) = , Im(F(s)) = − (19) 2 2 2 2 0 (x − t) + y 0 (x − t) + y To construct a regularized solution we discretize the problem, and formulate minimization problem having introduced a stabilization functional which constrains the set of minimizers. We reformulate the problem as an unconstrained minimization using the Lagrange multiplier. In case of a quadratic stabilization functional, the minimization problem and its Euler equation take the following form equivalent to Tikhonov regularization with regularization parameter α: 2 3 5 6 (20) minn Km − f 22 + α Lm2 , mα = K ∗ K + αL∗ L −1 K ∗ f. m∈R

We model bone as a cylinder filled with 3rd rank laminated composite made of cancellous bone and bone marrow with bone and marrow material properties

96

BONIFASI-LISTA, CHERKAEV 0.025 1.5 1

0.02

0.5

0.015 0

0.01 −0.5

0.005

−1 −1.5

0 −2

−0.005 −2.5 −3 0

10

20

30

40

50

60

70

80

90

100

−0.01 0

0.1

0.2

0.3

(a)

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

Figure 1. (a) The effective complex shear modulus over a range of frequency. (b) Reconstruction of η using Tikhonov regularization TABLE I.

Computed estimates for bone porosity

Technique / Porosity Tikhonov regularization, Im[F(s)] Tikhonov regularization, Re[F(s)]

70 %

75%

80 %

85%

90 %

95%

69.99 69.99

74.97 74.98

79.97 79.97

84.98 84.99

90.02 90.02

94.99 95.00

(Cowin, 2001). The effective shear modulus of these structures is calculated analytically and is used as a data set in reconstruction algorithm. Figure 1(a) shows data used in computations. Figure 1(b) shows function η for 3d rank laminated composite computed with Tikhonov regularization. The volume fraction of bone marrow is estimated for several values of bone porosity, the results are shown in Table 1. References Bergman D. J. (1978) The dielectric constant of a composite material - A problem in classical physics, Phys. Rep. C 43 377-407. Cherkaev E. (2001) Inverse homogenization for evaluation of effective properties of a mixture, Inverse Problems 17 1203-1218. Cowin S. C. (2001) Bone Mechanics Handbook, CRCPress. Golden K., Papanicolaou G. (1983) Bounds on effective parameters of heterogeneous media by analytic continuation, Comm. Math. Phys. 90 473-491. Muskhelishvili N. I. (1963) Some basic problems of the mathematical theory of elasticity, 22, Noordhoff ltd Groningen. Sanchez-Palencia E. (1980) Non-homogeneous Media and Vibration Theory, Springer-Verlag Berlin Heidelberg New York. Tokarzewski S., Telega J. J., Galka A. (2001) Torsional Rigidities of Cancellous Bone Filled with Marrow: The Application of Multipoint Pade Approximants, Engineering Transactions 49 135153.

BEAM VIBRATIONS WITH NON-IDEAL BOUNDARY CONDITIONS Hakan Boyacı Department of Mechanical Engineering, Celal Bayar University, 45140 Manisa, Turkey

Abstract. A simply supported damped Euler-Bernoulli beam with immovable end conditions is considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the boundaries are assumed to allow small deflections and moments. Approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique. Key words: stretched beam vibrations, non-ideal boundary conditions, method of multiple time scales

1. Introduction Beams are frequently used as design models for vibration analysis. In such analysis, types of support conditions are important and have direct effect on the solutions and natural frequencies. The real systems are considered to satisfy those ideal boundary conditions. However, small deviations from ideal conditions in real systems indeed occur. For example, a beam connected at ends to rigid supports by pins is modeled using simply supported boundary conditions which require deflections and moments to be zero. However the hole and pin assembly may have small gaps and/or friction which may introduce small deflections and/or moments at the ends. Similarly a real built-in beam may have very small variations in deflection and/or slope. To represent such behavior, non-ideal boundary condition concept has been proposed recently.Non-ideal boundary conditions are modeled using perturbations. Different beam vibration problems having non-ideal boundaries have been treated (Pakdemirli&Boyaci, 2001, 2002, and 2003). Here in this work, the idea is widened to a damped forced nonlinear simple-simple beam vibration problem in which the non-linearity is due to stretching effects. As non-ideality for a simple-simple support case, small variations at deflections and moments are allowed at both ends. Ideal and non-ideal frequencies as well as frequency response curves are contrasted. Combined effects of non-linearity and non-ideal boundary conditions on the natural frequencies and mode shapes are 97 .


98

BOYACI

examined using the method of multiple scales, a perturbation technique. While the stretching effect increases the frequencies, non-ideal boundary conditions may increase or decrease the frequencies. 2. Problem formulation and solution The model considered is an Euler-Bernoulli beam with immovable end conditions causing nonlinear stretching effects. The dimensionless equation is 1 w¨ + w = εw 2

#1

iv

w2 dx − 2εµw˙ + Fˆ cos Ωt

(1)

0

where w is the deflection, µ is the damping, t is the time, x is the spatial variable and F and Ω are the magnitude and frequency of the external excitation respectively. Dot denotes differentiation with respect to time t and prime denotes differentiation with respect to the spatial variable x. Here it is assumed that the beam is simply supported at both ends. However, the boundary conditions are not ideal and some slight variations occur in deflections and moments, hence w(0, t) = εα1 (t), w (0, t) = εβ1 (t), w(1, t) = εα2 (t), w (1, t) = εβ2 (t)

(2)

ε is a small perturbation parameter denoting that the variations in deflections and moments are small. An approximate solution of the below form is assumed w(x, t) = w0 (x, T 0 , T 1 ) + εw1 (x, T 0 , T 1 ) + . . .

(3)

where T 0 is the usual fast time scale and T 1 =εt is the slow time scale in the method of multiple scales. Only primary resonance case is considered and hence, the forcing term is ordered as Fˆ = εF. The time derivatives are defines as Substituting equation (3) into equations (1) and (2), separating at each order of ε, one has Order 1 D20 w0 + wiv 0 =0 w0 (0, T 0 , T 1 ) = 0, w (4) 0 (0, T 0 , T 1 ) = 0, w0 (1, T 0 , T 1 ) = 0, w0 (1, T 0 , T 1 ) = 0 Order ε D20 w1 + wIV = −2µD0 w0 − 2D0 D1 w0 +

1 1 w20 dx + F cos ΩT 0 w1 (0, T 0 , T 1 ) = α1 (T 0 , T 1 ) 2 w0

0 (0, T 0 , T 1 ) = β1 (T 0 , T 1 ) w 1 w1 (1, T 0 , T 1 ) = α2 (T 0 , T 1 ), w 1 (1, T 0 , T 1 )

= β2 (T 0 , T 1 )

(5)

BEAM VIBRATIONS WITH NON-IDEAL BOUNDARY CONDITIONS

99

The solution at the first order is w0 = (A(T 1 )eiωT 0 + cc)Y(x)

(6)

where cc stands for complex conjugates of the preceding terms. Substituting equation (6) into equation (4) yields the normalized solution √ Y(x) = 2 sin nπx, ω = n2 π2 n = 1,2,3, . . . (7) At order ε , one substitutes equation (6) into the right hand side of equation (5). The result is  

1   3 1 −2iωD1 A Y − 2iµωAY + A2 A¯ Y Y 2 dx + FeiσT1  = D20 w1 + wiv 2 2 1 eiωT 0

0

+ N.S .T. + cc

(8)

where N.S .T. stands for non-secular terms. It is also assumed that the external excitation frequency is close to one of the natural frequencies of the system; such that Ω = ω + εσ

(9)

Here σ is a detuning parameter of order 1. A solution of the below form is assumed w1 = ϕ(x, T 1 )eiωT 0 + W1 (x, T 0 , T 1 ) + cc

(10)

The first part of the solution is the one corresponding to secular terms and the second is the one corresponding to non-secular terms. Substituting equation (9) into equation (8) with boundary conditions yields 3 ϕiv − ω2 ϕ = −2iωD1 A Y − 2iµωAY + A2 A¯ Y 2 ϕ(0, T 1 ) = α1 A(T 1 ), ϕ(1, T 1 ) = α2 A(T 1 ),

#1 0

1 Y 2 dx + FeiσT1 2

ϕ (0, T 1 ) = β1 A(T 1 ), ϕ (1, T 1 ) = β2 A(T 1 )

(11)

(12)

In writing (11), the variations of deflections at the boundaries are considered to be of the same form as the time variations of the solutions and αi and βi are now constants. Since the homogenous problem has a non-trivial solution, the nonhomogenous problem (10) and (11) has a solution only if a solvability condition is satisfied (Nayfeh, 1981). The solvability condition requires √ 3 1 2in2 π2 (D1 A + µA) + n4 π4 A2 A¯ + 2nπKA − f eiσT1 = 0 (13) 2 2

100

BOYACI

where K = (β1 − β2 cos nπ − n2 π2 α1 + n2 π2 α2 cos nπ)and f =

1

FYdx

(14)

0

Substituting the polar form A(T 1 ) = 12 a(T 1 ) eiθ(T 1 ) into equation (12), separating real and imaginary parts, one obtains da = −n2 π2 µa + 12 f sin γ n2 π2 dT 1

dγ n2 π2 a dT = n2 π2 σa − 1

3 4 4 3 16 n π a

√

2 2 nπK a = γ

−

+ 12 f cos γ

(15)

where γ = σT 1 − θ. In the steady state case, = 0 and solving for the detuning parameter yields ) √ 3 2 2 2 2 f2 σ= n π a + K± − µ2 (16) 16 2nπ 4n4 π4 a2 For free undamped vibrations, non-ideal nonlinear natural frequencies are √    3 2  2  ωni = ω + ε  ωa + √ K  (17) 16 2 ω In the above relation, the first term in O(ε) is due to the nonlinearity and the second term is due to the non-ideal boundary conditions. If only the non-ideality term of order ε is to be considered, it may increase or decrease the frequencies depending on the mode number n and amplitudes of end variations that are α1 due to deflections and β1 due to moments. Also it is apparent that deflection effects become dominant compared to the moment effects as the mode number increases. Different cases are summarized in Table 1. In Figure 1a and 1b fundamental nonlinear frequencies versus amplitudes are contrasted for the ideal and non-ideal cases using the same parameter values except α1 and α2 . TABLE I.

Effect of non-ideal boundary conditions on the frequencies

n

Frequency

odd

frequencies increase if β1 + β2 > n2 π2 (α1 + α2 ) frequencies decrease if β1 + β2 < n2 π2 (α1 + α2 ) no change if β1 + β2 = n2 π2 (α1 +α2 )

even

frequencies increase if β1 + n2 π2 α2 ¿ β2 + n2 π2 α1 frequencies decrease if β1 + n2 π2 α2 ¡ β2 + n2 π2 α1 no change if β1 + n2 π2 α2 = β2 + n2 π2 α1

BEAM VIBRATIONS WITH NON-IDEAL BOUNDARY CONDITIONS

101

Figure 1a.,1b. Nonlinear frequencies versus amplitudes for ideal (dashed) and non-ideal (solid) cases for the first mode

a)α1 = α2 =0.1, β1 = β2 =10, ε=0.1, b)α1 = α2 =2.0, β1 = β2 =10, ε=0.1 One can also obtain amplitude-excitation frequency relation from equations (9) and (14) as ) √ 2 ε2 f 2 3 2 − ε2 µ2 (18) Ω = ω + ε ωa + ε √ K ± 16 4ω2 a2 2 ω In Figure 2a and 2b frequency response graphs for the fundamental nonlinear frequencies are compared for the ideal and non-ideal cases using the same parameter values except α1 and α2 . The approximate beam deflection to the first order are as follows w = a cos(Ωt − γ)Y(x) + O(ε) (19) where Y(x) is given in equation (9).

Figure 2a.,2b. Frequency response curves for ideal (dashed) and non-ideal (solid) cases for the first mode

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BOYACI

a) α1 = α2 =0.1, β1 = β2 =10, ε=0.1, µ=0.5,f=10, b) α1 = α2 =2.0, β1 = β2 =10, ε=0.1, µ=0.5,f=10

3. Concluding remarks Non-ideal boundary conditions are defined and formulated using perturbation theory. A sample problem of stretched damped beam with external excitation is treated. Approximate analytical solution of the problem is presented using the method of multiple scales. It is shown that small variations of deflections and moments at the ends may affect the frequencies of the response. Depending on the mode numbers and amplitudes of variations, the frequencies may increase or decrease. Deviations from the ideal conditions lead to a drift in frequencyresponse curves which may be positive, negative or zero depending on the mode number and amplitudes of variations.

References Nayfeh A. H. (1981) Introduction to Perturbation Techniques, New York. Pakdemirli M., Boyacı H. (2001) Vibrations of a stretched beam with non-ideal boundary conditions, Mathematical and Computational Applications 6 217-220. Pakdemirli M., Boyacı H. (2002) Effect of non-ideal boundary conditions on the vibrations of continuous systems, Journal of Sound and Vibration 249 815-823. Pakdemirli M., Boyacı H. (2003) Non-linear vibrations of a simple-simple beam with a non-ideal support in between, Journal of sound and Vibration 268 331-341.

A TWO-FRACTAL OVERLAP MODEL OF EARTHQUAKES Bikas K. Chakrabarti and Arnab Chatterjee Theoretical Condensed Matter Physics Division and Centre for Applied Mathematics and Computational Science, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India

Abstract. We introduce here the two-fractal model of earthquake dynamics. As the fractured surfaces have self-affine properties, we consider the solid-solid interface of the earth’s crust and the tectonic plate below as fractal surfaces. The overlap or contact area between the two surfaces give a measure of the stored elastic energy released during a slip. The overlap between two fractals change with time as one moves over the other and we show that the time average of the overlap distribution follows a Gutenberg-Richter like power-law, with similar exponent value. Key words: earthquake, fractals, Cantor sets, Gutenberg-Richter law

1. Introduction The earth’s solid outer crust, about 20 kilometers in average thickness, rests on the tectonic shells. Due to the high temperature-pressure phase changes and the consequent powerful convective flow in the earth’s mantle (a fluid of very high density), at several hundreds of kilometers of depth, the tectonic shell, divided into a small number (about ten) of mobile plates, has relative velocities of the order of a few centimeters per year (Gutenberg and Richter, 1954; Kostrov and Das, 1989; Scholz, 1990). Over several tens of years, enormous elastic strains develop on the earth’s crust when sticking (due to the solid-solid friction) to the moving tectonic plate. When sudden slips occur between the crust and the tectonic plate, these stored elastic energies are released in ‘bursts’, causing the damages during the earthquakes. Earthquakes occur due to fault dynamics in the lithosphere. A geological fault is created by a fracture in the rock layers, and is comprised of the rock surfaces in contact. The two parts of the fault are in very slow relative motion which causes the surfaces to slide. Because of the uniform motion of the tectonic plates, the elastic strain energy stored in a portion of the crust (block), moving with the plate relative to a ‘stationary’ neighboring part of the crust, can vary only due to the random strength of the solid-solid friction between the crust and the plate. A slip occurs when the accumulated stress exceeds the resistance due to the frictional 103 .


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force. The potential energy of the strain is thereby released, causing an earthquake. As mentioned before, the observed distribution of the elastic energy release in various earthquakes seems to follow a power law. The slip is eventually stopped by friction and stress starts developing again. Strain continues to develop till the fault surfaces again slip. This intermittent stickslip process is the essential characteristic feature of fault dynamics. The overall distribution of earthquakes, including main shocks, foreshocks and aftershocks, is given by the Gutenberg-Richter law (1944, 1954): log10 Nr(M > M) = a − b M

(1)

where, Nr(M > M) denotes the number (or, the frequency) of earthquakes of magnitudes M that are greater than a certain value M. The constant a represents the total number of earthquakes of all magnitudes: a = log10 Nr(M > 0) and the value of the coefficient b is presumed to be universal. In an alternative form, the Gutenberg-Richter law is expressed as a relation for the number (or, the frequency) of earthquakes in which the energy released E is greater than a certain value E: Nr(E > E) ∼ E −b/β , suggesting Nr(E) ∼ E −γ ,

(2)

for the number density of earthquakes, where γ = 1 + b/β. The value of the exponent γ is generally observed to be around unity (Knopoff, 2000); see also http://web.cz3.nus.edu.sg/˜chenk/gem2503 3/notes7 1.htm. One class of models for simulating earthquakes is based on the collective motion of an assembly of connected elements that are driven slowly, of which the block-spring model due to Burridge and Knopoff (1967) is the prototype. The Burridge-Knopoff model and its variants (Carlson et al., 1994; Olami et al., 1992) have the stick-slip dynamics necessary to produce earthquakes. The underlying principle in this class of models is self-organized criticality (Bak, 1997). Another class of models for simulating earthquakes is based on overlapping fractals, which will be discusses in details in the next sections. 2. Fractals A fractal is a geometrical object that displays self-similarity on all scales. For random fractals, the object need not exhibit exactly the same structure at all scales, but the same ‘type’ of structures appear on all scales. For example, the Black sea coastline measured with different length rulers will show differences: the shorter the ruler, the longer the ‘length’ measured but not exactly following the ratio of the inverse of the lengths of the rulers; greater than that! This is because more structures come into play at lower length scales; or in other words, the coastline is not really a ‘line’ but rather a fractal having dimension greater than unity. Looking at smaller and smaller length-scales, one can find self-similar structures (see Fig. 1).

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Figure 1. Fractal structure of Black sea coastline at Is¸ık (Istanbul), represented actually by a time series of stock price. This shows the remarkable self-similarity (examplified by the blow up of the segment in the box) involved in many such natural processes

Figure 2. Sierpinski gasket (at generation number n = 5), having fractal dimension log 3/ log 2 (when n → ∞)

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n=1 n=2 n=3 Figure 3. Construction process of Cantor set is shown (upto n = 3). The set becomes a fractal when n → ∞

One can easily construct such regular fractals as carpets or gaskets (see Fig. 2). In Fig. 2, a basic unit of equilateral triangle with an inner triangle obtained by joining the mid-points of each side and keeping the inner space void, one constructs a gasket. At each step, as the length of each side changes by a factor L = 2, the mass of the fractal changes by a factor M(= Ld f ) = 3, giving therefore the fractal dimension of the object to be d f = log 3/ log 2. This object in Fig. 2 of course represents a non-random fractal. However, a random fractal (as in Fig. 1) can be easily constructed if the void is not always at the center but at any of the 4 triangles at random for each generation; the (mass) dimension d f remain the same. A similar fractal of dimension log 2/ log 3 can be constructed as the Cantor set shown in Fig. 3. Starting from a set of all real numbers from 0 to 1 one removes the subset from the middle third, so that for each l = 3, the mass (size) of the set m = 2 and hence the dimension as n → ∞. This void subset can again be randomly chosen, giving a random Cantor set, with same d f . 3. Fractal overlap model of earthquake Overlapping fractals form a whole class of models to simulate earthquake dynamics. These models are motivated by the observation that a fault surface, like a fractured surface (Chakrabarti and Benguigui, 1997), is a fractal object (Okubo and Aki, 1987; Scholz and Mandelbrot, 1989; Sahimi, 1993). Consequently a fault may be viewed as a pair of overlapping fractals. Fractional Brownian profiles have been commonly used as models of fault surfaces (Brown and Scholz, 1985; Sahimi, 1993). In that case the dynamics of a fault is represented by one Brownian profile drifting on another and each intersection of the two profiles corresponds to an earthquake (De Rubeis et al., 1996). However the simplest possible model of a fault − from the fractal point of view − was proposed by Chakrabarti and Stinchcombe (1999). This model is a schematic representation of a fault by a pair of dynamically overlapping Cantor sets. It is not realistic but, as a system of overlapping fractals, it has the essential feature. Since the Cantor set is a fractal with a simple construction procedure, it allows us to study in detail the statistics of the overlap of one fractal object on another. The two fractal overlap magnitude changes in time as one fractal moves over the other. The overlap (magnitude) time


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Figure 4. (a) Schematic representations of a portion of the rough surfaces of the earth’s crust and the supporting (moving) tectonic plate. (b) The one dimensional projection of the surfaces form Cantor sets of varying contacts or overlaps (s) as one surface slides over the other

series can therefore be studied as a model time series of earthquake avalanche dynamics (Carlson et al., 1994). The statistics of overlaps between two fractals is not studied much yet, though their knowledge is often required in various physical contexts. It has been established recently that since the fractured surfaces have got well-characterized self-affine properties, the distribution of the elastic energies released during the slips between two fractal surfaces (earthquake events) may follow the overlap distribution of two self-similar fractal surfaces (Chakrabarti and Stinchcombe, 1999; Pradhan et al., 2003). Chakrabarti and Stinchcombe (1999) had shown analytically by renormalization group calculations that for regular fractal overlap (Cantor sets and carpets) the contact area distribution ρ(s) follows a simple power law decay: ρ(s) ∼ s−˜γ ; γ˜ = 1. (3) Study of the time (t) variation of contact area (overlap) s(t) between two wellcharacterized fractals having the same fractal dimension as one fractal moves over the other with constant velocity, has revealed some features which can be utilized to predict the ‘large events’ (Pradhan et al., 2004). Bhattacharyya (2005) has recently studied this overlap distribution for two Cantor sets with periodic boundary conditions and each having dimension log 2/ log 3. It was shown, using exact counting, that if s ≡ 2n−k (n is the generation number) then the probability ρ(s) ˜ to get an overlap s is given by a binomial distribution (Bhattacharyya, 2005) n−k k 1 2 n n−k ρ(2 ˜ )= ∼ exp(−r2 /n); r → 0, (4) n−k 3 3

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2 where r2 = 23 32 n − k . Expressing therefore r by log s near the maxima of ρ(s), ˜ one can again rewrite (4) as 2 (log s) ˜ ∼ exp − ρ(s) ; n → ∞. (5) n Noting that ρ(s)d(log ˜ s) ∼ ρ(s)ds, we find ρ(s) ∼ s−˜γ , γ˜ = 1, as in (3) as the binomial or Gaussian part becomes a very weak function of s as n → ∞ (Bhattacharyya et al.,, 2005). It may be noted that this exponent value γ˜ = 1 is independent of the dimension of the Cantor sets considered (here log 2/ log 3) or for that matter, independent of the fractals employed. It also denotes the general validity of (3) even for disordered fractals, as observed numerically (Pradhan et al., 2003; Pradhan et al., 2004). Identifying the contact area or overlap s between the self-similar (fractal) crust and tectonic plate surfaces as the stored elastic energy E released during the slip, the distribution (3), of which a derivation is partly indicated here, reduces to the Gutenberg-Richter law (2) observed.


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4. Summary We introduce here the two-fractal overlap model of earthquake where the average distribution of the overlaps between the surfaces, as one fractal (here, Cantor set) moves over the other gives the Gutenberg-Richter like distribution (3), with γ˜ = 1 exactly in the model. We note that this is an exactly solvable model of earthquake dynamics and the result for the distributions compare favorably with the observations. Acknowledgements We are grateful to P. Bhattacharyya, P. Chaudhuri, M. K. Dey, S. Pradhan, P. Ray and R. B. Stinchcombe for collaborations in various stages of developing this model. References Bak P. (1997) How Nature Works, Oxford Univ. Press, Oxford. Bhattacharyya P. (2005) Physica A 348, 199. Bhattacharyya P., Chatterjee A., Chakrabarti B. K. (2005) arXiv:physics/0510038. Brown S. R., Scholz C. H. (1985) J. Geophys. Res. 90, 12575. Burridge R., Knopoff L. (1967) Bull. Seismol. Soc. Am. 57, 341. Carlson J. M., Langer J. S., Shaw B. E. (1994) Rev. Mod. Phys. 66, 657. Chakrabarti B. K., Benguigui L. G. (1997) Statistical Physics of Fracture and Breakdown in Disorder Systems Oxford Univ. Press, Oxford. Chakrabarti B. K., Stinchcombe R. B. (1999) Physica A 270, 27. De Rubeis V., Hallgass R., Loreto V., Paladin G., Pietronero L., Tosi P. (1996) Phys. Rev. Lett. 76, 2599. Gutenberg B., Richter C. F. (1944) Bull. Seismol. Soc. Am. 34, 185. Gutenberg B., Richter C. F. (1954) Seismicity of the Earth, Princeton Univ. Press, Princeton. Knopoff L. (2000) Proc. Natl. Acad. Sci. USA. 97, 11880. Kostrov B. V., Das S. (1990) Principles of Earthquake Source Mechanics, Cambridge Univ. Press, Cambridge. Okubo P. G., Aki K. (1987) J. Geophys. Res. 92, 345. Olami Z., Feder H. J. S., Christensen K. (1992) Phys. Rev. Lett. 68, 1244. Pradhan S., Chakrabarti B. K., Ray P., Dey M. K. (2003) Physica Scripta T106, 77. Pradhan S., Chaudhuri P., Chakrabarti B. K. (2004) Continuum Models and Discrete Systems, Bergman D. J., and Inan E. Eds, Nato Sc. Series, Kluwer Academic Publishers, Dordrecht, pp.245-250; cond-mat/0307735. Sahimi M. (1993) Rev. Mod. Phys. 65, 1393. Scholz C. H., Mandelbrot B. B. Eds. (1989) Fractals in Geophysics, Birkhaüser, Basel. Scholz C. H. (1990), The Mechanics of Earthquake and Faulting, Cambridge Univ. Press, Cambridge.

DYNAMICS OF STRUCTURES WITH BISTABLE LINKS ∗ Andrej Cherkaev1 , Elena Cherkaev1 and Leonid Slepyan2 1 University of Utah, Department of Mathematics, Salt Lake City, UT 84112, USA 2 Tel Aviv Univ., Dept. of Solid Mechanics, Materials and Systems, 69978, Israel

Abstract. The paper considers nonlinear structures with bistable links described by irreversible, piecewise linear constitutive relation: the force in the link is a nonmonotonic bistable function of elongation; the corresponding elastic energy is nonconvex. The transition from one stable state to the other is initiated when the force exceeds the threshold; the transition propagates along the chain and excites a complex system of waves. Mechanically, the bistable link may consist of two rods joined at the ends, the longer rod initially being inactive. This rod starts to resist when large enough strain damages the shorter rod. The transition wave is a sequence of breakages, it absorbs the energy of the loading force, transforms it into high frequency vibrations, and distributes the partial damage throughout the structure. In the same time, the partial damage does not lead to failure of the whole structure, because the initially inactive links are activated. The developed model of dynamics of cellular chains allow us to explicitly calculate the speed of the transition wave, conditions for its initiating, and estimate the energy of dissipation. The dissipation or absorption of the energy can be significantly increased in a structure characterized by a nonlinear discontinuous constitutive relation. The considered chain model reveals some phenomena typical for waves of failure or crushing in constructions and materials under collision, waves in a structure specially designed as a dynamic energy absorber and waves of phase transitions in artificial and natural passive and active systems. Key words: phase transition, local instabilities, bistable links, nonlinear dynamics, protective structures

1. Introduction The paper is concerned with phase transition and dynamics of structures with nonmonotonic bistable stress-strain constitutive relations described by a two-branched diagram. These systems have multiple equilibria that correspond to local energy minima. Such inertial systems describe various natural phenomena, bistable-link structures are found in many engineering applications, specially designed, these structures are capable of absorption and dissipation of large amounts of energy. The bistable model describes an irreversible transition such as the damage propagation in foams, structural materials, and constructions. The simplest example ∗

Supported by ARO grant 41363-MA, NSF grant DMS-0072717, and The Israel Science Foundation grant 1155/04. 111 .


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is the falling domino. Another example is metallic open-cell foams. Even larger scales are used to describe demolition of a multilevel construction. Here, the transition means crashing of some links and compactification. The transition wave is a result of multiple sequential breaks of its elements (cells); after the transition, the cells regain their strength in a new state. Once initiated, the wave propagates through the structure which would be stable under an equal static load. The novel area of application of the bi-stable structures is design of materials absorbing large amounts of energy. While the theoretical limit for the energy absorbed in a material is the energy corresponding to the melting point, real structures are destroyed by a tiny fraction of this energy due to material’s instabilities and an uneven distribution of the stresses throughout the structure. The goal of design is to find a structure that absorbs maximal energy without rupture or breakage. The effect is achieved by using bistable breakable links [(Cherkaev et al., 2005; Slepyan et al., 2005)]. The key of the approach is a special structure or a geometric principle of assembling solid elements in a structure. The superior properties of these energy absorbing structures are due to their special morphology. The structures possess several length scales and their averaged constitutive relations are non-monotonic due to special spatial configuration of their elements. These “waiting element structures” allow propagation of waves of “partial damage” which delocalize and eventually disperse the energy of an impact. Bistable-bond chain model was considered in a number of works (see [(Slepyan and Troyankina, 1984, 1988); (Slepyan, 2000, 2001, 2002), (Ngan and Truskinovsky, 1999), (Puglisi and Truskinovsky, 2000), (Balk et al., 2001a, 2001b)], [(Charlotte and Truskinovsky, 2002)], [(Cherkaev et al., 2005; Slepyan et al., 2005)]. In the bistable chain, the transition wave is accompanied by a pronounced dissipation due to the excitation of structure-associated high-frequency waves. The more complex problem of breakable two-dimensional structure with waiting links was numerically investigated in [(Cherkaev and Zhornitskaya, 2004)]. The bistable chains from elastic-plastic material were studied in [(Cherkaev et al., 2005a)]. Transition waves in two-dimensional lattices was analyzed in [(Slepyan and Ayzenberg-Stepanenko, 2004)]. 2. Analytic description of the phase transition The dynamics of the chain is described by the equation M u¨ m (t) = T (um+1 − um ) − T (um − um−1 ),

m = −∞, . . . , ∞,

(1)

where um is the displacement of the m-th mass, t is time, M is the mass of the particle. A non-monotonic (and discontinuous) function T is the tensile force in the locally unstable link. One of two-branch constitutive diagrams that permit analytical solution, is shown in Fig. 1(a).

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Equivalent problem Consider an infinite bistable chain with an irreversible force– elongation relation shown in Fig. 1(a). The transition to the second branch of the diagram occurs at the moment t = t∗ when the elongation q first time reaches the critical value, q = q∗ . This nonlinear dependence can be modelled using an equivalent problem which considers the intact linear chain with the bonds corresponding to the first branch of the diagram. In the equivalent problem, at the moment t = t∗ (different for different bonds) a pair of external forces ∓P∗ is instantly applied to the left and right masses connected by this bond. These forces compensate for the difference between the branches P = P∗ +(1−γ)µ(q−q∗ ), (t > t∗ ) . Here µ is the stiffness of the bond. Thus, the parallel-branch problem can be considered as a linear one with these additionally applied external forces which model the nonlinearity. In a special case γ = 1, see Fig. 1(a), the value of the external forces is a constant. In this case, the problem is simplified, since the right-hand side of the first-branch equation is known in advance. In this case, if all the bonds to the left of a mass correspond to the second branch, while all the bonds to the right correspond to the first branch, the resulting external force is only applied to this mass. The other external forces compensate each other. In a general case, γ 1, the external force depends on the current elongation of the bond, and we have different equations from different sides of the transition wave front. Below we outline main steps in solving the equivalent problem. Steady-state formulation Assume that the transition wave propagates to the right with a constant speed, v > 0. The nondimensional time interval between the transition of neighboring bonds is equal to a/v, where a is the distance between two neighboring masses at rest. We assume that u˙ m+1 (t + a/v) = u˙ m (t). The motion of a mass is a function of a continuous variable, η = vt − x, where x is the continuous coordinate, and it is steady for any value of η. In the ‘steady-state’ regime we have: du(η) , η = vt − am . (2) u˙ m (t) = −v dη The transition wave speed v is unknown. Its value and conditions required for initiating the transition are determined by the solution of the problem. We are looking for v as a function of the intensity of an external source of energy which causes the transition wave. However, it is more convenient to consider an implicit problem and find the source intensity as a function of the speed v. The resulting piece-wise linear problem is solved analytically and/or numerically. Analytical solution Analytical solution is derived using the Fourier transform. This solution is used to estimate the total dissipation rate. The steps are as follows. We consider nondimensional equation for the bistable, parallel branch chain with

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an introduced relation qm (t) = um (t) − um−1 (t) and the external forces in the righthand side:   0 if (1, 1) or (2, 2)    ˆ − P (3) u¨ m (t) − qm+1 (t) + qm (t) =  ∗ if (1, 2)    Pˆ if (2, 1) ∗

where the pairs (1, 1) . . . , (2, 2) denote the state (the first branch (1) or the second one (2)) of the bonds connecting the mth mass with the left and the right neighbors, respectively. The steady-state version of (1) is the following v2

d2 q(η) +2q(η)−q(η−1)−q(η+1) = P∗ [2H(−η)− H(−η−1)− H(−η+1)] , (4) dη2

where H is the Heaviside function, and we assume M = 1. To complete the formulation we introduce conditions at infinity. We consider a semi-infinite chain, m = −N, −N + 1, ..., assuming number N to be large. We consider two types of excitation conditions which lead to the same solution. (1) Assume that a constant tension force P directed to the left, is applied to the end mass, m = −N, at the moment of time t = −N/v. (2) Alternatively, this mass is forced to move with a constant speed, u˙ −N = −w. We also assume that there is no energy flux from infinity on the right, as m → ∞. So that, only the waves with group velocities exceeding the transition front velocity v, can exist ahead of the transition front. Because of linearity of the equivalent problem, the strain q(η) can be represented as a sum of a homogeneous solution q0 of (4) (this corresponds to a zero wavenumber incident wave caused by an external action at η = −∞) and an inhomogeneous solution q(η) (that corresponds to waves excited by the right-hand side forces). The homogeneous solution corresponds to the energy brought from infinity; the inhomogeneous solution constructed under the causality principle, describes the energy carried from the transition front to ±∞. We consider the homogeneous part of the total solution, and calculate the source at infinity as a force acting on the end mass of the chain or the velocity of the the end mass. This is the sought relation between the source intensity and the transition wave speed expressed in terms of parameters of the bistable links. The incident wave moves to the right with unit speed which is larger than the speed of the transition front. This wave is characterized by q = q0 , u˙ = −q0 . To distinguish the solution of the inhomogeneous equation (4) we denote the corresponding elongations by q(η). Using the Fourier transform # ∞ F q (k) = q(η) exp(ikη) dη (5) −∞


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on η as a continuous variable we obtain qF (k) =

2P∗ (1 − cos k) , h(k) = (0 + ikv)2 + 2(1 − cos k) ikh(k)

(6)

Here 0 + ikv = lim s→+0 (s + ikv). The integrant contains poles as zeros of h(k) (the finite number of zeros and their values depend on the speed, v). According to the causality principle for a steady-state solution (see book of Slepyan [(Slepyan, 2002)])), these poles are placed outside the real axis if s > 0 approaching the axis in the limit, s → 0. The rule defines the half-space - positive or negative, containing the pole. In turn, this defines sign of the half-residual at the pole. The integral is thus represented as a sum of the half-residuals and a Cauchy principal value of the integral n P∗ v (Q2ν+1 cos(h2ν+1 η) − Q2ν cos(h2ν η)) + P ∗ 2(1 − v2 ) ν=0 # ∞ 1 − cos k 2P∗ V.p. − sin(kη) dk , 2 2 π 0 k[2(1 − cos k) − k v ]

q(η) = −

(7)

where v2 h2ν+1 , 2(sin h2ν+1 − v2 h2ν+1 ) v2 h2ν = , Q0 = 0 , 2(sin h2ν − v2 h2ν )

Q2ν+1 = Q2ν

(8)

and h2ν , h2ν+1 are real zeros of the function h(k). The obtained solution depends on the speed v which is still unknown. It can be determined using the condition at η → −∞ (m = −N) that the applied force P equal to the uniform part of the tensile force T , or that a given uniform part of the particle velocity −w. The corresponding relations depend only on the ratios P0 = P/q∗ , P0∗ = P∗ /q∗ , w0 = w/q∗ , but not on the parameters P, P∗ and w. In these normalized terms, the relation between the speed v and the applied force P become n 1 − 2v2 0 0 (Q2ν+1 − Q2ν ) = 1 − P0 . + P (9) P ∗ 2(1 − v2 ) ∗ ν=0 The relation between the speed v and the initial velocity −w is similar. Relation (9) gives the equation for finding the transition wave speed v. It contains two nondimensional parameters; parameter P0∗ defines the material properties, and P0 (or w0 ) describes the level of external excitation. Analytically calculated speed v of the transition wave for a bi-stable model was compared with numerical estimates in [(Slepyan et al., 2005)]. This comparison is shown in Figure 1(b).

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Figure 1. (a) The force–elongation dependence for the parallel-branch piecewise linear diagram: 1. T = µq (the first branch) and 2. T = µq − P∗ (the second branch). (b) Analytically calculated speed of the transition wave compared with numerical estimates shown by stars

The analytical technique is presented in details in [(Slepyan, 2002; Slepyan et al., 2005)]. The last work also deals with a more complicated case of non-parallel branches when γ 1. In this case, the Wiener-Hopf technique is exploited after taking the Fourier transform of q(η). An additional analysis is required to account for the variable elastic moduli, and for linkage of elements in a two-dimensional triangular lattice. It is possible to describe plane and crack-like waves of transition in the lattices using similar formalism [(Slepyan and Ayzenberg-Stepanenko, 2004)]. 3. Dynamic homogenization Description of the effective behavior of a bistable chain cannot be derived from Gibbs principle of minimum of energy, it requires corresponding “dynamic homogenization”. We are interested in the effective elongation of the chain due to a force applied to its end. The system is conservative, but a part of the work of the applied force is transformed to the kinetic energy of the phase transition. In open systems, a part of the energy is radiated by high frequency oscillations and waves. The adequate homogenization model should account for these energy transformations, which leads to the homogenized equation of the type: L(u) = ρü − ∇ · σ + Θ(u, u˙ , ∇u),

σ=

∂ Hd F(q) ∂q

(10)

Here the “pseudo-temperature” Θ(u, u˙ , ∇u) accounts for the energy stored in excited high-frequency waves which become invisible in the average description of the lower frequency part, and Hd F is an analog of the quasiconvex envelope of


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the nonconvex energy F: the dynamical average. The description can be obtained as a Γ-limit of the atomistic model taking into account the excited high-frequency waves. This high frequency part cannot be neglected because the kinetic energy of the system is bounded below from zero due to dynamical instabilities. 4. Numerical simulations Model of bistable bonds The discussed irreversible tensile force can be modelled using a damage function c(t) which introduces dependence on prehistory [(Cherkaev et al., 2005)]. The damage parameter c instantly jumps from zero to one at the first time moment t when the elongation q exceeds the critical value q∗ , 4 0 if maxt ∈[0,t) q(t ) < q∗ (11) c(t) = 1 otherwise The non-monotonicity of the force-elongation dependence corresponds to a nonconvex strain energy of the bond. The state of the bond under a time-dependent loading is described as T (q, c) = [1 − c(t)]T 1 (q) + c(t)T 2 (q)

(12)

where T 1 (q) is the dependence for the first stable region (the first branch) and T 2 (q) is that for the second branch which includes the unloading. The waiting-element structure A bistable link can be mechanically realized as a “waiting element” [(Cherkaev and Slepyan, 1995)]. A link, consists of two rods, one straight and one slightly curved, joined by their ends. We assume that the links are made of an elastic-brittle material. Using the introduced damage parameter c(t) we express the stress-strain relation in a rod as T (q, c) = (1 − c(t))H(q) µ q

(13)

where H(q) is the Heaviside function. The force-elongation dependence in the waiting rod is similar to the one in the basic link, but is shifted by τ. We assume that the waiting link starts to resist when the basic link is broken. The whole link is bistable, the tensile force T is the sum, T (q, cb , cw ) = T b (q, cb ) + T w (q, cw ) where T b , T w are the tensile forces in basic and waiting links described by (13) with the stiffness moduli µb and µw and the damage parameters cb and cw corresponding to the basic and waiting rods. The design parameters of the bistable link include: the critical elongations of the basic and waiting rods, q∗ and q∗∗ , the shift parameter τ (the gap between the branches of the stress-strain diagram is equal to τ − q∗ ) and the moduli µb and µw . In the following calculation, we use the representation µb = αµ, µw = (1 − α)µ

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where α is the relative thickness of the basic rod and µ is the stiffness modulus of the total cross-section of the element. In numerical simulations shown below, we assume that the waiting rod is stronger than the basic rod, or α < 0.5. Equations The dynamics of the chain is described by a system of difference– differential equations with respect to the displacements um of the masses and elongations qm = um − um−1 . The system has the form m m M u¨ m + γu˙ m = T (qm+1 , cbm+1 , cm+1 w ) − T (qm , cb , cw ) .

(14)

Here m is the index of the mass, γ is the coefficient of viscosity introduced to m stabilize numerical simulations, and cm b and cw are the damage parameters for the basic and waiting rods in the mth link. We consider the following model: A rested chain of N masses M bonded by the waiting-element links is impacted by a large mass M0 moving with initial velocity v0 . In simulations, we increase the mass M0 until the chain is broken. The goal is to optimize the parameters of the links to increase the amount of energy absorbed by the chain before breaking or to increase the value of the loading mass that can be withstood by the chain. The dynamics of the chain attached to a support, is described by the equations (14), condition for the mass in the root u0 = 0, the equation for the loading mass at m = N M0 u¨ N = −T (qN , cbN , cwN ), and the equations (11) for the damage indicators cbN and cwN where i = 1, . . . , N. The initial conditions are um = ma, m = 1, . . . , N u˙ m = 0, m = 1, . . . , N − 1,

u˙ N = v0

(15) (16)

In the numerical simulations, we use a chain of 32 elements. The initial speed v0 of the loading mass is 7% of the speed of the sound in material (the long wave speed & in the undamaged chain c = a µb /M). The additional elongation τ of the waiting link is larger than the limiting elongation q∗ , so that the basic link is completely broken before the waiting link is activated. Results of numerical simulations The trajectories of the nodes in a chain of regular and of bi-stable elements are shown in Figures 2(a) - 2(d). Figures 2(a) and 2(b) show trajectories of the knots of the chain without waiting elements. First of them shows the results of simulation of the chain impacted by a mass M0 = 100. The chain sustains the impact. However, being impacted by the mass M0 = 125, the chain is broken, see 2(b). Figures (c) and (d) show the bi-stable waiting element chain made of the same amount of material. The chain withstands


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much higher impact. Figure 2(c) shows the trajectories of the knots in the chain with waiting elements, α = 0.25, τ = 0.5, impacted by the mass M0 = 375: The chain sustains the impact. One can observe the elastic wave and two waves of a partial damage originated at both ends of the chain where the magnitude of the elastic wave is maximal. One can notice intensive oscillations of the masses after all links are partially broken. Trajectories of the knots in the chain with the same waiting elements and with a small dissipation, γ = 0.2, impacted by the mass M0 = 600 are shown in Figure 2(d). The chain sustains the impact. The dissipation has little influence on the waves propagation, but suppresses subsequent chaotic oscillations of the masses in the relaxed chain. The computer experiments show that structures with waiting links sustain the loading masses 5 - 6 times greater than the conventional structures. We stress that this result is achieved using the same amount of the same material, the only difference is the morphology of the structure. Modification of the model: Dissipation The dynamic of the chain with partially broken links indicates that the masses are involved in a chaotic motion after the basic links are broken. It is exactly the chaotic motion phase that leads to the final damage of the chain due to excessive elongation of a spring. Observing this phenomenon, we may question the adequacy of the model that does not take into account a small dissipation that is presented in a mechanical system; this dissipation would reduce the chaotic motion, especially in a long time range. We introduce a small dissipation into the numerical scheme; this practically does not influence the initial wave of the damage but reduces the intensity of afterward chaotic oscillations. The model with small dissipation gives even larger increase in the energy absorption capacity of the chain that can now sustain the twice larger loading mass. Notice that a small dissipation would have practically no influence on the behavior of chains made of a stable material, if the excitation is slow. However, the chain of unstable materials excite intensive fast-propagating waves no matter how slow is the impact, see [( Balk et al., 2001b)]. The intensity of the transition wave is determined not by the external excitation but by the inner structure of bi-stable elements. The wave is excited when the elongation of a bistable element is larger than the critical value and even a slow impact creates intensive waves which lead to significant energy dissipation. Large dissipation The increase of the dissipation coefficient allows for more orderly transition since the propagating and reflected waves are suppressed at a distance. We observe the wave of phase transition originated at the point of impact that propagate to the root of the chain. However, a second to the root link partially breaks when the weaken viscous-elastic wave reach the far end and reflects.

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5. Conclusion The results show a dramatic increase of the strength of the chain with waiting elements compared to the conventional design. The dissipation further increases the effect.

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References Balk A. M., Cherkaev A. V., Slepyan L. I. (2001a) Dynamics of chains with non-monotone stressstrain relations. I. Model and numerical experiments, J. Mech. Phys. Solids 49 131-148. Balk A. M., Cherkaev A. V., Slepyan L. I. (2001b) Dynamics of chains with non-monotone stressstrain relations. II. Nonlinear waves and waves of phase transition, J. Mech. Phys. Solids 49 149-171. Cherkaev A., Slepyan L. (1995) Waiting element structures and stability under extension, Int. J. Damage Mech. 4 58-82. Cherkaev A., Zhornitskaya L. (2004) Dynamics of damage in two-dimensional structures with waiting links, In: Asymptotics, Singularities and Homogenisation in Problems of Mechanics, A.B. Movchan editor 273-284, Kluwer. Cherkaev A., Cherkaev E., Slepyan, L. (2005) Transition waves in bistable structures I: Delocalization of damage, J. Mech. Phys. Solids 53 383-405. Cherkaev A., Vinogradov V., Leelavanichkul S. (2005a) The waves of damage in elastic-plastic lattices with waiting links: design and simulation, Mechanics of Materials accepted. Charlotte M., Truskinovsky L. (2002) Linear chains with a hyper-pre-stress, J. Mech. Phys. Solids 50 217-251. Ngan S. C., Truskinovsky L. (1999) Thermal trapping and kinetics of martensitic phase boundaries, J. Mech. Phys. Solids 47 141-172. Puglisi G., Truskinovsky L. (2000) Mechanics of a discrete chain with bi-stable elements, J. Mech. Phys. Solids 48 1-27. Slepyan L. I., Troyankina L. V. (1984) Fracture wave in a chain structure, J. Appl. Mech. Techn. Phys. 25 921-927. Slepyan L. I., Troyankina L. V. (1988) Impact waves in a nonlinear chain, In: ‘Strength and Viscoplasticity’ (in Russian), Journal Vol 301-305, Nauka.

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Slepyan L. I. (2000) Dynamic factor in impact, phase transition and fracture, J. Mech. Phys. Solids 48 931-964. Slepyan L. I. (2001) Feeding and dissipative waves in fracture and phase transition. II. Phasetransition Waves, J. Mech. Phys. Solids 49 513-550. Slepyan L. I. (2002) Models and Phenomena in Fracture Mechanics, Springer-Verlag. Slepyan L. I., Ayzenberg-Stepanenko M. V. (2004) Localized transition waves in bistable-bond lattices, J. Mech. Phys. Solids 52 1447-1479. Slepyan L., Cherkaev A., Cherkaev E. (2005) Transition waves in bistable structures II: Analytical solution, wave speed, and energy dissipation, J. Mech. Phys. Solids 53 407-436.

REAL-TIME SEISMIC MONITORING OF THE NEW CAPE GIRARDEAU (MO) BRIDGE AND RECORDED EARTHQUAKE RESPONSE Mehmet Çelebi USGS, Menlo Park, 94025, Ca, USA

Abstract. This paper introduces the state of the art, real-time and broad-band seismic monitoring network implemented for the 1206 m [3956 ft] long, cable-stayed Bill Emerson Memorial Bridge in Cape Girardeau (MO), a new Mississippi River crossing, approximately 80 km from the epicentral region of the 1811-1812 New Madrid earthquakes. Design of the bridge accounted for the possibility of a strong earthquake (magnitude 7.5 or greater) during the design life of the bridge. The monitoring network consists of a superstructure and two free-field arrays and comprises a total of 84 channels of accelerometers deployed on the superstructure, pier foundations and free-field in the vicinity of the bridge. The paper also introduces the high quality response data obtained from the network. Such data is aimed to be used by the owner, researchers and engineers to (1) assess the performance of the bridge, (2) check design parameters, including the comparison of dynamic characteristics with actual response, and (3) better design future similar bridges. Preliminary analyses of low-amplitude ambient vibration data and that from a small earthquake reveal specific response characteristics of this new bridge and the free-field in its proximity. There is coherent tower-cabledeck interaction that sometimes results in amplified ambient motions. Also, while the motions at the lowest (tri-axial) downhole accelerometers on both MO and IL sides are practically free-from any feedback from the bridge, the motions at the middle downhole and surface accelerometers are significantly influenced by amplified ambient motions of the bridge. Key words: earthquake, monitoring, real-time, response, ambient, frequency

1. Introduction In seismically active regions, acquisition of structural response data during earthquakes is essential to evaluate current design practices and develop new methodologies for analysis, design, repair and retrofitting of earthquake resistant structural systems, including lifelines such as bridges. This is particularly true for urban environments in seismically active regions. The New Madrid area, where the great earthquakes of 1811-1812 occurred, is a highly active seismic active region requiring earthquake hazard mitigation programs, including those related to investigation of strong shaking of structures and the potential for ground failures in the vicinity of structures (Nuttli, 1974; Woodward-Clyde Consultants, 1994). The Bill Emerson Bridge (here in after, 123 .


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the Cape Girardeau Bridge), in service since December 2003, is located approximately 80 km due north of New Madrid, Missouri. Design of the bridge (a) accounted for the possibility of a strong earthquake (magnitude 7.5 or greater) during the design life of the bridge and, as a result (b) was based on design response spectrum anchored to a zero-period acceleration (ZPA) of 0.36 g with a 10% probability of being exceeded in 250 years (Woodward-Clyde, 1994). Therefore, the Federal Highway Administration (FHWA), Missouri Department of Transportation (MoDOT), the Multi-disciplinary Center for Earthquake Engineering Research (MCEER), and the United States Geological Survey (USGS) collaborated in developing the seismic monitoring system for the bridge. For seismic monitoring of a bridge, three main categories in recording motions are sought: a.

instrumentation of the superstructure and pier foundations to capture and define (i) overall motion of the cable-stayed bridge, and motions of the (ii) two towers, to assess their translational and torsional behavior - relative to the caissons and deck levels and (iii) the deck, to assess the fundamental and higher mode translational (longitudinal, transverse and vertical) and torsional components, and (iv) extreme ends of the bridge and intermediate pier locations to provide data for the translational, torsional, and rocking soilstructure interaction (SSI) at the foundation levels as well as the horizontal and vertical spatial variation of ground motion.

b. instrumentation of the free-field in the vicinity of the structure including those related to downhole measurements and horizontal spatial arrays to assess the differential motions at the piers of the long span structure. c. ground failure arrays in the vicinity of the structure (outside the scope of this network). The general schematic of the bridge, shown in Figure 1, illustrates (a) overall longitudinal dimensions of the bridge and key elevations, and (b) key sensor locations alongside the bridge. The objective of this paper is to introduce details of the extensive, state-ofthe-art, real-time and broad-band seismic monitoring network deployed on and in the vicinity of the new Cape Girardeau Bridge. The real-time capability provides three basic advantages; (i) in addition to recording strong-motion events, it is possible to selectively record continuous real-time low-amplitude response data, on demand, with relative ease, (ii) use of the near real-time information can help make informed decisions related to the response and performance of the bridge (e.g. “health monitoring”) and (iii) maintenance of the system will be readily and easily enhanced as any malfunction of the sensors and related hardware will be detected via the real-time streamed information.

REAL-TIME SEISMIC MONITORING OF THE NEW CAPE GIRARDEAU 125

The figure illustrates (a) overall longitudinal dimensions of the bridge and key elevations, and (b) key sensor locations alongside the bridge, (c) general schematic of the seismic monitoring system with antennas for wireless communication of data between different locations. to and from the Data Acquisition Blocks 1 and 2 housed in Pier 2 and Pier 3 towers, and to the off-structure Central Recording System and (d) free-field arrays on both Missouri and Illinois sides.(Note: Shear wave velocities shown in fps. [1fps=0.3048m/s])

Figure 1.

The paper also presents preliminary sample analyses of low-amplitude data including motions (at ∼ 10−4 − 10−3 g levels) caused by a small earthquake that occurred 175 km away and ambient vibrations caused by tower-cable-deck interactions sources of which maybe traffic or wind. For example, site response effects or soil-structure interaction or tracing of motions from the free-field to the top of tower or reverse phenomenon caused by excitations at the free-field caused by ambient vibrations of the structure (tower-cable-deck interaction) can be clearly observed and a variety of computations can be made to provide insight into vibrational behavior of the bridge. However, in this introductory paper, while sample analyses using data from the network are presented, detailed analyses of each phenomena occurring at particular location of the bridge or the free-field sites (e.g. site response) are not included due to space constraints. Earlier publication (C ¸ elebi and others, 2004) related to this bridge did not include data analyses since data streaming started in March 2005.

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2. Instrumentation The detailed schematic of the bridge in Figure 2 shows key locations and orientations of the 84 channels of accelerometers deployed on the bridge and the two free-field arrays. Identifier two-character location codes are important for data management and for referencing motions this paper. All horizontal sensors oriented transverse (to the bridge) are more closely aligned to NS, and longitudinal sensors are more closely aligned to EW. Therefore, in this paper, directions of components of horizontal motions are hereafter referred to as NS and EW instead of lateral and longitudinal.

Figure 2. General schematic illustrating longitudinal dimensions of Cape Girardeau Bridge. Also shown are the locations and orientations of the accelerometers - both for the bridge and its piers and the free-field arrays on Missouri and Illinois sides. Location identifier codes (e.g. T1 for tower top location 1 at south end of tower top at Pier 2) are used for data management. Each arrow indicates one channel of accelerometer and its orientation

In addition to 84 channels of Kinemetrics EpiSensor1 accelerometers, additional monitoring hardware consists of Q3301 digitizers, and data concentrator

1 Citing commercial hardware throughout this manuscript does not imply endorsement of vendors or their products.


and mass storage devices (herein called Balers2 ) with wireless communication units (Figure 1). In each one of the hubs at Piers 2 and 3, combination of Q330 digitizers/recorder and Baler45 units constitute a multi-channel Data Acquisition Block (DAQ Block) that collects the analog signals from the accelerometers located throughout the bridge and then transmits the digitized data to the Central Recording System (CRS) using wireless radio transmittal via the antennas shown in Figure 1. The CRS merges the streamed data from the DAQs and records at location (on a pre-planned manner using a trigger algorithm to produce file events) and broadcasts the streamed data out using standard TCP/IP communications protocol. From CRS, all 84 channels of data are now being transmitted to Incorporated Research Institutions for Seismology (IRIS) via the Antelope3 software at the Central Recording System of the Cape Girardeau Bridge Seismic Monitoring System. From IRIS, data are disseminated online to the user community. Figure 2 also shows the general accelerometer deployment scheme for deck locations at the centerline (CL) and at the locations L1, L2, R1 and R2 of the cable-stayed deck. The exact sensor locations at these four locations are based on results of mathematical modal analyses (S. Dyke, written communication, 2001). It is noted herein that deck instrumentation at Pier 2 and 3 is on deck level at elevation 124.5 m (408.4ft). At both Piers 2 and 3, the deck is supported by the cables and does not rest on the piers. There are pot bearings where the edge beams rest on the pier cap. Therefore, there is a separate set of sensors at pier elevation 121.3 m (398 ft) (Figure 2). One of the two permanent surface and downhole free-field arrays is deployed at the Missouri [MO] side and the second at the Illinois [IL] side of the Mississippi River. The MO free-field array is approximately 100 m (∼ 399 ft) south of Bent 1 and the IL array is approximately 300m (∼ 900 ft) south-east of Pier 15 (Figures 1 and 2). Geotechnical characteristics of the boreholes that house the triaxial downhole accelerometers at defined depths from the surface are qualitatively and quantitatively shown in Figures 1 and 2. On the MO side, the two downhole accelerometers are at 9ft (∼3m) and 23 ft. (∼7.5m) from the surface. On the IL side, the two downhole accelerometers are at 47 ft (∼14.3 m) and 94 ft. (∼30m) from the surface. These two free-field arrays, intended to be without any feedback from the structure, are essential in providing the input ground motions that may be used as a surrogate for the various piers of the bridge and also for convolution and deconvolution studies of the free-field ground motion. However, as discussed later in this paper, the data acquired to date indicates that while the lowest 2

Baler is a data concentrator and mass storage unit. These units gather data and pass it to the next location as they are instructed to do so. In essence, a Baler serves as the brain and router of the data acquisition system. Baler-14 works with a single Q330 unit. Baler45 works with multiple number of Q330 units. In the case of Cape Girardeau instrumentation scheme, since there are multiple Q330 units, then only Baler45 units are used. 3 The Antelope software is licensed and installed at the site server by Kinemetrics. Citing commercial names does not imply endorsement of the vendor.

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(tri-axial) downhole accelerometers on both MO and IL sides (locations F3 and F6 in Figures 1 and 2) depict that there are practically free-from any feedback of motions of the bridge, the middle downhole and surface accelerometers are not. 3. Preliminary sample data analyses Preliminary analyses are performed for a 267 second interval of data that includes the earthquake (M=4.1) that occurred at 12:37:32 (UTC) on May 1, 2005 but, due to space limitations, only sample results are discussed. More detailed results are provided elsewhere (C ¸ elebi, 2005). The epicenter, 175 km from the bridge, was located at 6 km (4 miles) SSE (162◦ ) from Manila, AR (with epicental coordinates 35.830◦ N, 90.150◦ W) . The hypocentral depth was estimated to be 10 km (6 miles). Figure 3 shows full 267 seconds of only the NS component of acceleration data from MO and IL free-field arrays. The length of the record neither represents neither the duration of the earthquake nor the strong shaking caused by the earthquake. Such long duration shaking at this site from a small earthquake that occurred 175 km away is caused by crustal reverberations and surface waves (Hanks, 2005) and the network of broad-band instruments deliver strong signals even at small amplitudes (∼10-4 of g levels). The figure depicts (a) earthquake record of May 1, 2005 is between 30-130 seconds of the 267, this length of the ”earthquake” response caused by surface waves, (b) the motions at locations of the middle downhole accelerometers (F2 and F5) and of surface accelerometers (F1 and F4) are significantly amplified and are affected by (ambient) motions of the bridge when compared to the motions at lowest downhole accelerometer locations (F3 and F6), (c) when compared with the low-amplitude earthquake motions, the feedback from the bridge ambient motions are significant and occurs several times (exemplified by spikes) in the 267 second long record and (d) using the approximately 78-seconds of the earthquake portion of the 267 length of the data (approximately between 30-110 seconds), it is possible, as done later in the paper, to infer qualitative and quantitative site amplification that occurs at the freefield locations (even from such small amplitude motions caused by the distant small earthquake). Figure 4 shows NS time-history plots of motions at center of the deck (C2) , tower tops (T2 and T4) mid-tower locations (M2 and M4) side span deck (L2 and R6) and other pier locations and the free-field arrays. Compared to the ambient motions at deck center and other key structural locations (C2, T2, T4, L2, R6), the free-field motions due to the May 1, 2005 earthquake are much smaller. The total 267-second record reveals some important behavior of the deck and tower tops which are connected by cables under tension. Figures 5 shows displacement time histories in the EW (longitudinal) direction of the bridge at the north end of tower top locations (T2 at Pier 2 and T4 at Pier 3) and the north


267 seconds of free-field (NS component) acceleration data from surface and downhole deployments at both Missouri and Illinois sides. The earthquake of May 1, 2005 is best identified by the lower downhole accelerometers as they are the least affected by site response effects clearly seen at surface, and also by ambient vibration effects from the bridge reflected back to the surface and mid-downhole accelerometer locations. Short-duration spikes or bursts of energy, such as those preceding the P-Wave from the earthquake are likely due to traffic on the bridge

Figure 3.

end deck center location, C2. At approximately 0.33 Hz, vertical motions at C2 and EW motions at T2 are highly coherent and approximately 180◦ out of phase while vertical motions at C2 and EW motions at T4 are highly coherent but are in phase (0◦ phase angle) indicating that while deck center is going up, T4 is in phase and therefore moving eastward while T2 is 180◦ out of phase and therefore moving westward. Figure 6 shows similar plots for the (a) east-west motions at T2 and T4 which are perfectly coherent and (180◦ ) out of phase at 0.33 HZ and (b) north and south ends of deck center that are not vibrating perfectly in phase even though there is perfect coherency between the two. Thus, this implies that the amplified ambient motions of key locations of the structure (deck center, top of towers) are possibly due to time-variant cable action - that is, stored tensile energy in the cables vary and increase/decrease according to the displacements of the deck center and tower tops (as well as other locations where cables connect the deck to the towers). However, it is uncertain whether all cables on either north end or south end, and in particular those that are symmetrical with respect to center of the deck, remain in proportionally compatible tension. Thus, the cables store and release energy continuously. The amplified motions of the deck and tower tops may be caused by the cable actions. While detailed modes are not identified from the analyses of the low-amplitude data recorded, it is noted that the fundamental vertical mode of the bridge is approximately 0.33 Hz - similar to the 0.29 Hz frequency computed by modal analysis performed by Dyke (2001). Amplified

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Figure 4. The (NS) motions at free-field locations are shown relative to the piers and deck center and top of towers. The amplified motions of the structure between 100-150 seconds into the record are anomalies caused by cable vibrations of the structure and not by the much smaller earthquake between 30-100 seconds into the record

Figure 5. Displacement time-histories, cross-spectrum, coherency and phase-angle plots. [left]: EW motions between Tower Top at Pier 2 (Location T2) and vertical motions at center of deck, and [right]: EW motions between Tower Top at Pier 3 (Location T4) and vertical motions at center of deck

ambient motions of the bridge caused by cable vibrations or wind actions occur quite often as observed from the continuously streaming data but are not further presented herein due to space constraints. May 1, 2005 earthquake data of 78 second duration is extracted from the 267 second data. Figure 7 shows the time-histories (left) and corresponding amplitude


(left) Displacement time-histories, cross-spectrum, coherency and phase-angle plots of (left) EW motions between Tower Top at Pier 2 (Location T2) and Tower Top at Pier 3 (Location T4) and (right) vertical motions at the north end (C2) and south end (C1) of the center of the deck

Figure 6.

spectra (right) of the NS motions at west of deck center. It is seen that the frequencies associated with the tower have significant peaks around 1 and 2.2 Hz. Side span L2 has significant energy between 2-2.3 Hz and also at 1.3 Hz. Deck center displays several peaks, the lowest at 0.75 Hz. Figure 8 shows the time-histories and corresponding amplitude spectra of the EW motions at west of deck center. The Pier 2 tower locations, T2 and M2, exhibit significant peaks between 1.8 and 2.0 Hz and other significant peaks between 2.65-3.0 Hz.

Figure 7. Time-history plots of the NS accelerations (left) and corresponding amplitude spectra (right), at key locations to the west of the deck center, recorded during the May 1, 2005 earthquake

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Time-history plots (left) and corresponding amplitude spectra (right) of the EW accelerations at bent 1, mid-tower and tower top at Pier 2, and at pier 4 [location P6] and mid-tower and tower top of Pier 3 recorded during the May 1, 2005 earthquake

Figure 8.

Figure 9 shows vertical accelerations (left) and corresponding amplitude spectra (right) at various deck locations, downhole at Pier 2, at Bent 1 and surface and free-field array to the west of deck center. Figure 10 shows vertical accelerations (left) and corresponding amplitude spectra (right) at various deck locations, downhole at Pier 3, Pier 15 and surface and free-field array to the east of deck center. There is a much richer set of modes displayed in the amplitude spectra of the motions at different locations of the deck. The lowest frequency of the deck vertical motions is displayed at 0.33 Hz. The amplitude spectra and associated frequencies of the motions at the non-structural locations (free-field including downholes) and support locations of the structure (e.g. B1, L4, R4, D1) to the west of deck center are very similar but are considerably different than the spectra and frequencies of the motions at the non-structural locations (free-field and downholes) and support locations (Pier 15, Pier 3, Pier 4, Pier 5, downhole at Pier 3). This is attributed to different site conditions (different shear wave velocities depths to hard soil) at the (MO) west side versus the (IL) east side of the bridge. As previously stated, one of the advantages in having continuous monitoring and streaming of data with broad-band instruments is that recurring unusual responses and anomalies can be observed, interpreted and if necessary cautionary steps can be initiated. In addition to earthquake response, due to observance of amplified ambient motions in data, similar occurrences were searched in hours


Time-history plots (left) and corresponding amplitude spectra (right) of the vertical accelerations at various deck locations, downhole at Pier 2, bent 1 and surface and free-field array to the west of deck center recorded during the May 1, 2005 earthquake

Figure 9.

of data immediately before the earthquake of May 1, 2005 and found within 300 second long interval approximately 6 hours before the earthquake. Figure 11 shows a plot of this data from key locations of the bridge and also of the free-field locations on both MO and IL side of the bridge. Other occurrences of such amplified vibrations of the deck and towers are frequently observed in the continuously streaming data. Such random and irregular fluctuations in duration and amplitudes of motions of the deck and towers of the bridge are likely caused by cable vibrations particularly because the vertical motion of the deck center and the EW (longitudinal) motion of the towers are coherent and are generally in-phase for significant structural frequencies. Also noted in Figure 11 is that some of the motions that originate at the bridge deck and tower are reflected to the piers and bents and then to the free-field sites - reverse to that of an earthquake (e.g. motions starting at tower and deck travel to Bent1, then to F1, and then to F2 and are attenuated and possibly diminishes by the time they reach the lower downhole, F3). In other words, the structure is forcing the ground into motion. The frequency at ∼ 2 Hz observed in the time-history plots of accelerations at deck center and tower tops are considerably different than the 0.33 Hz frequency previously noted as belonging to the fundamental vertical mode of the structure and is likely that of higher vertical mode. The 10 Hz frequency superimposed on the 2 Hz motion is likely caused by cable vibrations.

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Figure 10. Time-history plots (left) and corresponding amplitude spectra (right) of the vertical accelerations at various deck locations, downhole at Pier 3, Pier 15 and surface and free-field array to the east of deck center recorded during the May 1, 2005 earthquake. Cause of ambient motions at E3 unknown

The ambient motions of the bridge deck and towers exhibit several frequencies as depicted in the amplitude spectra in Figure 12 plotted for both 0-20 Hz and 05 Hz frequency bands. It is seen in the figures that the NS spectra of tower top T2 location (at Pier 2) and T4 (at pier 3) are very similar. The same is true for the EW spectra. The 0.33 Hz frequency is observed in the EW spectra of the tower tops T2 and T4 as also in the vertical motions of the deck center, C2 but is practically absent in the spectra of NS motions. The figures show that there are several frequencies identifiable in the spectra. Different modes corresponding to the frequencies are not investigated herein.

4. Conclusions and recommendations In this paper, a new, integrated network of broad-band seismic instruments deployed on and in the vicinity of the new cable-stayed bridge in Cape Girardeau, MO is introduced. This is a significant accomplishment providing opportunities to acquire high quality data even at small amplitudes in the order of 10−4 − 10−3 g. Continuously recorded low-amplitude motions can facilitate assessment of dynamic characteristics of the structure and provide a basis for estimating levels of shaking during much less frequent stronger events (as in the New Madrid Seismic Zone). Preliminary analyses of low-amplitude data including an earthquake


(Left) Data from approximately 6 hours before the occurrence of the May 1, 2005 earthquake exhibits amplified motions of the deck and also feedback to the MO side free-field array. (Right) 90-100 second windows of the 300 second data exhibits motions from the bridge traveling to the free-field. The 0.5 second (2 Hz) period waveforms superimposed by approximately 0.1 s period waveforms (10 Hz) are clearly apparent and are likely caused by cable vibrations. Note the near absence of significant motions on the deepest borehole sensor (F3)

Figure 11.

demonstrates that (a) identification of response characteristics are successful and (b) amplified ambient vibrations caused by tower-cable-deck interactions are observed. However, their amplitudes are very small compared to amplitudes that may be experienced by stronger shaking. In the long-term, data from these smallamplitude vibrations provide opportunities to study for rupture scenarios, material fatigue and low-cycle fatigue. Signals of the data at low amplitudes are so good that, feedback from even ambient vibrations originating at the structure and traveling to the surface and downholes are observed in detail - indicating that the structure moves the ground during stronger ambient vibrations caused by towers-cable-deck interactions. It is hoped that the advance planning for instrumentation of the Cape Girardeau Bridge will set an example for future large projects in seismically active regions. In the future, whenever feasible, sufficient additional sensors (e.g. for soil-structure interaction (SSI) or liquefaction or wind related) can be integrated into the new system.

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Figure 12. Amplitude spectra of ambient motions show that the frequencies of the motions of the two towers are similar and the deck center vertical motions and EW components of towers exhibit several frequencies that are similar

References Çelebi M., Purvis R., Hartnagel B., Gupta S., Clogston P., Yen P., O’Connor J., Franke M. (2004) Seismic Instrumentation of the Bill Emerson Memorial Mississippi River Bridge at Cape Girardeau (MO) : A Cooperative Effort, 4th Int’l Seismic Highway Conf., Memphis, TN. January. Çelebi M. (2005) Real-Time Seismic Monitoring of the Cape Girardeau, MO, Bridge and Preliminary Analyses of Recorded Data: An Overview, paper in preparation. Dyke S. (2001) written communication. Hanks T. (2005) personal communication. Nuttli O. W. (1974) Magnitude-recurrence relation for Central Missippi Valley earthquakes, Seismological Society of America Bulletin 64, 1189-1207. Woodward-Clyde Consultants (1994) Geotechnical Seismic Evaluation Proposed New Mississippi River Bridge, (A-5076) Cape Girardeau, Mo, , Woodward-Clyde Consultants Report 93C8036500.

EARTHQUAKE RESPONSE OF MASONRY INFILLED PRECAST CONCRETE STRUCTURES Fuat Demir, Mustafa Türkmen, Hamide Tekeli and ˙Iffet C ¸ ırak Engineering Faculty, Department of Civil Engineering, SDÜ, Isparta, Turkey

Abstract. In design of precast concrete structures, frames are generally assumed to be two dimensional and analysed accordingly without taking into account participation of infill walls. However, it is well known that, the infill walls increase lateral stiffness and load carrying capacity of the frame structures. The objective of this study is to investigate the effects of infill masonry on structural behavior, performance and collapse mechanism of precast concrete structures. The diagonal strut model is adopted for modeling masonry infill. Load deformation relationship and collapse mechanism and of bare and infilled precast concrete structures are evaluated comparatively and results are given in figures. Key words: earthquake, infill masonry, precast concrete structure

1. Introduction Hinge connected precast concrete structures are ideal solutions to build factories, which requires open spaces without columns. Numerious buildings in industrial regions are constructed by using this type of structures. Precast concrete structures are also prefered for their economy and rapid production. In Turkey, industrialization is developing in the unprecedented speed. In respect of this development, the requirements for industrial structures are increased gradually. On the other hand a large part of the industrial regions are located on high risk earthquake zones. In recent earthquakes, large numbers of people have died and many of injured due to complete collapse of concrete buildings constructed with and without precast concrete structures. It is known that infill walls increase global stiffness and strength of the structure. However, this features of infills are generally not taken into account in the evaluation of the lateral load capacity of precast concrete structures. In this study, the effect of infill walls on the overall structural behavior and on the lateral load capacity of hinge connected precast concrete structures is investigated.

137 .


138

¨ DEM˙IR, TURKMEN, TEKEL˙I, ÇIRAK

2. Precast structures There is a great demand for industrial buildings in Turkey. In order to minimize the time span of the construction, use of precast elements is generally preferred. Turkey is located at around one of the most active fault zones in the world and is exposed frequently to destructive earthquakes .In Turkey, 92% of the territory, 95% of the overall population and 98% of the entire industrial facilities are located on seismically active zones (Tankut and Korkmaz, 2005). Most of the factories, under seismic risk, are constructed with precast elements. Since redundancy of precast structures is low, their lateral drifts under lateral loads are very large, compared to cast-in-place structures. Consequently the ductility and inelastic load capacity of precast structures are low. Many industrial facilities of precast concrete collapsed as a result of failures at beam to column connections. Particularly in Marmara Earthquake, a considerable amount of precast structures were damaged as a consequence of poor nature of the connection details. 3. Effects of masonry infills to the behavior of precast frame structures Masonry infills find widespread use as partition and external walls in precast buildings in Turkey. These masonry infills are generally constructed after completing of precast frames and they are rarely are included in analysis of the structural system subjected to vertical and lateral loads. Sometime precast panels are used instead of the infills. Masonary infills and precast panels are generally considered as non-structural elements. Although they are designed to perform architectural functions, masonry infills do resist lateral forces with substantial structural action. In addition to this, infills have a considerable strength and stiffness and they have significant effect on the seismic response of the structural system. There is a general agreement among researchers that infilled frames have greater strength as compared to bare frames. On the other hand, the presence of the infill also increases the lateral stiffness considerably. Due to the change in stiffness and mass in the structural system, the dynamic characteristics of the structural system change as well. Recent earthquakes of Erzincan, Düzce and ˙Izmit showed that infills have an important effect on the resistance and stiffness of buildings (Demir and Sivri, 2002) provided that they are constructed properly. It is not seldom that a number of structures have been strengthened by adding masonary infills in order to increase the lateral load carrying capacity of the structural system. Infill masonry effects not only the lateral load capacity and lateral displacement of the system, but also the damage pattern of the frames as well (Hong et al., 2002). Since precast structures generally have low rigidity, these buildings respond to seismic forces by swaying with them, rather than by attempting to resist them with rigid connections and low redundancy. Depending on the level of seismic forces, this can be an elastic response as well as plastic one. When the

EARTHQUAKE RESPONSE OF MASONRY

Figure 1.

139

Models for masonry. (a) Diagonal strut model, (b) continuum model

structure is subjected to seismic forces, low-level cracking which is distributed throughout walls comes into being, even the forces is small. This type distributed damage indicates the behavior of the structure is working. This working increases the energy dissipation of the structure (Langenbach, 2002-2003). However, the effect of the infills on the building under seismic loading is very complex and complicated. Since the behavior of the structural systems is highly nonlinear it is very difficult to predict it by analytical methods unless the analytical models are supported by experimental data. The effects of the infills on the analysis must be considered together with high degree of uncertainty related to the behavior, namely (Penelis and Kappos, 1997); 1. The variability of their mechanical properties, and therefore the low reliability in their strength and stiffness; 2. Their wedging condition, that is how tightly they are connected to the surrounding frame; 3.The potential modification of their integrity during the use of the building; 4. The non-uniform degree of their damage during the earthquake. In general, the presence of masonry infills affects the seismic behavior of the building as follow (Dowrick, 1987; Tassios, 1984) 1.Stiffness of the building is increased, the fundamental period is decreased and therefore the base shear due to seismic action is increased. 2.Distribution of the lateral stiffness of the structure in plan and elevation is modified. 3. Part of seismic action is carried by the infills, thus relieving the structural system. 4. Ability of the building to dissipate energy is substantially increased. In the conventional analysis of infilled frame systems, the masonry infill is modeled using either an equivalent diagonal strut model as given in Figure 1(a) or a refined continuum model shown in Figure 1 (b). The former is simple and computationally attractive, and generally preferred for simplicity but is theoretically weak.

140


First, identifying the equivalent nonlinear stiffness of the infill masonry using diagonal struts is not straightforward, especially when some openings, such as doors or windows, exist, in walls. Furthermore, it is also not possible to predict the damaged area of the infill either. The latter method based on continuum model can provide an accurate computational representation of both material and geometrical aspects of the problem, if the properties and the sources of nonlinearity of the infill carefully defined (Hong et al., 2002).

Figure 2. Compression diagonal strut model for representation of the infill stiffness.

As it is shown in Figure 1, the in-plane stiffness of infill wall is represented with an equivalent diagonal compression strut. The width of the strut Wef.depends on various parameters of the wall. However, for practical analysis it is generally assumed that (Negro and Colombo, 1996) & We f = 0.175 (λh H)−0.4 H 2 + L2 (1) 4 E i t sin 2θ (2) λh = 4Ec Ic Hi H and L are the height and length of the frame, Ec, and Ei are the elastic modulus of the column and of the infill wall, respectively; t is the thickness of the infill wall, is the angle of the diagonal strut, Ic is the modulus of inertia of the column and Hi is the height of the infill wall. 4. Numerical example A structure consisting of hinge connected precast concrete frames located in Campus of Suleyman Demirel University is investigated to study the effect of infill walls on the overall behavior and on the lateral load capacity. The infill walls exist along two longitudinal exterior edges. The structure is 64 m long, 21 m wide and 6.5 m high as shown in Figure 3. Quality of concrete is evaluated experimentally and found to be C30. The infill walls of thickness 110 mm exist within the frames. In the analysis, the infill walls are modelled by using the diagonal strut model.

EARTHQUAKE RESPONSE OF MASONRY

Figure 3. The precast concrete structure

141

Figure 4. Base shear-displacement relationship of the precast concrete structure

In order to study seismic behaviour of the frame structure a horizontal load is applied in the longitudinal direction at the top of the structure and a nonlinear pushover analysis is carried out. Flexural moment hinges are assigned to the bottom of the columns only. Default hinge properties based on ATC-40 and FEMA-273 criteria are provided (ATC-40 Report, 1996; FEMA-273, 1997). The base shear and the top displacement of the structure obtained from nonlinear pushover analyses in cases of the bare and uniformly infilled precast concrete structure are given in Figure 4. As it can be seen from the figure, the maximum base shear is about 500 kN for the bare frame and the maximum base shear is about 2200 kN for the infilled frame. Due to the existence of the infill walls, the base shear capacity of the infilled precast concrete frame increases more than four times and the displacement gets almost 60% smaller as given in Figure 4. The fundamental period of the bare and the infilled frames are 1.72 s and 1.63 s, respectively. As it is expected, the infill walls increase the base shear capacity, whereas they make the structures more stiffer. In the bare frame, the first plastic

Figure 5.

Collapse mechanism of bare and infilled precast concrete structure

hinges occur on the bottom of the columns, when the longitudinal load is about 450 kN. When the load reaches about 550 kN, the structure is in the level of immediate occupancy. The structure is in the level of life safety, when the load is 573 kN. However, a small increase of the loads brings the structure to collapse.

142


In masonry infilled frames, the first plastic hinges in the infill struts occur, when the load is 1224 kN. For 1993 kN the first plastic hinges on the bottom of the columns comes into being. 5. Conclusions The numerical results of the nonlinear pushover analysis show that the presence of nonstructural infill wall modifies the seismic response of the precast concrete structures to a large extend. The stability and integrity of precast concrete frames are enhanced with infill walls. Presence of infill walls also alters displacements and base shear of the precast concrete frame. If there is enough infill walls and infills cause a significant increase in lateral strength and stiffness. Although the numerical results reported base on a specific structural system, it is recommended that contributions of the infill walls should be included into structural analysis for obtaining more realistic results. References Demir F., Sivri M. (2002) Earthquake Response of Masonry Infilled Frames, ECAS International Symposium on Structural and Earthquake Engineering METU, Ankara. Dowrick D. J. (1987) Earthquake Resistant Design for Engineers and Architects, John Wiley & Sons, New York. Hong Hao, Guo-Wei M., Yong L. (2002) Damage RC Frames Subjected to Blasting Induced Ground Excitations, Engineering Structures 24 671-838. Langenbach, R. (2002-2003) Traditional Timber-laced Masonry Buildings that Survived the Great 1999 Earthquakes in Turkey and the 2001 Earthquake in India, While Modern Buildings Fell, Senior Disaster Recovery analyst, Federal Emergency Management Agency, USA. Negro P., Colombo A. (1996) Irregularities Induced by Nonstructural Masonry Panels in Framed Buildings, Engineering Structures 19 576-585. NEHRP Guidelines for the Seismic Rehabilitation of Buildings, FEMA 273, Developed by the Building Seismic Safety Council for the Federal Emergency Management Agency (Report No. FEMA 273), Washington, D. C., (1997). Penelis G. G., Kappos A. J. (1997) Earthquake-Resistant Concrete Structure, E&FN Spon, London. Seismic Evaluation and Retrofit of Concrete Buildings, 1, ATC-40 Report, Applied Technology Council, Redwood City, California, (1996). Tankut T., Korkmaz H. H. (2005) Performance of a precast concrete beam-to-beam connection subject to reversed cyclic loading, Engineering Structures 27 1392-1407. Tassios T. P. (1984) Masonry Infill and R/C Walls Under Cyclic Actions, CIB Symposium on Wall Structure, Invited State- of-the Art Report, Warsaw.

NONLINEAR WAVES IN FLUID-FILLED ELASTIC TUBES: A MODEL TO LARGE ARTERIES Hilmi Demiray Department of Mathematics, Faculty of Arts and Sciences, Işık University, Maslak, İstanbul, Turkey

Abstract. In the present work, by treating the arteries as a prestressed thin walled elastic tube of variable radius ,and the blood as an incompressible inviscid fluid, we have studied the propagation of weakly nonlinear waves in such a medium through the use of long wave approximation and the reductive perturbation method. The KdV equation with variable coefficient is obtained as the evolution equation. By seeking a progressive wave type of solution to this equation, we observed that the wave speed decreases with increasing inner radius while it increases with decreasing inner radius of the tube. Such a result is to be expected from physical considerations. Key words: solitary waves, variable coefficient KdV equation

1. Introduction Due to its applications in arterial mechanics, by employing various asymptotic methods, the propagation of small-but-finite amplitude waves has been investigated by (Johnson, 1970), (Hashizume, 1985), (Yomosa, 1987) and (Demiray, 1996, 1997) and various evolution equations are obtained depending on the balance between the nonlinearity, dispersion and dissipation. In obtaining such evolution equations, they treated the arteries as circularly cylindrical long thin tubes with constant radius. In essence, the arteries have variable radius along the tube axis. In the present work, by treating the arteries as a prestressed and thin walled elastic tube, we have studied the propagation of weakly nonlinear waves through the use of the long wave approximation and the reductive perturbation method. The variable coefficient KdV equation is obtained as the evolution equation. By seeking a progressive wave type of solution to this equation, we observed that the wave speed decreases with increasing inner radius while it increases with decreasing inner radius of the tube. Such a result is to be expected from physical considerations.

143 .


DEM˙IRAY

144

2. Basic equations and theoretical preliminaries We consider a circularly cylindrical long thin tube made of an isotropic elastic material, which is assumed to be a model for arteries, and subjected to an axial stretch ratio λz and to a variable inner static pressure P0 ∗ (Z ∗ ). In the course of blood flow, a dynamical radial displacement u∗ (z∗ , t∗ ) is superimposed on this initial static deformation. In reality, blood is an incompressible non-Newtonian fluid. However, for the sake of its simplicity in the analysis, we shall treat it as an incompressible inviscid fluid. Following (Demiray, 2004), the nondimensional field equations may be given as follows: 2

∂u ∂u ∂w + 2( f + )w + (λθ + f + u) = 0, ∂t ∂z ∂z ∂w ∂p ∂w +w + = 0, ∂t ∂z ∂z

p= −

∂2 u ∂Σ 1 m + 2 λz (λθ + f + u) ∂t λz (λθ + f + u) ∂λ2

∂ ( f + ∂u/∂z) 1 ∂Σ { }. 2 1/2 (λθ + f + u) ∂z [1 + ( f + ∂u/∂z) ] ∂λ1

(1)

where p is the total fluid pressure, u is the radial displacement, Σ is the strain energy density function of the tube, w is the axial fluid velocity, ( λz , λθ ) are the stretch ratios in the axial and the circumferential directions after static deformation, f (z) is a function charaacterizing the change of radius along the tube, m is a constant characterizing the inertia of the tube, λ1 and λ2 are the stretch ratios along the meridional and circumferential directions, respectively, and defined by λ1 = λz [1 + ( f +

∂u 2 1/2 ) ] , ∂z

λ2 = (λθ + f + u).

(2)

In equations (1), the first equation characterizes the conservation of mass, the second one the balance of linear momentum for fluid and the third one the balance of linear momentum of the tube in the radial direction. For our future purposes we need the power series expansion of the pressure in terms of the radial displacement u as

p = +p0 +

∂2 u¯ m ∂2 u¯ m ∂2 u¯ ∂¯u 2 α0 ∂2 u¯ ) −α +β u ¯ − u ¯ −α ( −(2α − )¯u +β2 u¯ 2 , 0 1 1 1 λθ λz ∂t2 ∂z λθ ∂z2 ∂z2 λ2θ λz ∂t2 (3)

NONLINEAR WAVES IN FLUID-FILLED ELASTIC TUBES

145

where p0 is the constant initial pressure, u¯ = u + f , and the coefficients α0 , α1 , β1 and β2 are defined by α0 = β0 =

1 ∂Σ , λθ λz ∂λθ

1 ∂Σ , λθ ∂λz

β1 =

α1 =

1 ∂2 Σ , 2λθ ∂λθ λz

1 ∂ 2 Σ β0 − , λθ λz ∂λ2θ λθ

β2 =

1 ∂ 3 Σ β1 − . 2λθ λz ∂λ2θ λθ

(4)

3. Long-wave approximation In this section we shall examine the propagation of small-but-finite amplitude waves in a fluid-filled elastic tube with variable radius, whose dimensionless governing equations are given in (1) and (3). For this, we employ the reductive perturbation method (Jeffrey and Kawahara, 1981). The nature of the problem suggests that we should consider it as the boundary value problem. For such problems, the frequency is specified and the wave number is calculated through the dispersion relation k = k(ω). Thus, it is convenient to introduce the following type of stretched coordinates ξ = 1/2 (z − gt),

τ = 3/2 z,

(5)

where is a small parameter measuring the weakness of nonlinearity and dispersion and g is the scale parameter to be determined from the solution. Solving z in terms of τ we get z = −3/2 τ. Introducing this expression for z into f (z) we have f (z) = h(; τ).

(6)

In order to take the effect of the variation of the radius into account, the order of h(, τ) should be h(, τ) = l(τ), where the function l(τ) is of order of 1. We shall further assume that the field quantities u, w and p may be expanded into asymptotic series as u = u1 (ξ, τ) + 2 u2 (ξ, τ) + ... w = w1 (ξ, τ) + 2 w2 (ξ, τ) + ... p = p0 + p1 (ξ, τ) + 2 p2 (ξ, τ) + ... .

(7)

Noting the coordinate transformation (5) and introducing the expansion (7) into the field equations (1) and (3), and setting the coefficients of like powers of equal to zero, the following set of differential equations are obtained O () equations: −2g

∂u1 ∂w1 + λθ = 0, ∂ξ ∂ξ

−g

∂w1 ∂p1 + = 0, ∂ξ ∂ξ

p1 = β1 [u1 + l(τ)].

(8)

DEM˙IRAY

146 O ( 2 ) equations: −2g

∂w1 ∂w2 ∂w1 ∂u1 ∂w1 ∂u2 + λθ + λθ + 2w1 + u1 + l(τ) = 0, ∂ξ ∂ξ ∂τ ∂ξ ∂ξ ∂ξ −g p2 = (

∂w1 ∂w2 ∂p2 ∂p1 + + + w1 = 0, ∂ξ ∂ξ ∂τ ∂ξ

mg2 ∂2 u1 − α0 ) 2 + β1 u2 + β2 [u1 + l(τ)]2 . λθ λz ∂ξ

(9)

4. Solution of the field equations From the solution of the set of equations (7) we obtain u1 = U(ξ, τ),

w1 =

2g [U + w¯ 1 (τ)], λθ

p1 = β1 [U + l(τ)],

(10)

provided that the following relation holds true g2 =

λθ β1 , 2

(11)

where U(ξ, τ) is an unknown function whose governing equation will be obtained later, 2gw¯ 1 (τ)/λθ corresponds to the steady axial flow resulting from the pressure β1 l(τ), which also causes the radial displacement l(τ) for the tube. Introducing the solution given in (10) and (11) into equation (9), we have −2g

∂U dw¯ 1 4g ∂w2 ∂U ∂u2 + λθ + 2g( + ) + (U + w¯ 1 ) ∂ξ ∂ξ ∂τ dτ λθ ∂ξ + −g

2g ∂U 2g ∂U − h1 (τ) =0 U λθ ∂ξ λθ ∂ξ l ∂w2 ∂p2 2g2 ∂U + + + ) ( ∂ξ ∂ξ λθ ∂τ dτ +

p2 = (

∂U 4g2 = 0, (U + w¯ 1 )) 2 ∂ξ λθ

mg2 ∂2 U − α0 ) 2 + β2 [U + l(τ)]2 + β1 u2 . λθ λz ∂ξ

Eliminating w2 between equations (12), we have 10g2 4g2 ∂U ∂U ∂U 8g2 2g2 + ( 2 + 2β2 )U + [ 2 w¯ 1 + 2 l(τ) + 2β2 l(τ)] λθ ∂τ ∂ξ ∂ξ λθ λθ λθ

(12)


+(

∂3 U 2g2 d mg2 [w¯ 1 + l(τ)] = 0. − α0 ) 3 + λθ λz λθ dτ ∂ξ

147 (13)

The equation (13) should even be valid when U = 0 (steady flow), which results in d [l(τ) + w¯ 1 (τ)] = 0, or w1¯(τ) = −l(τ). (14) dτ Introducing (14) into equation (13) yields the following Korteweg-de Vries equation with variable coefficient ∂3 U ∂U ∂U ∂U + µ1 U + µ2 3 + µ3 l(τ) = 0, ∂τ ∂ξ ∂ξ ∂ξ

(15)

where the coefficients µ1 , µ2 and µ3 are defined by µ1 =

5 β2 + , 2λθ β1

µ2 = (

m α0 − ), 4λz 2β1

µ3 = −

1 β2 + . λθ β1

By introducing the following coordinate transformation # τ l(s)ds, τ = τ, ξ = ξ − µ3

(16)

(17)

0

the variable coefficient KdV equation reduces to the conventional KdV equation ∂3 U ∂U ∂U + µ U + µ = 0. 1 2 ∂τ ∂ξ ∂ξ 3

(18)

This nonlinear evolution equation admits the following type of localized travelling wave solution (19) U = a sech2 ζ, wher a is the wave amplitude and the variable ζ is defined by # τ µ1 a µ1 a 1/2 ) [ξ − µ3 l(s)ds − τ]. ζ=( 12µ2 3 0

(20)

Here, it is seen from equation (20) that the trajectory of the solitary wave is not a straight line, it is rather a curve. In our subsequent analysis we need the explicit expressions of the coefficients α0 , β1 , , β3 . For that purpose we shall utilize the constitutive equation proposed by (Demiray, 1972) for soft biological tissues as Σ=

1 1 {exp[α(λ2θ + λ2z + 2 2 − 3] − 1}, 2α λθ λz

(21)

DEM˙IRAY

148

where α is a material constant. Introducing (21) into (4), the explicit expressions of the coefficients α0 , α1 , β1 , β2 , may be given as follows 1 1 (λz − 2 3 ) F, λθ λθ λz

α0 = β1 = [ β2 = [−

4 λ5θ λ3z

+2

α 1 (λθ − 3 2 )2 ] F, λθ λz λθ λz

10 α 1 11 α2 1 + (λ − )(1 + ) + 2 (λθ − 3 2 )3 ] F, θ 3 4 2 6 3 2 λθ λz λθ λz λθ λz λθ λz λθ λz λθ λz

(22)

where the function F is defined by F = exp[α(λ2θ + λ2z +

1 λ2θ λ2z

− 3)].

(23)

So far we have not said anything about the form of the function l(τ), which characterizes the variation of the radius. In what follows we shall study some special cases. (i) Linear Tapering: For this case the function l(τ) can be approximated by l(τ) = Φτ,

(24)

where Φ is the tapering angle. Introducing (24) into (20) we obtain ζ=(

1 1 β2 µ1 a µ1 a 1/2 τ]. ) [ξ − (− + )Φτ2 − 12µ2 2 λθ β1 3

(25)

As can be seen from equation (25), the trajectory of the wave is a parabola, which is the result of variable radius of the tube. In order to study the numerical illustration we need the value of the material constant α. The mathematical model that we have presented here was compared before by us (Demiray, 1976) with some experimental measurements by Simon et al (Simon et al., 1972) on canine abdominal artery with characteristics Ri = 0.31cm Ro = 0.35cm and λz = 1.6 and the value of material constant α was found to be α = 1.948. Using this value of material constant α and the initial deformation λz = 1.6 and λθ = 1.6 the values of (β2 /β1 ) and µ1 are calculated as β2 /β1 = 3.348, µ1 = 4.89. For this case the variable ζ becomes µ1 a 1/2 ζ=( ) (ξ − 1.35Φτ2 − 1.63aτ). (26) 12µ2 Choosing the values of Φ and a as unity, it becomes µ1 1/2 ) (ξ − 1.35τ2 − 1.63τ). ζ=( 12µ2

(27)


149

Here we note that ξ characterizes the time variable, whereas τ characterizes the space variable. The speed of propagation may be defined by vp =

1 dτ = . dξ 1.63a + 2.7Φτ

(28)

Here 1/1.63a corresponds to the wave speed for a tube of constant radius. As is seen from this expression, the wave speed decreases with distance parameter τ for Φ > 0 (expanding tube ) but increases for Φ < 0 (narrowing tube). This is to be expected from physical consideration. (ii) Stenosised Artery : For this case the function l(τ) can be approximated as l(τ) = −c sech2 τ,

(29)

where c is the magnitude of the stenosis. Introducing this expression (29) into equation (20) we obtain ζ=(

β2 1 µ1 a 1/2 ) [ξ − ( − )c tanh τ − 1.63aτ]. 12µ2 β1 λθ

(30)

Again, the profile of the travelling wave is bell-shaped and the speed of the solitary wave may be defined by vp =

1 [ µ31 a

+

( ββ21

−

2 1 λθ )csech τ]

.

(31)

Here, again, it is seen that the wave speed increases with distance τ from the center of the stenosis. 5. Conclusions The results of linearly tapered and stenosed arteries show that the mathematical model presented here yields the solitary wave solution, as observed by (McDonald, 1974) in his experimental measurements. Moreover, as it is to be expected from physical considerations, the wave speed decreases for expanding tube whereas it increases for narrowing tubes. References Demiray H. (1972) A not on the elasticity of soft biological tissues, J. Biomechanics 5 309-311. Demiray H. (1976) Large deformation analysis of some basic problems in biophysics, Bull. Math. Biology 38 701-711. Demiray H. (1996) Solitary waves in a prestressed elastic tube, Bull. Math. Biology 58 939-955. Demiray H., Antar N. (1997) Nonlinear waves in an inviscid fluid contained in a prestressed viscoelastic thin tube, Z. Angew. Math. Phys. 48 325-340.

150

DEM˙IRAY

Demiray H. (2004) Modulation of nonlinear waves in a fluid-filled elastic tube with stenosis, Int. J. Mathematics and Mathematical Sciences 60 3205-3218. Jeffrey A., Kawahara T. (1981) Asymptotic Methods in Nonlinear Wave Theory, Pitman, Boston. Johnson R. S. (1970) A nonlinear equation incorporating damping and dispersion, J. Fluid Mechanics 42 49-60. Hashizume Y. (1985) Nonlinear pressure waves in a fluid-filled elastic tube, J. Phys. Soc. Japan 54 3305-3312. McDonald D. A. (1974) Blood Flow in Arteries, The Williams and Wilkins Co., Baltimore. Simon B. R., Kobayashi A. S., Stradness D. E., Wiederhielm C. A. (1972) Re-evaluation of arterial constitutive laws, Circulation Research 30 491-500. Yomosa S. (1987) Solitary waves in large blood vessels, J. Phys. Soc. Japan 56 506-520.

THE NONLINEAR AXISYMMETRIC VIBRATIONS OF CIRCULAR PLATES WITH LINEARLY VARYING THICKNESS UNDER RANDOM EXCITATION Vedat Do˘gan ˙ ˙ Department of Aeronautical Engineering, Istanbul Technical University, Istanbul, Turkey

Abstract. In this study, the nonlinear axisymmetric behavior of circular isotropic plates with linearly varying thickness under random excitation is investigated. It is assumed that plate response is axisymmetric when plate is subjected to axisymmetric random loading. The Berger type nonlinearity is used to obtain the governing equations of motion for clamped circular plates. A Monte Carlo simulation of stationary random processes, single-mode Galerkin-like approach, and numerical integration procedures are used to develop nonlinear response solutions. Response time histories, root mean squares and spectral densities are presented for different random pressure levels. Parametric results are also presented. Linear responses are included to investigate the nonlinear effects. Key words: circular plates, nonlinear, random vibration

1. Structural formulation The nonlinear vibrations of plates and shells were extensively studied (Sathyamooorthy 1997, Nayfeh 2004). Circular plates are commonly used engineering structural components and their vibrations have been investigated by many researchers (Chang 1994, Dumir et al., 1984, Efstathiades 1971, Ghosh 1986, Hadian and Nayfeh 1990, Liu and Chen 1996, Ramachandran 1975, Sathyamooorthy 1996, Smaill 1990, Sridhar et al., 1975). The Berger type nonlinear equation of motion for axisymmetric vibrations in terms of normal deflection is given as (Ramachandran 1975) 4 7 ∂D ∂3 w 2 + ν ∂2 w 1 ∂w 2 3 + + D∇ w + − ∂r r ∂r2 r2 ∂r ∂r 4 7 ∂2 D ∂2 w ν ∂w ∂2 w ∂w 2 + 2 + pr (r, t) + w = ρh − ch − N∇ 2 2 r ∂r ∂t ∂r ∂r ∂t 4

151 .


(1)

˘ DOGAN

152 and

2 u Nh2 ∂u 1 ∂u = + (2) + 12D ∂r 2 ∂r r where ρ , c and h are the material density, viscous damping coefficient and plate thickness, respectively. The pr (r, t) is random pressure acting on the circular plate. It is assumed that radial displacement vanishes at the boundary (i.e. u(a, t) = 0 and also u(0, t) = 0 ), after Eqn. (2) is multiplied by rdr and integrated over the plate area (Ramachandran 1975), then Nh21 a2 8 9 1 α + log(1 − α) = − 2 12D1 α 2 where α is the taper ratio. Linearly varying thickness can be taken as r h = h1 1 − α a

a

# 0

∂w ∂r

2 rdr

(3)

(4)

and the bending stiffness

r 3 D = D1 1 − α (5) a where h1 and D1 are the thickness and flexural stiffness at r = 0, respectively. Eh3

1 The plate bending stiffness is D1 = 12(1−ν 2 ) where E and ν are the modulus of elasticity and the Poisson’s ratio. Solution for normal (transverse) deflection w(r, t) may be approximated by

w(r, t) =

N1

ηm (t)φm (r)

(6)

m=1

where ηm (t) is the generalized coordinate,φm (r) is the mth shape function which satisfies the normal boundary conditions, and N1 is the number of chosen shape function. For an approximation, a single mode is chosen in this study ' r 2 (2 (7) w(r, t) = η(t)φ(r) = η(t) 1 − a mth

Equation (7) clearly satisfies the clamped edge boundary conditions. Inserting Eqn.(7) into Eqn. (3) and Eqn.(1), then Eqn. (3) into Eqn. (1), with simulated external random pressure pr (t) (Shinozuka and Jan, 1972), Galerkin procedure is used to reduce the system into the following ordinary nonlinear differential equation: d2 η dη + µ1 η(t) + µ2 η3 (t) = λpr (t) (8) ρh1 2 + ch1 dt dt

VIBRATIONS OF CIRCULAR PLATES

153

where µ1 =

44D1 1680 − 315α2 (−9 + ν) − 288α (15 + 2ν) + 32α3 (−23 + 6ν) − 256α) (9) 18480α2 D1 (10) µ2 = − 4 2 , a h1 log(1 − α) + α

a4 (693

1160 (11) (693 − 256α) The numerical solutions of Eqn.(8) were carried out by using the Runga-Kutta fifth order method. Aluminum plate with h1 = 0.002m , a = 0.5m, α = 0.2 is considered, unless indicated otherwise. Responses are found at the center of the plate. The displacement response time histories for SPL=160 dB is given in Fig. 1. For selected geometry/material properties, RMS responses for different Sound Pressure Levels are presented in Fig. 2, which also includes linear response. Nonlinearity seems to gain importance for input loading above 120 dB. Effect of the taper ratio on the responses is shown in Fig. 3. Response spectral density is given in Fig. 4. Responses for different thickness to radius ratio are presented in Fig. 5. λ=

Figure 1.

Displacement response time histories for SPL=160 dB

˘ DOGAN

154

Figure 2.

Linear and nonlinear RMS responses for different Sound Pressure Level

Figure 3.

The taper ratio effect on the RMS responses

VIBRATIONS OF CIRCULAR PLATES

Figure 4.

Figure 5.

155

Response Spectral Density

RMS responses vs. Sound Pressure Levels for different radius/thickness ratios

156

˘ DOGAN

References Chang T. P. (1994) Random Vibration of a geometrically nonlinear orthotropic circular plate, Journal of Sound and Vibration 170 426-430. Dumir P. C., Nath Y., Gandhi M. L. (1984) Non-linear axisymmetric transient analysis of orthotropic thin annular plates with a rigid central mass, Journal of Sound and Vibration 97 387-397. Efstathiades G. J. (1971) A new approach to the large-deflection vibrations of imperfect circular disk using Galerkin’s procedure, Journal of Sound and Vibration 16 231-253. Ghosh S. K. (1986) Large Amplitude Vibrations of Clamped Circular Plates of Linearly Varying Thickness, Journal of Sound and Vibration 104 371-375. Hadian J., Nayfeh A. D. (1990) Modal Interaction in Circular Plates, Journal of Sound and Vibration 142 279-292. Liu C. F. Chen G. T. (1996) Geometrically Nonlinear Axisymmetric Vibrations of Polar Orthotropic Circular Plates, Int. J. Mech. Sci. 38 325-333. Nayfeh A. H., Pai P. F. (2004) Linear and Nonlinear Structural Mechanics, John Wiley Sons, New Jersey. Ramachandran J. (1975) Nonlinear vibrations of circular plates with linearly varying thickness, Journal of Sound and Vibration 38 225-232. Sathyamoorthy M. (1996) Large Amplitude Circular Plate Vibration with Transverse Shear and Rotatory Inertia Effects, Journal of Sound and Vibration 194 463-469. Sathyamoorthy M. (1997) Nonlinear Analysis of Structures, CRC Pres, Boca Raton, Florida. Shinozuka M., Jan C. M. (1972) Digital simulation of random processes and its applications, Journal of Sound and Vibration 25 111-128. Smaill J. S. (1990) Dynamic Response of Circular Plates on Elastic Foundations: Linear and NonLinear Deflection, Journal of Sound and Vibration 139 487-502. Sridhar S., Mook D. T., Nayfeh A. H. (1975) Non-linear Resonances in the forced responses of plates Part I: Symmetric Responses of Circular Plates, Journal of Sound and Vibration 41 359-373.

DYNAMIC ANALYSIS OF A HELICOPTER ROTOR BY DYMORE PROGRAM Vedat Do˘gan and Mesut Kırca Faculty of Aeronautics and Astronautics, Rotorcraft Research and Development ˙ ˙ Centre, Istanbul Technical University, Istanbul, Turkey

Abstract. The dynamic behavior of hingeless and bearingless blades of a light commercial helicopter which has been under design process at ITU (˙Istanbul Technical University, Rotorcraft Research and Development Centre) is investigated. Since the helicopter rotor consists of several parts connected to each other by joints and hinges; rotors in general can be considered as an assembly of the rigid and elastic parts. Dynamics of rotor system in rotation is complicated due to coupling of elastic forces (bending, torsion and tension), inertial forces, control and aerodynamic forces on the rotor blades. In this study, the dynamic behavior of the rotor for a real helicopter design project is analyzed by using DYMORE. Blades are modeled as elastic beams, hub as a rigid body, torque tubes as rigid bodies, control links as rigid bodies plus springs and several joints. Geometric and material cross-sectional properties of blades (Stiffness-Matrix and Mass-Matrix) are calculated by using VABS programs on a CATIA model. Natural frequencies and natural modes of the rotating (and non-rotating) blades are obtained by using DYMORE. Fan-Plots which show the variation of the natural frequencies for different modes (Lead-Lag, Flapping, Feathering, etc.) vs. rotor RPM are presented. Key words: bearingless, rotor dynamics, Dymore

1. Introduction There are basically four types of helicopter rotor hubs in use. These are the teetering design, the articulated design, the hingeless design, and the bearingless design (Schindler and Pfisterer, Chopra and Inderjit, Mil et al., 1967). One of the bearingless rotor designs is “Elastic Articulation” (EA) Rotor which was originally developed by Lockheed, California in the early 1960s and further developed by Tom Hanson. General overview of EA Rotor system is shown in Fig. 1. In EA rotor system, the flexbeam is responsible to carry out all blade motions (flapping, lead-lag, and feathering). The torque tubes give the pitching input to the rotor system without taking the tensional stresses by using the spherical pivot bearing at the root. A sweep angle exists between the blade c/4 line and the feathering axis. This sweep angle is the part of a system that aims to achieve automatic blade stabilization during the articulation of the system. 157 .


˘ DOGAN, KIRCA

158

Figure 1.

EA Rotor Exploded View (Hanson, 1998)

2. Analysis In this study, Hanson’s Rotor system (Hanson, 1998) shown in Fig. 2 is adapted as base. The main tasks related with designing a rotor are calculating the clearance between blade tip and tail boom, obtaining the mode shapes, plotting all the frequencies in the fan plot, manipulating the flexure and blade properties to find an optimum frequency placement to avoid resonance, investigating the ground and air resonance. DYMORE is a handy tool for the analysis of these types of problems. DYMORE is a multi-body dynamic analysis code developed by Prof. O.A. Bauchau at Georgia Institute of Technology (Bauchaou). The needed inputs for dynamic analysis from a preliminary design program output file include

Figure 2.

General view of the bearingless rotor (EA) system

DYNAMIC ANALYSIS OF A HELICOPTER ROTOR

Figure 3.

Figure 4.

159

Flexure-blade assembly

Flexure cross-section

gross weight, tip speed, number of blades, rotor diameter, blade chord, tip weight, forward sweep angle and blade weight. Present study centers around the fan plot which gives variations of the natural frequencies of rotating blades vs. rotor speed. A CATIA model of flexbeam-blade assembly is shown in Fig. 3 , Fig. 4 and Fig. 5 give the cross sections of flexbeam and blade at a station, respectively. Materials used in blades are also indicated in Fig. 5. Rotor blades and flexure are usually modeled as slender beams with heterogeneous cross sections. VABS (Variable Asymptotic Beam Section) is a computer program that implements a variable asymptotic method for computing the stiffness of a heterogeneous beam at a given cross-section.

˘ DOGAN, KIRCA

160

Figure 5.

Blade cross-section materials

VABS is capable of computing the full 6-by-6 stiffness matrix (all bending, extension, shear and torsion terms) with asymptotic accuracy. The sectional stiffness can then be used in 3-D flexible multibody analysis tools such as DYMORE . Following main steps are employed to obtain stiffness matrix: − Take cross-sections at a given (chosen) station (Fig. 7) − Import those sections to ANSYS − Create ANSYS areas in each cross section. − Generate a 2-D mesh within each area and nodal compatibilities between areas. − Write macros to define reference curves and element material orientations and 3-D material properties − Prepare VABS input file − Run VABS, and get VABS Output file. Output (Figure 8) includes “stiffness matrix” which can be readily used within DYMORE model. Steps are shortly repeated in Fig. 6.

Figure 6.

Flow diagram of the analysis

DYNAMIC ANALYSIS OF A HELICOPTER ROTOR

Figure 7.

161

Blade and flexure cross sectional stations from CAD model

To avoid resonance, natural frequencies must not cross 1P to 4P lines in the vicinity of 1P of rotor speed. If it does, as in Fig. 9, 2nd Lead-Lag frequency and 4P lines cross each other, resonance occurs. By changing geometric and material properties as well as other design parameters, “optimum” frequency placement can be achieved.

Figure 8.

VABS Output required by DYMORE

˘ DOGAN, KIRCA

162

Figure 9.

Fan-plot (Natural Frequencies vs. Rotor Speed) obtained by DYMORE

References Bauchaou O. A. DYMORE: User’s Manual and Theory Manual. Chopra, Inderjit. Helicopter Dynamics, University of Maryland, Center for Rotorcraft Education and Research. Hanson T. F. (1998) Designer Friendly Handbook of Helicopter Rotor Hubs. Mil M. L., et al. (1967) Helicopter Calculation and Design, Volume 2- Vibrations and Dynamic Stability, Moscow. Schindler R., Pfisterer. Impacts of Rotor Hub Design Criteria on the Operational Capabilities of Rotorcraft, AGARD-CP-354.

PARAMETRIC STIFFNESS EXCITATION AS A MEANS FOR VIBRATION SUPPRESSION ∗ Horst Ecker Vienna University of Technology,Institute of Mechanics and Mechatronics, A-1040 Vienna, Austria

Abstract. This contribution presents an overview of recent research activities, with the goal of utilizing parametric excitation as a novel method to suppress self-excited vibrations by amplifying the damping properties of the system. The basic idea of the method is discussed and different methods to analyze parametrically excited systems are reviewed. Three different examples are presented, where parametric stiffness excitation is applied and either stabilizes the system or improves the damping behavior. Key words: parametric resonance, self-excitation, stability of mechanical systems

1. Introduction If an excitation is applied to a mechanical system it will, in general, vibrate. Most of the time these vibrations are unwanted and require countermeasures if the vibration amplitudes exceed a certain level. Depending on the nature of the excitation, different strategies and methods of vibration reduction are in use. This contribution focuses on a rather new idea to reduce, and in some cases even cancel vibrations in mechanical systems by means of parametric excitation. In mechanical systems different types of excitation are observed. A frequently encountered type is the external excitation, also named forced excitation. An unbalance excitation as known from rotating machinery is one example, the excitation of a structure which is attached to a vibrating foundation is another. A mechanical system under the influence of an external excitation will perform forced vibrations. Vibration amplitudes are determined by the dynamical properties of the system and the amplitude of excitation. In the case of a linear system, vibration amplitudes are proportional to excitation amplitudes and damping properties of the system. Vibration reduction for the system in general is mainly either achieved by choosing these parameters appropriately or by tuning the system as to avoid resonances. ∗

This work was partly funded by the Austrian Science Fund under Grant P-16248 163

.


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Although mentioned in first place, the reduction of vibrations caused by external excitation will only play a minor role in this paper. We will almost exclusively deal with mechanical systems where two other types of excitation mechanisms are prevailing: Self-Excitation (SE) and Parametric Excitation (PE). As in most practical cases, self-excitation will be the source of unwanted vibrations. It will be shown that parametric excitation can be employed as a mean to suppress selfexcited vibrations. 2. Parametric excitation - past and presence Parametrically excited systems have been of interest since a long time, and date back as far as to the 19th century, when M. Faraday investigated sloshing liquids in a container and, about 30 years later, when E. Mathieu established the famous equation given his name. Since then, parametric excitation (PE) has attracted much interest, mainly because it may lead to a unique type of resonances, called parametric resonances. Given the available space, it is virtually impossible to even give a brief overview and just name the important contributions in this field of ongoing active research. From an application point of view and focussing on mechanical engineering two different aspects of parametric excitation can be extracted from the numerous references: how to avoid or reduce the effect of parametric excitation in a system and how to take advantage of PE, especially PE-resonances. The second aspect is much less popular and not very many applications of PE in this sense are known. Within this area, the idea prevails, to make use of the large amplitudes which will occur when a system is operated at a parametric resonance. For instance, recent works focus on micro-electromechanical systems (MEMS) as a possible application, see e.g. (Shaw, et al., 2004). However, not all PE-resonances lead to large amplitudes, since some of them may be non-resonant. This special case has not been studied at all, until Tondl found out about an interesting phenomenon associated with non-resonant parametric resonances (Tondl, 1998). In his paper Tondl shows early results obtained from analog computer simulations of an unstable, non-linear, parametrically excited system, see Fig. 1. The surprising detail in this result is a frequency interval of the PE, where the selfexcited vibration amplitudes of the system are completely suppressed. Since this occurs at the frequency of a parametric resonance, the phenomenon was named parametric anti-resonance. This pioneering work triggered research efforts at various places. It led to a growing number of contributions related to this phenomenon, only some of them shall be mentioned here. Analytical methods and bifurcation analysis have been applied in Utrecht by Verhulst and his students (Fatimah, 2002), (Abadi, 2003). Very comprehensive investigations, both analytically and numerically were carried out in Vienna by Dohnal (Dohnal, 2005) and this author (Ecker, 2003).

PARAMETRIC EXCITATION AND VIBRATION SUPPRESSION

165

Figure 1. Simulation result (obtained with an analog computer) of a self-excited system exhibiting vibration suppression near the parametric combination resonance frequency η0 = Ω2 − Ω1 . From (Tondl, 1998)

Parametric excitation of a more general type has been applied by Makihara in Tokio (Makihara et al., 2005) and valuable contributions have been made also by Nabergoj from Trieste (Nabergoj and Tondl, 2001). Last but not least, Tondl himself has continued to study the effect of vibration quenching by parametric excitation (Tondl et al., 2005) and e.g. has also investigated parametric damping and parametric mass excitation, see (Tondl, 2001). 3. Modelling systems with parametric stiffness excitation The generic equations of motion of a mechanical system with parametric stiffness excitation can be written in a rather general matrix form as M¨x + C + G(ν) + CZ (x) x˙ + K + N(ν) + KZ (x) x + KPE (t)x = Fex . (1) The vector of deflections is denoted x. For a linear, homogeneous system with constant system matrices, only the following matrices would be needed and therefore non-zero: mass matrix M, damping matrix C, stiffness matrix K. Parametric stiffness excitation (PSE) is introduced by matrix KPE (t) with time-periodic coefficients according to harmonic functions cos(ωt + pi j ). Only single-frequency PSE with frequency ω is considered for this system but multi-location parametric excitation is not excluded. Phase relations between different locations of PE are introduced by phase angles pi j KPE (t) = cos(ωt + pi j )PE .

(2)

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The number of degrees of freedom of the system determines the size of the system matrices. It is pretty obvious how to establish these matrices for simple two or three mass chain systems, as used in (Tondl, et al., 2005) and several other references by the author. Self-excitation can be introduced to the system by setting elements of the damping matrix C to a negative value. Negative damping is one of the common methods to represent the effect of flow-induced self-excitation (Blevins, 1977). Basic non-linear behavior can be represented by the additional stiffness and damping matrices CZ (x) and KZ (x), which may depend in an arbitrary way on vector x and also, if required, on x˙ . Matrix G(ν) is a function of a system parameter ν and is needed in mechanical systems to represent gyroscopic forces, which would depend on a rotational speed ν. Also frequently encountered in rotor systems are non-conservative forces N(ν)x, created by bearings and seals. Such forces usually increase with increasing rotor speed and may ultimately destabilize such a system. It is of particular interest to investigate rotor system, since the effect of a parametric anti-resonance could improve the performance of rotating machinery quite significantly. Finally, to take into account the effect of external forces, Fex appears on the right hand side of Eq.(1). 4. Parametric resonance frequencies It is widely known, see e.g. (Cartmell, 1990), that a system with parametric expr citation may exhibit Principle Parametric Resonances at frequencies η j/n and Parametric Combination Resonances at frequencies ηcrj±k/n for the PSE-frequency ω equal to: pr

η j/n =

2Ω j , n

ηcrj±k/n =

|Ω j ± Ωk | , n

( j k), ( j, k, n = 1, 2, 3, ...).

(3)

Symbols Ω j and Ωk denote the j-th (k-th) natural frequency of the system. The denominator n represents the order of the parametric resonance. Most of the time only first order resonances n = 1 of the lowest frequencies Ω1 , Ω2 are significant. The effect of a parametric anti-resonance can only occur for parametric combination resonances. It depends on the system, whether the difference type or the summation type is non-resonant and can be used to achieve vibration suppression. It can be shown (Ecker, 2003) that for a symmetric stiffness matrix K = KT parametric vibration suppression will occur for the difference-type combination resonance ηcr ( j−k)/1 = (Ω j − Ωk ) and that an interval of instability will be observed at the summation-type combination resonance ηcr ( j+k)/1 = (Ω j + Ωk ). To predict the appropriate PE-frequency for vibration suppression it is necessary to know the natural frequencies of the system. Therefore, the lower undamped


167

natural frequencies Ω1,2,3,... have to be calculated from system Eq.(1) by solving the eigenvalue problem for M¨x + Kx = 0. 5. Analysis of systems with parametric excitation In its most general version Eq.(1) defines a set of non-homogeneous, non-linear, time-periodic differential equations of second order. Also because of its basically unlimited complexity with regard to the size of the system there does not exist a single method to suit all kind of problems. Depending on the actual size of the problem, the presence of non-linearities and inhomogeneous terms, different methods are advantageous to be applied. Another factor is also if time series of system states of the original problem are sought or if only the local stability of such a solution is of interest. 5.1. NUMERICAL SIMULATION METHOD

The most direct method, which can be applied to virtually any kind of such problems is numerical simulation. By integrating the system equations in the time domain, starting from initial conditions, the solution is computed. Nonlinearities and large matrices only affect the computational speed, but would not prevent using simulation. Of course, appropriate integration methods have to be applied, to balance computational effort and accuracy. Nevertheless, CPU-time may become still a problem, when the stability of a system near the stability threshold shall be investigated and very slowly changing transients have to be followed. 5.2. ANALYTICAL METHODS

A number of analytical and semi-analytical methods have been developed to deal with time-periodic systems. Even trying to briefly introduce the most interesting ones would exceed the length of this overview by far. Therefore, only one method is explicitly mentioned, since it has been used by the author and his coworkers and others quite successfully in this context. This method is nowadays mostly called Method of Averaging (MoA). However, based on being promoted by Krylov, Bogoliubov and Mitropolski, in the past the method is also associated with these names. A rather detailed comparison of three distinctively different methods is presented in (Ecker, 2003). The Method of Averaging is applicable to a linear(ized) and homogeneous subset of Eq.(1) and will primarily provide information about the stability of the system. It can be implemented as a first order method, as well as for higher orders. However, deriving a first order solution can be already cumbersome for a low-dimensional system. This, and the need to identify a small parameter in the system, are the major disadvantages of this method. But to be fair, a price has to be

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paid with practically every of the analytical methods. A very recent and detailed presentation of AoM is found in (Verhulst, 2005). The application to PE-systems of various complexity is thoroughly discussed in (Dohnal, 2005). The advantages of analytical methods can be seen easily by the following example. Equations (3) and (4) are a simplified and normalized version of Eq.(1), which have been used by Tondl and others to investigate the stability of PE-excited two-degree of freedom systems. 5 6 u + Ω2 u = −ε Θu + cos ητQ c u , '

( u1 u= , u2

( Θ11 Θ12 Θ= , Θ21 Θ22

'

'

Ω2

(4)

' c ( ( c Ω21 0 c = Q11 Q12 . = , Q Qc21 Qc22 0 Ω22

(5)

By application of the Method of Averaging two necessary conditions are obtained for stability at a parametric excitation frequency η = (Ω2 − Ω1 ): Θ11 + Θ22 > 0, 1 Qc Qc > 0. Θ11 Θ22 + 4Ω1 Ω2 12 21

(6) (7)

Not only the stability of the system can be examined for a certain PE-frequency. It is also possible to calculate the frequency interval of stability in the vicinity of this frequency. The interval is defined as η0 + εσlo < η < η0 + εσhi . with σlo,hi

(Θ11 + Θ22 ) =∓ 2

−

Qc12 Qc21 − 1. 4Ω1 Ω2 Θ11 Θ22

(8)

(9)

It is interesting to note that first order averaging leads to exactly the same results as obtained in (Tondl, 1998) by a different method based on Floquet theory. In exchange for compact results, one has to accept, that accuracy is degraded as soon as the parameter ε ≮ 1 cannot be considered as small anymore, at least if only a first order approximation is used.


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5.3. NUMERICAL STABILITY ANALYSIS

The stability of the trivial solution x = 0 of system Eq.(4),(5) can be investigated also numerically by means of Floquet-theory, see (Verhulst, 2000). Floquet’s theorem postulates that for a system of first order differential equations y˙ = A(t) y,

A(t) = A(t + T ),

(10)

with a T -periodic matrix A(t) each fundamental matrix M(t) of the system can be represented as a product of two factors M(t) = Q(t) exp(tC),

(11)

where Q(t) is a T -periodic matrix function and C is a constant matrix. Stability of the time-periodic system can be determined either from the eigenvalues of the Floquet exponent matrix C or from the monodromy matrix M(T ), which is in fact the state transition matrix evaluated after a period T . The monodromy matrix can be calculated numerically by repeated integration of the system equations over one period T , starting from independent sets of initial conditions. It is convenient to use the columns of the identity matrix I as initial vectors to start from. By solving n initial value problems over one period T y˙ = A(t)y,

[y(0)1 , y(0)2 , ..., y(0)n ] = I,

t = [0, T ],

(12)

and by arranging the results as follows M(T ) = [y(T )1 , y(T )2 , ..., y(T )n ]

(13)

the monodromy matrix is obtained. Finally the eigenvalues of the monodromy matrix Λ = eig(M(T )), (14) are calculated numerically. The system is unstable if any of the eigenvalues are larger than one in magnitude 4 < 1 stable system (15) max(|Λ1 |, |Λ2 |, ..., |Λn |) > 1 unstable system. This procedure leads to a very efficient numerical method, which allows to examine the stability of a parametrically excited system much faster than with direct numerical integration. This method, however, cannot be used when nonlinearities are present and will not give vibration amplitudes. 6. Examples In this section three examples of mechanical systems with parametric excitation are presented, to demonstrate the capabilities of the proposed method. Due to

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x1

x2

c01

c02 m1

m2

k12 k01ec01

D gxd3

Figure 2. Two-mass system with self-excitation because of negative damping c02 < 0 and parametric stiffness excitation at k01 (t)

limited space, not all the details for every model can be included and therefore references are given where to find the full documentation for each example. 6.1. TWO-MASS SYSTEM

In Figure 2 a schematic is shown of a two-mass system with one stiffness element being periodically changed according to k01 (t) = k01 (1 + ec01 cos(ωt). It is assumed that the damping element c02 < 0 and therefore the system may exhibit selfexcited vibrations. To demonstrate the stabilizing effect of parametric excitation, the system was investigated for the critical value of parameter c02 at the stability threshold. In Fig. 3 the stable area is filled white and the unstable area is grey. This result was obtained by a numerical stability analysis. One can see easily that significantly lower values of the critical parameter c02 are possible at the combination resonance frequency of the system Ω2 − Ω1 . This indicates that PE at this frequency and with a sufficiently high amplitude will allow for an increase of the self-excitation parameter without becoming unstable. If the wrong frequency is chosen, however, then the effect is turned upside down, as one can see from the result in the vicinity of the parametric combination resonance of the summation type. The stability threshold, separating stable and unstable regions, is also plotted as a broken line, near the parametric resonances. These two lines have been obtained by analytical formulas for the interval of stability, similar to Eqs.(8) and (9). Note that also the interval of instability near Ω1 + Ω2 can be obtained with rather high accuracy, compared to the numerical solution. The full set of data for this example can be found in (Ecker, 2003).

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SE−Parameter c02

Ω2−Ω1

Ω1+Ω2

−0.1

−0.15

−0.2

2

2.5 3 3.5 Parametric excitation frequency ω

4

Figure 3. Typical stability map for a system as shown in Fig. 2. White areas indicate a stable system, grey areas instability due to self-excitation. Stability threshold is calculated numerically (solid line) and by analytic formulas (broken lines)

6.2. ROTOR SYSTEM

The results in this section are obtained from a rather elaborate model of a Jeffcottrotor with 4 degrees of freedom for the rigid disk and 2-dof at each of the bearing stations. Parametric stiffness excitation is created by time-periodic bearing forces, see (Ecker, 2003). Self-excitation forces are acting on the rotor due to internal damping of the shaft. To limit the vibration amplitudes when self-excitation

r

max(x )

1.5 1 1.5 0.5 0 1

1.25 1 1.5

2

0.75 2.5

3

3.5

4

0.5

Parameter ν

PE−frequency η

Figure 4. Parameter study for a rotor with internal damping and parametric stiffness excitation by the bearing forces

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occurs, progressive damping is introduced and consequently the rotor reaches a limit cycle, when becoming unstable. This nonlinear model can only be investigated by numerical simulation, when none of its features shall be neglected or approximated. On the other hand, the case where vibration suppression occurs can be seen directly from the results for the amplitudes, which vanish after a transient period. Figure 4 shows the deflection at the rotor disk station as a function of the rotor angular speed ν and the frequency of the simultaneous harmonic stiffness variation in both bearings. The amplitude of the stiffness variation is about 20% of its nominal value. The region where self-excitation does not create instability is the gray shaded area. Without any PE this region would reach almost to a rotor speed of ν = 1.0). Now with PE tuned to the right frequency, which would be near 1.7, a rather narrow stretch of stability and no amplitudes reaches up to ν = 1.45). This means that in this example the rotor speed could be increased by 50% if PE is employed to cancel self-excited vibrations. 2

max(|zr|)

1.5 1 0.5 0 0.4 0.2 0 u

e

0

0.5

1 Rotational speed ν

1.5

2

Figure 5. Rotor amplitudes as a result of unbalance ue , self-excitation due to internal damping and parametric excitation

The result in Fig. 5 shows primarily the unbalance resonance peak as a function of the unbalance parameter ue and the rotor speed ν. Parametric excitation is in effect and increases the onset of instability from about ν ≈ 1 to ν = 1.4 at zero rotor unbalance. Obviously and as expected, there is no significant influence of the PE on the resonance peak. This observation holds in both directions and therefore also no adverse effect due to parametric excitation on the unbalance response is

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PARAMETRIC EXCITATION AND VIBRATION SUPPRESSION 60

Force Ampl.

50 40 30 20 10 0

50

100

150

200 250 PE−Frequency ω

300

350

400

Figure 6. Stability map for cantilever beam with periodic axial force excitation. Shaded areas indicate instability. Enhanced damping is obtained in the center of the contour lines near ω = 140

observed. In fact, there is even a minor increase of the stability threshold, as the unbalance increases. 6.3. CANTILEVER BEAM WITH PERIODIC AXIAL FORCE EXCITATION

0.4

0.4

0.2

0.2

Tip Deflection

Tip Deflection

The last example shall point out, that parametric excitation, when adjusted to a non-resonant PE-resonance frequency, introduces additional damping to the system. This is basically the mechanism why self-excited systems can be stabilized by this method. The system investigated now is a uniform, continuous slender cantilever beam in an upright position. A vertical load of constant direction is applied to the tip of the beam and this load has a time-periodic component. Although the force is an external force, the system equations turn out to be those of a system with parametric excitation, see (Ecker et al., 2005). As there is no

0 −0.2 −0.4

0 −0.2 −0.4

0

0.2

0.4 0.6 Time

0.8

1

0

0.2

0.4 0.6 Time

0.8

1

Figure 7. Vibration signal at the tip of the cantilever beam. (Left) free oscillations without PE, (right) with optimal harmonic axial force excitation

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self-excitation in the system, instabilities are only created by resonant parametric excitations. See Fig. 6 for a stability chart of the system, where the parameters of the diagram are PE amplitude and frequency. The damping effect of parametric excitation can be observed as an area of reduced eigenvalues of the monodromy matrix, near ω ≈ 140 as indicated by the contour lines in Fig. 6. Taking the best combination of the force amplitude and the frequency one can study the decay of free oscillations of the free system. In Fig. 7 left, transversal vibrations at the tip of the beam decay only very slowly due to little structural damping. If parametric excitation is applied, however, the situation can be improved quite significantly, as one can see from Fig. 7 (right). References Abadi (2003) Nonlinear Dynamics of Self-excitation in Autoparametric Systems, Ph.D. Thesis, Utrecht University. Blevins R. D. (1977) Flow-induced vibration, Van Nostrand Reinhold, New York. Cartmell M. (1990) Introduction to Linear, Parametric and Nonlinear Vibrations, Chapman and Hall, London. Dohnal F. (2005) Damping of mechanical vibrations by parametric excitation, Ph.D. Thesis, Vienna University of Technology. Ecker H. (2003) Suppression of Self-excited Vibrations in Mechanical Systems by Parametric Excitation, Habilitation Thesis, Vienna University of Technology. Ecker H., Dohnal F., Springer H. (2005) Enhanced Damping of a Beam Structure by Parametric Excitation, Proc. of European Nonlinear Oscillations Conf. (ENOC-2005) Eindhoven, NL. Fatimah S. (2002) Bifurcations in dynamical systems with parametric excitation, Dissertation, Utrecht University. Makihara K., Ecker H., Dohnal F. (2005) Stability Analysis of Open Loop Stiffness Control to Suppress Self-excited Vibrations, J. Vibration and Control 11. Nabergoj R., Tondl A. (2001) Self-excited vibration quenching by means of parametric excitation, ˇ Acta Technica CSAV 46 107-211. Shaw S. W., Rhoads J. F., Turner K. L. (2004) Parametrically excited MEMS oscillators with filtering applications, Proc. 10th Conf. on Nonlinear Vibrations, Stability, and Dynamics of Structures Blacksburg, VA. ˇ Tondl, A. (1998) To the problem of quenching self-excited vibrations, Acta Technica CSAV 43 109-116. ˇ Tondl A. (2001) Systems with periodically variable masses, Acta Technica CSAV 46 309-321. ˇ Tondl A. (2002) Three-mass self-excited systems with parametric excitation., Acta Technica CSAV 47 165-176. Tondl A., Ecker H. (2003) On the problem of self-excited vibration quenching by means of parametric excitation, Archive of Applied Mechanics 72 923-932. Tondl, A., Nabergoj, R., Ecker, H. (2005) Quenching of Self-Excited Vibrations in a System With Two Unstable Vibration Modes, Proc. of 7th Int. Conf. on Vibration Problems (ICOVP-2005) Istanbul, Turkey. Verhulst F. (2000) Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag. Verhulst F. (2005) Methods and Application of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics, Springer.

A FORMULATION OF MAGNETIC-SPIN AND ELASTO-PLASTIC WAVES IN SATURATED FERRIMAGNETIC MEDIA ¨ A. Unal Erdem Department of Mech. E., Gazi University, 06570, Maltepe, Ankara, Turkey

Abstract. In this work our objective is to obtain a mathematical model for the formulation of propagation of magnetic spin waves as coupled with elasto-plastic waves in magnetically saturated ferromagnetic insulators . Elasto-plastic deformations are considered to be infinitesimal (classical theory). Key words: magnetic-spin, elasto-plastic waves, ferrimagnetic media

1. Introduction In view of the two significant shortcomings of the stress-space formulation , we shall adopt the strain-space formulation (Khan and Huang, 1995). At this point, prior to proceeding with plasticity , it will be useful to expose the results of the related portion of magneto-elasticity. For the case of infinitesimal strains, the constitutive equations for magnetically saturated insulators are given by (Eringen and Maugin, 1990). E ti j

=

∂Σ 1 = ci j k l εk l + κk i j µk + λi j k l µk µl =E t j i , κk i j = κk j i ∂ εi j 2

(1)

∂ Σ −1 = − ρ0 χi j µ j + κi j k ε j k + λi j k l εk l µ j , χi j = χ j i ∂ µi

(2)

bi = −ρ−0 1

(µ)

σi j = Ai k µ j, k ,

(Ai k = Ak i )

ti j =E ti j − ρo bi µ j (Balance o f angular momentum)

(3) (4)

1 ti j = ci j k l εk l + κk i j µk + λi j k l µk µl + κi k l εk l µ j + χi k µk µ j + λk l i m εk l µm µ j 2 (5) 175 .


176

ERDEM

1 1 1 ci j k l εi j εk l + χi j µi µ j +κi j k µi ε j k + λi j k l εi j µk µl +Ai j µk , i µk , j 2 2 2 (6) with , being the free energy density for the unit undeformed mass, µ ≡ Mρ magnetization per unit mass, ε infinitesimal strain tensor, t total strain tensor , b local magnetic induction, and σ(µ) magnetic spin stress tensor (Tiersten tensor). At the right-hand side of Eq. (6) we can see the elastic energy, magnetic anisotropy energy, piezomagnetic energy, magnetostriction energy, and magnetic spin-exchange energy, respectively. Σ ≡ ρ0 Ψ ≡

2. Formulation Strain-space formulation can now be summarized as follows: For any given loading history, the boundary between elastic and plastic strain-states is the current yield strain εY . With σe being a fictitious stress obtained by assuming a totally elastic deformation represented by σ e = E ε , in 1-D case, the current stress is given by σ = E ε − σP , where σ e = E εP (εP , current plastic strain).In 3-D case this is generalized to the expressions σP = C : εP σ = C : ε − εP = C : ε − σP

(7) (8)

Here Ck l m n stands for the stiffness tensor in the case of pure linear elastic deformations. Accordingly we have, σi j = E σi j − ρo bi µ j = ti j − σiPj σ j i

(9)

where ti j is given by Eq. (5). As for the term σiPj we proceed as follows: The yield surface, with the same order of magnitudes involved, will be given by ϕ = ϕ(εk l , µk , σkPl , αk l , κ) − λ ( µk µk − µ2S ) = 0 ,

(10)

where λ is the Lagrange multiplier which takes care of the restriction µk µk = const. , due to saturation. The relation d ϕ = 0 is to be satisfied on the yield surface (consistency condition). Here εk l is infinitesimal strain tensor, αk l and κ representing kinematic and isotropic hardening parameters, respectively. Evolution equations (Khan and Huang, 1995) consist of the Flow rule: d σP = d η

dϕ , or dε

•P

•

σ =η

dϕ , dε

(11)

Isotropic hardening: % dκ (2/3) dσP : d σP , d κ = N ε , µ , σP , α , κ : d σP ≈ dl

(12)

MAGNETIC-SPIN WAVES AND ELASTO-PLASTIC WAVES

# d σP d σP (2/3) : dt , κ = κ (l), l ≡ dt dt and kinematic hardening: d α = L ε , µ , σP , α , κ ≈ m C−1 : d σP (m , aparameter)

177

(13)

(14)

where ≈ signifies the assumption of theory to be that of infinitesimal strains. Use of evolution equations in the consistency condition d ϕ = 0 determines the value of dη as follows. Differentiating dϕ =

∂ϕ ∂ εk l

d εk l +

∂ϕ ∂ µk

d µk +

∂ϕ ∂ σkPl

d σkPl +

∂ϕ ∂ αk l

d αk l +

∂ϕ ∂κ

d κ − λ (2 µk d µk ).

We note that coefficient of Lagrange multiplier λ does not contribute here, since µk dµk is identically zero due to saturation restriction. Now using Eq.’s (11-14), we get 1 ∂ϕ ∂ϕ (15) dη = : d εk l + d µk . D ∂ εk l ∂ µk )   &   ∂ϕ ∂ ϕ 2 ∂ ϕ ∂ϕ ∂ ϕ ∂ κ −1 (∂ ϕ / ∂ ε) : (∂ ϕ / ∂ ε)  +m :C : + D ≡ −  P : ∂ε ∂α ∂ε 3 ∂κ ∂l ∂σ (16) With this value of dη flow rule takes the form of, ' ( ∂ ϕ ∂ϕ 1 ∂ϕ ∂ϕ P d σi j = d εk l + d µk (17) D ∂ εi j ∂ εk l ∂ εi j ∂ µk in which all the increments are fully independent. Total stress increment is given by Eq.’s (9), (5) and (17). From differentiation of Eq. (5), substituting from (17) yields * * + + 1 ∂ ϕ ∂ϕ 1 ∂ ϕ ∂ϕ d σi j = ci j k l − D ∂ εi j ∂ εk l d εk l + κk i j − D ∂ εi j ∂ µk d µk + (18) 1 2 λi j k l + 2 χi k δl j d (µk µl ) + κi k l d εk l µ j + λi m k l d µm εk l µ j Divergence of the total stress can be evaluated as σi j , i =

∂ σi j ∂ εk l

∂σ + ∂ (εkl iµj r )

εk l , i +

∂ σi j ∂ µk

(εkl µr ), i +

µk , i +

∂ σi j ∂ (µk µl )

∂ σi j ∂ (µm εkl µr )

(µk µl ), i

(µm εkl µr ), i

(19)

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ERDEM

* σi j , i = ci j k l − + 12

1 ∂ ϕ ∂ϕ D ∂ εi j ∂ εk l

+

* uk , l i + κk i j −

1 ∂ ϕ ∂ϕ D ∂ εi j ∂ µ k

+

µk , i (20)

λi j k l + 2 χi k δl j (µk µl ), i + κi k l εk l µ j + λi m k l µm εk l µ j ,i

,i

This last expression is to be used in the equations of motion for the ferrimagnetic material medium. ρ0

∂2 u j = ρ0 µk b j , k + σk j, k ∂ t2

(b, : Local magnetic induction),

(21)

along with the value of the local magnetic induction given by Eq. (2). As for the magnetic-spin equation of motion consistent with the magnetoelastic formulation, we have, ∂ µi 1 (µ) = γ εi j k µ j Hk + bk + σ (22) ∂t ρ0 r k , r in which the third term is supplied by (3). The unknowns are the fields u j and µi . Upon substituting the constitutive equations (2) and (3) into (22); and (2) and (20) into (21), respectively, we get equations of motions ∂ µi = γ εi j k µ j Hk − ρ−1 χ γ ε µ µ + κ u + λ µ u − A µ i j k j k p p k r p r , p k l r p l r, p r p k ,r p 0 ∂t (23) 2 ρ0 ∂∂ tu2i = −(χ j r µi µr, i + κ j r s µi ur, s i + λ j r s p µi µr u s, p + ,i

* + ci j k l −

1 ∂ ϕ ∂ϕ D ∂ ε i j ∂ εk l

+

* uk , l i + κk i j −

1 ∂ ϕ ∂ϕ D ∂ εi j ∂ µ k

+

µk , i

(24)

+ 12 λi j k l + 2 χi k δl j (µk µl ), i + κi k l uk, l µ j + λi m k l µm uk, l µ j ,i

,i

Quasi-magnetostatic field will be governed by ∇ · B = 0, ∇ × H = 0, B ≡ H + M = H + ρ0 µ

(25)

H = −∇Φ , ∇2 Φ = ρ0 µk, k or Φ, k k = ρo µk , k

(26)

Integrating Eq. (26)3 over the space variables and assuming that initial magnetization is the saturation magnetization of magnitude µ s ,we get $$ o $$ o $ $ (27) µk = µok + ρ−1 0 Φ, k ( µk ≡ initial magnetization, µ = µ s

MAGNETIC-SPIN WAVES AND ELASTO-PLASTIC WAVES

179

In the undisturbed state, µo is parallel to H due to magnetic saturation.). Now a magneto-elastoplastic plane wave, which is a disturbance of initial magnetization µo from the direction parallel to H, propagating in the direction of a unit vector n can be represented by, uk = uk (n · x − c t)

and Φ = Φ (nk xk − c t )

(28)

(x : spatial position on the wave front) By the use of Eq.’s (28) with the substituting of the spacial and temporal derivatives into Eq. (22), noting also the fact that propagation direction n is perpendicular to the initial saturation magnetization, namely, µoi ni = 0 , the magnetic spin equation of motion we get o n Φ = γ ε o µo n u κ γε n χ µ n + λ µ c Φ − ρ−1 i j k i k p p i j k i k r p k l r p j j p r 0 l (29) o µo n Φ u + γ εi j k ni λk l r p ρ−1 µ r j l p 0 with (.) and (.) denoting first and second derivatives with respect to the argument of the functions defined by Eq. (31). As for the displacement equation of motion, −1 ρo c2 uj = −χ j r ρ−2 0 nr Φ Φ − κ j r s ρ0 n s Φ ur +

o −2 2 + λ j r s p ρ−1 0 µ n p Φ u s + ρ0 nr n p Φ u s * + ci j k l −

1 ∂ ϕ ∂ϕ D ∂ εi j ∂ εk l

+

* nl ni uk + κk i j −

1 ∂ ϕ ∂ϕ D ∂ εi j ∂ µ k

+

ρ−1 0 nk ni Φ

o o −1 + 12 ρ−1 0 λi j k l + 2 χi k δl j µk nl + µl nk + 2 ρ0 nk nl ni Φ +κi k l ni nl µoj uk

+

κi k l ni nl ρ−1 0 n j Φ uk

+

(30)

κi k l ni nl ρ−1 0 n j Φ uk

o −1 o + λi m k l ni nl (µoj µom uk + ρ−1 0 µm n j Φ uk + ρ0 µm n j Φ uk 2

o −1 o −2 + ρ−1 0 µ j nm Φ uk + ρ0 µ j nm Φ uk + ρ0 nm n j Φ uk +2ρ−1 0 nm n j uk Φ Φ )

These two equations are strongly coupled and highly nonlinear. For further linearization, in (29) we ignore the term Φ ur as compared to the others, then take the derivative of both sides with respect to the common argument of Φ and solve this for Φ as a function of Φ and ur . Setting Φ = 0 (meaning to ignore its 3rd

180

ERDEM

variation near the others) and solving for Φ yields, γ εi j k ni µ0j n p κk r p + λk l r p µ0l = const. Φ = Rr ur , Rr ≡ −1 χ γ εi j k ni µ0j n p ρ−1 c k p 0 0

(31)

Now ignoring all the nonlinear terms in Eq. (30), we get an eigenvalue problem for a nonsymmetric matris Q, ) q 2 Q j r − qδ j r ur = 0, q ≡ ρ0 c , c = , (32) ρ0 1 ∂ ϕ ∂ϕ − κ Q j r ≡ ni nl [ (ci j r l − D1 ∂∂εϕi j ∂∂εϕr l + ρ−1 li j D ∂ εi j ∂ µ l R r 0 (33) 5 6 o o −1 o −1 + κi r l − λi m rl µm µ j + ρ0 λi j k l + χi k δl j + χi l δk j µk + 2 ρ0 nk Rr ] This is nothing but the acoustical tensor for this particular problem. 3. Conclusion The format of the solution is that, positive eigevalues q(i) of matrix Q yield possible wave speeds c(i) and corresponding eigenfunctions ur (i) . The displacement field ur is obtained by integrating twice over the phase variable z ≡ n · x−c t . Then µk (x , t) are determined from Eq.’s (33) and (29). Further linearization will, of p course, be necesseray when we introduce the yield function ϕ εk l , µk , εk l , αk l , κ , in terms of its arguments, explicitly. Also, most of the ferrimagnetic materials consist of the centrosymmetric cubic and uniaxial, that is, transversely isotropic crystals. Accordingly, various order tensors in the expression of Qi j are to be written in appropriate forms (Eringen and Maugin, 1990). On the other hand, Q is not a symmetric tensor, and the contribution of the terms involving the yield function ϕ will reflect the effects of plastic deformation, as well as elastic deformation. This explicit investigation necessitates a numerical work on the subject which we postpone until later for a separate study. References Eringen, A. C. and Maugin, G. (1990) Electrodynamics of Continua II, Fluids and Complex Media, Springer Verlag. Khan, A. and Huang, S. (1995) Continuum Theory of Plasticity, John Wiley.

EARTHQUAKE RESPONSE OF SUSPENSION BRIDGES Mustafa Erdik1 and Nurdan Apaydın2 1Boğaziçi University, Istanbul, ˙ Turkey 2 General Directorate of State Highways, 17th Division, Istanbul, ˙ Turkey

Abstract. Suspension bridges represent critical nodes of major transportation systems. Bridge failure during strong earthquakes poses not only a threat of fatalities but causes a substantial interruption of emergency efforts. Although wind induced vibrations have historically been the primary concern in the design of suspension bridges, earthquake effects have also gained importance in recent decades. This study involves ambient vibration testing and sophisticated three-dimensional dynamic finite element analysis and earthquake performance assessment of Fatih Sultan Mehmet and Boğaziçi suspension bridges in Istanbul under earthquake excitation. Nonlinear time history analysis of 3D finite-element models of Fatih Sultan Mehmet and Boğaziçi suspension bridges included initial stresses in the cables, suspenders, and towers in equilibrium under dead load conditions. Multi-support scenario earthquake excitation was applied to the structure. Suspension bridges are complex 3-D structures that can exhibit a large number of closely spaced coupled modes of vibration. The large number of closely spaced modes, spatially different ground motion characteristics, and the potential for nonlinear behaviour complicate the seismic response of suspension bridges. Spatial variation of ground motion exhibits itself with different ground motions at supports due to the wave passage, incoherence and local site effects. The source of nonlinear behaviour is due to geometric nonlinearity and the presence of cables that can only carry tensile forces. The natural frequencies of vibration and the corresponding mode shapes in their dead load and live load configurations are determined. Displacement time histories and stresses at critical points of the bridges are computed and their earthquake performance under the action of scenario earthquake are estimated. Key words: Earthquake, suspension bridge

1. Introduction The main structural elements of a suspension bridge are: the main cable system, the towers or pylons, the anchorage and the deck (or girder). Suspension bridges are flexible structural systems susceptible to the dynamic effects of traffic, explosions wind and earthquake loads. The assessment of the dead and live load effects on the structure, the response of the structure to wind and earthquake loads, and the subsequent design of an optimized lateral and vertical load resisting system, require sophisticated analytical methods. Although the computational power of today’s hardware and software eliminated most of the analytical difficulties encountered in the assessment of the 181 .


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ERD˙IK, APAYDIN

dynamic response of such structures, the following general issues still remain to be elaborated and developed for on the wind and earthquake response/performance and design of suspension bridges. • The structural system and its characteristics, • The nature of wind forces and earthquake forces, • The computer modelling and the limitations of the software utilized, • Wind tunnel and shaking table experiments • Strong Motion Instrumentation There exists two suspension bridges in İstanbul crossing the Bosphorus straits: the Bosphous and Fatih Sultan Mehmet Bridges. Past investigations on these two suspension bridges concentrated on: Analytical assessment of dynamic properties; Forced and Ambient Vibration Testing; Wind induced vibration measurements and Structural Vibration (Health) Monitoring. ˙ 2. Description of the suspension bridges in Istanbul Bosporus Suspension Bridge, commissioned in 1973, joins the European and Asian Continents through Ortakoy and Beylerbeyi districts of İstanbul. It is a gravity anchored suspension bridge with steel pylons and inclined hangers. The main span is 1,074 m (World rank 12th). It consist of one main and two side spans. Side spans are not suspended and rest on slender columns. The cost of the bridge amounted to USD 200 million. Fatih Sultan Mehmet Suspension Bridge, located about 5km north of the Bosphorus Bridge, also spans the Bosporus strait between Hisar (European Side) and Kavacık (Asian Side). It is a gravity-anchored suspension bridge with no side spans and with steel pylons and double vertical hangers The main span is 1,090 m (World rank 11th). It was completed in 1988 at a cost of USD 130 million. Both bridges were designed by Freeman Fox & Partners. Table 1 provides a summary of the structural characteristics of both bridges. Figures 1 and 2 respectively shows the general structural sections and elevations of the Bosporus and Fatih Sultan Mehmet Suspension Bridges. 3. Full scale testing Vibration characteristics and the dynamic properties of the suspension bridges are important design parameters controlling their wind and the earthquake safety. Full-scale dynamic testing of suspension bridges is a reliable method for the assessment of the free vibration characteristics (i.e. vibration mode shapes, frequencies and the associated damping rations), and the calibration of the analytical and finite element models. Furthermore the detection of the changes in these vibration characteristics are one of the tools used for structural health monitoring

EARTHQUAKE RESPONSE OF SUSPENSION BRIDGES

Figure 1.

183

Bosphorus Suspension Bridge General Structural Sections and Layout

purposes. The size and the very low frequencies of vibration of the suspension bridges disqualifies the use of forced vibration techniques (such as harmonic excitation or impulsive loading) commonly used for other bridges and viaducts.One of the best methods utilized to assess the dynamic characteristics of such massive structures is the measurement of the structural response to the ambient excitations, such as, wind and traffic noise. The structural vibrations caused by such excitations are termed the “Ambient Vibrations”. Since ambient vibration tests provide the “output-only” type measurements, the system identification techniques based on Fourier Spectral Analysis techniques are generally employed. In the Fourier Spectral Analysis technique the modal vibration frequencies and the shapes are obtained on the basis of Fourier Amplitude and Phase Spectra obtained at measurement points. The so-called “halfpower bandwidth” procedure using the Fourier Amplitude Spectra at the identified modal frequencies of vibration is employed to obtain the modal damping characteristics. Recent applications of system identification also include the so-called Natural Excitation Techniques paired with the Eigensystem Realization Algorithms (NexT-ERA), initially developed by James et al. (1993). The basic principle behind this method relies on the fact that the theoretical cross-correlation function between two response data has the same analytical form as the impulse response function of the bridge. This technique is claimed to be especially suited to

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ERD˙IK, APAYDIN

TABLE I. Comparison of the Structural Characteristics of the Bosporus and Fatih Sultan Mehmet Suspension Bridges. ˘ ˙IC ˙ SULTAN BOGAZ ¸ ˙I FATIH MEHMET Span Deck Width: Substructure Dead Load per Unit Length: Areas per Cable: Gross x-sectional area Deck Moment of Inertia Modulus of Elasticity of Cables Sag: Hanger Geometry: Clearance from the sea level

Main Span 1074 m Asian Side Span 255m European Side Span 231 m 28 m. (33.4 m total) 2 Steel Towers (165 m) 2 Anchorages Spread Foundation 143 kN/m 0.205 m2 0.85 m2 I xx =1.24 m4 Iyy =63.6 m4 193 kN/mm2 93 m+7.1 m Inclined 64 m

Main Span 1090 m 33.8 m (39.4 m total) 2 Steel Towers (107 m) 2 Anchorages Spread Foundation 216 kN/m 0.365 m2 1.26 m2 I xx =1.73 m4 Iyy =129.2 m4 205 kN/mm2 91 m+6.3 m Vertical (Double) 64 m

extract modal parameters from a structure with closely spaced modes. The “ambient vibration” tests which used the ambient and non-measurable excitation (such as wind and traffic) as the source have been successfully applied to several important suspension bridges all over the world such as Vincent Thomas (Abdel-Ghaffar and Housner 1978), Golden Gate (Abdel-Ghaffar and Scanlan 1985), Roebling (Ren et al., 2004) and Humber (Brownjohn et al., 1987). Similarly, in Turkey vibration tests were conducted in Bosporus Bridge (Petrovski et al., 1974; Tezcan et al., 1975; Brownjohn et al.,1989, Erdik et al.,1988; Beyen et al., 1994 and Kosar, 2003) and in Fatih Sultan Mehmet Bridge (Brownjohn et al.,1992 and Apaydın, 2002). Table 2 and Table 3 provides the empirical frequencies and general shapes associated with first several modes of vibration of Bogazici and Fatih Sultan Mehmet Suspension Bridges. 4. Wind induced vibration measurements The response of the Boğaziçi and Fatih Sultan Mehmet Suspension Bridges to strong winds on February 12-13, 2004 were recorded by the strong motion accelerograms installed in the north and south sides at the center of the main span of the bridges at every half hour for 5 minutes. The wind speeds ranged between 60 to 90 km/hr. The RMS value of the accelerations vary between 0.02g and 0.06g in the lateral and 0.1g and 0.3g in the vertical direction. Whereas peak


Figure 2.

185

Fatih Sultan Mehmet Suspension Bridge General Structural Sections and Layout

TABLE II. Experimental Modal Frequencies of Vibration of Boğaziçi Suspension Bridge Ranges in Kosar (2003) refer to light and heavy traffic conditions Mode Shape

Frequency (Hz) Erdik et al., 1988

Frequency (Hz) Kosar, 2003

First Lateral Symmetric First Vertical Asymmetric First Vertical Symmetric Second Vertical Asymmetric First Lateral Asymmetric

0.072 0.13 0.16 0.19 0.22

0.073-0.063 0.163-0.155 0.171-0.169 0.215-0.22

The damping ratios in these modes vary between 7% (first mode) and 4% (others).

accelerations, velocities and displacements reached respectively to 0.1g, 0.1m/s and 0.2m in the horizontal and 0.6g, 0.2m/s and 0.3m in the vertical direction (Kaya and Harmandar, 2004). Figure 3 provides the Fourier Amplitude Spectra in transverse, longitudinal and vertical directions at the center of the main span (north side) for the Boğaziçi Suspension Bridge. The peaks of the spectra can be associated with the modal frequencies of vibration.

ERD˙IK, APAYDIN

186

TABLE III. Experimental Modal Frequencies of Vibration of Fatih Sultan Mehmet Suspension Bridge Mode Shape First Lateral Symmetric First Vertical Asymmetric First Vertical Symmetric Second Vertical Asymmetric

Frequency (Hz) Apayd ın, 2002 0.068 0.102 0.118 0.154

Frequency (Hz) Brownjohn et al., 1992 0.076 0.108 0.125 0.145

Figure 3. Fourier Amplitude Spectra of wind response at mid-span of Boğaziçi Suspension Bridge (After Kaya and Harmandar)

5. Analytical studies In addition to the in-plane and sway vibrations of the cable subsystem and due to the simultaneous occurrence of lateral, vertical and torsional modes, dynamic behaviour of suspension bridges exhibit great vibrational complexities. Closed form solutions to the governing dynamic equilibrium equations in lateral direction are studied by Silverman (1957), Selberg (1957) and Moisseif (1933) before the computer era. Assuming inextensible cables and suspenders, perfectly flexible deck and rigid towers, the deformations of the deck follows the cable deflections. The frequencies of free vibration of a simply supported parabolic catenary with a mass distribution equal to that of the total axial mass distribution of the suspension bridge approximately yields the frequencies of vibration of the pure vertical modes of the bridge.


187

Pugsley (1949) provides closed form expressions for these frequencies in terms of cable length, cable sag, mass per unit length and horizontal tension. On the basis of such a simplified approach the frequencies of vibration for the first three vertical vibration modes can be calculated as 0.12, 0.16 and 0.23Hz (Beyen et al., 1994). These compare well with the empirical results listed in Table 1. A two dimensional finite element solution of the continuum approach was developed by Abdel-Ghaffar (1978) for the lateral vibration of suspension bridges. This simple approach is based on the Hamilton’s Principle with the basic assumptions of: (1) vibration amplitudes are sufficiently small to remain in linear ranges, (2) Coupling between lateral and torsional modes are ignored and (3) the cable, ends are immovable. Due to the high lateral stiffness of the towers, supports of the cable and the deck are also assumed to be immovable. Analytically derived energy expressions are then used to express the stiffness and consistent mass matrices for the finite element application. Erdik et al., (1988) computed the lateral modes of vibration of Boğaziçi Suspension Bridge using only eight elements for the span. The analytical results were fount to be in good agreement with the experimental values. The modal vibration frequencies and the shapes of the Bosporus Suspension Bridge were determined on the basis of a three-dimensional finite element model in Beyen et al., (1994) and Kosar (2003). Beyen et al., (1994) used LUSAS (1993) finite element program where, the bridge deck is modelled using equivalent shell elements and the effects of axial forces on the stiffness of the cables are considered through the use of equivalent frame elements. The model neglects the towers and the side spans. However, the boundary conditions are chosen to simulate the effect of the deck and cable supports at the towers. Apaydın (2002) utilized SAP-90 finite element analysis program (Wilson and Habibullah, 1989). To account for the geometric nonlinearity P-Delta effect is modeled using the geometric stiffness matrix, which incorporates the influence of axial forces on transverse stiffness. An estimate of the appropriate magnitude of tension was obtained from a static analysis of the bridge under self-weight. The resulting tension was then added to the suspension and hanger cables as a directly specified P-Delta axial force. The finite element model includes the following elements and boundary conditions: All portal beams and towers are represented as frame elements; Constraints are used to enforce certain type of rigid-body behavior, to connect together different parts of the model and to impose certain types of symmetry condition; Deck is defined as a frame element and body constraints are used to minimize relative displacement in the deck elements and the camber of the deck, geometry of deck and towers are defined in the model; Main cables and hangers are modeled with frame elements with hinged ends and; At the deck-tower connection the rocker frames are connected to the deck with a hinge with a moment release only in longitudinal direction. Kosar (2003) also utilized SAP-90 finite element analysis program (Wilson and Habibullah, 1989). The finite element model incorporates

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considerations similar to those of Apaydın (2002). At the deck-tower connections the rocker frames are connected to the deck with a hinge with a moment release only in longitudinal direction. To provide an example, the computed modal vibration shapes of the Boğaziçi Suspension Bridge are provided in Figure 4 for the first six lateral vibration modes. In connection with the earthquake performance and retrofit studies of the Boğaziçi and Fatih Sultan Mehmet Suspension Bridges, Japanese Bridge and Structure Institute (JBSI, 2004) conducted analytical studies to gain an understanding of the dynamic seismic behavior of the bridges. Figure 5 illustrates the vibration shape of the first three vertical modes of free vibration for the Boğaziçi Suspension Bridge.

Figure 4. Computed modal vibration shapes of the Boğaziçi Suspension Bridge for the first six lateral vibration modes (After Beyen et al., 1994)

6. Earthquake response analysis For the earthquake response of suspension bridges different ground motion at each support may need to be specified. Apaydın (2003) provides a comprehensive assessment of the response of the Fatih Sultan Mehmet Suspension Bridge under excitation by spatially varying (asynchronous and/or travelling) ground motion. Bridge responses to incoherent earthquake ground motions are due to a dynamic portion and differing support movements (denoted as “pseudo-static” displacements). It is computationally equivalent to decompose the total (absolute)


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Figure 5. Vibration shape of the first three vertical modes of free vibration for the Boğaziçi Suspension Bridge (After JBSI, 2004)

displacement into pseudo-static displacement and relative (vibrational) displacement resulting from inertial effects. Pseudo-static displacements are those resulting from the static application of support displacements at any time without inertial effects. For the analysis a unit displacement (in each direction-vertical, longitudinal, transverse) is applied to the bridge at one of the ground points and its static response displacement vector is calculated. To provide an example Apaydın (2003) studied the response of the bridge under spatially varying ground motion the ground motions obtained during 1979 Imperial Valley earthquake at the El Centro Array Stations 7 and 6 are applied respectively to the east and west tower and anchorage foundations. El Centro Array Stations 7 and 6 are located at each side of the Fault and represent extensive spatial variance in a distance comparable to the length of the bridge. The response of the Fatih Sultan Mehmet Suspension Bridge to travelling ground motion is analyzed using an artificial earthquake record as the input. Input motion is selected to represent a full cycle displacement pulse with a period of 1.2s. For this analyses, the ground motion is assumed to travel from Asia to Europe. The lateral response at the top of the European and Asian towers are illustrated in Figure 6 for a phase difference of 1.00 s (propagation velocity of about 1000m/s). 7. Structural health monitoring Structural health monitoring can be defined as the diagnostic monitoring of the integrity or condition of a structure. Early detection of damage or structural

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Figure 6. Lateral Displacement at the top of the Asian and European Towers of the Fatih Sultan Mehmet Suspension Bridge excited by a full cycle displacement pulse with a 1s phase difference

degradation prior to local failure can prevent catastrophic failure of the system. The system should be able to automatically detect, locate and assess structural damage anywhere within the bridge system (health monitoring), and to communicate the status (alerting) to responsible authorities. The diagnosis techniques generally includes finite-element modelling of the suspension bridge, measurement of strong wind, earthquakes and traffic loadings, analysis of the ambient vibration data through the use of the appropriate system identification techniques and selection of sensors and techniques to detect localized damage and defects. A vibration monitoring system encompassing 12 tri-axial acceleration transducers and a data acquisition unit is installed along the Asian half span of the Fatih Sultan Mehmet Suspension Bridge (Apayd ın and Erdik, 2001; Apaydin, 2002). A general layout of the system is illustrated in Figure 7. 8. Earthquake performance and rerofit investigations An earthquake response and performance analysis of the Boğaziçi and Fatih Sultan Mehmet Suspension Bridges was performed by Japanese Bridge and Structure Institute (JBSI, 2004) for the General Directorate of State Highways (Turkey) to determine the seismic vulnerability and retrofitting requirements of the bridges. In this connection comprehensive analytical studies were also conducted to gain an understanding of the dynamic seismic behavior of the bridges. The analysis was performed using detailed three-dimensional models that included geometric and material non-linearity and soil-structure interaction. The performance-based


Figure 7. Bridge

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Structural Health Monitoring System installed on Fatih Sultan Mehmet Suspension

seismic analysis and retrofit design provides the means for designing structures with measurable levels of structural response. It leads to structures with seismic load resisting systems of specified strength, stiffness and energy dissipation capacity. For earthquake resistant retrofit analysis and design the following criteria were considered: Under exposure to the Functional Evaluation Earthquake (FEE) Ground Motion the damage level will be minimal (essentially elastic performance) and the functionality of the bridge will continue without interruption. This ground motion refers to the high probability earthquakes that can affect the structure one or twice during its lifetime. This earthquake is generally associated with a 50% probability of exceedance in 50 years. Due to specific seismo-tectonic the FEE earthquake will be taken as the site-specific ground motion that would result from the Mw=7.5 scenario earthquake on the Main Marmara Fault. Under exposure to Safety Evaluation Earthquake (SEE) ground motion only repairable damage is allowed, such that, the damage can be repaired with a minimum risk of losing functionality without endangering and lives. In consideration of regional earthquake occurrences this level of ground motion will be associated with

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a site-specific probabilistic ground motion associated with a 2% probability of occurrence in 50 years. The performance criteria for both suspension bridges under SEE require that: No damage to main and hanger cables (within tensile strength) No slipping of cable clamps Light damage in Towers with no local buckling No damage at rocker bearings (within yield point) No damage due to collision between tower and deck No damage at expansion joints. Although both suspension bridges were originally designed for much lower earthquake loads under SEE earthquake (Peak Ground Acceleration of about 0.4g0.5g) they exhibited very good performance. The retrofit actions recommended for the main spans of both suspension bridges consisted of the reinforcement of the tower legs below the portal beams (for collision between deck and tower) and installation of falling down prevention devices (cables), and reinforcement of the rocker bearings. At the Asian and European joints of the approach viaducts (side spans) of the Boğaziçi Suspension Bridge the retrofit actions will encompass replacement of the elastomeric bearings, construction of new diaphragms, and installation of hysteretic dampers. 9. Comments on the earthquake response of the suspension bridges Following general comments on the dynamic characteristics and the earthquake response of the suspension bridges can be made: Suspension bridges are complex 3-D structures that can exhibit a large number of closely spaced coupled modes of vibration. Time domain analysis is the most rational method for the earthquake response analysis. The characterization and effects of variable support input motions and the influence of nonlinearities in suspension bridges are two areas which are not well understood and which will require significant additional research. For suspension bridges exposed to strong earthquake motions, significant changes in the initial bridge geometry will result in geometric nonlinearities as the stiffness of the suspension cables and deck system change appreciably with large displacements of the structure. Once the analysis is carried out under the deformed geometry (under dead load) the variation of geometry due to lateral earthquake loads are quite negligible and the total system can be treated without consideration of the any geometric non-linearity. Theoretical analysis of the free vibration of suspension bridges indicate that the modes of the structure can be divided into three groups with respect to the dominance of the respective displacements. In the first group the displacements


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of the deck is important (Deck Modes), in the second the displacements of the cables are important (Cable Modes) and in the third group the displacements of the towers become important (Tower Modes). The deck modes can also be classified as symmetric and asymmetric or as lateral, vertical and torsionally dominant. A relatively large number of modes may be necessary to obtain a reasonable representation of the response. Studies reported herein and the similar ones indicate that finite element analysis techniques can be used satisfactorily for the analytical assessment of these mode shapes, frequencies and the participation factors. Earthquakes have a complicated interaction with the whole structure. For example vertical vibrations of these structures are induced to a great extent by the lateral shaking in addition to the vertical excitation. In vertical modes of vibration the earthquake induced cable tensions may reach considerable values. In coupled vertical-tensional vibrations, either large amplitude vertical or large amplitude torsional vibration may dominate the motion. The first symmetric torsional modes appear to dominate the torsional response because of its high participation factor. It becomes necessary to consider asynchronous excitation at the each and every contact point of the structure with the ground. This strong ground motion should be specified as three-axial and all possible kinds of wave fields (especially SH, Rayleigh and Love) should be considered in the analytical techniques used in their simulation. Under multi-support excitation at both of the towers in the lateral direction, different peak displacements are observed. For the travelling ground motion, as the phase difference increases, the displacements of the deck in the lateral direction become smaller. At the towers, as the phase difference increases, peak displacements in the lateral direction increase. The two towers may move in different directions. For suspension bridges located in the fault near-field where large ground displacement pulses and deterministic wave radiation patterns are known to be a possibility, special ground motion simulations should be considered. References Abdel-Ghaffar A. M., Housner G. W. (1978) Ambient Vibration Tests of Suspension Bridges,, Journal of Engineering Mechanics Division, ASCE, No.EMS, Proc. Paper 140b5, 104 983-999. Abdel-Ghaffar A. M., Scanlan R. H. (1985) Ambient Vibration Studies of Golden Gate Bridge: I-Suspended Structure, J. Engrg. Mech., ASCE, No. EM4, 111 463-482. Apaydin N., Erdik M. (2001) Structural Vibration Monitoring System for the Bosporus Suspension Bridges in Strong Motion Instrumentation for Civil Engineering Structures (NATO Science Series), Ed.By. M.Erdik et al., Springer-Verlag New York, LLC, 373. Apaydin A. (2002) Seismic Analaysis of Fatih Sultan Mehmet Suspension Bridge, Ph.D. Thesis, Department of Earthquake Engineering, Bogazici University, Istanbul, Turkey.

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Beyen K., Uckan E., Erdik M. (1994) Ambient Vibration Investigation of the Bogazici Suspension Bridge, Istanbul, Turkey, Proc. Earthq. Res. Construction and Design, Balkema, Rotterdam. 915-922. Brownjohn J. M. W., Dumanoglu A. A., Severn R. T., Taylor C. A. (1987) Ambient Vibration Survey of the Humber Suspension Bridge and Comparison with Calculated Characteristics, Proc. Instn. Civ. Engrs. 83 561-600. Brownjohn J. M. W., Dumanoglu A. A., Severn R. T. (1992) Ambient Vibration Survey of the Fatih Sultan Mehmet (Second Bosphorus Suspension Bridge), Earthquake Eng. Struct. Dyn. 21 907-924. Erdik M., Uckan E. (1988) Ambient Vibration survey of the Bogazici Suspension Bridge, Report No: 89-5, Department of Earthquake Engineering, Bogazici University, Istanbul, Turkey. Japanese Bridge and Structure Institute - JBSI (2004) Project Reports and Basic Design Documents for the project entitled “Seismic Reinforcement Of Large Scale Bridges In Istanbul”, prepared for the General Directorate of State Highways, Turkey. Kaya Y., Harmandar E. (2004) Analysis of Wind Induced Vibrations on Bogazici and Fatih Sultan Mehmet suspension Bridges, Internal Report, Department of Earthquake Engineering, Bogazici University, Istanbul, Turkey. Kosar U. (2003) System Identification of Bogazici Suspension Bridge, M. Sc. Thesis, Department of Earthquake Engineering, Bogazici University, Istanbul, Turkey. LUSAS (1993) Finite Element Analysis Computer Program, Finite Element Analysis Ltd., Surrey, U.K. Petrovski J., Paskalov T., Stojkovich A., Jurokovski D. (1974) Vibration Studies of Istanbul Bogazici Suspension Bridge, Report OIK 74-7, Institute of Earthquake Engineering and Engineering Seismology, IZIIS, Skopje, Yugoslavia. Ren W. X., Harik I. E., Blandford G. E., Lenett M., Baseheart T. M. (2004) Roebling suspension bridge. II: Ambient testing and live-load response, Journal of Bridge Engineering 9 119-126. Tezcan S., Ipek M., Petrovski J., Paskalov T. (1975) Forced Vibration Survey of Istanbul Bogazici Bridge, Proc. 5th ECEE, Istanbul Turkey 1.2. Wilson L. E., Habibullah A. (1989) SAP90: A Series of Computer Programs for the Static and Dynamic Analysis of Structures, Computers and Structures Inc., Berkeley, CA., USA.

THE MICROWAVE SENSOR OF SMALL MOVING FOR THE ACTIVE CONTROL OF VIBRATIONS AND CHAOTIC OSCILLATIONS MODES ∗ V. A. Fedorov1 , S. M. Smolskiy1 , A. V. Mizirin1 , V. V. Shtykov1 , A. V. Fomenkov1 and S. M. Kaplunov2 1 Moscow Power Engineering Institute (Technical University), Russia 2 Mechanical Engineering Research Institute of Russian Academy of Sciences, Russia

Abstract. In the report results of practical development of the modeling vibrations test bench and the experimental check of a new class of vibrations and the midget moving sensors are given on the basis of the microwave radar, capable to fix the object moving in space with the high linear resolution up to 10-6 m. Such sensors was named “Pulsar” and authors for the first time used it for remote non-contact registration of fluctuations of a person body sites surface under influence of various physiological processes - breath, pulse blood capillary filling, clonus, acoustic vibrations etc. Simultaneously with the sensor development laboratory researches on modeling vibrations test bench were carried out with the excitation by various complex signals. At work with biological objects the majority of the signals registered by microwave radartracking sensor “Pulsar” and caused by physiological processes, belong to a class of the signals most adequately described by the theory of nonlinear dynamic chaos (Glass and Mackey, 1991), (Moon, 1987). Therefore authors have put before themselves the purpose of creation of the computer analysis method of the signals received on the “Pulsar” sensor output at its any application, including at research of functional conditions of various biological objects. The offered report is devoted to results of vibrations studying in mechanical and biological objects with the help of the radar-tracking sensor of small movings. The specified purpose is achieved with the help of computer mathematical model of chaotic signals, transfer of these signals on the vibrations test bench activator and registration of the vibrations amplitude by remote non-contact “Pulsar” system. Key words: microwave, active control, chaotic oscillation

1. The basic preconditions of researches At present chaotic signals draw the increasing attention of researchers in the diversified areas of a science and engineering. Systems, developing such signals, are studied as with the help of classical methods of the nonlinear oscillations theory, and with the help of the new approaches using such concepts as chaotic process, “strange attractor”, a bifurcation mode, a set fractional dimension etc. Therefore ∗

Not presented 195

.


196 FEDOROV, SMOLSKIY, MIZIRIN, SHTYKOV, FOMENKOV, KAPLUNOV development of devices and methods of precision registration and the analysis of small moving with the help of the sensors, which are not deforming these small movings, is the important and actual problem of researchers. Most fully these requirements are answered with the non-contact sensors working in a microwave range. Such sensors do not depend on object of studying, do not render physical influence on it, can function on appreciable distance from an object during measurement, they are compact, have small power consumption, are capable to work continuously long time. They are especially perspective at research of functional conditions of alive objects, including the person as for them the usual clothes are not an obstacle and besides researches can be carried out on real objects during their active vital phase and in real time. It is necessary to note, that microwave sensors are easily integrated to a computer which can undertake a significant role in processing the radio signals reflected from object and in the automatic analysis of measurements results with the analysis results output in real or quasi-real time. During researches authors have applied the complicated chaotic signal which distinctive feature is its rather complex mathematical description and technically complex formation. Physiological processes in alive organisms are so complex, that very seldom it is possible to give their unequivocal theoretical interpretation. In these conditions in an arsenal of the researcher it is important to have mathematical models of physiological processes with which help it would be possible to receive a picture of complex interactions of separate sub-systems of an organism. The present report is devoted exactly to this question. 2. Computer modeling of chaotic process One of ways of chaotic process modeling with the help of a computer is a process examination on low frequency. This way has the next two problems. The first one is a complexity of behavior in time of the signals reflected from biological objects with account of their individual specificity and characteristic features for different patients. The second one is a necessity of transfer of the low-frequency chaotic process, modeled by a computer, in microwave area (on the carrying frequency of “Pulsar” equal 60 GHz) with the subsequent allocation, a filtration and an amplification of the signal low-frequency Doppler component, that is necessary at synthesis of mathematical model of real physiological processes. These signal transformations demand high linearity and a minimization of noise parameters of a signal passing path. Such necessary quality requirements of send-receive and amplifying paths of “Pulsar” sensor were successfully achieved. Since the reflected signals with chaotic structure from biological objects are difficult to simulate, it was necessary to fulfil a measurement technique on

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mechanical vibrations test bench, having excited its mechanical vibrations by a signal, synthesized on a computer with the help of mathematical model of chaotic process. In the theory of nonlinear fluctuations there are examples of mathematical modeling of signals with chaotic dynamics (Chernobaev, 2002), (Tomashevskiy, 2003). As such examples systems with phase control which represent on structure the system of automatic control with a phase feedback can quite serve. For experiments with “Pulsar” the computer model of chaotic fluctuations was used on the basis of the theory of electronic amplifiers with phase control (Chernobaev, 2002). At first the opportunity of chaotic signals generating with the help of ordinary electronic amplifiers was analyzed. According to it we shall enter, following (Chernobaev, 2002), the general name for researched system: the amplifier with phase control (PCA), generalized block diagram of which is presented on Figure 1. In this figure it is shown, that on an input of an amplifying path the input quasiharmonic signal u1 (t) is acting (here amplitude and phase functions U1 (t), ϕ(t) are assumed slow, i.e. poorly varying for the period of carrying frequency ωc ), and on an PCA output a target quasi-harmonic signal u2 (t) is developed . The main reason causing of an oscillations phase deviation ϕ2 on an PCA output compared to a phase ϕ1 of an input signal, the central frequency detuning ξosc of the oscillatory circuit, included in phase discriminator PD, is concerning of an input signal frequency. In the PCA model submitted by the block diagram on Figure 2, the detuning ξosc plays the role of the basic destabilizing factor. In a mode of the output phase stabilization the PCA operates on a principle of an automatic control system, reducing the current mismatch of an output phase ϕ2 relative to the input one ϕ1 which is measured with the help of phase discriminator (PD). Produced in FD the mistake signal as slowly varying voltage e(t) is used by Figure 1. Structural scheme of an amplifier with phase a control circuit (CC) for control regulation of tuning frequency of amplifying path with the help of the special frequency controller (FC), for example, a varactor. Let’s take advantage, following (Chernobaev, 2002), mathematical models of an examined path as the symbolical abbreviated differential equations of the 3-rd

198 FEDOROV, SMOLSKIY, MIZIRIN, SHTYKOV, FOMENKOV, KAPLUNOV order minimally necessary for occurrence in examined system of dynamic chaos interesting us. Let on a system input the signal of a voltage operates u1 (t) = U1 (t) cos [ωc t + ϕ1 (t)] = Re[U1 (t)e jωc t e jϕ1 (t) ],

(1)

and on an output a signal with the same frequency operates u2 (t) = U2 (t) cos [ωc t + ϕ2 (t)] = Re[U2 (t)e jωc t e jϕ2 (t) ].

(2)

Let’s assume, that complex amplitudes of input U1 (t)e jϕ1 (t) and output U2 (t)e jϕ2 (t) signals are the functions of time slowly varying for the period of high frequency T c = 2π/ωc , and we shall connect these complex amplitudes through symbolical transfer factor of an amplifying path: U2 (t)e jϕ2 (t) = K[p, ξ? , ξ(t)] U1 (t)e jϕ1 (t) ,

(3)

where p ≡ d/dt is the differentiation operator of slowly varying functions, ξosc is the own frequency detuning of the PCA oscillatory system concerning frequency of an input signal, ξ is the resulting detuning inserted in PCA from FC side. Expression in the operational form for inserted detuning will look so: ξ = ΩW(p)F(ϕ1 , ϕ2 , U1 , U2 ),

(4)

where F is the normalized PD characteristic, W(p) is the transfer factor of the control circuit, dependent on the differential operator, Ω is the area of stable frequency detuning of system. Expressions (3) and (4) are the differential equations which have been written down in the operational form which describe the PCA operation, and the equation (3) is the complex one, i.e. it breaks up to two real equations. Three real equations allow the determination of three unknown time functions of a problem: a phase and amplitude of an output signal and the inserted frequency detuning. Studying of transient modes has the big importance at revealing dynamic properties of system as really the amplitude U1 (t) and a phase ϕ1 (t) of an input signal bear the helpful information on vibrations of mechanical system or on a functional condition of the patient for our device. Research of transients is meant a supervision over processes of an establishment in time of amplitude U2 (t) and a phase ϕ2 (t) of the output oscillations, occuring in the constructed models at various initial conditions and at values of systems parameters. Such dynamic processes are usually displayed in “phase space” of system with the indication of coordinate axes and grid where time is the independent parameter varying along integrated curve of the initial differential equations system. It appears that in the described system at quite certain combinations of parameters there can be repeated chaotic oscillations. The received mathematical


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model of this chaotic process has permitted to design and make measuring vibrations test bench with chaotic processes and to carry out measurements, using the following procedure. First with the help of a computer and the appropriate mathematical methods the low-frequency chaotic process in the digital form with use of the equations (1)(4) is modeled. Parameters of system and, hence, parameters of chaotic process can be changed during measurements. Then with the help of a computer sound card and the special software this digital process is transformed on an exit of digital-to-analog converter (DAC) to a real electric signal after that this signal is passed through the linear amplifier and moves the vibrations test bench activator. In result this test bench starts to make low-frequency fluctuations according to a signal received on mathematical model. At the second stage modeling vibrations test bench, excited by a low-frequency chaotic signal, is irradiated from distance of 1-2 m with the microwave shortrange radar “Pulsar” working in a millimeter wave band. The signal reflected from a vibrating surface, which phase changes under the law set by synthesized chaotic process, accepted by the radar antenna and due to the subsequent frequency transformation the initial chaotic signal can be transferred in any area of a spectrum. Taking into account, that properties of a chaotic signal of computer model can be changed at the request of the experimenter, we receive the mechanism of technical formation of a wide class of regular and chaotic processes and their carry to any frequency area necessary for experiments, i.e. a way of creation of some generator of real chaotic signals. Thus, the first object in view is achieved - to create a method of a chaotic signal formation by computer means and to compare its key parameters (for example, the time form, auto correlation function of process and its phase portrait) to the same parameters received as a result of signal processing in “Pulsar” radar. Thus it is supposed that at concurrence of the specified characteristics of initial chaotic process and characteristics of process, past through all radar transformation blocks, a validity of carry of chaotic process in a radio range it will be proved and remote sensor “Pulsar” can be with good reason used as the non-contact sensor of the signals carrying the undistorted information on physiological processes in an alive organism. 3. Displays of a functional condition of complex system to a phase plane Researches on studying behavior of complex objects including alive systems (Glass and Mackey, 1991) with application of the mathematical approach of nonlinear dynamic processes have shown, that in a normal functional condition in characteristics of their behavior is present an irregular component with a high degree of complexity. On the other hand, at aging and deterioration of any system the

200 FEDOROV, SMOLSKIY, MIZIRIN, SHTYKOV, FOMENKOV, KAPLUNOV well defined periodicity of characteristics accompanying with some “decrease” of a randomness degree and a complexity degree of dynamics of investigated parameters are shown. Thus, in the complex hierarchical system, normally functioning, characteristics of its behavior can be submitted by chaotic processes. In complex system which parameters have deviated norm as have shown experiments, the obvious periodic component gradually superseding chaotic characteristics of its behavior is shown. The method of phase space allows the description of a phase portrait of analyzed dynamic system if one spatio-temporal sequence as any parameter is known only, describing behavior of system. The analysis of a phase portrait permits to determine the type or prominent features of system dynamics. Authors carry out the description of a measurement method of the vibrations test bench oscillations amplitude with the help of the remote radar-tracking sensor of a millimeter range which construction is discussed below. This test bench simulates complex oscillations in space according to the given mathematical model of oscillatory (chaotic) process and an active test bench surface which points are oscillating and investigating, makes moving to space with chaotic movement parameters. To register the moving character and amplitude of test bench surface points by a traditional way, it is necessary to arrange moving sensors on a researched surface. For this purpose strain gauges (accelerometers) are usually used. In a case when it is necessary to place some several strain gauges, there are complexities with fastening contact conductors and gauges on an active surface of test bench. Authors offer to use as researched model of chaotic vibrations a usual powerful loudspeaker with big diffuser in the sound coil of which the chaotic process of an electric current generated by a sound card of a personal computer is created and it creates with usage of the mathematical model a researched chaotic signal. Under action of a chaotic sound current a loudspeaker diffuser makes chaotic vibrations which further are fixed by any moving sensor. Authors used as measuring system the microwave radar-tracking sensor of high resolution on space (device “Pulsar”) which can be arranged on distance from 0.1 up to 5.0 m from a surface of the modeling vibrations test bench and to investigate dynamic properties of any chosen point on it. The output information signal of the radar-tracking sensor through a cable or the radio channel goes to a computer on which monitor it is possible to observe in a real time mode results of measurement process and the analysis of registered moving of a test bench surface. 4. Initial preconditions of mathematical modeling In laboratory experiment with research of vibrations test bench parameters the remote millimeter range radar-tracking sensor of spatial moving “Pulsar” developed


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in MPEI with the resolution on space about 1 micron was used. The block diagram of this experiment is given on Figure 2. The personal computer on this circuit forms in software the mathematical model of chaotic process and directs an output electric signal on the vibrations test bench driver. The device “Pulsar” irradiates a vibrating element, perceives the reflected radio signal and at first transfers by a radar a signal to the necessary area of a spectrum, and then processes a signal in the same computer and creates on a monitor screen a phase portrait of the chaotic process accepted by a radar. The method of phase space is widely used at the analysis of complex nonAntenna Radiation diagram Excitaition system linear dynamic systems as Radar A for shaking table it gives an evident picture Model shaking table of system behavior. Besides well developed apThe program which is carrying The converter of function A(t) out registration of voltage U(t) into voltage U(t) proaches of this method Analysis software of a vibrating Mathematical model of chaotic can be used for interpresignal signals generator tation of research results, Software for vibrating process for example, a system loPersonal computer phase portrait drawing cal stability. On Figure 3 one of possible mathematical solutions of the differential Figure 2. The block diagram of the experiment equations is given, describing behavior of an output signal of the circuit, shown on Figure 1, as the diagram of time function for a case when the amplitude and the phase of the output process have obviously expressed chaotic dynamics. On Figure 4 the auto correlation function of an output signal in this complex mode is given, and on Figure 5 the phase portrait of this process is shown.

Figure 3. A chaotic process in time

Figure 4. Auto correlation function

202 FEDOROV, SMOLSKIY, MIZIRIN, SHTYKOV, FOMENKOV, KAPLUNOV The given characteristics illustrate dynamic properties of computer mathematical model of oscillatory system as the nonlinear differential equations of 3-rd order. The mathematical solution of these equations, received on a personal computer, can be transformed into chaotic voltage U(t) with the help of the built-in standard computer sound card and through digital-to-analog converter (DAC) can be sent on an input of the linear activator resulting in movement the modeling vibrating test bench (as shown in Figure 2). Thus, real chaotic fluctuations of the test bench are activated. The next problem was a registration of vibrations test bench fluctuations with the help of the remote microwave radar-tracking moving sensor “Pulsar” [5], which represents the small-sized send-receive radar device of Doppler type. Its block diagram is given on Figure 6.

Figure 5. The phase portrait in chaotic mode

Figure 6. The block diagram of “Pulsar” radar

“Pulsar” device operates in this application as follows. The transmitter antenna radiates in a test bench direction a narrow directed signal with 60 GHz frequency. At this time the test bench is already excited by the chaotic signal received with the help of computer mathematical model of chaotic oscillations, and oscillates in a chaotic mode. The radio transmitter signal is reflected from activated test Figure 7. Time signal at Pulsar output bench and gets in the reception antenna of a microwave measuring instrument. On an output of a reception path of device “Pulsar” the low-frequency in-phase and quadrate components are developed (Figure 6) which are digitized and move on a computer (Figures 2 and 6). The special computer


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software developed for “Pulsar” registers test bench fluctuations and makes the analysis of these fluctuations by a technique of the chaotic fluctuations analysis. Characteristics of the signal, reflected from test bench, and a phase portrait of test bench oscillatory process are the results of this analysis. Phase processing of the signal reflected from the test bench, permits to receive the high resolution on space and to register thus the minimal vibrations amplitude about 1 micron. For confirmation of an opportunity of chaotic process carry on frequency in a radio range it is enough to compare its characteristics (Figure 3-5) for computer mathematical modeling to characteristics and the phase portrait, received at the analysis of an output signal of a radar-tracking measuring instrument “Pulsar”.

Figure 8. Auto correlation function

Figure 9. A phase portrait of the output process

Comparing in pairs Figures 3 and 7, 4 and 8, 5 and 9, we find out, that diagrams of correlation functions of the modeling and measured processes almost ideally coincide, that confirms identity of both processes. Some differences in the figures concerning mathematical model and received on measured signal speak about a difficulty of full synchronization on time of a modeling signal and the measured signal among themselves. So, a comparison of results shows that computer modeling of initial processes in a mode of chaotic oscillations and experimental results on carry of these processes through a sound card and the activator on modeling test bench, which fluctuations are analyzed by precision radar-tracking approach, gives a similar picture on a radar output. 5. Conclusion Thus, as a result of the carried out researches we receive the following important results. 1. The new radar tracking millimeter device “Pulsar” is developed as a new non-contact device for registration of mechanical vibrations as small moving of any type of the diversified alive and lifeless objects.

204 FEDOROV, SMOLSKIY, MIZIRIN, SHTYKOV, FOMENKOV, KAPLUNOV 2. New opportunities of modeling of complex chaotic processes with the help of a computer and carry of this process to area of real radio signals with the aid of the radar-tracking signal transformation are offered. 3. The developed method of mathematical synthesis of signals with chaotic phase change can be used for generating and the analysis of the signals adequately reflecting dynamics of physiological processes of alive organisms at various functional conditions of the patients, giving the researcher an opportunity to analyze on a computer a course of patients illness and its treatments results. References Chernobaev, V. G. (2002) Chaotic oscillations generators on the base of phase synchronization systems, (in Russian)., Ph.D. dissertation. MPEI. Moscow. Fedorov, V. A. and Krokhin, L. A. (1993) The method of arterial pulse and breathing frequency registration and the device for Doppler radar location., Russian patent 2000080. Fedorov, V. A., Driamin, M. Yu., Smolskiy, S. M., Shtykov, V. V., Kaplunov, S. M. (2000) New technology of measurement and analysis in medical diagnostics with application of nonlinear dynamic models, International Conference on Advances in Structural Dynamics, 1151-1158. Glass, L., and Mackey, M. (1991) From Clocks to Chaos. The Rhythms of Life (in Russian), MIR Publisher, Moscow. Moon, F. C. (1987) Theoretical and Applied Mechanics., Cornell University. Ithaca, New York. Tomashevskiy, A. I. (2003) Chaotic oscillations formation in amplifying paths with phase control, (in Russian)., Ph.D. dissertation. MPEI. Moscow.

FORCED OSCILLATIONS OF TUBE BUNDLES IN LIQUID CROSS-FLOW ∗ T. N. Fesenko and V. N. Foursov Mechanical Engineering Research Institute of Russian Academy of Sciences, Russia

Abstract. The stiffness of tube system for industrial heat-exchangers is usually increased by application of intermediate supports with corresponding tube-to-support gaps. The cases are frequent when intensive hydrodynamically excited vibrations and tube wear can arise at the location of tube-to-support contacts. In this case under the conditions of tube-to-support vibrocontacts the mechanism of wear during sliding impact takes place. Such a wear is more intensive one than a pure fretting wear at constant contact parameters. The present report considers an algorithm to solve a problem of tube bundle forced oscillations in liquid cross flow with taking into account the intermediate supports and main design parameters. The model of oblique impact with normal and tangential components of the reaction forces is used which takes into consideration tube-to-support impacts. The impulsive reactions of supports are included in the right sides of nonlinear forced oscillation equations. The problem of forced oscillations is solved by use of the known force factors. The present model takes into account the vortex and hydroelastic mechanisms of oscillation excitation in cross-flow. Key words: forced oscillation, tube bundle

1. Introduction Increase in heat-exchanger durability is connected with development of investigation methods for structure vibrations of heat-exchangers subjected to the action of a coolant flow. In designing heat-exchangers it is of importance to predict levers of tube displacements, stresses as well as the quantities affecting tube vibrowear at the location of tube-to-intermediate support contacts. The purpose of the present article is to propose algorithm permitting to determine tube displacements, vibrovelocities, contact loadings, sliding path, time of contact for heat-exchanger tubes, with taking into account their real intermediate supports with clearances, i.e. nonlinear elastic couplings for example for a case when tube-to-baffle clearances exist. ∗

Not presented

205 .


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FESENKO, FOURSOV

2. Basis of mathematical model Tube bundles are subjected to simultaneous action of several types of excitation so they must apparently be included into mathematical models of tube bundle vibration . The model used in the present paper is applicable for predicting response of tube bundle cell under the action of vortex and hydroelastic excitation mechanism. The model in full measure takes into account the results on this problem obtained by R.D. Blevins, Y.N. Chen, S.S. Chen, F. Axisa et al. Hydrodynamic forced (g, h) which depend on tube motion are determined by the expression of a form (Chen, 1978): k ∂2 u ∂2 w ∂u ∂w gi = αi j ∂t j + σi j ∂t2 j + α i j ∂t j + σ i j ∂t j + α i j u j + σ i j w j j=1 (1) k ∂2 u ∂2 w ∂u ∂w τi j ∂t2 j + βi j ∂t2 j + τ i j ∂t j + β i j ∂t j + τ i j u j + β i j w j , hi = j=1

where u j , w j are tube displacements in y- and x- directions, αi j , σi j , τi j , βi j are matrices of attached masses, α i j , σ i j , τ i j , β i j , α i j , σ i j , τ i j , β i j are matrices of hydrodynamic stiffness. All these coefficients depend on reduced flow velocity, mutual arrangement of tubes, cylinder oscillation amplitudes and characteristics of an approach flow. In practice it is often admissible to use linearized equations of motion. In this case the coefficients of hydrodynamic forces are thought to be independent on oscillation amplitudes. About calculation of hydrodynamic force coefficients we shall say in detail below. To describe tube bundle vibration we use equations for small bending oscillations of rods. The following expressions for the lift force and drag force acting on i-th tube are taken: fiL = 0, 5ρQ v2 DCy sin ω s t + φiL (2) fiD = 0, 5ρQ v2 DC x 1 + sin 2ω s t + φiD , φiL , φiD − describe phase dependence for tubes. The values of Sh, C x , Cy were taken in accordance with the data in (Makhutov et al., 1985, Belocerkovskiy et al., 1983) where Sh and Cy are shown to depend on a relative pitch of the bundle; ω s = 2πSh v/D is frequency of vortex shedding. In terms of the abovementioned general equations of motion for tube bundles subjected to the action of cross flow will be k 4 ∂2 u ∂2 w ∂u ∂2 ui i + m + Ei Ii ∂∂zu4i + 2εi mi ∂u αi j ∂t2 j + σi j ∂t2 j + α i j ∂t j + i ∂t2 ∂t j=1 N ∂w j y + σ i j ∂t − α i j u j − σ i j w j = 0, 5ρQ v2 DCy sin ω s t + φiL + Ril (t) δ (z − zl ) l=1

(3)

FORCED OSCILLATIONS OF TUBE BUNDLES

207

k 4 ∂2 u ∂2 w ∂u ∂2 wi i + m + τi j ∂t2 j + βi j ∂t2 j + τ i j ∂t j + Ei Ii ∂∂zw4 i + 2εi mi ∂w i ∂t2 ∂t j=1 N ∂w β i j ∂t j −τ i j u j − β i j w j = 0, 5ρQ v2 DC x 1+?sin 2ω s t+φiD + Rilx (t) δ (z − zl ) , l=1

(4) y where Rilx and Ril are projections onto x- and y-axes of reactions of l-th intermediate support for i-th tube. 3. Calculated analysis of tube bundle vibrations Instead of finite-element discretization of the Equations (3), (4) which is usually used by many investigators (Makhutov et al., 1989) it is expedient to project them onto a basis of natural oscillation modes of the related linear system. The main advantage of such an approach is saving of calculation time since it is sufficient to construct solutions only at the locations of supports with clearances and at some intermediate points. As far as in most real heat-exchanger structures all the tubes are of the same length and have identical type of boundary conditions in x- and y-directions the modal functions for the tubes vibrating in x- and y-directions will be identical. Thus let: ∞ ui (z, t) = αim Φm (z) m=1 (5) ∞ bim Φm (z) , wi (z, t) = m=1

where Φm (z) is m-th mode of natural oscillations of the related system in vacuum, i.e. #L Φm (z) Φn (z) dz = δmn . 0

If we project the equations (3), (4) onto this basis we obtain the following matrix equations: ¨ + [C2 ]{A} ˙ + [C3 ]{A} + [C4 ]{ B} ¨ + [C5 ]{ B} ˙ + [C6 ]{B} = {P1 } [C1 ]{A} ¨ ˙ ¨ ˙ + [D6 ]{A} = {P2 }. [D1 ]{ B} + [D2 ]{ B} + [D3 ]{B} + [D4 ]{A} + [D5 ]{A} These equations may be written as a single equation: [M]{S¨ } + [C]{S˙ } + [K]{S } = {F}, where:

'

C1 [M] = ' D4 C3 [K] = D6

( ' ( C2 C5 C4 , [C] = , D1 ( 4 D5 7D2 4 7 C6 P1 A , {F} = , {S} = D3 P2 B

(6)

208

FESENKO, FOURSOV

M, C, K are matrices of masses, damping and stiffness correspondingly and S, F are vectors of displacement and external loading. Matrices M, C, and K have dimensions 2k×2k; k is a number of tubes in a bundle. To solve the problem stated we used the Wilsons method of step-by-step integration (Bathe and Wilson, 1972). 4. Determination of hydrodynamic coupling matrixes For determination of matrix elements M, C, K the following assumptions are introduced: − length of tubes is considerably higher than radius and tube-to-tube spaces; − plane flow; − ideal and incompressible liquid; − vortex-free flowing over bodies. Hydrodynamic force is determined by distribution of pressure about the surface of moving profile; the pressure is connected with flow potential Cauchy-Lagrange integral. In terms of these assumptions we obtained the following expressions for coefficients of added masses, hydrodynamic damping and hydrodynamic forces at the first approximation (Nikolayev and Smirnov, 1989) αi j = −2πρQ D2 ε2i j cos2γi j , σi j = −2πρQ D2 ε2i j sin2γi j , τi j = −2πρQ D2 ε2i j sin2γi j , βi j = −2πρQ D2 ε2i j cos2γi j , i j, αii = βii = πρQ D2 ; τii = σii = 0;

(7)

α i j = 4πρQ Dv2 ε3i j cos3γi j , σ i j = 4πρQ Dv2 ε3i j sin3γi j , τ i j = 4πρQ Dv2 ε3i j sin3γi j , β i j = 4πρQ Dv2 ε3i j cos3γi j , i j, β ii = σ ii = τ ii = β ii = 0;

(8)

β i j = α i j = −24πρQ v2 ε4i j sin4γi j , σ i j − τ i j = −24πρQ v2 ε4i j cos4γi j , α ii = β ii = 24πρQ v2 ε4i j sin4γi j , ji σ ii = −τ ii = 24πρQ v2 ε4i j cos4γi j .

(9)

ji


209

To investigate small oscillations of elastic tube bundles we took the values for parameters and in the expressions for the coefficients in stationary configuration. A momentary configuration is taken in terms of.

Figure 1

When it is necessary one can use the second approximation for the predetermined coefficients. 5. Forces of contact interaction between tube and support In real heat exchangers tubes are being spaced by use of spacer grids with clearances. Taking into consideration impact of tubes with ring constraints we took the model of oblique impact with normal and tangential components of support reaction force. Description of contact interaction in a normal direction includes that energy of dissipation at an impact is not taken into account, and expression for normal forces will be of a form: RilN (t) = −C[ril (t) − δil ]η[ril (t) − δil ] , where ril (t) is radial displacement at l-th support; δil is clearance at l-th support; η is Heavyside’s function. To calculate tangential forces of oblique impact we used dry friction hypothesis, according to which the normal force by coefficient of friction and is directed countermotion Rilτ = fT RilN . The total reaction for i-th tube at l-th intermediate support is determined by geometric summation of forces il + R ilτ . il = R R N An expression for projection of support reactions onto axis x and y in the equations (3) and (4) will be of a form: Rilx (t) = −C[ril (t) − δil ] (cosϕ + fT sinϕ) η (ril − δil ) , y Ril (t) = −C[ril (t) − δil ] (sinϕ + fT cosϕ) η (ril − δil ) ,

210

FESENKO, FOURSOV

where φ is phase angle, C is equivalent stiffness for bodies of a contact pair. Satisfactory results are obtained if the following stiffness assessment is used: . & C = 1, 9 Eh2 D h/D , where h is thickness of a tube wall. Contact duration for every impact depends on value of C. As a rule these durations are very short, less than 10−3 s. However demand of high accuracy for that assessment is not often necessary, due to the fact that influence of the processes is averaged in considerably higher time scales being determined by bending oscillations of tubes. 6. Results As it is known from the experiments (Makhutov et al., 1985), only interactions between the nearest tubes of a bundle are of essential importance, which is a feature of short-range action of hydrodynamic couplings. For heat-exchanges with great number of tubes used in practice one can distinguish a system fragment consisting of 4 – 7 tubes being nearest to i-th tube. We shall call this system including i-th tube “a cell” of bundle and shall take into account the interaction of i-th tube only with the tubes of this cell. In cross sections of real bundles tubes displacement symmetry often takes place i.e. for all the inner tubes of the bundle geometry of the corresponding typical cells is identical. This symmetry can give analogous symmetry in hydrodynamic couplings. In this case all the internal tubes of the bundle are under the identical condition of flow over bodies, so hydrodynamic couplings in the entire typical cell will be identical. For large bundles with a regular layout of cross-section it is sufficient to perform investigations of tube vibrations on model bundle with geometrically similar typical cell and consisting of a smaller number of tubes. In conclusion we present the computational results for vibration of 3 different cells of a bundle (Fig. 2). Each tube is hinged at the ends, Intermediate supports have coordinates z1 = 0,515 m; z2 = 1,215 m; they have clearances δ = 0,0005 m. At midspan of tubesystems there is a cross flow of steam.


Figure 2

211

Figure 3

The main parameters are the following: length of tubes l = 2, 13 m Young’s modulus E = 1,1•105 MPa Internal diameter d = 0,0176 m External diameter D = 0,019 m. The tubes are displaced relatively to the centre of openings at intermediate supports; the eccentricity down the flow is taken to be e = 0,005 m. The spacing T between the tubes is taken to be equal to 0,028 m for all the cells shown in Fig. 2. Fig. 4-6 presents the obtained dependences of maximum displacements along and across the flow for three various types of cells. The first maxima of displacements (Fig. 4-6) correspond to the first natural frequencies of a bundle. The right boundary value of approach flow velocity on graphics corresponds to initiation of hydroelastic instability induced by tube-to-tube interaction.

Figure 4

Figure 5

212

FESENKO, FOURSOV

Figure 6

In this case flow velocity is equal to 9,2 m/s for a cell of three tubes (Fig. 2a), to 9,4 m/s for a cell of 6 tubes (Fig. 2b) and to 8,0 m/s for a cell of 7 tubes (Fig. 2c). As it can be seen the difference in values of critical velocity is not great as well as the difference in oscillation amplitude level which in fact confirms a possibility to study tube bundle vibrations on the cell consisting of a small number of tubes. In our case it was a cell of three tubes. References Bathe, K. I., Wilson, E. L. (1972) Large eigen value problems in dynamics analysis, J. Eng. Mech. Div. 98, 1471-1485. Belocerkovskiy, S. M., Kotovskiy, V. N., Nisht, M. I., Fedorov, V. M. (1983) Mathematical modeling of unsteady separated flow of circular cylinder (in Russian), News of AS of USSR. Mechanics of Liquid and Gas 4, 138-148. Chen, S. S. (1978) Cross flow Induced Vibrations of Heat Exchanger Tube Bundles, Nuclear Engineering Design 47, 67-86. Makhutov, N. A., Kaplunov, S. M., Prouss, L.V. (1985) Vibration and longevity of the marine power equipment (in Russian), Sudostroenie, Leningrad, USSR. Makhutov, N. A., Fesenko, T. N., Kaplunov, S.M. (1989) Dynamic of Systems in a Liquid Flow and Structure Durability, Intern. Conf. EAHE, Prague, 148-156. Nikolayev, N. Ya., Smirnov, L.V. (1989) Determination of boundary of hydroelastic excitation of oscillation for the single-row tube bundle streamlined by cross flow (in Russian), Inter-HighSchool Collection, Gorky, USSR, 38-45.

THE EXACT AND APPROXIMATE MODELS FOR THE VIBRATING PLATE PARTIALLY SUBMERGED INTO A LIQUID

George V. Filippenko and Daniil P. Kouzov Institute of Mechanical Engineering of Russian Academy of Sciences, Russia

Abstract. The problem of free oscillations of the plate partially submerged into the layer of liquid is considered in the rigorous mathematical statement. The exact analytical solution of the problem is constructed. The eigen frequencies and the eigen functions of vibrating plate basing on analyses of exact solution are calculated. The influence of liquid’s level on eigen frequencies and on eigen functions is analysed. Key words: plates, membranes, acoustic field, wave propagation, boundary-contact problems

1. Introduction The problem of oscillations of elastic constructions partially submerged into the water is one of the actual problems of modern technics. Ships, oil platforms, sea airports are the examples of such bodies. However the exact calculation of such bodies vibrations is rather complicated. So it is useful to explore the possible oscillations in these objects taking as an example more simple mechanical systems. The aim of this work is to build the exact solution of free oscillations problem of the most simple mechanical model of this class - the plate partially submerged into liquid. Then we compare the exact solution for plate with the solution of the approximate model. The related problems of acoustic waves passing through elastic body have been described in articles (Romilly, 1964)–(Belinsky, 1984). In these articles, the plate overlapped the waveguide channel without passing over its bounds. Free oscillations of thin elastic body protruding above the surface are described below. The model is shown on Fig. 1. The plate is orthogonally rigidly fixed to the bottom of water reservoir. It crosses the layer of liquid and protrudes over its upper free surface. We restrict ourselves to two-dimensional model, considering the processes independent on the coordinate z.

213 .


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FILIPPENKO, KOUZOV

The similar problem solved for the vibrating membrane one can find in (Filippenko and Kouzov, 2001). Our considerations are based on the precise analytical representations for acoustic field in the fluid and for the field of bending displacements of the plate. In order to get this representation the rigorous mathematical statement of this problem is given. It is considered to be a boundary-contact value problem in a medium, i.e. the problem that besides conditions on the bounds of the medium (on the bottom, on the free surface, on the plate) - has contact conditions at the points of the plate fixing with the bottom and at the point of submerging of the plate into the water.

y l H z x Physical model

Figure 1.

We satisfy these conditions at the final stage of the solution’s construction. In this case that leads to the finite system of linear algebraic equations for the determination of several constants. The size of this system is defined by the number of boundary-contact conditions. This number does not depend on the number of normal modes of the waveguide we take into consideration for numerical calculations. 2. The exact statement of the plate oscillations problem Let’s start considering a layer of an ideal compressible fluid, where the acoustic pressure P(x, y) is described by the Helmholtz equation (∆ + k2 )P(x, y) = 0,

(−∞<x Silicon has cubic anisotropy in its principal material coordinates. The solution coordinate system used for FEM calculations for silicon structures fabricated from (100) wafers is obtained by a rotation of principal material Cartesian coordinate system of 45◦ around the < 100 > direction, or z-axis and the resulting material model is linear elastic constant orthotropic. An APDL programming language based ANSYS macro is written for easy and quick FEM simulations. Two non-dimensional parameters n = a/b and k = L/2 max(a, b) are defined. The flexure dimensions are adjusted to keep torsional natural frequency of the micromechanical scanner around 20KHz. For our study, we fixed the torsion frequency to be near 20KHz and used the following mirror dimensions in the analytical and FEM calculations using ANSYSTM :

TORSIONAL VIBRATION OF MICROMECHANICAL SCANNERS

253

TABLE I. Linear elastic material coefficients of silicon and steel Steel 2.03 × 1011 2.03 × 1011 2.03 × 1011 0.293 0.293 0.293 7.85 × 1010 7.85 × 1010 7.85 × 1010

Ex Ey Ez ν xy ν xz νyz G xy G xz Gyz

TABLE II.

Silicon 1.7 × 1011 1.7 × 1011 1.30 × 1011 0.064 0.36 0.36 5.1 × 1010 7.94 × 1010 7.94 × 1010

Mirror and Flexure dimensions used for FEM comparison 1.5mm 1.5mm 0.3mm between 114 and 473µm between 92 and 194µm between 227 and 4732µm

Dm Lm tm a b Lf

15 different (k, n) combinations for flexure width ratio values of k = [1, 2, 3] and flexure length ratios of n = [2, 4, 6, 8, 10], exploring a large portion of the flexure design space. The constant mirror dimensions and varying flexure dimensions used for calculations are given in Table 2. TABLE III.

Comparison of results

Scanner Model Number

SL28

E201-30

CB02

Mirror Shape

round

square

round

Flexure width ratio, k = a/b

1/16

2.1

1.9

Flexure length ratio, n

6.7

19.6

6.2

Torsion Formula (Hz)

251

1168

7985

Torsion Experimental (Hz)

270

1069

8450

Torsion ANSYS (Hz)

250

N/A

N/A

254

KAN, UREY, S¸ENOCAK

4. Experimental results Three different comp-driven micromechanical scanners are tested to validate analytical derivations. The tests are performed at Ko University Optical MicroSystems Laboratory by a Laser Doppler Vibrometer (LDV). The test results are compared with ODF discrete analytical formulation and 3D FEM simulations in Table 3. 5. Conclusions We derived discrete and continuous mathematical models for torsional free vibration of micromechanical scanners. The calculation of torsional stiffness using approximate energy methods is presented. The approximate and exact torsional stiffness of rectangular cross-section are given. Analytical derivations are compared with 3D FEM simulations. The analytical and FEM results are partly verified with test results. The FEM simulations and test results have shown that discrete ODF mathematical model is a good and reasonable representation of physical system and the need to continuous model for a better flexure inertia calculation is little. References Dickensheets D. L., Kino G. S. (1998) Silicon micromachined scanning confocal optical microscope, J. Microelectromechanical Systems 7, 38-47. Garnier A. et al. (2000) A Fast Simple and Robust 2-D Micro-Optical Scanner Based on Contactless Magnetostrictive Actuation, Proc. MEMS’2000, Miyazaki, Japan 715-720. Kan C. (2004) Vibration of Micro Scanners, MS Thesis, Istanbul Technical University Lin L. Y., Goldstein E. L. (2002) Opportunities and challenges for MEMS in lightwave communications, J. Sel. Top. in Quantum Elect. 8 163-172. Meirovitch L. (1967) Analytical Methods in Vibrations, Macmillan, New York. Timoshenko S. P., Goodier J. N. (1951) Theory of Elasticity, McGraw-Hill, New York. Meirovitch, L. (1997) Principles and Techniques of Vibrations, Englewood Cliffs, New Jersey. Urey H. (2003) Retinal Scanning Displays, Encyclopedia of Optical Engineering, Ed. Driggers R., Dekker 3 2445-2457. Weaver W. Jr., Timoshenko S. P., Young D. H. (1990) Vibration Problems in Engineering, Wiley, New York. Young W. C., Budynas R. G. (2002) Roark’s Formulas for Stress and Strain, McGraw Hill, New York, 7th Edition.

PHYSICAL MODELING OF STATIONARY AND IMPULSIVE PROCESSES FOR LARGE-SCALED CONSTRUCTIONS OF FLUID ELASTIC SYSTEMS S. M. Kaplunov, N. A. Makhutov, V. I. Solonin and N. V. Shariy∗ IMASH RAN, MGTU by N.Bauman, OKB GIDROPRESS Russian Federation

Abstract. The most applicable method to decide the tasks of practical analysis of structure dynamic response under stationary and nonstationary excitation is physical modeling. The basic points of method are in the simulation of necessary excitation by means of similar medium propelling or of shaking tables with corresponding physical models of structure according to the special relations between main parameters or by accelerogramm (at the second case). The complicated constructions of power equipment are studied in the paper. In this case practice of straight physical modeling confirms the effectiveness of large-scale models application which are produced from the same as prototype material and provided with use of the same technology. Such modeling gives the opportunity to conduct rather representative row of experimental investigations to forecast dynamic behavior and longevity of prototype object. Key words: impulsive processes, large-scaled constraction fluid, elastic systems

1. Introduction Ever growing production and operation of the same type systems of machines and structures undergoing force action of medium flows gave rise to physical modeling of complex dynamic processes and interactions (Alaboujev et al., 1968; Bolotin, 1979; Kaplounov and Makhutov, 1999; Shoulman, 1976; Frolov et al., 2002). Modeling plays a specific role in investigation of the phenomena related to flow separation including self-excited oscillations (Kaplounov and Makhutov, 1999; Frolov et al., 2004). A pure theoretical approach and mathematical models developed recently (Frolov et al., 2002; Frolov et al., 2004). permits to reduce to minimum necessity in experimental data, but nobody can manage without them completely. Aero-hydroelastic damping of oscillations (ξ) and three-dimensional-time distributions of pressure pulsations (p’) can only be obtained experimentally as well as clamping conditions and mechanical damping. To some extent the semi-empirical ∗

Not presented 255

.


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methods (Alaboujev et al., 1968; Kaplounov and Makhutov, 1999; Frolov et al., 2002), are also based on test results. That is why the experiment is the main way to knowledge and investigation. Thanks to the experiment validity used hypotheses and models are checked and some coefficients are set. Experimental studies performed on models have some essential advantages over the prototypes (Kaplounov and Makhutov, 1999; Frolov et al., 2002; Frolov et al., 2004). The considered direction of physical modeling in aero-hydroelastic systems is based on the main points of the similarity theory (Alaboujev et al., 1968). 2. General approach The structures considered are complex systems consisting of elements in the form of shells, tubes, bars, plates with partial or full perforation and with a wide range of clamping condition characteristics (Fig. 1). The known experience of investigation allows us to call force action of flow on the structure as a random stationary process with features of ergodicity and with peFigure 1. Shell-tube heat-exchanger (scheme, 1 - case, riodic components of load2 - input-output, 3 - tube-bundle, 4 - tube-sheets) ing. Separate components of special interest are those of vortex and vibroacoustic origin (Kaplounov and Makhutov, 1999; Frolov et al., 2002). That is why it is necessary to take into account correlation couplings of pressure pulsations and velocities at similar points of the system. Under separated flow self-excited oscillations of elements or their groups can be observed at qualitative change of damping nature which in fact is connected with changes of hydrodynamic conditions of flow. The main principles of hydroelastic interaction modeling can be obtained on the basis of physical processes analysis at the constant density ρ2 of flow medium (subscripts 1 and 2 – structure and medium correspondingly), Poisson’s coefficient for metal of structure µ1 and temperature of coolant Tc at identical points of the prototype and model structures. The influence of critical phenomena of heat exchange is not taken into account. The liquid is supposed to be incompressible. Firstly we study steady vibrations or periodic processes when the initial conditions are out of consideration.

LARGE-SCALED CONSTRUCTIONS OF FLUID ELASTIC SYSTEMS

257

At the interface of two subsystems (structure and liquid) the equilibrium condition must be considered as the equality of the corresponding components of stress tensors in the liquid and in deformed solid. For dynamic stresses of the corresponding direction the relation σ1v = p2v and σ ¯ 1v = p2v are valid (p2 pressure in liquid). The relations of parameters pv = p p /pm type are equal to the corresponding scale (subscripts p and m – prototype and model correspondingly). The simultaneity condition at the boundary is determined by the equality of normal interface velocities for the solid v1 and the liquid v2 independently on the boundary layer of liquid related to the added mass (Frolov et al., 2002). The corresponding condition of materials properties provides identical meaning of relation between dynamic characteristics of the model and the full-scale system. It is expressed in a ρ1 /ρ2 = idem form. If one takes into account that the medium (construction material) density can be determined through the elastic modulus E and velocity of sound c - ρ = E/c2 then the expression for the condition of mediums properties correspondence in two - component system is E1 c22 /E2 c21 = idem (E2 for liquid is k2 ). Condition of the contact similarity in static has a form by N.Prigorovsky (1985) (E1v + E2v ) lv2 = 1 + E12p Fv where F is force, E12p = E1p /E2p , subscripts 1, 2 relate here to the contacting parts of the system. The condition is satisfied by corresponding choice of model material and by choice of linear and force scales lv , Fv . In descriptions of dynamic characteristics of contact in the similar systems the relation of test duration on full-scale objects and on a model corresponds to the time scale tv = lv and it is also provided by the choice of materials, mediums, quality of surface and scales lv , Fv . In a complex dynamic model the vibroimpact process is supposed to be accompanied by oblique impacts. Here according to Newton’s second hypothesis (Frolov et al., 2002) as dynamic contact similarity we suppose proportionality of tangential and normal components of the impact impulse. If it is necessary stress in the contact may be taken into account on the basis of the known models. 3. Main physical (criteria) model Considering all the main physical features of the phenomena, including buffeting, vortex resonance, hydro elastic self-sustained oscillations and galloping we get the main equation: σ1y = F(d, ρ1 , ρ2 , E, E’, k2 , p¯ 2 , p2 , p2a , Fc , ∆ s , b s , l s , kvw , L s ,¯v2 , v, v2a , R, h, ν, t, ξ, f, µ,L, εσ , K’ s , ψ, q), (1) where d, L – diameter and length, k2 - liquid compression modulus, E’ – dynamic elasticity modulus, p¯ and p - average and pulsating magnitudes of parameter; Fc – contact force; ∆ s , b s , l s – tube-support gap, width of support and length of span,

258


kvw - vibrowear coefficient, L s - total length of conjugate surface slip, R and h – shell radius and thickness, ν - liquid kinematic viscosity coefficient, ξ - damping coefficient, f – frequency, εσ - relative deformation, K’ s - parameter of roughness; ψ, q – angle of flow and tube-bundle density. Here x and y are directions along and perpendicular to flow. In accordance with π-theorem of similitude theory (Alaboujev et al., 1968) for a standard deviation of relative amplitude (σ ¯ 1y = σ1y /d) we obtain: Π’1 = σ1y /d = ϕ(Π’2 , Π’3 ,. . . ,Π’28 )

(2)

The whole procedure of getting the relation (2) is shown in Table 1 in (Frolov et al., 2002). At the first stage in the Expression (1) completeness of the characteristic parameters complex is assessed on the basis of analysis of known complexes in (Alaboujev et al., 1968; Kaplounov and Makhutov, 1999; Frolov et al., 2002; Frolov et al., 2004) taking into account mathematical models in (Alaboujev et al., 1968; Bolotin, 1979; Kaplounov and Makhutov, 1999; Shoulman, 1976; Frolov et al., 2002; Frolov et al., 2004; Abe et al., 1978). The formal approach applied to find a complex of fundamental criteria permits to obtain a traditional form in addition to the known simple algebraic transformations of nondimensional values. Correctness of transition from one main criteria complex to another is also determined by their independence. Analysis of determining parameters and criteria completeness was obtained by authors in (Frolov et al., 2002). For 20 years we have been studying a problem of determining dynamic stresses in the structures of water-water reactors (high-head model), liquid-metal-water reactors (low-head model), multispan tube heat exchangers (vibrowear model) and of different types heat exchangers vibroacoustical processes (Frolov et al., 2002; Frolov et al., 2004) (vibroacoustical model). During all these investigations we successfully used the general criteria model as a basis (Table in (Frolov et al., 2002)). In all cases the equivalent transformation for initial fundamental criteria determinant was applied. In such a way we received the traditional criteria form which permitted us to develop a further analysis of hydrodynamic processes similarity by use of available experimental data bank (Alaboujev et al., 1968; Kaplounov and Makhutov, 1999; Shoulman, 1976; Frolov et al., 2002; Frolov et al., 2004). Equivalence of the transition was determined by independence condition satisfaction. 4. Added masses analysis If we apply a model medium imitator of different density we must keep in mind the influence of the density on frequency scaling. It can be done in accordance with


259

Figure 2. Eigenfrequency parameter dependence on added mass coefficient χm and relative medium density ρ2v for shell systems

general formula of the corresponding oscillation modes in “elastic shell-liquidrigid shell” system (Frolov et al., 2004), where f12 is frequency of structure-andadded mass; æ or χ is coefficient of added mass. If one uses the models of the same material as full-scale ones (ρ1v = 1) frequency scaling is to be made according to the relations as in Fig. 2. The figure shows that as medium density in full-scale system decreases its corresponding natural frequencies rise (curves ρ2v ¡ 1). It is necessary to point out that construction eiugenfrequencies depend on added mass effect which is as well as damping are relevant for tube-bundles with small clearances in intermediate supports and for coaxial shells in liquid. 5. Assessment of physical modeling error On the whole the total error of modeling depends on the accuracy of similarity criteria reproduction which in its turn automatically includes the errors due to the taken assumptions, to the measuring process and to the relative influence of the constant parameters reproduction. That is why to solve the engineering problems by similarity method it is

Figure 3. Comparison of autocorrelation functions of input pressure fluctuations for VVER-440 model and prototype

260


sufficient to highlight main typical processes. So modeling of the phenomena to be investigated is always approximate to some extent. According to V.A. Venikov’s results in (Frolov et al., 2002) consideration of approximated similarity gives us a criterial equation of regression on the basis of statistical analysis of criteria relations. A relative effect of deviation for separate criteria magnitudes is determined here in terms of standardized coefficients of regression for this equation and according to it one finds the necessary accuracy of the similarity criteria reproduction. Physical modeling approximation and efficiency of implementation are assessed in each separate case on the basis of the known experience of experimental studies as it was made by M.Wambsganss (1977) and D.Gorman (1982). For the low-head criteria model (criteria No. 2,3,5,7,15,17-25 in Table in (Frolov et al., 2002)) we find ε0 as general modeling approximation. Final value of modeling relative error is about 15-20%, that is comparable with typical error of experiment. 6. Modeling of impulsive processes Described experimental-and-numerical approach is based on straight decision of dynamic equations or with use of the pseudo dynamic testing method in case of impulsive loading when we can determine the restoring force and solve dynamic equations. The basic points of method are in the simulation by means of shaking tables with special physical models of structure according to the special accelerogramms (recalculated or really characteristic for prototype one) (Shoulman, 1976; Abe et al., 1978). As we study unsteady vibrations or processes the initial conditions are into consideration. The general reception to define similitude conditions of stressed-strained construction condition under the impact (Alaboujev et al., 1968; Shoulman, 1976) consists of ratios AD = æ A st ; σD = æ σ st ; RD = æ R st

(3)

where æ – dynamics coefficient; A st , σ st , R st – static displacement, strain and force reaction, correspondingly. In this case considered system (Alaboujev et al., 1968) consists of rigid body A with weight P without accounting of deformation, which falls from the height H on the stable fixed elastic system B. Dynamics coefficient (Alaboujev et al., 1968) is nondimensional magnitude. That is why similitude under the shaking excitation may be provided with some limitation by observance of similitude conditions in static action of loading with condition of dynamics coefficient identity æ = idem.

(4)


261

Next step to traditional of seismic durability when the initial magnitudes modeling 2 of acceleration X¨ 0 = ddtX2 and displacement X 0 are known. In this case the next 0 conditions ¨ t Xc kX ωX mX¨ = idem, = idem, = idem, = idem, P0X V P0X P0X

(5)

km Am k p A p mm amax m m p amax p = , = P0m P0p P0Xm P0X p gives us the opportunity to define the connection between coordinates and time for the model and prototype for any directions (P0 - force amplitude, k – rigidity, c – damping, m – mass, ω – frequency, a – acceleration, t - time). Obviously that instead of usual coordinates we can apply the characteristic ones namely oscillations amplitude A, maximum acceleration amax and period of the oscillations T. The last one will be defined as: ω p T p = ωm Tm . In dynamic analysis equilibrium condition of a system is expressed while considering inertia force, damping, resistance and applied force (Frolov et al., 2004; Abe et al., 1978; Chen et al., 2000) as [M] {w} ¨ + [C] {w} ˙ + {R} = {P} (6) where [M] and [C] are the mass and damping matrix respectively, {P} is the external load vector; {w} ¨ and {w} ˙ are the respect acceleration and velocity vectors. Modern experimental methods give a possibility to receive rather correct results on the structure models or model - fragments with use of shaking tables and different investigation technique with FEM or FSEM application. The investigation procedure (Frolov et al., 2004) is the next: digital accelerogram - accelerogram modeling with use of similarity theory - pc - code/analogue - analogue signal tape-recording - transformation of seismic force influence accelerogram according to corresponding scale. If we can determine the restoring force {R}, we can solve equation (6). That is the main point of the pseudodynamic testing method. In our case the mass, damping and stiffness matrix differs from (6) by additional hydroelasticity interaction coefficients (Frolov et al., 2002) and become asymmetric in flow.

262


Figure 4. Seismic stand (shaking table – 1, vibrators – 2, model – 3, pneumo-support - 4)

Figure 5. Simulated realizations of seismic excitation (240 points, step of figure procedure for a model 1:20 is 0.00095 s; a and – accelerograms No. 1 and 2)

Figure 6. Calculation overloading coefficients kg of reactor model (Frolov, 2004) in dependence of supporting scheme ( a – reactor model with main elements 1-10, b – model with two supports, c – one support model, 1 and 2 – calculation for accelerograms No. 1 and 2 correspondingly for and ; 3 – experimental data (pointed by crosses))


263

During pseudo dynamic test especially for inelastic behavior of construction (Chen et al., 2000) the restoring force {R} is directly measured by use of modern interface by National Instruments (LabView-5-7) from the model at each time step and this force is substituted into equation (6) to solve the solution for the next time increment. In such a way combines the reality of shaking table tests with calculation approach. The main advantage here is over quasi-static method in fact that the real restoring force of the model is used in the calculation instead of the hypothetical model. 7. Conclusion Wide spread physical modeling approach may considerably decrease the costly prototypes experiments to the minimum necessary to verify only some basic parameters magnitudes in three or four similar points of the structure. Besides only physical modeling gives us the opportunity to investigate carefully the phenomenon in the whole, particularly in the critical conditions that can’t be performed in prototype. References Alaboujev, P. M., Geronimus, V. B., Minkevich, L. M. and Shekhovzev, B. A. (1968) Theory of similitude and dimensions. Modeling., Moscow, Vishay Shkola. Bolotin. V. V. M. (1979) Handbook “Vibration in technique” in 6 V. (in russian), Mashinostroenie. Kaplounov, S., Makhutov, N. (1999) Physical modeling of Dynamics and Strength of AeroHydroElastic System Structures, 3rd EAHE Conf., Czech. Rep., Prague 215-222. Shoulman, S. G. (1976) Handbook Calculation of hydraulic structure seismic strength with account of influence of water medium. M.,Energiy, (in russian). Frolov, K. V., Makhutov, N. A., Kaplounov, S. M. (2002) Dynamics of hydroaeroelastic system structures Ed. By S. Kaplunov and L. Smirnov. M. (in russian), Nauka. Frolov, K., Dragounov, J., Makhutov, N., Kaplunov, S. (2004) Dynamics and Strength of Light Water Reactors Ed. By N. Makhutov. M. (in russian), Nauka. Abe, T., Fujikawa, T., Kurohashi, M., Inoue, Y. (1978) An Approximate Analytical Method for Seismic Response of Structure Having Elastoplastic Behavior, Proc. Inst. Cont. “Vibration in Nuclear Plant”, Keswick, U. K. paper 9:4. Chen, A., Lam, E. S. S. and Wong, Y. L. (2000) Verification of a Pseudodynamic Testing System, Proc. of the Int.Conf. on Advances in Structural Dynamics (ed. By J. M. Ko and Y. L. Xu), Hong Kong, China II. 851-859.

THERMAL STRESSES AND NONLINEAR THERMAL DEFORMATION ANALYSIS OF SHALLOW SHELL PANEL B. Karmakar1 , P. Biswas2∗ , R. Kahali3 and S. Karanji4 Scholar, Von Karman Society, Jalpaiguri, India 2 Founder Member (ICOVP), Old Police Line, Jalpaiguri, India 3 Dept. of Physics, P. D. Women’s College, Jalpaiguri, India 4 Dept. of Maths, University of North Bengal, Darjeeling, India

1 Research

Abstract. Modern aerospace structures such as high-speed spacecrafts, missiles and engineering and nuclear structures are often subjected to severe thermal loads and reveal a clearly nonlinear response. In such situations the associated strains and stresses are usually determined from Von Karman coupled nonlinear partial differential equations extended to thermal loading in terms of transverse displacement and stress function. Although citations of several authors may be made who have employed the method, only a few is mentioned here which contains several cross-references (Chang and Jen, 1986), (Biswas, 2001) and (Mansfield, 1982). The purpose of the present paper is to further generalize the equations for the case of a simplysupported rectangular panel under thermal loading. Applications of Galerkin’s Procedure ultimately lead to a cubic equation involving several parameters. Key words: thermal stress, shallow shell, thermal loading

1. Basic governing equations Following (Donnell, 1976) and (Timoshenko and Woinowisky-Krieger, 1959) and with usual notations, basic governing equations for transverse displacement (w) and stress function (F) can be derived in the forms: α1 E 2 (∇ MT ) = 1−ν   ∂2 w 2 4  − ∇ F = Eh  ∂x∂y

D∇4 w +

∂2 F ∂2 w ∂2 F ∂2 w ∂2 F ∂2 w · + · − 2 · ∂x∂y ∂x∂y ∂y2 ∂x2 ∂x2 ∂y2  Eh ∂2 w ∂2 w ∂2 w  2 − α · − · E ∇ N  t T R ∂x2 ∂x2 ∂yz 

(1)

(2)

where MT and NT are given by (Nowachi, 1962). ∗

Note: Keeping main essence, the Paper is substantially reduced to five pages. Interested readers may contact biswas [email protected] for more details.

265 .


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KARMAKAR, BISWAS, KAHALI, KARANJI

2. Rectangular panel simply-supported at the edges To approach the particular problem concerning a rectangular panel simplysupported at the edges origin being located at one corner of the shell in the middle surface. Let a, b be the length and peripheral width of the shell and are taken as the x and y axes, z - axis being normally downwards. The distribution of temperature in the direction of z axis is taken as linear in the form (Nowachi, 1962). T (x, y, z) = τ0 (x, y) + zτ(x, y) where

T1 + T2 T1 + T2 , τ(x, y) = 2 h h h T 1 = T x, y, , T 2 = T x, y, − 2 2

τ0 (x, y) = and

(3) (4) (5)

Since MT is constant one can express it in the form of the Fourier series. MT =

α

α

amn sin

m=1.3... m=1.3...

mπy mπx sin a b

T where amn = 16M . mnπ2 The deflection w is assumed in the form πy πx sin w = w0 sin a b

(6)

(7)

which satisfies the boundary conditions for simply-supported edges: w=0= w=0=

∂2 w ∂x2 ∂2 w ∂y2

+ +

MT D(1−ν) MT D(1−ν)

atx = 0, a aty = 0, b

(8)

Since NT is a constant and appears in the boundary conditions for inplane displacements, we take ∇2 (NT ) = 0 in a equitation (1) from which the stress function is obtained in the form 2 2 Ehw2 2 2πy b2 F(x, y) = A x2 + B y2 + 32 0 ab2 cos 2πx + cos 2 a b a +

Ehw 0 Ra2 π2 12 + a

1 b2

πy sin πx a sin b

(9)

SHALLOW SHELL PANEL

267

where, A and B are arbitrary constants to be determined from inplane boundary conditions. In accordance with the conditions occurring in airplane structures, the shell is considered rigidly framed, all edges remaining unaltered after deformation. The elongations of the shell in the directions of x and y are independent of y and x respectively. [(Timoshenko and Woinowisky-Krieger, 1959), P - 426] one gets the constants A and B as: ( ( ' ' Ehw2 π2 1 Ehw2 π2 ν ν 1 Eα1 NT Eα1 NT A = 5 0 26 2 + 2 − B = 5 0 26 2 + 2 − (10) 1−ν 1−ν 8 1−ν b a 8 1−ν b a Applying Galerkin’s procedure in equation (1) a Cubic Equation is obtained for central deflections in the following non-dimensional form: w 3 w 2 w 0 0 0 − C4 = 0 − C2 + C3 (11) C1 . h h h where, C1 , C2 , C3 and C4 are known constants. Median surface membrane stresses N x and Ny are given by: (N x ) a , b 2 2

Eh Ny a

b 2,2

Eh

= C5

= C7

w 2 0

h w 2 0

h

− C6

− C8

w 0

h w 0

h

where, C5 , C6 , C7 and C8 are known constants.

−

αt (T 1 + T 2 ) 2(1 − ν)

(12)

−

αt (T 1 + T 2 ) 2(1 − ν)

(13)

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KARMAKAR, BISWAS, KAHALI, KARANJI

3. Numerical computations TABLE I. Exhibits the variations of non-dimensional central deflections (w, h) for a square panel (b/a = 1) and for b/a = 2 considering a/h = b/h = 10 and αt = 11.9 × 10/◦ C (Steel) w/h

(T 1 + T 2 = 10◦ ) (T 1 − T 2 )/h

(T 1 + T 2 = 50◦ ) (T 1 − T 2 )/h

(T 1 + T 2 = 100◦ ) (T 1 − T 2 )/h

0

0

0

0

.2

.008168(R/h=10)∗

.007734(R/h=10)∗

.007295(R/h=10)∗

.009248(R/h=20)∗

.008814(R/h=20)∗

.008375(R/h=20)∗

.008732(R/h=10)∗∗

.008293(R/h=10)∗∗

.007853(R/h=10)∗∗

.009838(R/h=20)∗∗

.009398(R/h=20)∗∗

.008958(R/h=20)∗∗

.025690(R/h=10)∗

.024820(R/h=10)∗

.023940(R/h=10)∗

.026120(R/h=20)∗

.025250(R/h=20)∗

.024375(R/h=20)∗

.030582(R/h=10)∗∗

.029700(R/h=10)∗∗

.028820(R/h=10)∗∗

.031024(R/h=20)∗∗

.030140(R/h=20)∗∗

.029260(R/h=20)∗∗

.052654(R/h=10)∗

.051350(R/h=10)∗

.050035(R/h=10)∗

.053626(R/h=20)∗

.052320(R/h=20)∗

.051007(R/h=20)∗

.069162(R/h=10)∗∗

.067840(R/h=10)∗∗

.066520(R/h=10)∗∗

.070150(R/h=20)∗∗

.068830(R/h=20)∗∗

.067520(R/h=20)∗∗

.096900(R/h=10)∗

.095160(R/h=10)∗

.093410(R/h=10)∗

.098630(R/h=20)∗

.096880(R/h=20)∗

.095140(R/h=20)∗

.036040(R/h=10)∗∗

.034280(R/h=10)∗∗

.132523(R/h=10)∗∗

.137810(R/h=20)∗∗

.136050(R/h=20)∗∗

.134290(R/h=20)∗∗

.115801(R/h=10)∗

.113660(R/h=10)∗

.111466(R/h=10)∗

.142830(R/h=20)∗

.140660(R/h=20)∗

.138460(R/h=20)∗

.191080(R/h=10)∗∗

.188880(R/h=10)∗∗

.186680(R/h=10)∗∗

.218710(R/h=20)∗∗

.216510(R/h=20)∗∗

.214310(R/h=20)∗∗

.4

.6

.8

1.0

∗

(results for a square panel when a/b = 1)

∗∗

(results for a square panel when b/a = 2)

SHALLOW SHELL PANEL

269

4. Observations In brief, it is observe from the above table for numerical results that deflections increase with increase of temperature along the thickness of the panel and such increase is higher for shell geometries other than a square panel. It is also observed that when T 1 + T 2 increases deflection decreases with the increase of (T 1 − T 2 )/h for any shell geometry. References Biswas, P. (2001) Analytical and Computational Approach on Some Nonlinear Thermoelastic Plate and Shell problems, 5th ICOVP Moscow (IMASH). Chang, W. P., and Jen, S. P. (1986) Nonlinear Flexural Vibrations of Heated Orthotropic Plates, Int. J. Solid & Structures 22, 267-281. Donnell, L. H. (1976) Beams, Plates and Shells, Mc. Graw Hill Pub. Co., New York. Mansfield, E. H. (1982) On the Large-Deflection Vibrations of Heated Plates, Proc. Royal Society (London) A 379, 15-39. Nowachi, W. (1962) Thermoelectricity, Addison Wesley Pub. Co., New York. Timoshenko, S., and Krieger, S. W. (1959) Theory of Plates and Shells, Mc. Graw Hill Pub. Co.

NONLINEAR THERMAL VIBRATIONS OF A CIRCULAR PLATE UNDER ELEVATED TEMPERATURE B. Karmakar1 , S. B. Karanji2 , R. Kahali3 and P. Biswas4 1 Vibration Research Group, Von Karman Society for Advanced Study and Research, Residence : Deshbondhu Para, Jalpaiguri-735101, West Bengal, India 2 Dept. of Maths, University of North Bengal, Darjeeling, India 3 Dept. of Physics, P. D. Women’s College, Jalpaiguri, India 4 Principal (Retired), A. C. College of Commerce and Head, Vibration Research Group, Von Karman Society for Advanced Study and Research, Old Police Line, Jalpaiguri - 735 101, West Bengal, India.

Abstract. Following a new approach, proposed and employed by earlier authors, nonlinear thermal vibrations of a circular plate under elevated temperature have been analysed. Key words: thermal stress, shallow shell, thermal loading

1. Introduction Problems of Mechanics of Thermal Vibrations have been analysed by many authors by using the classical von Karman field equations and Berger’s approximation extended to the dynamic case with the inclusion of thermaI-loading (1973, 1980, 1983 and 1999). Merits (advantages) and demerits (disadvantages) of the two methods have been discussed in the literature by many authors. In this paper “a new approach” proposed by Banerjee and Dutta (1981) and employed by Sinharay and Banerjee (1985) and others (1995) will be applied to derive the basic governing equations with the inclusion of thermal loading in the dynamic case and to make an emperical study of the nonlinear thermal vibrations of a circular plate with clamped immovable edges. The basic governing equations so derived have been solved by Galerkin’s procedure. Numerical computations have been presented graphically and followed by observations and discussions.

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2. Derivation of governing field equations We consider a circular plate of radius a and subjected a temperature distribution (Nowacki, 1962) (1) T (r, z) = φ0 (r) + zφ(r), The potential energy due to bending and stretching undergoing large deflections in the absence of any external mechanical load, may be expressed in the form  2 #a  2 2 3 2ν ∂ω ∂2 ω 1 ∂ω 12 2 2 D  ∂ ω + + + 2 e1 + 2(ν − 1)e2  rdr (2) V=  2 r ∂r ∂r2 r ∂r ∂r2 h 0

3

Eh where D = flexural rigidity of the plate = 12(1−ν 2 ) , ω is Central deflection, ν is Poisson’s ratio, E is Young’s modulus, u is inplane displacement and e1 , e2 are middle surface first and second strain invariants given respectively by  2 2    ∂u u 1 ∂ω u   ∂u 1 ∂ω   + + + , e2 = +  e1 = (3)   ,  ∂r r 2 ∂z r ∂r 2 ∂r 

Putting the expressions for el and e2 into equation (2) and rearranging one gets 2 4 7 #a  2 2 2  D 2ν ∂ω ∂2 ω 1 ∂ω 12 2  ∂ ω 2 u  V= + + + 2 e¯ 1 + (1 − ν ) 2  rdr (4)  2 r ∂r ∂r2 r ∂r ∂r2 h r 0

where

2 ∂u 1 ∂ω u + (5) +ν e¯ 1 = ∂r 2 ∂r r The kinetic energy K. E. and WT , the energy contribution due to heating effect in the plate are given by 2 2      ∂ω  ρh  ∂u  rdrdt (6) K.E. =     ∂t  2 ∂t  S

h/2

α ET t WT = − ¯ 1 − z∇2 ω dxdydz [Basuli 1968] 1−ν e S −h/2 e¯ 1 NT − ∇2 ωMT rdrdθ [Expressed in polar coordinates] =−

(7)

S

where (Nowacki, 1962) αt E NT = 1−ν

#h/2 T (r, z)dz = −h/2

αt E φ0 (r)h, 1−ν

(8)

NONLINEAR THERMAL VIBRATIONS OF A CIRCULAR PLATE

αt E MT = 1−ν

#h/2 zT (r, z)dz = −h/2

αt Eh3 φ(r), 12(1 − ν)

273

(9)

4 2 Following the ‘new approach’ (1981), the term (1 − ν2 ) ur2 is replaced by 4ł ∂ω ∂r or in the expression (3) where ł is a factor depending on the Possion’ s ratio of the plate material forming the Lagrangian L = K.E. − V − WT and applying Euler’s variational equations one gets the following two differential equations in the decoupled form NT h2 C f (t)rν−1 h2 + (10) e¯ 1 = 12D 12D ρh

2 4 7 7 4 2 ∂ ω ν ∂ω ∂2 ω 6łD ∂ω ∂2 ω + ∇2 MT = 0 ∇2 ω + 2 2 − C f (t)rν−1 + D∇4 ω − 2 + 2 2 ∂r r ∂r ∂t h ∂r ∂r

(11)

where C is a constant of integration and f (t) is some unknown function of time. 3. Method of solution For free vibrations MT = 0 and NT 0, [Mazumdar, et al., (1981)] For a clamped circular plate of radius a, the deflection function ω satisfying the boundary conditions ω=0=

∂ω at r = aand u = 0 at r = 0, a ∂r

is assumed in the form r2 r4 ω = AF(t) 1 + 2P 2 + Q 4 , P = 1 = Q a a

(for clambed edges)

(12)

(13)

‘A’ being the amplitude of vibrations. Considering (5), (10) and (13) and integrating over the area of the plate one gets r2 u =

( ' NT h2 rν+1 C f (t)h2 r2ν 8A2 F 2 (t) ν+3 P2 2PQ r2 Q2 r4 + C1 r + − − + 12D ν + 1 12D 2ν ν+3 a4 a2 ν + 5 a4 ν + 7

(14)

where C1 is a constant of integration. Considering the in-plane boundary conditions u = 0 at r = 0, a, the constant C1 is eliminated and one finally gets 1 2 1 2NT ν 1−ν 192A2 F 2 (t)a−(ν+1) − + − a Dν (15) c f (t) = 2 ν+3 ν+5 ν+7 ν+1 h

274


Applying Galerking’s Procedure in equation (11) and eliminating c f (t) with the help of (15), one gets the well -known time-differential equation in the form: .. where

. F(t) + αF(t) + βF 3 (t) = 0

(16)

   2560a2 ν NDT D  320   − α= 4 3 (ν + 1)(ν + 3)(ν + 5)(ν + 7)  a ρh ' ( * 2 196608ν 256ł A D + β = 10 4 2 2 2 35 h a ρh (ν + 3) (ν + 5) (ν + 7)

(17) (18)

18) In the limiting case in the absence of temperature, the result (16) is in exact agreement with that found in the literature (1985, 1995) The solution of equation ˙ (16) with the normalized conditions F(O) = 1, F(0) = 0 is given by Nash and Modeer (1959) in the form 2

F(t) = Cn(ω t, R), where ω = α + β, R2 =

β 2(α + β)

(19)

Cn being the Jocobian Elliptic Function. The corresponding time - period is T = 4K/ω where K is the complete elliptic integral of the first kind. The usual linear time - period is given by 2π T= Ω0 √ where Ω0 is obtained from (16) by dropping the nonlinear term so that Ω0 = α. Relative time-period is given by 1 β − 2 2K 1+ T /T = π α

(20)

4. Numerical results and discussions Numerical results have been computed and presented graphically with the nonA dimensional amplitude h along the horizental axis and the relative time periods T T along the vertical axis as shown in figures 1-14 considering the following set of values αt = 1.2 × 10−5 , ν = 0.3, ł = 2ν2 [Ref. (1981)] 5. Observations and discussions

Figure-3 is the combination of Figures 1 and 2. From this figure one can observe that, if the temperature parameter remains fixed at the level of 250K, relative


275

time periods diminish with the increase of non-dimensional amplitudes fr both the cases of aspect ratios 10 and 20. Figure-6 is the combination of Figures 4 and 5. From this figure one observes that, if the temperature parameter is kept fixed at the level of 500K, reverse situation develops, i.e., time periods diminish with the increase of non-dimensional amplitudes fr both the cases of aspect ratios 10 and 20. (i) relative time-period decreases with the increase of non-dimensional amplitude when aspect ratio is 10. (ii) relative time-period increases with the increase of non-dimensional amplitude when aspect ratio is taken as 20. Similar situation also exists for higher temperature levels 750K and 1000K as shown in Figures 9 (combination of Figures 7 and 8) and 12 (combination of Figures 10 and 11). So there should be a transition phase at certain temperature level beyond 250K as aspect ratio changes from 10 to 20. Figure-13 is the combination of Figures 1,4,7 and 10 for aspect ratio 10 at temperature levels 250K, 500K, 750K, 1000K whereas Figure 14 is the combination of 2, 5, 8, and 11 for aspect ratio 20 at temperature levels 250K, 500K, 750K, 1000K. These figures represent what have been presented in Figures 1-12.

Figure 1. Shows variations of ratio ah = 10, and φ0 = 250k

T T

vs

A h

for aspect


T T

vs

A h

for aspect

276


Figure 3. Shows comparative variations of vs Ah for fixed φ0 = 250k and ah = 10, 20

T T


T T

vs

A h

for aspect


T T

vs

A h

for aspect



T T

vs

A h

for aspect


T T

vs

A h

T T

for aspect



Figure 11. Shows variations of TT vs aspect ratio ah = 20, and φ0 = 1000k

A h

T T

Figure 10. Shows variations of TT vs aspect ratio ah = 10, and φ0 = 1000k

for


Figure 13. Shows comparative variations of T vs Ah for fixed ah = 10, and φ0 = T 250k, 500k, 750k, 1000k

277

A h

for

T T

Figure 14. Shows comparative variations of T vs Ah for fixed ah = 20, and φ0 = T 250k, 500k, 750k, 1000k

278


References Banerjee, B., Dutta, S. (1981) A new Approach to an Analysis of Large Deflection of Elastic Plates, Int. J. Nonlinear Mech. 16, 47-52. Basuli, S. (1968) Large Deflection of Elastic Plates under Uniform Load and Heating, Indian J. of mech. And Maths 6, 22. Biswas, P. (1980) Large Delection of heated Orthotropic Cylindrical shallow shell, Defence Science J. 30, 87-92. Biswas, P. (1983) Nonlinear Vibrations of Heated Elastic Plates, Indian J. of Pure and Appl. Maths. 14, 119-1203. Jones, R., Mazumdar, J., Chaung, Y. K. (1980) Vibrations and Buckling of Plates at Elevated temperature, J. Solids and Structures 16, 61-70. Mondal, U. K., Biswas, P. (1999) Nonlinear Vibrations of Heated Elastic shallow Spherical under Linear and Parabolic Temparature Distributions, J. Apl. Mechs (ASME) 66, 814-815. Nowacki W. (1962) Thermoelectricity, Pergamon Press, Oxford. Pal, M. C. (1973) Static and Dynamic Nonlinear behaviour of Heated orthotropic Circular Plates, Int. J. Nonlinear Mech. 8, 480-494. Paliwal, D. N., Kanagasabapathy, H., Gupta, K. M. (1955) Vibrations of an Orthotropic Shallow Spherical Shell on a Pasternak Foundation, J. of Comp. Structure 33, 135-142. Sinharay G., Banerjee, B. (1985) A new Approach to Large Deflection Analysis of Spherical and Cylindrical Shells under Thermal Load, Mech. Res. Comm 12, 53-64.

FLEXURAL-TORSIONAL COUPLED VIBRATION ANALYSIS OF A THIN-WALLED CLOSED SECTION COMPOSITE TIMOSHENKO BEAM BY USING THE DIFFERENTIAL TRANSFORM METHOD ¨ ¨ Metin Orhan Kaya and Ozge Ozdemir ˙ Faculty of Aeronautics and Astronautics, Istanbul Technical University, Maslak, ˙Istanbul, Turkey

Abstract. In this study, a new mathematical technique called the Differential Transform Method (DTM) is introduced to analyse the free undamped vibration of an axially loaded, thin-walled closed section composite Timoshenko beam including material coupling between the bending and torsional modes of deformation, which is usually present in laminated composite beams due to ply orientation. The partial differential equations of motion are derived applying the Hamilton’s principle and solved using DTM. Natural frequencies are calculated, related graphics and the mode shapes are plotted. Key words: composite Timoshenko beam, bending torsion coupling, differential transform method

1. Introduction A composite thin-walled beam with length L, cross sectional dimension B and wall thickness d is shown in Fig. 1. The dimensions are assumed to be d ≺≺ B so the terms related to the warping stiffness and the warping inertia are small enough to be neglected. The bending motion in the Z direction, torsional rotation about the X axis and rotation of the cross section due to bending alone are represented by w (x, t), ψ (x, t) and θ (x, t), respectively. A constant axial force, P, acts Figure 1. Configuration of an axially loaded comthrough the centroid of the cross posite Timoshenko beam section which coincides with the X axis. P, is positive when it is compressive as in Fig. 1. 279 .


¨ ˙IR KAYA, OZDEM

280 2. Formulation

The governing undamped partial differential equations of motion are derived for the free vibration analysis of the beam model represented by Fig. 1. Applying the Hamilton’s principle, the following equations of motion are obtained 5 6 (1) − ρI θ¨ + EIθ + kAG w − θ + Kψ = 0 5 6 (2) − µw¨ − Pw + kAG w − θ = 0 − I s ψ¨ − P (I s /µ) ψ + Kθ + GJψ = 0

(3)

Here, µ = ρA is the mass per unit length; I s is the polar mass moment of inertia; K and kAG are flexure-torsion coupling rigidity and shear rigidity of the beam, respectively. The boundary conditions at x = L and x = 0 for Eqs. (1-3) are as follows 6 5 (4) EIθ + kψ δθ = 0 5 6, (5) −Pw + kAG w − θ δw = 0 , (6) − (PI s /µ) ψ + Kθ + GJψ δψ = 0 A sinusoidal variation of w (x, t) ,ψ (x, t) and θ (x, t) with a circular natural frequency ω is assumed and the functions are approximated as w (x, t) = W (x) eiωt ,

ψ (x, t) = ψ (x) eiωt ,

θ (x, t) = θ (x) eiωt

(7)

The following nondimensional parameters can be used for simplification. ξ = x/L,

W (ξ) = W/L,

r2 = I/AL2 ,

()∗ = d()/dξ,

()´ = d()/Ldξ

(8)

Substituting Eqs.(7) and (8) into Eqs.(1-3), the dimensionless equations of motion are obtained as follows A1 θ¯ ∗∗ + A2 θ¯ + A3 W ∗∗ + A4 ψ¯ ∗∗ = 0

(9)

B1 W ∗∗ + B2 W + B3 θ¯ ∗ = 0

(10)

C1 ψ¯ ∗∗ + C2 ψ¯ + C3 θ¯ ∗∗ = 0

(11)

where the nondimensional coefficients are given by µL4 r2 ω2 kAGL2 kAGL2 K − , A3 = , A4 = , (12) EI EI EI EI P L2 µ 2 PI s I s L2 2 K , B2 = ω , B3 = −1, C1 = 1 − , C2 = ω , C3 = B1 = 1 − kAG kAG GJµ GJ GJ A1 = 1, A2 =

COMPOSITE TIMOSHENKO BEAM

281

3. The differential transform method The differential transform method is a transformation technique based on the Taylor series expansion and is used to obtain analytical solutions of the differential equations. In this method, certain transformation rules are applied to the original functions. As a result, the differential equations and the boundary conditions of the system are transformed into a set of algebraic equations and the solution of these algebraic equations gives the desired solution of the problem. A function f (x) , which is analytic in a domain D, can be represented by a power series with a center at xo , any point in D. The differential transform of the function is given by 1 dk f (x) F [k] = (13) k! dxk x=x0 where f (x) is the original function and F [k] is the transformed function. The inverse transformation is defined as f (x) =

∞

(x − x0 )k F [k]

(14)

k=0

Combining Eqs. (13) and (14) and expressing f (x) by a finite series, we get m (x − x0 )k dk f (x) f (x) = (15) k! dxk x=x0 k=0 Here, the value of m depends on the convergence of the natural frequencies (Ho and Chen, 1998). Theorems frequently used in the transformation procedure are introduced in Table I and Table II (Ozdemir and Kaya, 2005). 4. Formulation with DTM Applying DTM to Eqs.(9-11),the following transformed equations of motion are obtained. Here, we use ψ and θ instead of ψ and θ. A1 (k + 2) (k + 1) θ [k + 2] + A2 θ [k] + A3 (k + 1) W [k + 1] + A4 (k + 2) (k + 1) ψ [k + 2] = 0

(16)

B1 (k + 2) (k + 1) W [k + 2] + B2 W [k] + B3 (k + 1) θ [k + 1] = 0

(17)

C1 (k + 2) (k + 1) ψ [k + 2] + C2 ψ [k] + C3 (k + 2) (k + 1) θ [k + 2] = 0

(18)

¨ ˙IR KAYA, OZDEM

282

TABLE I.

Basic theorems of DTM

Original Function

Transformed Function

f (x) = g (x) + h (x)

F [k] = G [k] + H [k]

f (x) = λg (x)

F [k] = λG [k] F [k] = kl=1 G [k − l] H [l]

f (x) = g (x) h (x) f (x) = g (x) /h (x)

F [k] =

G [k + n]     0 if k n F [k] = δ (k − n) =    1 if k = n

f (x) = xn

TABLE II.

Basic theorems of DTM

x=0 f (0) = 0 d f (0)/dx = 0

F [0] = 0 F [1] = 0

(k+n)! k!

f (1) = 0

x=1 ∞

k=1

∞

d f (1)/dx = 0

d2 f (0)/dx2 = 0

F [2] = 0

d2 f (1)/dx2 = 0

d3 f (0)/dx3 = 0

F [3] = 0

d3 f (1)/dx3 = 0

k=1

∞

k=1

∞

k=1

F [k] = 0

kF [k] = 0

k (k − 1) F [k] = 0

k (k − 1) (k − 2) F [k] = 0

Applying DTM to Eqs.(4-6), the boundary conditions are obtained as follows ξ=0 ξ=1

⇒

⇒

θ [0] = W [0] = ψ [0] = 0

(19)

(k + 1) θ [k + 1] + A4 (k + 1) ψ [k + 1] = 0

(20)

5. Results and discussion An illustrative example, taken from (Baner jee, 1998), is solved and the results are compared with the ones in literature and the mode shapes are plotted. Variation of the first five natural frequencies of the above example with respect to the axial force is introduced in Table III. Here, the superscript γ represents the natural frequencies calculated when coupling is present (K 0). In this table, it is noticed that the natural frequencies decrease as the axial force varies from tension to compression. Furthermore, it is seen that the coupled natural frequencies are lower than the uncoupled ones. The effects of the axial force P and the rotary inertia parameter, r , on the first four natural frequencies are introduced in Fig. 2.

COMPOSITE TIMOSHENKO BEAM

283

Here, the solid and the dashed lines represent the frequencies of Timoshenko and Euler beams, respectively. As expected,the natural frequencies decrease with the increasing rotary inertia parameter, because Timoshenko effect, which is more dominant on higher modes, decreases the natural frequencies. However, since the fourth mode is torsion, r makes only a slight change in the fourth mode. Mode shapes of the beam under the effect of the compressive axial force, (P = 7.5) and coupling effect are introduced in Fig. 3. As it is seen, the first three normal modes are bending modes while the fourth normal mode is the fundamental torsion mode.

Figure 2.

Effects of the axial force and the rotary inertia parameter on the natural frequencies

Figure 3.

Normal mode shapes of the composite beam with bending-torsion coupling

¨ ˙IR KAYA, OZDEM

284

TABLE III. Natural frequency variation with the axial force NaturalFrequencies P = −7.5

P=0

P = 7.5

Present

Li et al.

Presentγ

Baner jeeγ

Presentγ

Baner jeeγ

Present

Li et al.

40.975

40.970

37.106

37.100

30.747

30.750

28.064

28.060

224.259

224.250

197.672

197.700

189.779

189.800

210.162

210.160

598.668

598.660

525.665

525.600

518.791

518.800

586.519

586.510

647.595

647.590

648.495

648.600

648.269

648.300

647.228

647.220

1125.711

1125.710

992.878

-

986.199

-

1113.950

-

References Banerjee J. R. (1998) Free Vibration of Axially Loaded Composite Timoshenko Beams Using the Dynamic Stiffness Matrix Method, Computers and Structures 69 197-208. Ho S. H., Chen C. K. (1998) Analysis of General Elastically End Restrained Non-Uniform Beams Using Differential Transform, Applied Mathematical Modeling 22 219-234. Li J., Shen R., Hua H., Jin X. (2004) Bending-Torsional Coupled Vibration of Axially Loaded Composite Timoshenko Thin-Walled Beam With Closed Cross-Section, Composite Structures 64 23-35. Ozdemir O., Kaya M. O. (2005) Flapwise Bending Vibration Analysis of a Rotating Tapered Cantilevered Bernoulli-Euler Beam by Differential Transform Method, Journal of Sound and Vibration (in press).

ESTIMATION OF MICROSTRETCH ELASTIC MODULI BY THE USE OF VIBRATIONAL DATA Ahmet Kırıs¸1 and Esin ˙Inan2 1 Faculty of Science and Letters, Istanbul ˙ ˙ Technical University, 34390, Istanbul, Turkey 2 Faculty of Arts and Sciences, I¸ ˙ sık University, 34398, Istanbul, Turkey

Abstract. In the present work, a nonlinear wave theory is used for the estimation of the material properties of a “microstretch” medium. For this purpose a thin plate is considered and triplicate Chebyshev polynomial series are used as admissible functions to ensure the satisfaction of geometric boundary conditions of the plate. The Ritz technique is applied to derive the frequency equation of the microstretch plate and an optimization procedure is performed by minimising the least square “distance” between computed natural frequencies from the energy method and measured natural frequencies. To realize the optimization procedure, a genetic algorithm is used to estimate the elastic moduli of microstretch medium. Key words: microstretch, plate vibration, Ritz technique, genetic algorithm, inverse problem

1. Introduction The linear theory of elasticity is adequate for the vibration analysis of plates made of hard materials like steel. But it is well known that material response to external stimuli depends on the motions of its inner structures, so the linear theory of elasticity is unable to explain the behavior of many materials having complex microstructure like composite materials, polymers and porous media, etc. The influence of material microstructure results in the development of new type of waves which is not found in the classical theory of elasticity for the case of vibrations characterized by high frequencies and small wave-lengths. Eringen’s microstretch elasticity which is included microstructural rotations and expansions provides a good model for such materials and reveal some high frequency branches of wave spectrum which is not appears in the classical theory of elasticity (Eringen, 1999). The wave propagation in materials modeled with microstretch theory is investigated by (Singh and Kumar, 1998), (Tomar and Garg, 2005) and (Inan, 1990). In this study, first we investigated the vibration of microstretch plates to estimate the overall material properties of microstretch plates. Here we used Chebyshev polynomials as admissible functions which satisfy the geometric boundary 285 .


286

KIRIS¸, ˙INAN

conditions of plate and have been used successfully by (Zhou et al., 2002) for vibration analysis of three dimensional rectangular plates in the linear theory of elasticity. The Ritz technique which is most commonly used method has some special advantages such as high accuracy. Second, an optimization procedure is performed by minimising the least square “distance” between computed natural frequencies from the energy method and measured natural frequencies obtained from experiments, and finally, we used a genetic algorithm to solve the inverse problem for identification of the overall properties of microstretch plate. Genetic Algorithms (GAs) are evolutionary programs that manipulate a population of individuals and based on the evolution theory of Darwin to explain creation of life. He proposed that the nature of living creatures changed over the years to result in stronger specimens. The stronger specimens would then dominate. An initial population of individuals (solutions) is generated randomly and these then undergo evolution by means of reproduction, crossover and mutation of individuals until a suitable solution is found. GAs have acquired great success in inverse problems and GAs are widely adopted as optimization techniques (Mota Soares et al., 1993) and (Chiroiu and Munteanu, 2002). In the present work, we had to limit Chebyshev polynomials with 5x5x1 for microstretch case while we could use 10x10x2 polynomials for classical case because of the computational difficulties and incapacity of the computer memory. In spite of this, it is shown that the applied GA has great success for the determination of the overall properties of the microstretch plate. 2. Kinetic and potential energies of a microstretch plate The linearized constitutive equations for a microstretch elastic solid are given by (Eringen, 1999) as tkl = λ εmm δkl + (µ + κ) εkl + µ εlk + λ0 ψ δkl , mkl = α γmm δkl + β γkl + γ γlk , mk = a0 ψ,k

(1)

where tkl is the stress tensor, mkl is the couple stress tensor and mk is the microstretch vector, λ and µ are Lamé constants, κ, α, β, and γ are the micropolar constants and λ0 , λ1 and a0 are the microstretch constants, and ρ is the mass density. kl , γkl and γk are the linear strain measures of microstretch elasticity with the following geometric relations εkl = ul,k + ∈lkm φm ,

γkl = φl,k ,

γk = 3ψ,k .

(2)

The linear elastic strain energy density of an isotropic microstretch medium is given by 1, W = tkl εkl + mkl γlk + mk γk + (s − t) ψ (3) 2

287

ESTIMATION OF MICROSTRETCH ELASTIC MODULI

where s = mkk and t = tkk and the kinetic energy per unit mass for an isotropic microstretch medium is given by K=

1 1 3 ˙ ρ u˙ k u˙ k + ρ j φ˙ k φ˙ k + ρ j ψ˙ ψ. 2 2 2

(4)

We consider a rectangular microstretch plate of thickness h, length a and width b. The Cartesian coordinate system is located at the middle of the plane denoted by Ω and the z axis normal to the plane. Substituting the constitutive equations (1) and the geometric relations (2) into (3-4), we can write the linear strain energy V and the kinetic energy T of the rectangular microstretch plate in integral form as a/2

b/2

h/2

V=

W dz dy dx

−a/2 −b/2 −h/2 a/2

b/2

h/2

= 12 −a/2 −b/2 −h/2

λ Λ21 + (2µ + κ) Λ2 + 2µ Λ3 + (µ + κ) Λ4 + α Λ25 +(β + γ) Λ6 + 2βΛ7 + γ Λ8 + a0 Λ9 + 2 λ0 ψ Λ1 + λ1 ψ2 dz dy dx

(5)

and T=

ρ 2

a/2

b/2

h/2

*

−a/2 −b/2 −h/2

∂ u1 2 ∂t

+3 j

+

∂ u 2

∂ ψ 2 ∂t

2

∂t

+

∂ u 2 + 3

∂t

+ j

*

∂ φ1 2 ∂t

+

∂ φ 2 2

∂t

+

∂ φ 2 + 3

∂t

dz dy dx (6)

where Λ1 = ε11 + ε22 + ε33,

Λ2 = ε211 + ε222 + ε233 ,

Λ4 = ε212 + ε213 + ε221 + ε223 + ε231 + ε232 , 2 + γ2 + γ2 , Λ6 = γ11 22 33

Λ3 = ε12 ε21 + ε13 ε31 + ε23 ε32,

Λ5 = γ11 + γ22 + γ33,

Λ7 = γ12 γ21 + γ13 γ31 + γ23 γ32,

2 + γ2 + γ2 + γ2 + γ2 + γ2 , Λ8 = γ12 13 21 23 31 32

Λ9 = ψ2,1 + ψ2,2 + ψ2,3 .

(7)

3. Ritz Method with Chebyshev polynomials Assuming a harmonic-time dependence, the periodic displacement, microrotation and microstretch components of the microstretch plate undergoing free vibration can be written in terms of the amplitude functions as follows {u(x, y, z, t), φ(x, y, z, t), ψ(x, y, z, t)} = {U(x, y, z, t), Φ(x, y, z, t), Ψ(x, y, z, t)} ei ω t (8)

KIRIS¸, ˙INAN

288

where ω denotes the natural frequency of the microstretch plate. For convenience and simplicity, we introduce the following non-dimensional parameters; ξ=

2x , a

η=

2y , b

ζ=

2z h

(9)

into the potential and kinetic energies (5) and (6), but we do not give their nondimensional forms in here for the shake of brevity. Each amplitude functions of the equation (8) can be written in the form of triplicate series of Chebyshev polynomials which ensure the satisfaction of the geometric boundary conditions as {U1 (ξ, η, ζ), U2 (ξ, η, ζ), U3 (ξ, η, ζ), Φ1 (ξ, η, ζ), Φ2 (ξ, η, ζ), Φ3 (ξ, η, ζ), Ψ(ξ, η, ζ)} =

∞ ∞ ∞ 2

3 Ai jk , Bi jk , Ci jk , A¯ i jk , B¯ i jk , C¯ i jk , A¯ i jk Pi (ξ) P j (η) Pk (ζ)

i=1 j=1 k=1

(10) where Pi (χ) is the one-dimensional i Chebyshev polynomial which can be written in terms of cosine functions as follows, , Pi (χ) = cos (i − 1)arccos(χ) . (11) th

Substituting the series expansions (10) into the nondimensional potential and kinetic energy expressions and using the results in the maximum energy functional of the microstretch plate defined as Π = Vmax −T max and minimizing the functional Π with respect to the coefficients of the Chebyshev polynomials, i.e. ∂Π ∂Ai jk ∂Π A¯ i jk

= 0,

∂Π ∂Bi jk

= 0,

∂Π ∂Ci jk

= 0,

∂Π ∂A¯ i jk

= 0,

∂Π ∂ B¯ i jk

= 0,

∂Π ∂C¯ i jk

= 0, (12)

= 0,

(i, j, k = 1, . . . , ∞)

can be written as the following governing eigenvalue equation in matrix form    [KU1 U3 ] 0 [KU1 Φ2 ] [KU1 Φ3 ] [KU1 Ψ ]   [KU1 U1 ] [KU1 U2 ]       [KU U ]T [KU U ]  [KU2 U3 ] −[KU1 Φ2 ] 0 [KU2 Φ3 ] [KU2 Ψ ]   1 2 2 2       T T   [KU3 U3 ] −[KU1 Φ3 ] −[KU2 Φ3 ] 0 [KU3 Ψ ]   [KU1 U3 ] [KU2 U3 ]     T T 0 −[KU1 Φ2 ] −[KU1 Φ3 ] [KΦ1 Φ1 ] [KΦ1 Φ2 ] [KΦ1 Φ3 ] 0        [K  T  0 −[KU2 Φ3 ]T [KΦ1 Φ2 ]T [KΦ2 Φ2 ] [KΦ2 Φ3 ] 0    U1 Φ2 ]      [KU Φ ]T [KU Φ ]T   0 [KΦ1 Φ3 ]T [KΦ2 Φ3 ]T [KΦ3 Φ3 ] 0   1 3 2 3     T T T  [KU Ψ ] [KU2 Ψ ] [KU3 Ψ ] 0 0 0 [KΨΨ ] 1

ESTIMATION OF MICROSTRETCH ELASTIC MODULI

  0 0 0 0 0   [M1 ] 0    0 [M ] 0   0 0 0 0  1        0    ] 0 0 0 0 0 [M 1         2   0 0 [M2 ] 0 0 0  −Ω  0     0 0 0 [M2 ] 0 0   0     0  0 0 0 0 [M ] 0   2    0 0 0 0 0 0 [M3 ] 

3 2  Ai jk       2 3   0            B     i jk         0 2 3                 C i jk     0    2 3      ¯     0 = Ai jk       2 3            0 ¯     B     i jk        2 3     0         ¯     C i jk       3 2   0    A¯  i jk

289

(13)

√ where M1 = 14 D0,0 G0,0 H 0,0 , M2 = jM1 , M3 = 3 jM1 and Ω = a ω ρ. The ii j j kk elements of the stiffness sub-matrices and mass sub-matrices are given by 1 0,0 0,0 1,1 D G H , α21 i i j j k k a κ α1 0,0 0,0 1,0 1,0 0,1 0,1 0,0 1,0 0,0 KU1 U2 = α1 (λD i i G j j Hk k + µD i i G j j Hk k ), [KU1 φ2 ] = − 2 2 α2 D i i G j j Hk k , G0,0 H 0,1 + µD0,1 G0,0 H 1,0 ), G0,0 H 0,0 , [KU1 U3 ] = αα12 (λD1,0 [KU1 ψ ] = a2 λ0 D1,0 ii j j kk ii j j kk ii j j kk α2 G1,0 H 0,0 , G1,0 H 0,1 + µD0,0 G0,1 H 1,0 ), [KU2 U3 ] = α12 (λD0,0 [KU1 φ3 ] = a2 2κ α1 D0,0 i i j j ii j j kk k k ii j j kk 1 0,0 0,0 1,1 2 1,1 0,0 0,0 [KU2 U2 ] = α1 (λ + 2µ + κ)D i i G j j Hk k + (µ + κ) α2 D i i G j j Hk k 1 aκ a G0,0 H 0,0 , [KU2 φ3 ] = − 2 2 D1,0 G0,0 H 0,0 , [KU2 ψ ] = 2 λ0 α1 D0,0 G1,0 H 0,0 , +(µ + κ)D1,1 ii j j kk ii j j kk ii j j kk α G0,0 H 1,0 [Kφ1 φ3 ] = α1 (αD1,0 G0,0 H 0,1 + βD0,1 G0,0 H 1,0 ), [KU3 ψ ] = a2 λ0 αα12 D0,0 ii j j kk ii j j kk 2 ii j j kk 1 2 0,0 1,1 1,1 0,0 0,0 0,0 [KU3 U3 ] = α1 α2 (λ + 2µ + κ)D i i G j j Hk k + (µ + κ)D i i G j j Hk k 2 a2 G0,0 H 0,0 , G0,0 H 0,0 + 4 κ D0,0 G0,0 H 0,0 [Kφ1 φ1 ] = (α + β + γ)D1,1 +(µ + κ)D1,1 ii j j kk ii j j kk ii j j kk 1 G1,1 H 0,0 + 2 D0,0 G0,0 H 1,1 ), [Kφ1 φ2 ] = α1 (αD1,0 G0,1 H 0,0 + βD0,1 G1,0 H 0,0 ), +γ α21 (D0,0 ii j j kk ii j j kk ii j j kk α1 i i j j k k 0,0 1,1 0,0 G0,0 H 0,0 + (µ + κ)α2 [KU1 U1 ] = (λ + 2µ + κ)D1,1 1 D i i G j j Hk k + ii j j kk

1 G1,1 H 0,0 + γ 2 D0,0 G0,0 H 1,1 + γ D1,1 G0,0 H 0,0 [Kφ2 φ2 ] = α21 (α + β + γ)D0,0 ii j j kk ii j j kk ii j j kk α 2 G0,0 H 0,0 , + a4 κ D0,0 i i j j k k [Kφ3 φ3 ] = α21 α12 (α 2

1

α21 0,0 1,0 0,1 α2 (αD i i G j j H k k

G0,1 H 1,0 , + βD0,0 i i j j k k G0,0 H 1,1 + γ D0,0 G1,1 H 0,0 + γ D1,1 G0,0 H 0,0 + β + γ)D0,0 ii j j kk ii j j kk ii j j kk α2 a2 2 0,0 0,0 1,1 0,0 0,0 0,0 0,0 1,1 0,0 + 4 κ D i i G j j Hk k , [Kψψ ] = a0 D i i G j j Hk k + α1 D i i G j j Hk k + α12 D0,0 G0,0 H 1,1 ii j j kk

[Kφ2 φ3 ] =

2

2

0,0 + a4 λ1 Dii0,0G0,0 j j Hkk ,

here the summation convention is not valid for indices with bar and where, D s, s¯ = ii

1

4

−1

(s, s¯=0, 1,

7 s¯ d s Pi (ξ) d Pi (ξ) dξ, dξ s dξ s¯

G s, s¯ = jj

i,i, j, j, k,k=1,...,∞).

1

−1

4

7 d s P j (η) d s¯ P j (η) dη, dη s dη s¯

1 H s, s¯ = k k −1

4

7 s¯ d s Pk (ζ) d Pk (ζ) dζ, dζ s dζ s¯

(14)

290

KIRIS¸, ˙INAN

4. Genetic algorithm and results To obtain the material properties of the microstretch plate, we constructed an optimization problem which minimizes the least square distance between the experimental natural frequencies and computed natural frequencies calculated from the eigenvalue problem. We assume that the first 20 experimental frequencies are (Chiroiu and Munteanu, 2002): {96.54, 115.33, 202.67, 365.65, 432.97, 489.11, 543.3, 774.92, 995.87, 1076.44, 1320.49, 1341.81, 1456.47, 1559.23, 1678, 1805.4, 1988.3, 2290.45, 2654, 2876} (Hz). The constructed optimization problem is solved by using a GA. We assumed that micro inertia and the density of the microstretch plate is given as j = 6.25 10−7 m2 , ρ = 1189 kg/m3 and the plate dimensions are 0.160 × 0.240 × 0.076m (length, width, height). We have nine material constants shown by {λ, µ, κ, α, β, γ, a0 , λ0 , λ1 } for a microstretch body and each of them can be considered as a property of an individual. So, we can consider that i. individual corresponds to a solution like {λ, µ, κ, α, β, γ, a0 , λ0 , λ1 }i . A population of individuals are called as “Generation”. We assume the population of each generation is 300. After the first 516 generations with the crossover probability % 75 and the mutation probability % 25, the material properties of the microstretch plate are found as λ = 7459MPa, µ = 6175MPa, κ = 34.4MPa, α = 2.3GN, β = 4.3GN γ = 6.1GN, λ0 = 3.1GN, λ1 = 4GN, a0 = 0.2GN. During computations, a P4 1.4GHz computer with 512MB RAM was used and the elapsed time was 453.42 minutes. References Chiroiu V. and Munteanu L. (2002) Estimation of micropolar elastic moduli by inversion of vibrational data, Complexity International 9, 1-10. Eringen, A. C. (1990) Microcontinuum Field Theories: Foundations and Solids, Springer-Verlag. Inan, E. (1990) Elastic waves in damaged material, In Proc. of Fourth ICOVP-99. A, 149-160. Calcutta, India. Mota Soares, C. M., Moreira F. M., Araujo A., and Pedersen P. (1993) Identification of material properties of composite plate specimens, Composite Structures 25, 277-285. Singh, B. and Kumar R. (1998) Wave propagation in a generalized thermo-microstretch elastic solid, Int. J. of Engineering Science 36, 891-912. Tomar, S. K. and Garg M. (2005) Reflection and transmission of waves from a plane interface between two microstretch solid half-spaces, Int. J. of Engineering Science 43, 139-169. Zhou, D., Cheung, Y. K., and Au F. T. K. (2002) 3-D vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method, Int. J. of Solids and Structures 39, 6339-6353.

VIBRATIONS OF A CIRCULAR MEMBRANE SUBJECTED TO A PULSE ¨ Onder Korfalı and ˙Ismail Burak Parlak Galatasaray University, ˙Istanbul, Turkey

Abstract. In this paper we have studied the vibrations of a circular membrane subjected to a pressure pulse. The exact solution of the problem is given in terms of Bessel functions using a Laplace transformation. Key words: circular membrane, pressure pulse

1. Introduction For the vibrations of a circular membrane of radius a we must solve the twodimensional wave equation (Dettman, 1969) 1 ∂2 ξ (1) c2 ∂t2 in the interior of the membrane; that is, 0 ≤ r < a, 0 ≤ θ ≤ 2π, subject to the boundary condition ξ = 0 on the edge, r = a. We shall specify zero initial ˙ θ, 0) = 0. With the displacement, ξ(r, θ, 0) = 0 and zero initial velocity, that is, ξ(r, introduction of polar coordinates, the wave equation becomes ∇2 ξ =

∂2 ξ 1 ∂ξ 1 ∂2 ξ 1 ∂2 ξ + + = ∂r2 r ∂r r2 ∂θ2 c2 ∂t2

(2)

For a circular membrane ∂ξ ∂2 ξ =0 , =0 (3) ∂θ ∂θ2 Consequently we have ξ = ξ(r, t) together with ξ(a, t) = 0. Using Eqs. 3 the Eq. 2 reduces to 2 ∂2 ξ Z 1 ∂ξ 2 ∂ ξ )+ = c ( + 2 2 r ∂r ρ ∂t ∂r where Z is force per unit area and

291 .


(4)

292

KORFALI, PARLAK

T (5) ρ T being the tension in the membrane and ρ, the density of the membrane material. With the introduction of c2 =

Z = P0 δ(t)

(6)

where δ(t) is impulse function we have 2 ∂2 ξ P0 δ(t) 1 ∂ξ 2 ∂ ξ )= − c ( + 2 2 r ∂r ρ ∂t ∂r

(7)

∂ξ )t=0 = 0 ∂t

(8)

with initial conditions ξ(r, 0) = 0

and

(

and B.C. ξ(a, t) = 0

(9)

2. Laplace transformation ¯ s) and using Eqs. 8 we have With L{ξ(r, t)} = ξ(r, ∂2 ξ ∂ξ ∂2 ξ 2¯ ¯ s) ) } = s and L{ } = s2 ξ(r, ξ(r, s) − sξ(r, 0) − ( t=0 ∂t ∂t2 ∂t2 Using Eq. 10 together with L{δ(t)} = 1, Laplace transformation of Eq. 7 is L{

¯ s) s2 ¯ s) 1 ∂ξ(r, ∂2 ξ(r, ¯ s) − P0 ] = 0 + − 2 [ξ(r, 2 r ∂r ∂r c ρs2 Now using the transformation ¯ s) − P0 y¯ (r, s) = ξ(r, ρs2 we obtain

(10)

(11)

(12)

∂2 y¯ (r, s) 1 ∂¯y(r, s) s2 − 2 y¯ (r, s) = 0 + (13) r ∂r ∂r2 c which is Bessel’s equation of zeroth order with argument ir( cs ). We know that ¯ s) is finite, consequently y¯ (0, s) must be finite too. The ξ(0, t) is finite, hence ξ(0, solution of differential Eq. 13 which is finite at r = 0 is s y¯ (r, s) = AJ0 (ir ) (14) c

VIBRATIONS OF A CIRCULAR MEMBRANE SUBJECTED TO A PULSE 293

¯ s) = 0. Using To determine the constant A we have ξ(a, t) = 0, consequently ξ(a, Eq. 12 ¯ s) − P0 = − P0 y¯ (a, s) = ξ(a, ρs2 ρs2

(15)

Using Eq. 15 together with Eq. 14 s P0 y¯ (a, s) = AJ0 (ia ) = − 2 c ρs

or

A=−

P0 2 ρs J0 (ia cs )

(16)

Substituting this into Eq. 14 we have s P0 J0 (ir c ) y¯ (r, s) = − 2 ρs J0 (ia cs )

(17)

Now using Eq. 12 together with Eq. 17 we obtain ¯ s) = ξ(r,

P0 J0 (ia cs ) − P0 J0 (ir cs )

(18)

ρs2 J0 (ia cs )

3. Inverse Laplace transformation ¯ s). The inNow we will find the inverse transformation (Spiegel, 1965) of ξ(r, ¯ s). ¯ s) is determined as the sum of residues of e st ξ(r, verse transformation of ξ(r, Therefore we have to calculate the residues of ¯ s) = [ e st ξ(r,

P0 J0 (ia cs ) − P0 J0 (ir cs ) ρs2 J0 (ia cs )

]e st

(19)

This has a second order pole at s = 0 and simple poles at the roots of equation J0 (ia

sn )=0 c

(20)

Putting iasn = λn c

where

n = 1, 2, . . . , ∞

(21)

we have simple poles iλn c λn c =− ia a where λn ’s are determined from J0 (λn ) = 0. sn =

The residue R0 at the double pole s = 0 is:

(22)

294

KORFALI, PARLAK s s d 2 P0 J0 (ia c ) − P0 J0 (ir c ) st {s [ ]e } s→0 ds ρs2 J0 (ia cs )

R0 = lim

(23)

Now taking the derivative with respect to s and using the well-known formules (Mathews and Walker, 1965; Selby, 1974) J0 (x) = −J1 (x) ,

J0 (0) = 1

J1 (0) = 0

and

(24)

we find that each of the terms within brackets below vanishes.

R0 = {[

P0 J0 (ia cs ) − P0 J0 (ir cs ) st ]te } s=0 ρJ0 (ia cs )

+{ +{

[− iac P0 J1 (ia cs ) + P0 irc J1 (ir cs )]ρJ0 (ia cs )e st ρ2 J02 (ia cs ) [P0 J0 (ia cs ) − P0 J0 (ir cs )]ρ iac J1 (ia cs )e st ρ2 J02 (ia cs )

} s=0

} s=0

(25)

Consequently R0 = 0. Now we will calculate the residues at simple poles sn . Residue at s = sn is lim (s − sn )[

s→sn

P0 J0 (ia cs ) − P0 J0 (ir cs ) ρs2 J0 (ia cs )

]e st

(26)

Using Eq. 20 this reduces to = lim −(s − sn )[ s→sn

P0 J0 (ir cs ) ρs2 J0 (ia cs )

]e st

P0 e st J0 (ir cs ) −(s − sn ) ] [ lim ] = [ lim s→sn J0 (ia s ) s→sn ρs2 c

(27) (28)

= 00

and applying Hospital’s rule P0 J0 (ir scn ) sn t P0 J0 (ir scn )e sn t 1 e ] = iaρ = [ ia sn ][ sn 2 ρs2n c J1 (ia c ) c sn J1 (ia c )

(29)

Using ia this reduces to

sn = λn c

and

ir

sn r = λn ( ) c a

(30)

VIBRATIONS OF A CIRCULAR MEMBRANE SUBJECTED TO A PULSE 295

=

P0 J0 (λn ar )e−

iλn c a t

iaρ i2 λ2n c2 J1 (λn ) c a2

=

iP0 J0 (λn ar )e−

iλn c a t

(31)

ρ ac λ2n J1 (λn )

and passing to summation ξ(r, t) =

iλn c ∞ iP0 J0 (λn r )e− a t

a

(32)


n=1

or in trigonometric form ξ(r, t) =

∞ iP0 J0 (λn r )[cos(− λn ct ) + i sin(− λn ct )] a

n=1

a c 2 ρ a λn J1 (λn )

a

(33)

4. Conclusion Taking finally the real part of Eq. 33 we have the series solution of the problem ξ(r, t) =

∞ P0 J0 (λn ar ) sin( λnact ) n=1


(34)

which satisfies Eq. 8 and Eq. 9, the initial conditions and boundary condition respectively. Using the initial condition, ( ∂ξ ∂t )t=0 = 0 we have from Eq. 34 (

∞ r P0 J0 (λn a ) ∂ξ )t=0 = =0 ∂t ρ n=1 λn J1 (λn )

(35)

and consequently the important result ∞ J0 (λn ar ) ≡0 λ J (λ ) n=1 n 1 n

for

0≤r≤a

(36)

where λn ’s (n = 1, 2, . . . , ∞) are the roots of J0 (λ) = 0. For r = 0 using J0 (0) = 1 this simplifies to ∞ n=1

1 =0 λn J1 (λn )

(37)

another property of Bessel functions. Taking the first term of rapidly converging series of Eq. 34 we have the approximate solution

296

KORFALI, PARLAK

ξ(r, t)

P0 J0 (λ1 ar ) sin( λ1act ) ρ ac λ21 J1 (λ1 )

(38)

which satisfies the initial condition, ξ(r, 0) = 0 and the boundary condition, ξ(a, t) = 0 but not the other initial condition, ( ∂ξ ∂t )t=0 = 0. Now inserting λ1 2.40 and J1 (λ1 ) 0.5191 from (Selby, 1974) into Eq. 38 and rearranging we have ξ(r, t)

P0 J0 (2.40 ar ) sin( 2.4ct a ) ρ ac (2.40)2 (0.5191)

(39)

and finally ξ(r, t)

P0 J0 (2.4 ar ) sin( 2.4ct a ) c 3ρ a

(40)

as the approximate solution of the problem. 5. Symbols J0 J1 P0 ξ

:Zeroth order Bessel function. :First order Bessel function. :Magnitude of the pressure pulse. :Vertical displacement of the membrane.

References Dettman, J. W. (1969) Mathematical Methods in Physics and Engineering, McGraw-Hill Book Company. Mathews, J. and Walker, R. L. (1965) Mathematical Methods of Physics, W. A. Benjamin, Inc. Selby, S. M. (1974) Standard Mathematical Tables, CRC Press. Spiegel, M. R. (1965) Theory and Problems of Laplace Transforms, Schaum’s Outline Series, McGraw-Hill Book Company.

A METHOD OF DISCRETE TIME INTEGRATION USING BETTI’ S RECIPROCAL THEOREM Nahit Kumbasar ˙ Faculty of Civil Engineering, Istanbul Technical University, Maslak 34469, ˙Istanbul, Turkey

Abstract. There are several known algorithms for the numerical integration of the equation of motions in structural dynamics. However, efforts have been made recently to obtain more efficient and accurate methods of numerical integration. This paper aims to obtain a simple and efficient algorithm, based on Betti’s reciprocal theorem. Key words: Betti’s reciprocal theorem

1. Introduction The differential equation of motion for a multi-degree of freedom system is of the form ¨ + CX ˙ + KX = F MX (1) in which M, C, K and F represent mass, damping, stiffness and force in matrix form. M, C and K are constant for linear systems. Superscript dots denote differentiation with respect to time. This equation may be uncoupled for some types of damping, using modal analysis, into a series of differential equations that are similar to the equation for single degree of freedom system: m x¨ + c x˙ + kx = f

(2)

Several algorithms are known for the numerical integration of equation (1). A survey of these direct time integration methods is given by (Dokanish and Subbaraj, 1989a,b). More elaborate methods have also been published such as those by (Chang, 1997) and (Golly and Amer, 1999). In general, these algorithms consider displacement, velocity (and some of them acceleration) as specified at time t and calculate their values at time t + ∆t. Most of these methods estimate the variation of the displacement in cubic form, such as the well known Newmark-β and Wilson-θ methods (Wilson et al., 1973). Some of these methods use the weighted residual method instead of the equation of motion itself (1). The requirements of an algorithm for discrete time integration are stated by (Hilbert et al., 1977) as: 297 .


298

KUMBASAR

1. To be unconditionally stable when applied to linear problems. 2. To possess a numerical dissipation that can be controlled by a parameter other than the time step. In particular no numerical dissipation should be possible. 3. To possess a numerical dissipation that does not affect lower modes too strongly. These requirements are essentially needed to get rid of spurious effects for structural systems analysed using finite elements. However, they decrease the accuracy of the integration procedures. The parameters β and θ in Newmark and Wilson methods may be chosen to satisfy the above conditions. In this paper, a simple and effective procedure, obtained using Betti’s reciprocal theorem, is presented. Since the velocity and the acceleration are not included in this procedure, the memory size needed is less than those of the known methods. Numerical accuracy and computing time are also better, but the proposed method is only conditionally stable. It is shown that an unconditionally stable procedure may be found by including velocity and acceleration, which gives similar results as the Newmark method. 2. General description Numerical integration of differential equation of motion is mostly performed by the use of algorithms based on the finite difference methods. (Cakiroglu, 1979) proposed an improved finite difference method for structural systems, making use of the Betti’s reciprocal theorem. Differential equation of motion for a single degree of freedom system may be interpreted as the differential equation of equilibrium for a shear beam on elastic foundation. Displacements of a shear beam are governed by the equation GAu = −p(x)

(3)

where GA is the shear rigidity of the beam, u is the deflection, p(x) represents the external load and ( ) represents differentiation with respect to the x coordinate. Consider a shear beam, with GA = m, on an elastic foundation producing a distributed reaction proportional to the deflection and another reaction proportional to the slope, with spring constants k and c reciprocally. The differential equation of equilibrium for this beam is mu + cu + ku = −p(x)

(4)

This equation is similar to equation (2). The main difference between this equation and the ordinary shear beam equation is the type of the foundation spring constants, which are not physically meaningful compared to the accustomed ones. However, one may prefer to solve the equation (4) that represents the equilibrium

A METHOD OF DISCRETE TIME INTEGRATION

299

of an elastic system, instead of equation (2), since one may use Betti’s reciprocal theorem to obtain a solution in finite difference form. Considering three consecutive points of the shear beam with equal distances h, a displacement pattern, triangular at these points, is assumed as the first system (Figure 1). The actual displacement, which is the solution of the differential equation, is the second system. Corresponding loads are shown at the bottom of each deflection form. Application of Betti’s reciprocal theorem on these systems yields a relation among the three consecutive displacements, similar to a finite difference equation.

Figure 1.

− mh ui−1 +

Figure 2. 2m h ui

− mh ui+1 −

k.h 12 (ui−1

+ 10ui + ui+1 ) + c(ui−1 − ui+1 )/2

(5)

h = − 12 (pi−1 + 10pi + pi+1 )

A parabolic variation is assumed for the external load and the actual displacement to evaluate the integrals for this equation. Making use of the two previously computed displacement values, one can have an approximate value for the third point’s displacement. Each time, the relation among the three displacements contains only one unknown. To start the algorithm, the first two displacements can be obtained using initial conditions. The integration algorithm is thus formulated. ui+1 =

(−kh2 − 12m + 6ch)ui−1 + (−10kh2 + 24m)ui + h2 (pi−1 + 10pi + pi+1 ) kh2 + 12m + 6ch

(6) Numerical examples, based on this simple procedure, yields the results, with less computing time, better than those of the classical methods such as linear acceleration or Newmark-β methods. It is clear that one can assume different type of deflections instead of the single triangle in Figure 1. An asymmetrical triangle may be chosen for variable mesh size. Some interesting alternatives will be shortly presented. One of the alternatives is to consider four consecutive points and assume an antisymmetrical variation as shown in Figure 2. It yields 1 (−kh2 − 12m + 6ch)ui−2 + (−9kh2 + 36m − 6ch)ui−1 ui+1 = kh2 +12m+6ch (7) 2 2 +(9kh − 36m − 6ch)ui + h (pi−2 + 9pi−1 − 9pi − pi+1 )

300

KUMBASAR

which is quite different from (6), but gives exactly the same results. Since it necessitates first three ordinates at the outset it is not preferable. Another alternative is to include velocities at each point. Considering four consecutive points and an antisymmetrical variation for the first system as before, we may use a fourth degree polynomial for the actual displacement in terms of ui−2 , vi−2 , ui−1 , vi−1 , and vi . Application of Betti’s theorem through Figure 2 yields ui−2 (−12 mh − 2c) + ui−1 (12 mh + 2c) + vi−2 h(−4 mh − 56 c) + vi−1 h(−10 mh + +vi h(2 mh +

kh 2

+ 76 c) +

h 12 (pi−2

kh 2

− 73 c)

+ 9pi−2 − 9pi − pi+2 ) = 0

(8) from which the velocity vi maybe computed. The displacement of the point i may be found as h ui = ui−2 + (vi−2 + 4vi−1 + vi ) 3 This procedure is found to be unconditionally stable, but not so efficient, as will be seen in numerical examples. Yet another alternative is to find an unconditionally stable procedure that possesses numerical dissipation. To this end, the scheme of variation in Figure 1 is used including the velocity v and the acceleration a of the first mesh point. The displacement and the velocity of midpoint are expressed in terms of parameters of previous point, similar to Newmark method: vi+1 = vi + (1 − α)ai h + αai+1 h ui+1 = ui + vi h + (1/2 − β)ai h + βai+1 h

(9)

A third degree polynomial for the actual displacement is formed as: u = ui+1 + vi+1 t + ai+1

t2 t3 + (ai+1 − ai ) 2 6h

(10)

Application of Betti’s theorem through Figure 1 yields −ui kh − vi h(kh + c) + ai h2 [(α − β − 13 ) mh − ( 12 − β)kh − ( 23 24 − α)c] +ai+1 h2 [(β − α − 23 ) mh − (β +

1 12 )kh

− (α +

1 24 )c]

+

+ 10pi + pi+1 ) = 0 (11) from which a2 and through (9) u2 and v2 may be obtained. Similar to Newmark-β method, this procedure is unconditionally stable for (α = 0.5, β = 0.25), but not much improved. h 12 (pi−1

3. Multi-degree of freedom systems Extension of the procedure obtained for single degree of freedom systems to multi-degree of freedom systems is straightforward. One may consider a different


301

shear beam for each degree of freedom displacement, connected to 2n springs, to take into account the internal and damping forces, n being degree of freedom. Thus one may obtain each of the equations of motion. This will correspond to replace the scalar quantities m, c and k, with structural mass, damping and stiffness matrices M, C and, K. 4. Numerical examples Numerical examples are arranged to check and compare the efficiency of the proposed method with some well known methods. Some simple problems are considered in order to obtain the theoretical solution easily. 4.1. SINGLE DEGREE OF FREEDOM SYSTEM SUBJECTED TO HARMONIC LOAD

The theoretical solution of an undamped single degree of freedom system subjected to Cosine load: mü + ku = p0Cost is known as u=

p0 k(1 −

2 ω2

)

(Cost − Cosωt) where

ω2 =

k m

Three different numerical integration procedures are applied to solve the given differential equation. Time step length h is chosen as 0.01sec, one tenth of the natural period which is generally suggested for numerical integrations. The last 20 steps of the 100 step numerical results are presented in Figure 3, for finite difference method, Newmark-β method, and the first alternative (Eq.6) of the proposed method as well as the theoretical solution. In Figure 4, numerical results of the second (Eq.8) and third alternative (Eq.11) are displayed with those of Newmark-β method and the theoretical solution. The difference between third alternative and Newmark-β method is imperceptible. All are quite different from the theoretical solution, especially in the last steps.

302

KUMBASAR

Figure 3.

Figure 4.

Figure 5.


303

4.2. THREE DEGREES OF FREEDOM SYSTEM SUBJECTED TO HARMONIC LOADS

The theoretical solution of an undamped three degrees of freedom system subjected to harmonic loads at each mass, may also be formed. Since the fundamental period of the system is found to be 0.5sec, the time step length is chosen as 0.05sec for this case. In Figure 5, numerical results of the first alternative are displayed with those of Newmark-β method and the theoretical solution as the last 20 steps of 100 time steps. The presented diagrams show that for the suggested time step size, while the results of the suggested method is quite close, those of Newmark method tends to deviate from the theoretical solution. References Chang, S. Y. (1997) Improved numerical dissipation for explicit methods in pseudo dynamic tests, Earthquake Engineering and Structural Dynamics 26, 917-929. Çakıro˘glu (1979) A computation method of discrete system for structures (in Turkish), Proceedings of First National Congress of Mechanics. Dokanish, M. A, Subbaraj, K. A. (1989) Survey of direct time integration methods in computational structural dynamics. I. Explicit Methods, Int. Computers and Structures 32, 1371-1386. Dokanish, M. A, Subbaraj, K. A. (1989) Survey of direct time integration methods in computational structural dynamics. II. Implicit Methods, Int. Computers and Structures 32, 1387-1401. Golly, B. W. and Amer, M. (1999) An unconditionally stable time stepping procedure with algorithmic damping: a weighted integral approach using two general weight functions, Earthquake Engineering and Structural Dynamics 28, 1345-1360. Hilbert H. M., Hughes T. J. R., Taylor R. L. (1977) Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engineering and Structural Dynamics 5, 283-292. Wilson, E. L., Farhoomand, I., Bathe, K. J. (1973) Nonlinear dynamic analysis of complex structures, Earthquake Engineering and Structural Dynamics 1, 241-252. Zienkiewitz, O. C, Wood, W. L., Taylor, R. L. (1980) An alternative single step algorithm for dynamic problems, Earthquake Engineering and Structural Dynamics 8, 31-40.

BRIDGES IN VIBRATED GRANULAR MEDIA Anita Mehta S N Bose National Centre for Basic Sciences, Block JD, Sector 3, Salt Lake, Calcutta 700098, India

Abstract. We study a particular consequence of the dynamics of vibrated granular media, which is the spontaneous formation of stable bridges. Here we examine their geometrical characteristics, and compare the results of a simple theory with those of independent simulations of three-dimensional hard spheres. Our conclusion is that bridges are the signatures of spatiotemporal inhomogeneities, the carriers of the so-called ‘force chains’. Key words: granular media, bridges

1. Introduction Granular media are often referred to as a special state of matter; they flow like liquids, pack like solids, and in the fluidised state, can behave like gases (Mehta, 2006). This results predominantly from the fact that granular media are athermal, i.e., grains of sand do not move spontaneously in response to the kinetic energy provided by room temperature. For sand to flow, it needs to be externally perturbed; a particularly convenient form of external perturbation is vibration. We have studied structural properties of vibrated sand extensively in the past, details of which can be found in (Mehta, 1994; Mehta and Halsey, 2003; Mehta and Barker, 1991): our studies concern, among other things, the dependence of volume fraction, coordination number and correlation functions of displacement and position, on the intensity of vibration. These and other structural details, as well as their dependence on dynamics, are of course of crucial importance to engineering applications, which is indeed where the study of granular materials first began (Brown and Richards, 1966). In this article, we focus on recent work on the formation and dynamics of bridges, which are cooperative structures that are ubiquitous in granular media. 2. Definitions Under imposed vibrations, grains can fall independently under gravity to a point of stability; when instead, two or more grains fall together to a position of mutual 305 .


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support, bridges (Mehta et al., 2004) are formed. These can be stable for arbitrarily long times, since the Brownian motion that would dissolve them away in a liquid is absent in granular systems (which for simplicity and variety, we will often to as ‘sandpiles’ in what follows) – as mentioned above, grains are simply too large for the ambient temperature to have any effect. As a result, bridges can affect sandpile dynamics: our studies (Mehta and Barker, 1991) have shown that in the last stages of compaction in densely packed sand, voids can only be flushed out of the system by the gradual collapse of long-lived bridges. Bridges are also responsible for jamming in granular processes, for example, as grains flow out of a hopper (Brown and Richards, 1966). We first define a bridge in more quantitative terms. Consider a stable packing of hard spheres under gravity, in three dimensions. Each particle typically rests on three others which stabilise it, in the sense that downward motion is impeded. A bridge is a configuration of particles in which the three-point stability conditions of two or more particles are linked; that is, two or more particles are mutually stabilised. Bridges thus cannot be formed sequentially. While it is impossible to determine bridge distributions uniquely from a distribution of particle positions, we are able via our algorithm to obtain the most likely positions of bridges in a given scenario (Mehta et al., 2004). We now distinguish between linear and complex bridges via a comparison of Figures 1 and 2. Figure 1 illustrates a complex bridge, i.e., a mutually stabilised cluster of five particles (shown in green), where the stability is provided by six stable base particles (shown in blue). Of course the whole is embedded in a stable network of grains within the sandpile. Also shown is the network of contacts for the particles in the bridge: we see clearly that three of the particles each have two mutual stabilisations. Figure 2 illustrates a seven particle linear bridge with nine base particles. This is an example of a linear bridge. The contact network shows that this bridge has a simpler topology than that in Figure 1. Here, all of the mutually stabilised particles are in sequence, as in a string. A linear bridge made of n particles therefore always rests on nb = n + 2 base particles. For a complex bridge of size n, the number of base particles is reduced (nb < n + 2), because of the presence of loops in their contact networks. An important point to note is that bridges can only be formed sustainably in the presence of friction; the mutual stabilisations needed would be unstable otherwise! Although our Monte Carlo simulations (described below) do not contain friction explicitly, our configurations indirectly include this: in particular, the coordination numbers lie in a range consistent with the presence of friction (Mehta and Barker, 1991; Edwards, 1998; Silbert et al., 2002).

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A five particle complex bridge, with six base particles (left), and the corresponding contact network (right). Thus n = 5 and nb = 6 < 5 + 2

Figure 1.

Figure 2. A seven particle linear bridge with nine base particles (left), and the corresponding contact network (right). Thus n = 7 and nb = 9 = 7 + 2

3. Simulation details In the following, we give details of the simulation algorithm which enables us to model vibrated sand (Mehta and Barker, 1991): its main modelling ingredients involve stochastic grain displacements and collective relaxation from them. This algorithm has three distinct stages. (1) The granular system is dilated in a vertical direction (with free volume being introduced homogeneously throughout the system), and each particle is given a random horizontal displacement; this models the dilation phase of a vibrated sandpile.

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(2) The assembly is compressed in a uniaxial external field representing gravity, using a low-temperature Monte Carlo process. (3) Individual spheres in the assembly are stabilised using a steepest descent ‘drop and roll’ dynamics to find local potential energy minima. Steps (2) and (3) model the quench phase of the vibration, where particles relax to locally stable positions in the presence of gravity. Crucially, during the third phase, the spheres are able to roll in contact with others; mutual stabilisations are thus allowed to arise, mimicking collective effects. The final configuration has a well-defined contact network (Mehta and Barker, 1991) where each sphere is supported by a uniquely defined set of three other spheres. The simulation method recalled above builds a sequence of static packings. Each new packing is built from its predecessor by a random process and the sequence achieves a steady state, where structural descriptors such as the mean packing fraction and the mean coordination number fluctuate about well-defined mean values. The steady-state mean volume fraction Φ typically evolves to values in the range Φ ∼ 0.55 – 0.61, depending on the shaking amplitude; the mean coordination number is always Z ≈ 4.6 ± 0.1. It is known that for frictionless packings in d dimensions, Z = 2d (= 6 for d = 3) (Donev et al., 2004), while for frictional packings the minimal coordination number is Z = d + 1 (= 4 for d = 3) (Edwards, 1998); our configurations thus clearly correspond to those generated in the presence of friction. This is confirmed by the results of molecular dynamics simulations of sphere packings in the limit of high friction, which yield a mean coordination number slightly above 4.5 (Silbert et al., 2002). Each of our configurations includes Ntot ≈ 2200 particles. Segregation is avoided by choosing monodisperse particles: a rough base prevents ordering. A large number of restructuring cycles is needed to reach the steady state for a given shaking amplitude: about 100 stable configurations (picked every 100 cycles in order to avoid correlation effects) are analysed, corresponding to Φ = 0.56 and Φ = 0.58. From these configurations, and following specific prescriptions, our algorithm identifies bridges as clusters of mutually stabilised particles (Mehta et al., 2004). Figure 3 illustrates two characteristic descriptors of bridges used in this work. The main axis of a bridge is defined using triangulation of its base particles as follows: triangles are constructed by choosing all possible connected triplets of base particles, and the vector sum of their normals is defined to be the direction of the main axis of the bridge. The orientation angle Θ is defined as the angle between the main axis and the z-axis. The base extension b is defined as the radius of gyration of the base particles about the z-axis; note that this is distinct from the radius of gyration about the main axis of the bridge.

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Figure 3. Definition of the angle Θ and the base extension b of a bridge. The main axis makes an angle Θ with the z-axis; the base extension b is the projection of the radius of gyration of the bridge on the x-y plane

4. Bridge sizes and diameters: When does a bridge span a hole? In the following, we present statistics for both linear and complex bridges. While we recognise that bridge formation is a collective dynamical process, we adopt an ergodic viewpoint (Edwards, 1994) here. Inspired by polymer theory (Doi and Edwards, 1986), we visualise a linear bridge as a random chain which grows as a continuous curve, i.e. ‘sequentially’ in terms of its arc length s. (For complex bridges, this simplification is not possible in general – a direct consequence of their branched structure). This replacement of what is in reality a collective phenomenon in time by a random walk in space is somewhat analogous to the ‘tube model’ of linear polymers (Doi and Edwards, 1986): both are simple but efficient effective pictures of very complex problems. We first address the question of the length distribution of linear bridges. We define the length distribution fn as the probability that a linear bridge consists of exactly n spheres. We make the simplest and the most natural assumption that a bridge of size n remains linear with some probability p < 1 if an (n + 1)th sphere

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is ‘added’ to it: this leads to the exponential distribution fn = (1 − p)pn .

(1)

The exponential distribution above can also be derived by means of a continuum approach. Here, a linear bridge is viewed as a continuous random curve or ‘string’, parametrised by the arc length s from one of its endpoints. We assume also that such a bridge disappears at a constant rate α per unit length, either by changing from linear to complex or by collapsing. The survival probability S (s) of a linear bridge upto length s thus obeys the rate equation S˙ = −αS and falls off exponentially, according to S (s) = exp(−αs). Consequently, the probability distribution of the length s of linear bridges reads f (s) = −S˙ (s) = α exp(−αs), a continuum analogue of (1). This is in good accord with the results of independent simulations, which exhibit an exponential decay of linear bridges of the form (1), with α ≈ 0.99 (Mehta et al., 2004), which is clearly seen until n ≈ 12. Around n ≈ 8, complex bridges begin to predominate; these have size distributions which show a power-law decay: fn ∼ n−τ . (2) with τ ≈ 2 (Mehta et al., 2004). We have also measured the diameter Rn of linear and complex bridges of size n, which is such that R2n is the mean squared end-to-end distance. Our data on diameters and size distributions (Mehta et al., 2004) indicate that linear bridges in three dimensions start off as planar self-avoiding walks, which eventually collapse onto each other because of vibrational effects; on the other hand, complex bridges look like 3d percolation clusters. Another issue of interest to us is the jamming potential of a bridge. A measure of this, in the case of a linear bridge, is its the base extension b (see Figure 3); this is the horizontal projection of the ‘span’ of the bridge. Our simulation results (Mehta et al., 2004) indicate that three-dimensional bridges of a given length have a fairly characteristic horizontal extension, making it relatively easy to predict whether or not they would ‘jam’ a given hole. In order to compare our simulations with experiment (To et al., 2001; Liu et al., 1995), we plot in Figure 4 the logarithm of the probability distribution of base extensions p(b) against the (normalised) base extension b/b. This figure emphasises the exponential tail of the distribution function, and also shows that bridges with small base extensions are unfavoured. We note that this long tail is characteristic of three-dimensional experiments on force chains in granular media (Liu et al., 1995). The sharp drop at the origin as well as the long tail in Figure 4, are observed in normal force distributions obtained via molecular dynamics simulations of particle packings (Erikson, 2002), in the limit of strong deformations. Realising that the measured forces propagate through chains of particles, we use

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this similarity to suggest that bridges are really just long-lived force chains, which have survived despite strong deformations. We suggest also that with the current availability of 3D visualisation techniques such as NMR (Fukushima and Seidler, 2003), bridge configurations might be an easily measurable and effective tool to probe inhomogeneities in shaken sand.

0.4

Log(p(b))

0.0 -0.4 -0.8 -1.2 -1.6 0.0

0.5

1.0 1.5 b/

2.0

2.5

Figure 4. Distribution of base extensions of bridges, for Φ = 0.58. The logarithm of the normalised probability distribution is plotted as a function of the normalised variable b/b, where b is the mean extension of bridge bases

5. Turning over at the top; how linear bridges form domes Recall that a linear bridge is modelled as a continuous curve, parametrised by its arc length s. We here focus on its most important degree of freedom, the tilt with respect to the horizontal; the azimuthal degree of freedom is neglected. Accordingly, we define the local or link angle θ(s) between the direction of the tangent to the bridge at point s and the horizontal, and the mean angle made by the bridge from its origin up to point s, also with the horizontal: # 1 s Θ(s) = θ(u) du. (3) s 0 The local angle θ(s) so defined may be either positive or negative; it can even change sign along the random curve which represents a linear bridge. Of course,

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the orientation angle Θ measured in our numerical simulations is positive by construction, being defined as the angle between the main bridge axis and the z-axis (see Figure 3). (Note that by simple geometry, this ‘zenith angle’ made by the bridge axis with the vertical, equals the mean angle made by the basal plane of the bridge with the horizontal). Our simulations show that the mean angle Θ(s) typically becomes smaller and smaller as the length s of the bridge increases. Small linear bridges are almost never flat (Mehta et al., 2004); as they get longer, assuming that they still stay linear, they get ‘weighed down’, arching over as at the mouth of a hopper (Brown and Richards, 1966). Thus, in addition to our earlier claim that long linear bridges are rare, we claim further here that (if and) when they exist, they typically have flat bases, becoming ‘domes’. We use these insights to write down equations to investigate the angular distribution of linear bridges. These couple the evolution of the local angle θ(s) with local density fluctuations φ(s) at point s (with denoting a derivative with respect to s): θ = −aθ − bφ2 + ∆1 η1 (s), φ = −cφ + ∆2 η2 (s).

(4) (5)

The effects of vibration on each of θ and φ are represented by two independent white noises η1 (s), η2 (s), such that ηi (s)η j (s ) = 2 δi j δ(s − s ),

(6)

whereas the parameters a, ..., ∆2 are assumed to be constant. The phenomenology behind the above equations is the following: the evolution of θ(s) is caused, in our effective picture, by the sequential addition of particles to the bridge at its ends. The fluctuations of local density φ at a point s are caused by collective particle motion (Mehta et al., 1996). The first terms on the right-hand side of (4), (5) say that neither θ nor φ is allowed to be arbitrarily large. Their coupling via the second term in (4) arises as follows: if there are density fluctuations φ2 of large magnitude at the tip of a bridge, these will, to a first approximation, ‘weigh the bridge down’, i.e., decrease the angle θ locally. Reasoning as above, we therefore anticipate that for low-intensity vibrations and stable bridges, both density fluctuations φ(s) and link angles θ(s) will be small. Accordingly, we linearise (4), obtaining thus an Ornstein-Uhlenbeck equation: θ = −aθ + ∆1 η1 (s).

(7)

Let us make the additional assumption that the initial angle θ0 , i.e., that observed for very small bridges, is itself Gaussian with variance σ20 = θ02 . The angle θ(s) is then a Gaussian process with zero mean for any value of the length s. Its

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correlation function can be easily evaluated to be (Uhlenbeck and Ornstein, 1930):

θ(s)θ(s ) = σ2eq e−a|s−s | + (σ20 − σ2eq )e−a(s+s ) .

(8)

It follows from this that the variance of the link angle is: θ2 (s) = σ2eq + (σ20 − σ2eq )e−2as ,

(9)

We see from the above that orientation correlations decay with a characteristic length given by ξ = 1/a; also, in the limit of an infinite bridge, the variance ∆2

θ2 relaxes to σ2eq = a1 (Mehta et al., 2004). Thus, as the chain gets longer, the variance of the link angle relaxes from its initial value of σ20 (i.e. that for the initial link) to σ2eq for infinitely long chains. Given the above, it can be shown that the mean angle Θ(s) will also have a Gaussian distribution. Its variance can be derived by inserting (8) into (3): Θ2 (s) = 2σ2eq

−as )2 as − 1 + e−as 2 2 (1 − e + (σ − σ ) . eq 0 a2 s2 a2 s2

The asymptotic result 2

Θ (s) ≈

2σ2eq

≈

(10)

2∆21

, (11) as a2 s confirms our earlier statement that the longest bridges form domes, i.e. they have bases that are almost flat. Each such bridge can be viewed as consisting of a large number as = s/ξ 1 of independent ‘blobs’ of length ξ; this result suggests, yet again, strong analogies between linear bridges and linear polymers (Doi and Edwards, 1986). The result (11) has another interpretation. As Θ(s) is small with high probability for a very long bridge, its extension in the vertical direction reads approximately Z = z(s) − z(0) ≈ s Θ(s), (12) so that Z 2 ≈ s2 Θ2 (s) ≈ 2(∆1 /a)2 s. Switching back to a discrete picture of an n-link chain, we have Zn ∼ n1/2 . (13) The vertical extension of a linear bridge is thus found to grow with the usual random-walk exponent 1/2, in agreement with experiments on two-dimensional arches (To et al., 2001). Our observations on horizontal extensions of three- dimensional bridges have yielded (Mehta et al., 2004) a non-trivial exponent νlin ≈ 0.66. Putting all of this together, our results predict that long linear bridges are domelike; also, they are vertically diffusive but horizontally superdiffusive. Evidently, jamming in a three-dimensional hopper would be caused by the planar projection of such a dome.

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We now compare the results of this simple theory with data on bridge structures obtained from independent numerical simulations of shaken hard sphere packings (Mehta and Barker, 1991). Figure 5 confirms that the mean angle is Gaussian to a good approximation, while Figure 6 shows the measured size dependence of the variance Θ2 (s). The numerical data are found to agree well with a common fit to the first (stationary) term of (10) - the ‘transient’ effects of the second term of (10) are too small to be significant at our present accuracy. We thus conclude that our simple theory captures the principal structural features of linear bridges.

Figure 5. Plot of the normalised distribution of the mean angle Θ (in radians) of linear bridges of size n = 4, for both volume fractions. The sin Θ Jacobian has been duly divided out, explaining thus the larger statistical errors at small angles. Full lines: common fit to (half) a Gaussian law

6. Discussion In the above, we have studied the kinetics of bridge formation in sandpiles, as well as examined their structure. Their classification into linear and complex types enables a detailed examination of their geometrical, as well as their stability and spanning, properties. Such information, while useful in a physics context, is no less useful in an engineering one, from the point of view of the design of real bridges: it would be interesting to see such a relationship concretised from an engineering viewpoint.

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315

Plot of the variance of the mean angle of a linear bridge, against size n, for both volume fractions. Full line: common fit to the first (stationary) term of (10), yielding σ2eq = 0.093 and a = 0.55. The ‘transient’ effects of the second term of (10) are invisible with the present accuracy Figure 6.

Another issue that our study clarifies is that of inhomogeneities in granular systems. It is well known that forces are not transmitted homogeneously in sandpiles (Mehta and Halsey, 2003); they follow typically stringlike paths (Behringer, 2005) depending on the geometry of the underlying granular configuration. The configurational properties of these force chains have been found to be in close agreement with those of the linear bridges above, leading to the suggestion that the latter are the geometrical objects which sustain the major stresses in the sandpile. Our tentative conclusion is thus that long-lived bridges are natural indicators of sustained inhomogeneities in granular systems. References Barker G. C., Mehta A. 1992 Phys. Rev. A 45 3435. Barker G. C., Mehta A. 1993 Phys. Rev. E 47, 184. Behringer R. P. 2005 Nature, to appear. Brown R. L., Richards J. C. 1966 Principles of Powder Mechanics (Oxford: Pergamon). de Gennes P. G. 1999, Rev. Mod. Phys. 71 S374. Doi M., Edwards S. F. 1986 The Theory of Polymer Dynamics (Oxford: Clarendon). Donev A. et al. 2004 Science 303 990.

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Edwards S. F. 1994 in Granular Matter: An Interdisciplinary Approach, ed. Mehta, A. (Springer, New York). Edwards S. F. 1998 Physica 249 226. Erikson J. M. et al. 2002 Phys. Rev. E 66 040301. Fukushima E., Seider G. T. 2003 see chapters in Challenges in Granular Physics edited by Mehta A and Halsey T C (Singapore: World Scientific). Jaeger H. M., Nagel S. R., Behringer R. P. 1996 Rev. Mod. Phys. 68 1259. Liu C. H. et al. 1995 Science 269 513. Mehta A., Barker G. C. 1991 Phys. Rev. Lett. 67 394. Mehta A. 1994 in Granular Matter: An Interdisciplinary Approach, 1994 ed. Mehta A., (Springer, New York). Mehta A., Luck J. M., Needs R. J. 1996 Phys. Rev. E 53 92. Mehta A., Halsey T. C. 2003, see, e.g. Challenges in Granular Physics edited by Mehta A and Halsey T C, 2003, (Singapore: World Scientific). Mehta A., Barker G. C., Luck J. M. 2004 JSTAT P10014 Mehta A. 2006. The Physics of Granular Materials, 2006 (Cambridge University Press, Cambridge), to appear. Mueth D. M., Jaeger H. M., Nagel S. R. 1998 Phys. Rev. E 57 3164. O’Hern C. S. et al. 2002 Phys. Rev. Lett. 88 075507. Silbert L. E. et al. 2002 Phys. Rev. E 65 031304. To K., Lai P. Y., Pak H. K. 2001 Phys. Rev. Lett. 86 71. Uhlenbeck G. E. and Ornstein L. S. 1930 Phys. Rev. 36 823.

SOLUTIONS FOR DYNAMIC VARIANTS OF ESHELBY’S INCLUSION PROBLEM Thomas M. Michelitsch,1 Harm Askes,1 Jizeng Wang2 and Valery M. Levin3 1 Department of Civil and Structural Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom 2 Max-Planck Institute for Metals Research, Heisenbergtrasse 3, D-70569 Stuttgart, Germany 3 Instituto Mexicano del Petroleo, C.P.07730, Mexico D.F., Mexico

Abstract. The dynamic variant of Eshelby’s inclusion problem plays a crucial role in many areas of mechanics and theoretical physics. Because of its mathematical complexity, dynamic variants of the inclusion problems so far are only little touched. In this paper we derive solutions for dynamic variants of the Eshelby inclusion problem for arbitrary scalar source densities of the eigenstrain. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method which covers both the time and frequency domain is especially useful for dynamically transforming inclusions of any shape. Key words: dynamic variants of Eshelby inclusion problem, Helmholtz potentials, retarded potentials, wave equation, Helmholtz equation

1. Introduction Due to the mathematical complexity of the dynamic variant of Eshelby problem there are only a few cases for which closed form solutions exist, namely for homogeneous spherical inclusions (Mikata and Nemat Nasser 1990). For homogeneous ellipsoidal inclusions the solution of the dynamic Eshelby problem was reduced recently to simple surface integrals (Michelitsch et al., 2003). 2. Dynamic Eshelby inclusion problem We consider a homogeneous material with elastic constants Ci jrs . The constitutive relations are ∗ σi j = Ci jrs (rs − rs )

317 .


(1)

318

MICHELITSCH, ASKES, WANG, LEVIN

where σ, , ∗ , C denote stress, strain, eigenstrain and the tensor of elastic constants, respectively, with 1 (2) rs = (∂r u s + ∂ s ur ) 2 The inclusion S is assumed to undergo a uniform time dependent transformation with a spatially non-uniform eigenstrain ∗ which has the form ∗ (r, t) = f (t)ρ(r) 0

(3)

and with a ”density” function ρ given by ρ(r) = Θ s (r)χ(r)

(4)

where r = (xi ) = (x, y, z) denotes the space point and Θ s (r) denotes the characteristic function of inclusion1 and 0 is a symmetric and constant tensor. In (4) we have introduced the scalar function χ(r) which characterizes the spatial variation of the eigenstrain. The function f (t) is an arbitrary function of time2 which describes the transformation evolution of the eigenstrain. We assume the absence of external body forces. The displacement field then can be written as (Wang et al., 2005) 0 ul (r, ξ) = −Ck jrs rs ∂ jGkl (r, ξ)

(5)

where ξ denotes a set of geometric characteristics of the inclusions (for instance semi axes in case of an ellipsoidal inclusion) and G(r, ξ, t) =

1 1 g2 (r, ξ, t)1 + ∇ ⊗ ∇{h1 (r, ξ, t) − h2 (r, ξ, t)} µ ρm

(6)

in the case of an elastic isotropic medium. The functions gi (r, ξ, t) and hi (r, ξ, t) are determined by (∆ −

1 ∂2 )gi (r, ξ, t) + δ(t)ρS (r) = 0 c2i ∂t2

(7)

and ∂2 hi (r, ξ, t) = gi (r, ξ, t) (8) ∂t2 are causal functions (retarded potentials) being nonzero only for t > 0, i.e. after the δ(t) excitation. The strain becomes 1 2

Θ s (r) = 1, r ∈ S and Θ s (r) = 0, r S ∞ It is only assumed that −∞ f (t)dt < ∞.

DYNAMIC ESHELBY INCLUSION PROBLEMS

319

0 (Pi jkl (r, ξ))(il) il (r, ξ) = −Ck jrs rs

(9)

where (li) indicates symmetrization with respect to the subscripts il. Moreover it is denoted Pi jkl (r, ξ) = ∂i ∂ jGkl (r, ξ)

(10)

Using (9) we define the dynamic Eshelby tensor S analogously to statics by 0 il (r, ξ) = S ilrs (r, ξ)rs

(11)

where the dynamic Eshelby tensor S is given by S ilrs (r, ξ) = −Ck jrs (Pi jkl (r, ξ))(il)

(12)

This relation covers both, the time and frequency domain. It holds for arbitrary density functions ρS (r) and hence for arbitrary shapes of the inclusion. Pi jkl is a spatially non-uniform tensor function inside an ellipsoidal inclusion with homogeneous eigenstrain. When we expand Pi jkl in the frequency domain into a series with respect to frequency ω, the zero order in ω corresponds to the static Eshelby tensor. 3. Results and discussion The retarded potential defined in (7) is given by the convolution # g(r) = gˆ (r − r )ρS (r )d3 r In the frequency domain gˆ (R, β) = tion3

eiβR 4πR denotes the Helmholtz δ(t− Rc ) 4πR denotes the retarded

(13) Green’s func-

and in the time domain gˆ (R, t) = Green’s funcFor the numerical evaluation of the δ-function in the time domain Green’s function we make use of

tion4 .

2

δ(t − Rc ) e−(t−R/c) /(4) = lim √ →0+ 8πR π 4πR

(14)

To compute integral (13) we use the Gauss-Chebyshev quadrature formula (e.g. Press et al. (1992)). 3 4

gˆ (R, β) is defined by (∆ + β2 )ˆg(r, β) + δ3 (r) = 0. 2 gˆ (R, t) is defined by (∆ − c12 ∂t∂ 2 )ˆg(r, t) + δ3 (r)δ(t) = 0.

320


In Figs. 1 the retarded potential of a homogeneous solid sphere is drawn according to a source density δ(t)Θ(a − r) (a = inclusion radius). It is available in closed form (Wang et al., 2005) = gin Θ(a − r) + gout Θ(r − a)

g(r, a, t)

gin (r, a, t) = c2 tΘ(a − r − ct) + gout (r, a, t) =

c 4r

c 4r

a2 − (ct − r)2 Θ r2 − (ct − a)2

(15)

a2 − (ct − r)2 Θ a2 − (ct − r)2

where gin and gout denote the potentials of the internal space (r < a) and the external space (r > a), respectively. Figure. 1 show that the excitation δ(t)Θ(a − r) generates an outgoing spherical wave package of wavelength 2a which remains constant due to the absence of dispersion. Due to the finite propagation velocity c the wave package arrives at an external point r at (r − a)/c which is the runtime form the closest source point located on the boundary of the sphere. Moreover a remarkable superposition effect takes place in the internal space at spacepoint r for t < (a − r)/c: In this time range the potential increases linearly in time (for 0 < t < (a − r)/c is gin = c2 t). In Fig. 2 the real part g (frequency domain) for an inhomogeneous ellipsoidal inclusion of density (ρS = Θ(1 − P)χ, χ = 1, x, x2 , resp.) is drawn. The symmetry x ↔ −x of χ = 1, x2 is reflected by the potential. Moreover the effect of breaking this symmetry is shown for χ = x. In the frequency domain spatial oscillations occur due to a finite frequency (β = ωc ). The frequency domain representation (real part) of g for a homogeneous triangular prismatic inclusion is shown in Figs. 3. The prismatic inclusion covers the region |x| < 1, 0 ≤ y ≤ 1 − |x|, |z| < 1. 0.6 ct=a/2

1

0.5

a=1,c=1,γ=0

0.4 g

0.5

g

0.3 ct=a

0

0.2 ct=2a ct=3a

0.1

4

ct=4a

ct=5a

3

ct=0 0

ct/a

2 1

−0.1 0

1

2

3

4

5

6

0

0

1

2

3

4

5

6

r

r

Figure 1. Time evolution of the retarded potential of a solid spherical source with radius a = 1 (Eq. (15)) for different times t (c = 1)

321

DYNAMIC ESHELBY INCLUSION PROBLEMS 0.01

0 −0.01

g

−0.02

−0.03 −0.04 x

−0.05

ρ=e ρ=1

2

−x

−0.06

−0.07 −2

ρ=e

−1.5

−1

−0.5

0 x

0.5

1

1.5

2

Helmholtz potentials (frequency domain β = ωc ) for ellipsoidal sources for differ3 xi2 ent densities ρS = Θ(1 − P)χ(x), P2 = , χ = {1, exp (x), exp (−x2 )}, resp., vs. r=(x,0,0). a2 i=1 i (a1 = 1, a2 = 0.7, a3 = 0.4, β = 10) Figure 2.

Figure 3. Cross section of the Helmholtz potential of a prismatic source of unit density vs. r=(x,y,0) (β = 10)

4. Conclusion The dynamic variant of the Eshelby tensor of a three-dimensional infinite isotropic medium with a spatially inhomogeneous, dynamically transforming inclusion with eigenstrain (3) was derived. In the time domain the problem was reduced to determine the retarded potentials defined by (7). Examples of inclusions of different

322


shapes and densities were considered such as spheres, ellipsoids, cubes and triangular prisms. The dynamic Eshelby tensor is a main cornerstone for the solution of wide range of dynamical problems in theoretical physics. References Eshelby J. D. (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. Lond. A 241 376-396. Levin V. M., Michelitsch T. M., Gao H. (2002) Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities, Int. J. Solids Structures 39 5013-5051. Michelitsch T. M., Levin V. M., Gao H. (2002c) Dynamic potentials and Green’s functions of a quasi-plane piezoelectric medium with inclusion, Proc. Roy. Soc. Lond. A 458 2393-2415. Michelitsch T. M., Levin V. M., Gao H. (2003) Dynamic Eshelby tensor and potentials for ellipsoidal inclusions, Proceedings of the Royal Society of London A 459 863-890. Mikata Y., Nemat-Nasser S. (1990) Elastic field due to a Dynamically Transforming Spherical Inclusion, J. Appl. Mech. ASME 57 845-849. Press W. H., Teukolsky S. A, Vetterling W. T., Flannery, B. P. (1992) Numerical Recipes in Fortran 77: the Art of Scientific Computing (Second Edition), Cambridge University Press. Pao Y.H. (Ed.) (1978) Elastic Waves and non-destructive testing of materials, AMD-Vol. ASME 29. Wang J., Michelitsch T. M., Gao H., Levin V. M. (2005) On the solution of the dynamic Eshelby problem for inclusions of various shapes, International Journal of Solids and Structures 42 353-363.

NOTE ON LARGE DEFLECTIONS OF CLAMPED ELLIPTIC PLATES UNDER UNIFORM LOAD Subrata Mondal Jalpaiguri Govt. Engineering College, Jalpaiguri, West Bengal, India

Abstract. Large deflections of Clamped Elliptic Plates under Uniform Load have been investigated by many authors among which the work of Nash and (Nash and Cooley, 1959) needs special mention. As a result of bending the edge of the plate tends to decrease and if the boundary of the plate is immovable, the plate is subjected to tension along the edges. The question then arises as to whether the divergence of Nash’s results from the Linear Theory is mainly due to this induced radial tension or to using a non Linear Theory. The object of this paper is to investigate this point. Key words: eliptic plates, large deformations

1. Analysis Let us consider a clamped thin Elliptic Plate of major axis 2a, minor axis 2b and thickness h. The plate is also immovable, i.e., the radial movement of the plate is prevented. If T 0 be the radial tension per unit circumferential length, the differential equation for the normal displacement W takes the following form T0 2 q ∇ W= D D

∇4 W −

(1)

where q is the uniform load, D is the flexural rigidity of the plate and ∇2 =

∂2 ∂2 + ∂x2 ∂y2

A particular integral of (1) can be written as W0 = −

q (x2 + y2 ) 2 4Dα

(2)

where α2 = TD0 assumed constant. For the complementary function of (1) we rewrite (1) in the following form ∇2 (∇2 − α2 )W = 0 323 .


324

MONDAL

Let W = W1 + W2 . Thus we have,

and

∇2 W 1 = 0

(3)

∇2 W 2 − α 2 W 2 = 0

(4)

Let us now introduce elliptic co-ordinates (ξ, η) defined by x + iy = d cosh(ξ + iη) where 2d is the interfocal distance of the ellipse. Now we get, α W1 = C2m cosh 2mξ cos 2m η (5) m=0

W2 =

α

C2mCe2m (ξ, −q1 )Ce2m (η, −q1 )

(6)

m=0

where Ce2m (η, −q1 ) and Ce2m (ξ, −q1 ) are Mathieu function and Modified Mathieu 1 function of the first kind of order 2m and q4 = α2 d2 Thus the general solution determining the normal displacement W can be put as W = W1 + W 2 + W 0 Using the approximation proposed by (Nagdi, 1955) we can write the final solution as W = C1Ce0 (ξ, −q1 )Ce0 (η, −q1 ) −

qd2 (cosh 2ξ + cos 2η) + C2 8Dα2

(7)

If the boundary of the plate be clamped at ξ = ξ0 , we have W = 0 at ξ = ξ0 and ∂w ∂ξ = 0 at ξ = ξ0 . Using the above boundary conditions, the equations to determine the two unknown constants C1 and C2 , multiplying these equations by Ce0 (η j , −q1 ) and integrating with respect η from 0 to 2π and using the orthogonality relations and normalization (Mclachlan, 1947), we get C1 and C2 Here A0 (0) and A2 (0) are the first two Fourier Coefficient in the expression of Ce0 (η, −q1 ). Thus W is completely determined. =

T 0 d2 (1 − ν2 ) sinh 2ξ0 Eh

(8)

It is to be noted that in evaluating the integrals in the left hand side of (8) we have used the results (Mclachlan, 1947) Ce0 (ξ, −q1 ) = Ce0

(η, −q1 )

=

α

(−1)r A2r (0) cosh 2rξ

r=0 α r=0

(−1)r A2r (0) cosh 2rη

LARGE DEFLECTIONS OF CLAMPED ELLIPTIC PLATES

Where, A2r (0) are the Fourier Coefficient. Finding tions have been carried out determining

α2

=

∂w ∂w ∂ξ , ∂η

325

term by term integra-

T0 D.

2. Numerical results Numerical results have been carried for the ellipse a = 2b. To determine the deflection at any point of the ellipse one has to start with assumed values of αd (Here αd has been assumed to be αd = 1, 2, 3 . . .) in the equation (16) leading to qa4 the particular values of the load function Eh4 . qa4 will determine finally central deflections Wh0 from These values of αd and Eh4 equation (11). If αd is known, α a is also known for the ellipse a = 2b. Load vs. central deflection is graphically shown.

Figure 1.

326

MONDAL

3. Conclusion From the load deflection curve it is clear that the results of the present study are in very good agreement with those given by (Nash and Cooley, 1959). Thus the large deflection terms have little effect within the range considered. Acknowledgement The Author gratefully acknowledges the receipt of Financial Support from State Project Facilitation Unit, West Bengal to enable him to attend the ICOVP 7th Conference to be held at the ISIK University, Istanbul, Turkey during September 5 - 9, 2005. References Mclachlan, N. W. (1947) Theory and application of Mathieu Function. Nagdi, P. M. (1955) Journal of Applied Mechanics 22, 89-94. Nash, W. A., and Cooley, I. D. (1959) Large Deflection of elliptic plates, ASME 26-2. Timoshenko S. P., Woinowisky-Krieger, (1959) Theory of plates and shells, 2n d Edn. McGraw-Hiil, Newyork.

LOCALISED DEFECT MODES AND A MACRO-CELL ANALYSIS FOR DYNAMIC LATTICE STRUCTURES WITH DEFECTS A. B. Movchan, S. Haq and N. V. Movchan Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK

Abstract. This paper presents an analysis of inertial lattice structures with periodically distributed defects. Such structures exhibit stop bands for certain ranges of frequencies, prohibiting propagation of waves of such frequencies through the structure. In the engineering literature, structures of this type are referred to as “passive mass dampers”. In particular, we consider a problem of accumulation of damage within the vibrating microstructure. The eigensolutions of the corresponding spectral problem, set for a sufficiently large macro-cell, are sought in the form of Bloch-Floquet waves. A criterion is introduced for formation of the damage pattern within the macro-cell. The damaged macro-cell shows new spectral properties, which are illustrated with the use of dispersion diagrams. A particular attention is given to highly localised vibration modes associated with the presence of defects. The other type of damaged structures discussed in the paper involves layered materials with small variation of density (due to change in porosity from layer to layer). We describe an analytical algorithm which is used to analyse the spectral properties of such structures and to determine the position of stop bands for Bloch-Floquet waves.

1. Introduction The paper considers periodic discrete structures with defects and analyses how the accumulation of damage within such structures influences their spectral properties. In particular, we are interested in existence or otherwise of the so-called stop bands, intervals of frequencies for which no propagating modes are possible. This paper was motivated by earlier publications (Balk, 2001-Zalipaev, 2002). In particular, (Balk et al., 2001a, 2001b) present analysis of non-linear waves in lattice structures with non-monotonic stress-strain law. The paper (Martinsson and Movchan, 2003) and the book (Movchan et al., 2002) give the details of spectral analysis for inhomogeneous lattice structures and asymptotic models relating high-contrast continuum composites and lattice structures. The structure of the present paper is as follows. In Section 2 we employ a macro-cell analysis to study a two-dimensional bi-atomic lattice consisting of particles connected by pairs of springs. We assume that when the deformation of the link connecting the neighbouring particles reaches a certain level, one of 327 .


328

MOVCHAN, HAQ, MOVCHAN

the springs breaks thus changing the stiffness of the link and creating a defect in the lattice. The eigensolutions of the corresponding spectral problem in a macrocell are sought in the form of Bloch-Floquet waves. The corresponding dispersion diagrams show the existence of vibration modes localised in the vicinity of the region occupied by defects. In Section 3 we consider a periodic lattice structure modelling a porous layered medium. The damage in the lattice is introduced by slightly varying the porosity of each layers with a period which differs from the period of the lattice, so that the damage is accumulated over a macro-cell whose size can be quite large compared to the size of the elementary cell in the original (undamaged) lattice. We analyse the spectral properties of the damaged lattice and obtain analytic estimates for the position and width of stop bands as well as frequencies of the localised modes. 2. Accumulation of defects in a two-dimensional bi-atomic lattice Consider a bi-atomic square lattice which consists of particles of mass m(1) and m(2) connected by pairs of parallel springs of equal stiffness. The position of each particle within the lattice is described by a multi-index n = (m, n) ∈ Z2 , the stiffness coefficient of the elastic link between the neighbouring particles n and n is denoted by Cnn , and the distance between the particles n and n is l. We consider Bloch-Floquet waves in a square macro-cell generated by N 2 unit cells, and denote the amplitude of the displacement by u(n). The equations of motion then have the form Cnn (u(n) − u(n )) = 0, (1) m(n)ω2 u(n) + l n where ω is the radian frequency of time-harmonic (out-of-plane) vibrations, and the summation is taken over all nearest neighbours n of the particle n. For any particle n within the macro-cell the Bloch-Floquet conditions have the form u(n + m) = u(n)eiK.m ,

(2)

where K = (K1 , K2 ) is the Bloch vector, m = p1 a1 + p2 a2 . The multi-index p = (p1 , p2 ) has integer components, and a1 , a2 are the basis translation vectors that define the shape and the size of the macro-cell. The system (1) can be written in the matrix form ω2 Mu = σ(K)u, (3) where σ(K) is the N 2 × N 2 stiffness matrix, and M is the diagonal matrix of mass. This system has non-trivial solutions if and only if det{Mω2 − σ(K)} = 0.

(4)

LOCALISED DEFECT MODES AND A MACRO-CELL ANALYSIS FOR 329

Equation (4) is the dispersion equation which links the radian frequency ω and the Bloch parameter K. 2.1. DEFECTS IN THE MACRO-CELL

Defects in the lattice are created by reducing the stiffness coefficients of elastic links between the particles. Since the stiffness matrix σ(K) of the macro-cell changes with the presence of defects, solutions of the dispersion equation (4) will also change. For the sake of illustration, we assume that the two springs connecting the neighbouring particles are of the same stiffness, but one of the springs is characterised by a lower value of critical stress compared to the other spring. When the strain reaches the critical value, this spring breaks and hence the elastic link between the two particles reduces its stiffness (see Fig. 1). σ

(Stress)

Single spring connection

σcritical

One spring is being broken

ε (Strain) 0

ε critical

Double spring connection

Figure 1. Stress-strain relation

We analyse evolution of damage within the lattice structure for the case of dω standing waves when d|K| = 0. For numerical simulations we adopt the following procedure: − We first solve the dispersion equation (4). − According to (3) we then find an eigenvector u corresponding to standing waves within the macro-cell, the Bloch vector is chosen as K = KA (see Fig. 2). − Then we analyse the strain matrices for the elastic links within the macrocell, find the weakest link where the deformation attains its maximum value, break one spring in the corresponding double connection and thus create a defect.

330

MOVCHAN, HAQ, MOVCHAN y

π d

B

−π d

C

A π d

x

−π d Figure 2. Irreducible Brillouin zone in the reciprocal lattice

− Equations (3) and (4) are then analysed for the new discrete system with a defect and the steps 1, 2 and 3 are repeated again. − All single springs connections remain intact. The iterations stop when deformations within double spring links does not exceed those in the single spring connections. The next section presents the results of numerical simulations. 2.2. NUMERICAL SIMULATIONS FOR A LATTICE WITH DEFECTS

In the numerical example, we consider a square macro-cell consisting of 36 atoms. The masses of particles within the bi-atomic structure are m(1) = 4, m(2) = 1. We choose the Bloch-Floquet eigensolutions that represent standing waves; in the present numerical simulation, the Bloch vector K is the position vector of the point A in Fig. 2. In Fig. 3 we show a localised eigenmode in the lattice structure with defects. The corresponding dispersion curve (marked by ×) is nearly flat, and it represents a standing wave. In Fig. 4 the dispersion diagrams and the structures with defects are shown for different stages of damage. The standing waves, studied in this example, are aligned along the vertical coordinate axis. For a damaged macro-cell, we observe a high frequency band gap associated with the presence of defects. After 18 iterations we arrive to the structure shown in Fig. 4d, which contains weak layers of single-spring connections along the vertical coordinate axis. In Fig. 5 we plot projections of the dispersion curves onto the ω-axis. Each vertical line corresponds to one iteration in the series, single dots represent


2.5

X 2

1.5

w 1

0.5

0 −0.6

A

B −0.4

−0.2

0

A

C 0.2

0.4

0.6

0.8

1

1.2

1.4

(a)

(b)

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 6 5

6 4

5 4

3 3

2

2 1

1

(c) Figure 3. Localised vibration mode is shown in diagram (c); the corresponding dispersion curve is marked by × in the diagram (a), the damaged links are circled in diagram (b)

localised modes associated with the presence of defects. As expected, the admissible range of frequencies is reduced with the decrease of the overall stiffness within the macro-cell. 3. High-contrast stratified structure with defects For a high-contrast stratified medium, the weak layers are replaced by springs, and stiff porous layers are replaced by rigid solids. The porosity is assumed to be different from layer to layer, and a small variation of porosity results in the perturbation of the mass density. The corresponding structure is shown in Fig. 6. Here we consider a type of damage where defects are generated via the change of porosity of the layers within the array. In the unperturbed state, an elementary cell of periodicity contains two porous layers connected by elastic springs. The perturbed periodic structure is characterised by a larger period, and the corresponding macro-cell contains more than two layers of different porosity. For such structures, we study the transmission properties.

332


2.5

2

1.5

w 1

0.5

0 −0.6

B

A −0.4

−0.2

0

C 0.2

0.4

0.6

0.8

A 1

1.2

1.4

(a)

(b)

2.5

2

1.5

w 1

0.5

0 −0.6

B

A −0.4

−0.2

0

C 0.2

0.4

0.6

(c)

0.8

A 1

1.2

1.4

(d)

Figure 4. Dispersion diagrams for different stages of the iterative procedure and the corresponding structures with defects. The parameters are m(1) = 4, m(2) = 1, l = 1 and d = 0.5. Diagrams (a), (b) correspond to the 10th iteration, and diagrams (c), (d) to the 18th iteration

3.1. MACRO-CELL OF THE PERTURBED STRUCTURE

Let m(κ) , κ = 1, 2, be mass densities of the layers in an elementary cell of the unperturbed structure, of period T (old) = 2. Let f (1) and f (2) be bounded periodic functions of period T ( f ) ; the perturbation of mass densities of the layers is defined by ( j)

mn = m( j) + f ( j) (n), j = 1, 2,

(5)

where is a small non-dimensional parameter. For example, we can choose f (1) (x) = |cos(2x − 1)πα|, f (2) (x) = |cos2xπα|, where α = L/N is rational (L, N ∈ Z), and T ( f ) = 1/(2α). The period T (new) = NT (old) may be large, compared to the original size of the elementary cell. For time-harmonic vibrations of radian frequency ω, the equations of motion for the

LOCALISED DEFECT MODES AND A MACRO-CELL ANALYSIS FOR 333 2.5

2

1.5

w 1

0.5

0

0

2

4

6

8

N

Figure 5. defects

10

12

14

16

18

20

(iterations)

The structure of the spectrum in various stages of two-dimensional square lattice with

Figure 6.

Porous layered structure with a periodic porosity function

perturbed problem are (m(1) + f (1) (n))ω2 u(n) =

6 c 5 u(n) − v(n ) , l n

(m(2) + f (2) (n))ω2 v(n) =

6 c 5 v(n) − u(n ) , l n

(6) ( j)

where u(n) and v(n) are amplitudes of vibration of layers of mass density mn , n are coordinates of nearest neighbours to n, and c is the stiffness coefficient. At the boundary of such a macro-cell we set the Bloch-Floquet conditions u(n + N + 1) = u(n)e2iKNl , v(n − 1) = v(n + N)e−2iKNl ,

(7)

where K is the Bloch parameter. In the matrix form, the equations of motion can be written as Mω2 u = Σ(K)u,

(8)

334


where Σ(K) is the stiffness matrix, and M is the diagonal matrix of mass densities. The corresponding dispersion equation, which relates ω and K, is given by det{Mω2 − Σ(K)} = 0.

(9)

3.2. TRANSMISSION MATRICES

Here we consider a macro-cell of the size T (new) containing 2N particles, as described above. Using the results of (Felbacq et al., 1998), (Lekner, 1994) we develop an analytical approach for analysis of transmission properties of such a macro-cell and link it with the structure of band gaps on the dispersion diagram associated with equation (9). 3.2.1. Unperturbed structure Let

v(n) F(n) = , n = 1, 2, 3, . . . , N. u(n + 1) − v(n)

Then, for a macro-cell consisting of N elementary cells, we can write F(N + 1) = T N F(1).

(10)

Here T is the transmission matrix for the elementary cell,   m(1) ω2 m(1) ω2   1 − 2 − a  , T =  m(1) m(2) ω4 (ma(1) +m(2) )ω2 (1) 2 (2) 2 m ω m ω − (2 − )(1 − ) − 1 2 a a a a where a = cl . The Bloch-Floquet conditions (7) imply (T N − βI)F(1) = 0,

(11)

where β = e2iNlK . A non-trivial solution exists if det{T N − βI} = 0, which is equivalent to

β2 − tr(T N )β + 1 = 0.

(12)

When |tr(T N )| > 2, the waves do not propagate through the layered structure. 3.3. PERTURBED STRUCTURE

Perturbation of porosity across the stratified structure leads to increase of the width of the macro-cell. Analysis of the trace of the transmission matrix of this macro-cell enables us to determine the frequencies of non-propagating waves.


Let the transmission matrices τ j of the cells, included in the large macro-cell, be defined by τ j = T + A j, where the matrices A j have small entries. For the macro-cell, the transmission matrix is given by N
z0 ), the following relations can be used instead of (4): = E s −Et s x (σ,δs) z0 −z 2 Et z0 =σi E s −Et sy (σ,δs) z0 −z E s δεy − Et z0 σ2i = E s −Et s xy (σ,δs) z0 −z 2 3 E s δε xy − Et z0 σ2

δs x = E s δε x − δsy = δs xy =

(7)

i

where σ and ε are stress and strain functions and the following definitions apply: Π (σ, δs) = Π (s, δσ) = σ x δs x + σy δsy + 3σ xy δs xy = Et σi δεi

(8)

In the region of elastic strain and in the zone of unloading (z < z0 ), the variation of the stress and strain relations are written in the form: δs x = Eδε x ,

δsy = Eδεy

,

δs xy =

2 Eδε xy 3

(9)

where E is the Young’s modulus. In the region of elasto-plastic strains, the stresses σ x , σy , σ xy have different expressions for z > z0 and for z < z0 . Hence the integrals in the expressions for the forces and moments must be split into two parts. For the region z0 ≥ z ≥ −h/2 we take δs x , δsy , δs xy according to (7) and for the region h/2 ≥ z ≥ z0 , according to (9). Finally, should no effect of elastic unloading take place ( z¯0 = 1, the following equations are obtained: E s h3 9

'

( 1 3ϕts ∂4 w ∂4 w ∂4 w ∂2 w h ∂2 Φ ∂2 w + + 2 + t − + ρh = 0 (10) + T 0 4 4 ∂x4 ∂x2 ∂y2 ∂y4 ∂x2 R ∂x2 ∂t2

where the following definitions apply: ϕts = Et /E s

446

SOFIYEV, SCHNACK

4 1 ∂4 Φ E s ∂2 w 3 1 1 ∂4 Φ ∂ Φ + + + 3 − + =0 R ∂x2 4 4ϕts ∂x4 ϕts ∂x2 ∂y2 ϕts ∂y4

(11)

3. Solution of the basic equations Assuming the cylindrical shell to have hinged supports at the ends, the solution of the equation set (10, 11) is sought in the following form: m1 x ny m1 x ny cos , Φ = ζ (t) sin cos (12) R R R R where m1 = mπR/L, m is the half wave length in the direction of the x-axis, n is the wave number in the direction of the y-axis, ξ (t) and ζ (t) are the time dependent amplitudes. Substituting expressions (12) in the equation set (10, 11) and applying Galerkin’s method and eliminating ζ (t) , the following differential equation is obtained: w = ξ (t) sin

d2 ξ(τ) + {Λ1 − Λ2 τ} ξ(τ) = 0 (13) dτ2 where the dimensionless time parameter τ satisfies 0 ≤ τ ≤ 1, Λ1 and Λ2 are the parameters depending on the material properties, loading speed and the characteristics of the shell. The function satisfying the initial condition ξ (0) = ξ,τ (0) = 0 is in the following: , ξ (τ) = Aξ1 (τ) = Ae pτ τ (p + 2) / (p + 1) − τ

(14)

where A is found from the condition of transition to the static condition. The values of p will be determined numerically (Sofiyev and Schnak, 2004). For a cylindrical shell, the plastic buckling under axial compression is mainly axisymmetric, by taking this factor into consideration and substituting Eq. (14) in (13) then applying the Ritz type variational method to equation (13), for the static critical load and dynamic critical load the following expression is obtained: 2h (E s Et )1/2 3R  1/2   36T 02 Ω1 ρ R4  Ω0 E s h   = 2 + 4 + 3 R  (Ω0 E s )3 h4  T crs =

T crd =

T 0 tcr h

(15)

(16)

where Ω0 , Ω1 are the double integral expressions and depending on τ and p. The expression (15) is firstly obtained in (Gerard, 1957).

447

ELASTO-PLASTIC CYLINDRICAL SHELLS

4. Numerical results and discussion In Table 1 the experimental results in (Lee, 1962, Mao and Lu, 2001, and Ore and Durban, 1992) and the calculated results with the present study are compared. The calculationss were performed with simple support cylindrical shells, made of Al 3003-0 with the material parameters E = 7.0 × 101 0(Pa), N = 4.1, σy = 2.362 × 107 (Pa), ρ = 2.77x103 kg/m3 . Ramberg-Osgood Equation, characterizes Al 3003-0 shells. The comparison shows that the deformation theory gives good results but the flow theory predicts much too high critical static loads. Their results obtained by using the deformation theory correspond well with those from the present study. TABLE I. Comparison of the critical static stress (MPa) with experimental and analytical results Geometry of Lee Ore and Durhan Mao and Lu Present the shell study R/h

L/R

Experimental

Deform. theory

Flow theory

Deform. theory

Flow theory

Deform theory

9.36 19.38 29.16

4.21 4.10 4.06

96.87 78.60 64.74

88.49 74.09 67.06

162.33 122.03 103.98

89.71 74.87 67.70

165.46 124.25 106.0

88.21 72.68 64.91

TABLE II. Variations of the critical parameters for elastic and elasto-plastic buckling of cylindrical shells with different R/h and T 0 (R/L = 0.25) Elas. buck. Elas.-plas. buck. Elas. buck. Elas.-plas. buck. Al 3003-0 (T 0 = 5×108 Pa×m/sec) T crd /T crs T crd (MPa) R/h 50 75 100 R/h 50 75 100

883.02(p=66) 605.68(p=39) 468.48(p=27)

89.50(p=6) 87.48(p=5) 86.65(p=4)

1.042 1.072 1.106

1.612 1.785 1.941

Al 3003-0 (T0 = 10×108 Pa×m/sec) 903(p=42) 630(p=25) 495(p=18)

108.36(p=4.2) 108.21(p=3.6) 108.18(p=3.3)

1.067 1.114 1.167

1.953 2.216 2.451

In Table 2 are given the variations of the critical parameters for the elastic and elasto-plastic buckling of cylindrical shells with different R/h and T0 . In the elasto-plastic buckling, the values of the dynamic critical load are considerably lower than the corresponding dynamic critical load in the elastic buckling. With an increase of the ratio R/h, the values of the dynamic critical load in the elastic and elasto-plastic buckling of the shell decrease, however, the values of the ratio T crd /T crs increase. When the ratio of R/h is increased, the values of the dynamic critical load are decreased in a more acute way in the elastic buckling, but this decrease is less in the elasto-plastic buckling. Furthermore, when the ratio of R/h is increased, the values of the ratio dynamic critical load to static critical are

448

SOFIYEV, SCHNACK

increased more slowly in the elastic buckling, but this increase is more important in the elasto-plastic buckling. The increase of the ratio R/h affects more the values of the ratio dynamic critical load to static critical in the elasto-plastic buckling and this effect are even increased in aluminum shells. With an increase of the loading speed the values of the dynamic critical load and the values of the ratio T crd /T crs increase, whereas, that on the p decreases. 5. Conclusions The dynamic buckling of a cylindrical shell subject to a uniform axial compression, which is a linear function of time, is examined. The material of the shell is incompressible and the effect of the elastic unloading is not taken into consideration. Initially, employing small elasto-plastic deformation theory, the fundamental relations and Donnell type dynamic stability equation of a cylindrical shell have been obtained. Then, applying the Galerkin method and Ritz type variational method the critical parameters have been found. Finally, carrying out some computations, the effects of the varying loading speed and of the varying radius to the thickness ratios on the critical parameters of the shells has been studied. References Gerard, G. (1957) Plastic stability theory of thin shells, Journal of Aeronautic Science 24 264-279. Hill, R. (1983) The Mathematical Theory of Plasticity, (Chapter 2), Oxford University Press, London. Ilyushin, A. A. (1947) The elasto-plastic stability of plates. National Advisory Commitee for Aeronautics, Technical Memorandum, No 1188, Washington. Karagizova, D., Jones, N. (2002) On dynamic buckling phenomena in axially loaded elasto-plastic cylindrical shells, Int. J. Non-linear Mech. 37 1223-1238. Lee, L. N. H. (1962) Inelastic buckling of initially imperfect cylindrical shells subjected to axial compresion, Journal of Aerospace Science 29 87-95. Mao, R., Lu, G. (2001) Plastic buckling of cicular cylindrical shells under combined in-plane loads, Int. J. of Solids and Structures 38 741-757. Ore, E., Durban, D. (1992) Elasto-plastic buckling of axially compressed circular cylindrical shells, Int. J.Mech. Sci. 34 727-742. Sofiyev, A. H., Schnack, E. (2004) The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading, J. of Eng. Struct. 26 1321-1331. Wang, A. W., Tian, W. Y. (2003) Twin-characteristic-parameter solution of axisymmetric dynamic plastic buckling for cylindrical shells under axial compression waves, Int. J. of Solids and Structures 40 3157-3175. Wolmir, A. S. (1967) Stability of Deformable Systems, Moscow: Nauka. Wolmir, E. A. (1975) The Behavior of cylindrical panels under dynamic axial compression in elastoplastic zone (in Russian), X International Conference on Theory of Plates and Shells 352-355, Kutaisi-Moscow.

DYNAMIC BUCKLING ANALYSIS OF IMPERFECT ELASTICA ¨ Umit Sönmez ˙ Mechanical Engineering, Istanbul Technical University, Gümüs¸suyu, 80191, ˙ Istanbul, Turkey

Abstract. Dynamic buckling analysis of a pinned-pinned flexible beam attached to a sliding mass is investigated using nonlinear Elastica Theory. Initially straight flexible buckling beam having pinned end boundary conditions loaded at one of its end with curved imperfection is considered. Large deflection analysis of flexible beam is studied using nonlinear elastica theory. Imperfection analysis of the flexible beam is investigated considering imperfection as an initial curvature. Dynamic response of the flexible beam is studied using numerical simulation procedures under step loading conditions; assuming this member buckles in its first mode. Key words: buckling dynamics, large deflection theory, imperfection analysis, elastica

1. Introduction Hoff and Brooklyn (1951) investigated the motion of a perfectly elastic initially slightly curved column using small deflection theory. Prathap and Varadan (1978) studied the large amplitude vibration of a pinned beam taking into account displacement terms up to cubic order. Dynamic response of a tip-loaded elastica with an end-mass is investigated (Snyder and Wilson, 1990). Cameron Boyle and al. (2003) designed and synthesized a compliant constant force robot arm using polynomial fits to elastica. 2. Large deflection analysis of flexible beams The geometry of the initially circular shaped beam is shown in Fig. 1. One end of the beam is fixed and the other end is subjected horizontal applied load P. The initial shape of the curved beam is a circular arc with radius ρ, and the curved beam angle α. Under the influence of horizontal P load, the beam end undergoes horizontal and vertical deflections (h0-h) and (b0-b) respectively. Nonlinear deflection theory of curved beams originally derived by (Nordgren, 1956) and later investigated by (Sathyamoorty, 1998). These equations have not been used before in literature to investigate the imperfection buckling analysis. Large deflection analysis of curve beams may be summarized as follows. Assuming a slender beam 449 .


¨ SONMEZ

450

with an inextensional neutral axis, using the Bernoulli-Euler relation: Change in the beam’s curvature is proportional to beam’s bending moment. The following relation might be obtained

EI(

dθ 1 − ) = P(b − y) ds ρ

(1)

Differentiating above equation with respect to length coordinate s and assigning a2 = P/EI, considering differential arc length ds2 = dx2 + dy2 Eq.1 reduces to d2 θ = −a2 sin θ ds2 Curved beam subjected to a horizontal load Figure 1.

(2)

Integrating Eq. (2) once and applying boundary conditions at the free end (θ = θ0 ), the following equation might be obtained

2 1 dθ 1 = a2 (cos θ − cos θ0 ) + ρ2 (3) 2 ds 2 1 2 At the beam fixed end (θ = 0) and dθ ds = ρ + a b. Using above Eq. (3) and further simplifying, the normalized vertical deflection of curved beam might be obtained as a quadratic polynomial with an unknown end angle θ0 , 2 b 2 b 2 + 2 2 (4) − 2 2 (1 − cos θ0 ) = 0 ρ a ρ ρ a ρ Introducing a new variable β = cos θ0 − 0.5a2 ρ2 , curvature √ β such that; cos 0.5 dθ dx dθ = dx ds = dx cos θ = 2a(cos θ − cos β) and integrating the following as relation may be obtained # cos θ 1 dθ (5) h= √ (cos θ − cos β)0.5 2a dθ ds

Above integral relation can be converted to an integral equation containing elliptic integrals of first and second kind with amplitude γ0 and modulus k if variables transform as (1 − cos θ) = 2k2 sin2 γ = (1 − cos β) sin2 γ h 1 , ( )= 2E(γ0 , k) − F(γ0 , k) = 0 ρ aρ

(6)

BUCKLING DYNAMICS OF ELASTICA

451

Where F(γ0 , k) and E(γ0 , k) are elliptic integrals of first and second kind; and γ0 is given by sin2 γ0 = (1 − cos θ0 )/(1 − cos θ0 + (1/2a2 ρ2 ))

(7)

Using inextensibility criteria L = ρα the following relationship could be found, # 1 dγ = F(γ0 , k) (8) aρα = 2 (1 − k sin2 γ)0.5 The above equations may be solved numerically to obtain nonlinear load deflection characteristics of a curved beam subjected to a horizontal load. The load deflection characteristic of a fixed-free beam with length 0.5L subjected to a horizontal load is same as the pin-pin beam with length L due to the symmetry. The forces acting on a simple pinned-pinned segment are collinear along the line between the two pin joints. Moreover deflections of the flexible pinnedpinned segment are along this line. Pin ends do not carry moments therefore pinned-pinned flexible beam is a two-force member. 3. Load deflection plots of curved beams Load deflection plots of curved beams with three different subtending angles are shown in Fig. 2. Three different subtending angles α = 0.1rad, α = 0.01rad and α = 0.001rad are considered in this investigation. Load Deflection Characteristics of Imperfect Beam 25

Normalized Load

20

15

10

5

0

Figure 2.

0.001 0.01 0.1 0

0.2

0.4 0.6 Normalized Deflection

0.8

1

Normalized load-deflection plots of the imperfect beams

While the angle decreases also the imperfection decreases and the beams become more straight. Load deflection plots are obtained from solution of highly nonlinear Eq. (4) and Eq. (6) that takes into account large deflections and rotations. Deflection solutions are obtained until the pin ends touch each other. Imperfection of α = 0.001 beam is not noticeable to a naked eye when the curve beam is plotted. Normalized load (p = PL2 /EI) and normalized deflection

¨ SONMEZ

452

(u = y/L) results are plotted in Fig. 2. While 0.001 and 0.01 beams exhibit obvious critical buckling loads; 0.1 beam does not have one, but the load-deflection-slope change rate switches from a decreasing one to an increasing one at a certain load, this load could be considered as the critical load for comparison purposes. All three beams show softening spring characteristics with very little change in deflections until the critical buckling load, and then hardening spring characteristics with large change in deflections. 4. Dynamic response of imperfect curved beams Dynamic response of imperfect beams under a step loading condition; attached to a sliding mass at one end, pinned at both ends (as shown in Fig. 4) are investigated solving corresponding nonlinear equation of motion numerically. Fourth order Runge-Kutta routine in MATLAB (ODE45) is used to simulate the equation of motion.

Figure 3.

Free body diagram of a slider attached to a curved beam

The slider’s equation of motion may be written as, mÿ + c˙y + F NL = P

(9)

Where m is the slider’s mass, c is the viscous damping coefficient and F NL nonlinear load deflection characteristics of corresponding flexible curve beam segment and y is the axial displacement of the curve beam. Step loads are applied to a steel flexible beam with elasticity modulus E = 2.07 ∗ 1011 N/m2 , cross sectional area A = 16mm ∗ 0.1mm and length L = 10cm. Flexible beam cross section dimension is chosen from the commonly used standard steel spring strips described in the Handbook of Spring Design (SMI, 1991). This beam deflects elastically until the maximum deflection occurs; when the pinned ends touch each other. The stress at the middle of the beam (maximum stress location) should be kept below the yield stress of steel spring σmax = 800MPa and σyield = 1000MPa). Slider

BUCKLING DYNAMICS OF ELASTICA

453

mass m = PEuler /g is used in all examples (where g is the gravitational constant g = 9.81N/m2 ). 5. Curved beam under the influence of a step input Dynamic response of the curved beams is obtained under the influence of a step load with magnitudes 0.5PE , and 1.0PE , with no damping, where PE is the corresponding Euler buckling load of initially straight beam (perfect beam with the same cross sectional area and length). When 0.5PE is applied to the imperfect beams, the deflections are quite small, because 0.5PE is half of the buckling critical buckling load of the perfect beam and therefore beams does not buckle. The deflection responses oscillate around the static equilibrium points with the following magnitudes and their ratios: y001max = 1.387E −7m, y01max = 1.312E −5m and y1max = 1.293E − 3m. Curved beam length L = 100mm, y1max /y01max = 99 and y1max /y001max = 9300. The imperfect beams vibrate along the pinned-pinned axis with the following frequencies: f001 = 976.6Hz > f01 = 97.6Hz > f1 = 9.76Hz. These frequencies are exact multiples of each other f001 / f1 = 100 and f01 / f1 = 10. These results are confirmed with the linear theory results, where there is an inverse proportional relationship between frequencies and corresponding beam angles for small deflections.

Figure 4.

Dynamic response for step input 0.5PE

Step input PE causes curved beams to oscillate with amplitudes:0.6cm,1cm, 4cm, these displacements are large deflections and curved beam vibrations have the following frequencies: f001 = 1.26Hz < f01 = 1.71Hz < f1 = 1.76Hz. The order of the vibration frequencies of large deflections are reversed that of the small ones. 6. Conclusions The dynamic buckling response of imperfect straight beams with pinned ends is investigated using elastica theory for curved beams. The results show significant

¨ SONMEZ

454

Figure 5.

Dynamic response for step input PE

differences in vibration frequencies and deflections under the influence of same applied load as the beam imperfection parameter, very small subtending curved beam angle, changes. If a step load smaller than the critical Euler buckling load is applied, the vibration frequencies of the curved beam increase as the beam imperfection decreases. However the same is not true for the loads greater than the Euler buckling load, the vibration frequency order reverses. The vibration frequencies increases as the imperfection increases and the frequencies are of the same order (close to each other). Similar observations are made regarding to deflections: Step input loads less than Euler bucking load cause small deflections with distinct magnitudes, greater than buckling load causes large deflections of the same order. References C. Boyle C., L. L. Howell, S. P. M. and Evans, M. S. (2003) Dynamic Modeling of Compliant Constant-Force Compression Mechanisms, Mechanisms and Machine Theory 38, 1469–1487. Conway, H. D. (1956) The Nonlinear Bending of Thin Circular Rods, Journal of Applied Mechanics 23, 7–10. Hoff, N. J. and Brooklyn, N. Y. (1951) The Dynamics of the Buckling of Elastic Columns, Journal of Applied Mechanics pp. 68–75. Prathap, G. and Varadan, T. K. (1978) The Large Amplitude Vibration of a Hinged Beam, Journal of Applied Mechanics 45, 959–961. Sathyamoorty, M. (1998) Nonlinear Analysis of Structures, CRC Press, New York. SMI (1991) Handbook of Spring Design, Spring Manufacturers Institute. Snyder, J. M. and Wilson, J. F. (1990) Dynamic of the Elastica with End and Follower Loading, Journal of Applied Mechanics 57, 203–208.

SPECIFICATION OF FLOW CONDITIONS IN THE MATHEMATICAL MODEL OF HYDRAULIC DAMPER 2 and Radek Mat˘ ˘ ejec2 Rudolf Svoboda1 , Jan Skliba 1 Techlab Ltd, Praha, Czech Republic 2 Technical University of Liberec, Czech Republic

Abstract. Hydraulic dampers represent one of the basic instruments for absorption of vibration in dynamic systems. The damper is substituted either by its hydraulic velocity characteristic, or directly by a mathematical model as a dynamic subsystem. The standard damper model does not provide satisfactory results especially those concerning the strokes of all four damper valves. To improve these results it is necessary to simulate the flows through valves more precisely and, last but not least, to set adequately correct values to all essential parameters of both mechanical and hydraulic parts of the damper. In the paper is presented a new, corrected formula for discharge flow coefficients based on the measurements of flow characteristics of throttle elements of the damper with constant as well as variable slot width. The experimental equipment used for identification process is described as well. Key words: micromechanical scanners, torsional frequency, torsional rigidity, orthotropic material

1. Introduction Hydraulic damper modelling forms a part of theoretical and applied mechanics which is not getting publicity that would deserve. Leaving aside the works of corporate research and development which are usually not made public, during the last thirty years we can name just a few results published in prestigious journals or international conferences (Weigel et al., 2002). Nevertheless, thanks to the intense progress in computer techniques, computer times needed for damper model simulations on digital computers are presently comparable with former simulations on hybrid computers (Skliba, 1983) and the time is coming when the mathematical model of hydraulic damper will work in real-time dynamic systems. Therewith relate possibilities of damper model improvements consisting in application of powerful modern computing methods on the one hand and in exploitation of experimental results and their implementation into the model on the other hand. It will be possible to use the improved mathematical model of hydraulic damper either separately for damper designing and development purposes or as a component of the precise model of a dynamic system (for instance a vibroisolating 455 .


˘ ˘ SVOBODA, SKLIBA, MATEJEC

456

system) in which the damper is applied and where it is not sufficient to substitute the damper by its static velocity characteristics. 2. Problem formulation Owing to the intended utilizations of dynamic model of hydraulic damper in calculations of real-time dynamic systems it is unacceptable to solve damper dynamics in full complexity using finite element methods for calculations of mechanical damper parts movements and finite volumes method for description of flows inside the damper. To ensure reasonable computer time all mechanical parts of the damper are modelled using concentric parameters and the flows through the valves are modelled on the bases of laws of hydraulics. Therefore, the present dynamic models of hydraulic damper deal with pressures concentrated into separate regions (e.g. regions above the piston, under the piston and in the accumulator, supply channels of valves etc.); the flows among regions of the damper are supposed to be one-dimensional, the liquid is compressible, in the state of unsaturated solution (it does not contain free air), viscose.

Figure 1.

A damper cross-section view and corresponding scheme of damper model

MATHEMATICAL MODEL OF HYDRAULIC DAMPER

457

The standard mathematical model of the damper consists of: a) systems of differential equations describing equations of motion of valves, b) equations of dynamic balance of flows among the damper regions, c) equations describing evolution and destruction of columns of saturated vapors, d) algebraic equations describing dependence of discharge flow coefficients on corresponding Reynolds numbers. In case of enhanced model of hydraulic damper where the flows through valves are modeled more precisely and pressure gradients in supply channels are respected, equations for unknown flows Q j and pressure PC, j in supply channel of j-th valve ( j = 1, ..., 4) must be added to the model. But in many cases even this enhanced models do not provide fully acceptable results. The computed valve strokes do not fully correspond to experimentally obtained strokes and the courses of transient valve vibrations can differ from the reality too. Moreover, on certain combinations of definition parameters some zones with strong vibrations of self-excited type can occur in the solution. These vibrations arise in steady parts of both damper expansion and rebound phases where all transient phenomena have gone off long ago and working valves are fully open. It yields from analysis of linearized hydraulic damper model that the system is not stable for a wide range of damper definition parameters and that self-excited valve vibrations are generated by the oil flowing through the opened valve slot. Because the instability is strongly influenced by this flow the correct determination of valve definition parameters is crucial for description of vibration phenomena. 3. Specification of flow conditions From the point of view of mathematical damper model fidelity the replacement of empirical values used till this time by experimentally verified input data should be as broad as possible. Especially exact determination of flow definition parameters of both pressure and discharge valves is decisive for true description of flow phenomena in the damper. On the bases of precise experiments it was found that the standard formula used for flow description through the valve slot ) √ ν Q.dH Q = α. S O ∆P α = α(Re), Re = ρ S 0ν (S 0 slot area, ∆P pressure gradient, ρ oil density, α dimensionless discharge coefficient dependent on the Reynolds number, that represents a correction of

458


the flow regarding to the actual valve design, dH hydraulic diameter, ν kinematic viscosity) does not hold exactly. It results from evaluated experiments that both asymptotic value of the discharge coefficient and the steepness of its course in the start-up zone depend not only on Reynolds number but also on the valve stroke. This dependency on actual working conditions is especially strong on transient conditions of valve closing and opening. The results of regress analysis show that the identified dependency can be treated in the following approaches: a) To replace the area S 0 (x) of the valve slot by a reduced slot area S red (x) b) In the formula for discharge flow coefficient α = α0 exp(α1 .Re) to respect the dependency on the valve stroke x, so α0 = α0 (x), α1 = α1 (x). c) To replace the exponential formula for discharge flow coefficient by the relation α1 (x)Re α(Re, x) = F(Re, x) = α00 (x) + α0 (x) 1+α , 1 (x)Re which better complies with measured data. These results implemented in the damper model provide to improve further findings concerning the problems of flow-induced self-excited vibrations of damper valves. 4. Experimental equipment A special test stand (Figure 2) intended for determination of flow characteristics of throttle valve elements was constructed in laboratory of Technical university of Liberec. The main parts of the test arrangement form: a)source of pressurized liquid, b) working cylinder, c) generator of alternate liquid flow and d) the test chamber. For the purpose of flow characteristics identification the following values must be measured during the measurements: − pressure gradients caused by the flow through the throttle elements, − temperatures of flowing oil, − flow area of throttle elements with variable slot (typically valves). These quantities are necessary for determination of dependence of the pressure gradient on the volume flow through this element, from which the discharge coefficients are further identified. The assembly of the piston with the discharge and back valves used for experimental purposes is shown on Figure 4; the points for pressure and temperature measurement are presented on the configuration scheme on Figure 3. With respect


Figure 2.

459

Test chamber and alternate flow generator

to the piston and valves dimensions an application of expensive miniature pressure and stroke sensors was inevitable.

Figure 3. Measuring points, (pressure and temperature)

Figure 4. Components of the piston, discharge valve and back valve

The main difficulty, which is not satisfactorily solved till this time, consists in measurement of strokes of valve plates. Valve strokes attain tens of microns


460

only and must be therefore measured with accuracy to microns, which is a hardly viable problem in itself. Moreover, it deals with installation of stroke sensors into the areas with severe local conditions (high pressure up to 3M pa, temperature up to 250◦C). For the present an alternative method of stroke measurement on the bases of plate stops is used; however, the method is too work and time consuming and not very precise.

Figure 5.

Measured characteristic of the discharge valve

In Figure 5 are presented typical courses of pressure gradients and static characteristics of the discharge valve which were obtained in the process of measurements. On the bases of these (and another) measured static damper characteristics and on the bases of other supplementary flow characteristics (flow area, oil temperature, viscosity and modulus, etc.) the required flow discharge coefficients of the tested throttle element in dependence on Reynolds number were determined. 5. Conclusions It is evident that the first stage of the damper improvements will be crowned by experimentally verified model of hydraulic damper, which is oil-filled in the state with pulled out piston rod, where the pressure does not fall under the limit of atmospheric pressure and with presumption that working liquid has unchanging concentration of free and dissolved air. From the point of view of mathematical damper model fidelity the replacement of empirical values used till this time by experimentally verified input data should be as broad as possible. Especially exact determination of flow definition parameters of both pressure and discharge valves is decisive for true description of flow phenomena in the damper. The next damper improvements complying with the variable air content in liquid and with the possibility of formation of cavitation will demand heavy effort in future yet.


461

Acknowledgement This work has been supported by Ministry of Education of Czech Republic in the frame of the project No. 4674788501. References Derbaremdiker A. D. (1985) Hydraulic dampers of transport machines (in Russian), Mashinostroienie, Moscow, Russia. Lang H. L. (1978) A study of the characteristics of automotive hydraulic dampers at high stroking frequencies, Doctor dissertation, University of Michigan. ˇ Skliba J. (1983) Zur Problematik der Modellierung eines hydraulishen Dampfer, Proceedings of International Conference Dynamics of Machines, Praha-Liblice, Czech Republic. ˇ Skliba J., Svoboda R. (2000) Pressure distribution in a hydraulic damper, 8th International Conference on the theory of Machines and Mechanisms, 623-628 Liberec, Czech Republic. ˇ Skliba J., Svoboda R. (2004) On the necessity of hydraulic damper model specification, Proceedings of International Conference Flow Induced Vibration, Paris, France. ˇ Svoboda R., Skliba J. (2003) Identification of stiffness of hydraulic damper valves, Proceedings of 6th International Conference on Vibration Problems, 78-80 Liberec, Czech Republic. Weigel M., Mach W., Riepl A. (2002) Nonparametric shock absorber modelling based on standard test data, Vehicle Syst. Dynamics 38 415-432.

NONLINEAR TRANSIENT ANALYSIS OF RECTANGULAR COMPOSITE PLATES Hakan Tanrıöver and Erol S¸enocak ˙ Faculty of Mechanical Engineering, Istanbul Technical University, Mechanical ˙ Engineering Department, Gümüs¸suyu, TR-34437, Istanbul, Turkey

Abstract. The geometrically nonlinear analysis of laminated composite plates under dynamic loading is considered. Galerkin method with the use of Newmark’s scheme in association with Newton-Raphson method is applied to obtain the dynamic nonlinear response of the plates. First order shear deformation theory based on Mindlin’s hypothesis and von Kármán type geometric nonlinearity are utilized. The governing differential equations are solved by choosing suitable polynomials as trial functions to approximate the plate displacements. The solutions are compared to that of Chebyshev series, finite strips and finite elements. A very close agreement has been observed with these approximating methods. In the solution process, analytical computation has been done wherever it is possible, and analytical-numerical type approach has been made for all problems. Key words: Galerkin method, laminates, shear deformation, large deflection, transient analysis

1. Introduction Various numerical techniques can be utilized to investigate the geometrically nonlinear dynamic response of laminated composite plates. Reddy (1983) applied the finite element method (FEM), Chen et al. (2000) developed a semi-analytical finite strip method (FSM) and Nath and Shukla (2001) used Chebyshev series (CS) technique for the nonlinear transient analysis of laminates. Note that the first order shear deformation theory (FSDT) based on Mindlin’s hypothesis and von Kármán type geometric nonlinearity were employed in these works. In the present paper geometrically nonlinear transient analysis of laminated composite plates is performed using the Galerkin method (GM). FSDT based on Mindlin’s hypothesis with von Kármán nonlinearity is utilized and the dynamic nonlinear analysis is performed through using the Newmark method in association with the Newton-Raphson method. In the solution process, computations have been carried out analytically wherever it is possible and analytical-numerical type approach has been made for all cases. Suitable polynomials are chosen as trial functions to approximate the plate displacements.

463 .


¨ TANRIOVER, S¸ENOCAK

464 2. Governing equations

Consider a rectangular laminated plate with dimensions a, b and uniform thickness h. The origin of the coordinate system is chosen to coincide with the center of the midplane of the undeformed plate. The plate is assumed to be subjected uniform transverse pressure qo , and it is constructed of finite homogenous orthotropic layers perfectly bonded together. Under the assumptions of first order shear deformation theory based on Mindlin’s hypothesis; let u, v, w denote the displacements at an arbitrary point of the plate in the x, y, z directions and u0 (x, y), v0 (x, y), w0 (x, y) are the displacements at a corresponding point of the midplane of the plate in the x, y and z directions respectively. Then the displacement field of the first order theory is of the form (Mindlin, 1951): u = u0 + φ x z, v = v0 + φy z, w = w0 ,

(1)

where φ x and φy are the rotations of a transverse normal about the y and x axes respectively. The corresponding total strains could be expressed as follows: ε x = u0,x + 12 w2,x + κ x z, εy = v0,y + 12 w2,y + κy z, γ xy = u0,y + v0,x + w,x w,y + κ xy z, γ xz = w,x + φ x , γyz = w,y + φy .

(2)

where differentiations are denoted by comma. Midplane curvatures and twist of the plate are the following: κ x = φ x,x , κy = φy,y , κ xy = φ x,y + φy,x . For a plate with an arbitrary number of layers, the constitutive relations are 7 ' (4 0 7 4 7 ' (4 0 7 4 γyz A44 A45 N A B Qy ε =K , = , Qx A45 A55 M B D κ γ0xz

(3)

(4)

where N and M are the resultant forces and moments conjugate to ε0 and κ respectively. Q x and Qy are transverse forces and the parameter K is shear correction factor taken as 5/6 in the analyses. Five governing equations of motion for the plate can be written as follows in the general form (Reddy, 1997): R1 = N x,x + N xy,y − I0 u0,tt − I1 φ x,tt = 0, R2 = N xy,x + Ny,y − I0 v0,tt − I1 φy,tt = 0, R3 = Q x,x + Qy,y + (w,x N x + w,y N xy ),x +(w,x N xy + w,y Ny ),y + qo − I0 w,tt = 0, R4 = M x,x + M xy,y − Q x − I2 φ x,tt − I1 u0,tt = 0, R5 = M xy,x + My,y − Qy − I2 φy,tt − I1 v0,tt = 0.

(5)

NONLINEAR TRANSIENT ANALYSIS

465

For the Galerkin approach, the displacements of the plate are approximated in the form shown below: 0

u =

N M

0

amn ( t )Umn (x, y), v =

m=0 n=0

w= φx =

N M

N M

bmn ( t )Vmn (x, y),

m=0 n=0

cmn ( t )Wmn (x, y),

m=0 n=0

dmn ( t )S mn (x, y), φy =

m=0 n=0

N M

N M

(6)

emn ( t )T mn (x, y),

m=0 n=0

where amn , bmn , cmn , dmn and emn are unknown functions of time, Umn , Vmn , Wmn , S mn and T mn are the trial functions, and M and N are the number of terms in x and y directions respectively. Herein, polynomials are used as trial functions, which are chosen to satisfy the geometric boundary conditions, where as natural boundary conditions are not satisfied. In this case, simultaneous approximation is made to the solutions of differential equations and to the boundary conditions.

+b/2 +a/2 Umn R1 dxdy −b/2 −a/2

+a/2

+b/2 − −b/2 Umn N x | x=±a/2 dy − −a/2 Umn N xy |y=±b/2 dx = 0,

+b/2 +a/2 Vmn R2 dxdy −b/2 −a/2

+b/2

+a/2 (7) − −a/2 Vmn Ny |y=±b/2 dx − −b/2 Vmn N xy | x=±a/2 dy = 0,

+b/2 +a/2 Wmn R3 dxdy = 0, −a/2

+1

−b/2 +b/2 +a/2 S mn R4 dxdy − −1 S mn M x | x=±a/2 dy = 0, −b/2 −a/2

+1

+b/2 +a/2 T R dxdy − T M| dx = 0. mn 5 −b/2 −a/2 −1 mn y y=±b/2 3. Solution procedure In the application of the Galerkin method the geometrical boundary conditions are satisfied by choosing appropriate trial functions. Trial functions are weighted polynomials given as follows: Umn = Φ1 xm yn , Vmn = Φ2 xm yn , Wmn = Φ3 xm yn , S mn = Φ4 xm yn , T mn = Φ5 xm yn ,

(8)

where Φi (i = 1, . . . 5) denote the weight functions. Substituting Eq. (6) into Eqs. (7), nonlinear equations in terms of unknown coefficients amn , bmn , cmn , dmn and emn are obtained. These equations are solved by employing the Newton-Raphson methodology.


466 TABLE I. tions

Boundary conditions and corresponding weight func-

SS–1

u0 = v0 = w = M x = φy = 0 at x = ±a/2, u0 = v0 = w = My = φ x = 0 at y = ±b/2. Φi = (x2 − a2 /4)(y2 − b2 /4) (i = 1, . . . 3), Φ4 = (y2 − b2 /4), Φ5 = (x2 − a2 /4).

SS–2

N x = v0 = w = M x = φy = 0 at x = ±a/2, Ny = u0 = w = My = φ x = 0 at y = ±b/2. Φ3 = (x2 − a2 /4)(y2 − b2 /4), Φi = (y2 − b2 /4) (i = 1, 4), Φi = (x2 − a2 /4) (i = 2, 5).

Two different boundary conditions are considered and shown in Table 1. Note that whole plate models are analyzed in all cases presented here. To integrate Eqs. (7), Newmark’s direct integration scheme (Newmark, 1959) is employed. In the Newmark scheme the first time derivative of the displacement ˙ and the solution U are approximated at (n + 1) time step (i.e., at time field U t = tn+1 ≡ (n + 1)∆t) by the following expressions: ˙ n + ∆t[(1 − γ)U ¨ n + γU ¨ n+1 ], ˙ n+1 = U U 2 ˙ n ∆t + ∆t [(1 − 2β)U ¨ n + 2βU ¨ n+1 ], Un+1 = Un + U 2

(9)

where γ and β parameters are chosen as 1/2 and 1/4 respectively in all of the present analyses. The method gives nonlinear equations in terms of unknown coefficients amn , bmn , cmn , dmn and emn in each time step. These equations are solved by employing the Newton-Raphson methodology. Note that the evaluations of integrals are symbolically computed by using a commercial computer math code MathematicaT M (Wolfram, 1988). 4. Numerical examples Geometrically nonlinear transient analyses of an orthotropic and a cross-ply laminate are accomplished as a verification of the present technique. The orthotropic laminate is under SS–2 type boundary condition. The material, geometric and loading data are taken from (Chen et al., 2000) and given below. E1 = 525000 N/mm2 , E2 = 21000 N/mm2 , G12 = G13 = G23 = E2 /2, ν12 = 0.25, a = b = 250 mm, h = 5 mm, ρ = 800 kg/m3 , qo = 1 N/mm2 . Here, the time step ∆t is taken as 10 µsec. Results of GM (M = N = 5), FSM (Chen et al., 2000) and commercial FEM program ABAQUS are given in Fig. 1.

467

NONLINEAR TRANSIENT ANALYSIS

In the ABAQUS analyses presented here 10 × 10 mesh with S4 type elements is used. 4

GM FSM ABAQUS

3 center stress

3 center deflection

4

GM FSM ABAQUS

2

1

0

2

1

0

0.25

0.5

0.75

1 time

1.25

1.5

1.75

2

(a)

0.25

0.5

0.75

1 time

1.25

1.5

1.75

2

(b)

Figure 1. Response of an orthotropic laminate in time (millisec): (a) Central deflection (w/h). (b) Central stress (525 σ x /E1 )

Nonlinear transient analysis of an unsymmetric cross-ply [0◦ /90◦ /0◦ /90◦ ] laminate is considered. The cross-ply laminate is under SS–1 boundary condition. Material properties, geometry, loading data and related normalized variables of the plate are taken from (Nath and Shukla, 2001) and given below. E1/E2 = 25, G12 = G13 = E2 /2, G23 = E2 /5, ν12 = % 0.25, 4 q a 22 a = b, ξ = a/h = 10, q = Eo h4 = 125, w = wh , τ = t I4A h2 ξ 2 2

0

Here, the time step ∆τ is taken as 0.1. Results of GM, CS techniques (?) and ABAQUS are given in Fig. 1. 5. Conclusions Geometrically nonlinear transient analysis of composite plates based on FSDT is performed by using Galerkin approach with the use of Newmark’s scheme in association with Newton-Raphson method. The choice of trial functions is crucial to approximate the two dimensional displacement field. The present solution methodology may be used to solve dynamic large deflection analysis of the laminates in an easy and effective way with the help of a symbolic math package. The method is found to determine closely the displacements with a few number of terms. The results are compared to that of known other approximating methods (Chebyshev series and finite strips), and commercial FEM code ABAQUS. A very good agreement is observed.


468

center deflection

1.25

GM 3 GM 5 CS A

1 0.75 0.5 0.25 0 0.25 5

10

15

20 time

25

30

35

40

Figure 2. Center deflection (w ) of the cross-ply laminate in time (τ). GM 3: M = N = 3 in GM, GM 5: M = N = 5 in GM, CS: Chebyshev series results. A: ABAQUS results

References Chen, J., Dawe, D. J., and Wang, S. (2000) Nonlinear transient analysis of rectangular composite laminated plates, Composite Structures 49, 129-139. Mindlin, R. (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics 18, 31-38. Nath, Y. and Shukla, K. (2001) Non-linear transient analysis of moderately thick laminated composite plates, Journal of Sound and Vibration 247, 509-526. Newmark, N. M. (1959) A Method of Computation for Structural Dynamics, Journal of the Engineering Mechanics Division 8, 67-94. Reddy, J. N. (1983) Geometrically Nonlinear Transient Analysis of Laminated Composite Plates, AIAA 21, 621-629. Reddy, J. N. (1997) Mechanics of Laminated Composite Plates, New York, CRC Press. Wolfram, S. (1988) Mathematica: A System for Doing Mathematics by Computer, Redwood City, CA.

ON THE “RESONANCE” VALUES OF THE DYNAMICAL STRESS IN THE SYSTEM COMPRISES TWO-AXIALLY PRE-STRETCHED LAYER AND HALF-SPACE Fatih Tas¸çı,1 ˙Ibrahim Emiro˘glu1 and Surkay D. Akbarov2 1 Faculty of Chemistry and Metallurgy, Department of Mathematical Engineering, ¨ Davutpasa Campus, No:127, 34210 Esenler, Istanbul, ˙ YTU, Turkey 2 Inst. of Math. and Mech. of National Academy of Sciences of Azerbaijan, Baku, Azerbaijan

Abstract. This paper is devoted to the study of the influence of the pre-stretching of a covering layer on the “resonance” values of a stress. The investigations are performed within the framework of the piecewise homogeneous body model with the use of the Three Dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies (TLTEWISB). Numerical results are presented for some selected materials. Key words: elastic waves, bi-axially pre-stretch layer, resonance

1. Introduction In the present paper the investigations started in the papers by (Akbarov and Ozaydın, 2001a), (Akbarov and Ozaydın, 2001b), (Akbarov and G`‘ uler, 2005), (Güler and Akbarov, 2004), (Emiroglu et al., 2004) are developed for the forced vibration of the half-space covered with the bi-axially pre-stretched layer. In this case the attention is adressed to the determination of the “resonance” values of the frequency of the external point-located force. The investigations are carried out within the framework of the piecewise homogeneous body model with the use of the Three Dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies (TLTEWISB). Numerical results are presented for concrete selected materials. The influence of the pre-stretching of the covering layer on the absolute maximum values of the interface stresses under “ resonance” frequency is analysed. Throughout the study, repeated indices indicates summation over their ranges. However underlined repeated indices are not summed.

469 .


˘ TAS¸ÇI, EM˙IROGLU, AKBAROV

470

2. Formulation of the problem Consider the half-space covered by a bi-axially per-stretched layer. For generality, we assume that the half-space is also bi-axially pre-stressed. With the covering layer we associate a lagrangian coordinate system, Ox1 x2 x3 , which is in the undeformed state, would coincide with a cartesian coordinate system. Note that the covering layer and the half-space occupy the regions {−∞ < x1 < +∞, −h1 ≤ x2 ≤ 0, −∞ < x3 < +∞} and {−∞ < x1 < +∞, −∞ < x2 ≤ −h, −∞ < x3 < +∞}, respectively. We assume that, before contact, the layer and the half-space are prestressed separately in the directions of Ox1 and Ox3 axes, and homogenous initial stress states appear in both materials. The values related to the layer and half-space are denoted by upper indices (1) and (2), respectively. The values related to the initial stresses are denoted by upper indices (m), 0 where m = 1, 2. The linearly elastic material of the layer and the half-space are to be taken homogenous and isotropic. The initial stresses in the layer and the half-space are determined within the framework of the classical linear of elasticity as follows (m),0 σ11 = const1m ,

(m),0 σ33 = const3m , σi(m),0 =0 j

f or

i j 11; 33.

(1)

According to [Guz 2004], the equations for the small initial deformation state version of the TLTEWISB are ∂σ(m) ij ∂x j

+

(m),0 ∂ σ11

2 u(m) i ∂x12

+

(m),0 ∂ σ33

2 u(m) i ∂x32

=

2 (m) (m) ∂ ui ρ0 , ∂t2

i; j = 1, 2, 3; m = 1, 2. (2)

In Eq. (2), ρ(m) 0 denotes the density of the m-th material in the natural state. For an isotropic compressible material one can write the following mechanical relations. (m)

(m)

σi j = λ(m) θ(m) δi j + 2µ(m) εi j ,

(m) (m) θ(m) = ε(m) 11 + ε22 + ε33

(3)

where λ(m) , µ(m) are Lame’s constants and (m) ∂u(m) 1 ∂ui j ( = + ). ε(m) ij 2 ∂x j ∂xi

(4)

We assume that the following complete contact conditions exist between the layer and the half-space. (2) (1) (2) u(1) i | x2 =−h = ui | x2 =−h , σi2 | x2 =−h = σi2 | x2 =−h , i = 1, 2, 3.

(5)

In the free face plane of the covering layer, the following conditions are satisfied (1) (1) iwt σ(1) (6) 32 | x2 =0 = σ12 | x2 =0 = 0, σ22 | x2 =0 − P0 δ(x1 )δ(x3 )e

TWO-AXIALLY PRE-STRETCHED LAYER AND HALF-SPACE

471

In addition to these, we also assume that as x2 → −∞ there is no reflection, which means all waves travel in the negative x2 direction. In other words, we assume that (2) |u(2) i |, |σi j | < M = constant,

as

x2 → −∞.

(7)

This completes the formulation of the problem. Thus, within the framework of the equations (1)-(4), the contact conditions (5), the boundary conditions (6), and the assumption (7), we investigate the forced vibration caused by time-harmonic point-located force in the bi-axially pre-stressed half-space covered by the biaxially pre-stretched layer. (m),0 (m),0 It should be noted that in the case where σ11 = 0, σ33 = 0(m = 1, 2), the problem transforms to the corresponding formulation of the classical linear theory. 3. Method of solution We prefer to apply the double integral (Fourier transformation) method for the solution of the considered problem. We attempt to use Lame’s representations for displacements (Eringen and Suhubi; 1975) u1 =

∂φ ∂ψ3 ∂ψ2 + − , ∂x1 ∂x2 ∂x3

u2 =

∂φ ∂ψ ∂ψ3 + − , ∂x2 ∂x3 ∂x1

u3 =

∂φ ∂ψ2 ∂ψ1 + − (8) ∂x3 ∂x1 ∂x2

In the hypothesis of time-harmonic motion with circular frequency ω, every field g(x1 , x2 , x3 , t) can be expressed as g(x1 , x2 , x3 )eiωt and the following equations are obtained for φ, ψ1 , ψ2 and ψ3 from (2)-(4): (m),0

(m),0

∇2 φ(m) +

(1) σ33 ∂2 φ(m) ∂2 φ(m) (C2 )2 2 (m) + + Ω φ = 0, λ(m) + 2µ(m) ∂x12 λ(m) + 2µ(m) ∂x32 (C1(m) )2

(m) ∇ 2 ψn

where Ω =

σ11

(m),0

+

ωh (m) , c2 c(1) 2

σ11

µ(m)

(m),0

(m) σ33 ∂2 ψ(m) (C2(1) )2 2 (m) ∂ 2 ψn n + (m) + (m) Ω ψn = 0, µ ∂x12 ∂x32 (C2 )2

∂ψ(m) ∂ψ(m) ∂ψ(m) 1 + 2 + 3 = 0, ∂x1 ∂x2 ∂x3 % % (m) (m) = (µ(m) /ρ(m) , c = (λ(m) + 2µ(m) )/ρ0 . 0 1

(9)

From Eqs. (9), we employ the double Fourier transformation with respect to the coordinates x1 and x3 : # +∞ # +∞ f13F (s1 , x2 , s3 ) = f (x1 , x2 , x3 )e−i(s1 x1 +s3 x3 ) dx1 dx3 . (10) −∞

−∞


472

(m) After applying the transformation (10) to Eqs. (9), the functions φ(m) 13F and ψn13F are determined as follows: (1)

(m) γ1 φ(1) 13F = A1 (s1 , s3 )e

(s1 ,s3 )x2

(1)

−γ1 + A(1) 2 (s1 , s3 )e (2)

(2) γ1 φ(2) 13F = A1 (s1 , s3 )e (1) ψ(1) n13F = B1n (s1 , s3 )e

γ2(1) (s1 ,s3 )x2

(s1 ,s3 )x2

(s1 ,s3 )x2

, (1)

−γ2 + B(1) 2n (s1 , s3 )e (2)

(2) γ2 ψ(2) n13F = B1n (s1 , s3 )e

,

(s1 ,s3 )x2

,

(s1 ,s3 )x2

(11)

where (m),0

(m),0

(γ1(m) )2

=

s21 (1

σ11

+

)+ λ(m) + 2µ(m)

=

s21 (1

s23 (1

+

+

σ11

µ(m)

(c2 )2 2 ) − Ω , (m) λ(m) + 2µ(m) (c1 )2 (m),0

(m),0

(γ2(m) )2

σ33

)+

s23 (1

+

σ33

µ(m)

)−

2 (c(1) 2 )

(c(m) 2 )

Ω2 ,

(12)

(m) From last equation in (9) the unknowns B(m) 12 and B22 are expressed throught (m) (m) (m) B(m) 11 , B13 , B21 and B23 . Thus, we obtain from eqs. (5) and (6) the algebraic sys(m) tem of equations for unknowns A(m) 1 , ..., B23 . After determining these unknowns (m) we get the expressions for u(m) n13F , σn j13F and εn j13F from relations (8),(3) and (4). The original unknowns that were sought can now be represented as # +∞ # +∞ 1 (m) (m) (m) (m) (m) (m) {un , σn j , εn j } = 2 {un13F , σn j13F , εn j13F }ei(s1 x1 +s3 x3 ) ds1 ds3 (13) 4π −∞ −∞

The algorithm for calculation of the integral (13) are presented in the papers by (Emiroglu et al., 2004). Therefore we do not consider this algorithm here. 4. Numerical results and discussions The following materials are selected for numerical consideration: Steel (shortly St) with properties ρ0 = 7.86x10−3 kg/m3 , v = 0.29, c1 = 5890m/s, c2 = 3210m/s; Aluminium (shortly Al) with properties ρ0 = 2.7x10−3 kg/m3 , v = 0.35, c1 = 6420m/s, c2 = 3110m/s, where ρ0 , v, c1 and c2 denote the density, Poisson’s ratio, the speed of dilatation and distortion waves, respectively. The numerical investigation is carried out for following two cases: Case I: (Al+St), Layer= Aluminium, Halfspace= Steel, Case II: (St+Al), Layer=Steel, Halfspace=Aluminium. We in(2) troduce the notation σ22 = σ(1) 22 (0, −h, 0) = σ22 (0, −h, 0). We shall evaluate the numerical results obtained only for the stress σ22 . The validity of the algorithm

TWO-AXIALLY PRE-STRETCHED LAYER AND HALF-SPACE

473

and programmes are established as in (Emiroglu et al., 2004). We introduce the (1),0 (1) /µ for the estimation of the initial stretching of the covering parameter η = σ11 (1),0 (1),0 (2),0 (2),0 layer, and assume that σ11 > 0, σ33 = σ11 = σ33 = 0 and 0 < Ω ≤ 3.

Figures 1 and 2 show the graphs of the dependencies between σ22 h2 /P0 and Ω for the cases I and II respectively for various values of η. According to these graphs the behavior of the considered system comprises a covering layer and half-space under forced vibration is similar to the forced vibration of the system comprises a mass, a spring and a dashpot.

474


The values of Ω for which the absolute values of σ22 have its maximum we name as the “resonance” value of Ω. It follows from the numerical results that the absolute values of σ22 for “resonance” frequency decrease with increasing of initial stretching of the covering layer. 5. Conclusion In the present paper within the framework of the piecewise homogeneous body model with the use of the TLTEWISB the forced vibration of the half-space covered by the bi-axially pre-stretched layer is studied. It is established that the behavior of considered system is similar to the behavior of the system comprises a mass, spring and dashpot. The “resonance” values of the frequency of the pointlocated external force is determined for the concrete selected materials. It is shown that the absolute values of the interface normal stress arising under “resonance” frequency of the external force decrease with the pre-stretching of the covering layer. References Akbarov S. D., Ozaydin O. (2001a) The effect of initial stresses on harmonic stress field within a stratified half-plane, Europ. J. Mechnanics A/Solids 20, 385-396. Akbarov S. D., Ozaydin O. (2001b) On the Lamb’s problem for a prestressed stratified half-plane, Int. Appl. Mech. 37(10), 138-142. Akbarov S. D., Guler C. (2004) Dynamical (harmonic) interface stress field in the half-plane covered by the pre-stretched layer under a strip load, J.Strain Analysis 40(3), 225-235. Akbarov S. D., Guler C. (2005) Dynamical (harmonic) interface stress field in the half-plane covered by the prestretched layer under a strip load, J. Strain Analysis 40(3), 225-235. Emiroglu I., Tasci T., Akbarov S. D. (2004) Lamb’s problem for a half-space covered with a twoaxially pre-stretched layer, Mechanics of Composite Materials 40(3), 227-236. Gladwell G. M. L. (1968) The calculation of mechanical impedances related with the surface of a semi-infinite elastic body, Jour. Sound Vibr. 8, 215-219. Guler C., Akbarov S. D. (2004) Dynamic (harmonic) interfacial stress field in a half-plane covered with a prestretched soft layer, Mechanics of Composite Materials 40(5), 379-388. Guz, A. N. (2004) Elastic waves in bodies with initial (residual) stresses, Kiev: “A. C. K.” -672p. Lamb H. (1904) On the propagation of tremors over the surface of an elastic solid, Philosophical Transactions of the Royal Socety A203, 1-42.

OPTIMIZATION OF VIBRATING ARCHES BASED ON GENETIC ALGORITHM ¨ Nildem Tays¸i, M. Tolga Gög˘ u¨ s¸, Mustafa Ozakc ¸a Department of Civil Engineering, University of Gaziantep, 27310, Gaziantep, Turkey

Abstract. The present work is concerned with the structural shape optimization of vibrating arches. Natural frequencies and mode shapes are determined using curved, variable thickness, C(0) continuity Mindlin-Reissner finite elements. The whole shape optimization process is carried out by integrating finite element analysis, cubic spline shape and thickness definition, automatic mesh generation and Genetic Algorithm (GA). An example is included to highlight various features of the optimization procedure. Key words: arches, free vibration, optimization, genetic algorithm

1. Introduction The predominant emphasis and the most significant progress in the field of the optimal design of structures have been concerned with shape optimization in static situations (Tadjbakhsh, 1981; Uzman et al., 1999; Hinton et al., 1992). However, relatively few authors have concentrated on the shape optimization of vibrating structures using GA; see for example resent surveys by (Schittkowski et al., 1994; Arora et al., 1999). One possible explanation is that the dynamic response of structures is often considered of less significance than the static response. Nevertheless, the optimal design of structures subjected to dynamic loading can be extremely useful leading to improved dynamic characteristics. The underlying objective of the present study is to develop a robust and reliable shape optimization tool for vibrating arches based on Mindlin-Reissner theory and integrating ‘locking free’ finite element analysis, cubic splines to define the arch geometry, automatic mesh generation and GA. The type of optimization considered is maximization of the natural frequency by constraining the volume of the arch material (weight).

475 .


476

¨G ˘ US, ¨ OZAKC ¨ TAYS¸˙I, GO ¸A

Figure 1.

Structure chart of genetic algorithm

2. The structure of optimization using genetic algorithm Figure 1 illustrates the structure chart for the GA (Tays¸i, 2005). All the necessary data will be read in and the process of the GA will start for the first generation. The initial population will be generated randomly. The constraint violation may be computed so that the objective functions can be modified. By applying some statistics procedures, the average, the maximum and the fittest design will be found, and the convergence criteria will be checked. If the convergence is achieved the GA process will be terminated otherwise the GA process will resume by storing the best individual which will be used in the next generation. By producing the mating pool, the creation of the next population is started by applying the crossover operator, and the GA process will proceed continuously until convergence, or the maximum generation is achieved.

OPTIMIZATION OF VIBRATING ARCHES BASED ON GA

477

2.1. DEFINITION OF FITNESS FUNCTION AND SELECTION OF OBJECTIVE AND CONSTRAINT FUNCTIONS

The objective function is a mathematical function expressed in terms of the design vector s, which quantifies (in a mathematical sense) the worth of any design. It is a criteria which has to be chosen for comparing the different alternative acceptable designs and for selecting the best one. The choice of the objective and constraint functions is governed by the nature of the problem. Typical objective functions f (s) are weight minimization or fundamental frequency maximization under frequency or weight constraint. Weight minimization: Summing for the number of elements ne , the individual masses (the unmodified objective functions) for individual i and population nz are calculated from: f (s)i =

ne

ρAl

f or : j = 1, . . . , ne and i = 1, . . . , nz

(1)

j=1

Fundamental frequency maximization: In the case of the frequency maximization, objective function f (s) for the structure is computed as the f (s)i = ω p

(2)

ω is the associated frequency. The details of the volume and fundamental frequency computation is presented for arch structures in (Tays¸i, 2005). In order to complete the formulation of the problem, some restrictions must be imposed on the values of the design variables for the mathematical model to be meaningful. The constraints in an optimization problem can be geometric constraints setting a fixed volume for the structure throughout the entire optimization process. Alternatively, there are behavior constraints imposing limiting values on the frequency. Fundamental frequency constraint: the constraint from the fundamental frequency for each individual are calculated by: ci =

ωi ωinitial

(3)

where, ωi is the individual fundamental frequency to be compared against the initial fundamental frequency ωinitial . The constraint violation violi is obtained from the fundamental frequency constraints ci : violi = ci − 1.0 (4) where, ωi and individual fundamental frequency must be larger than the initial fundamental frequency values of ωinitial , without incurring constraint violation violi .

478


Weight constraint: Structural weight is kept constant by using the target weight WT , which is initial weight of structure. ci =

Wi WT

f or : Wi > WT

WT f or : Wi ≤ WT Wi The constraint violation violi is obtained from the weight constraints ci : ci =

violi = ci − 1.0

(5) (6)

(7)

where, and individual weight Wi must lie within the initial weight values of WT , without incurring constraint violation violi . This constraint is used together with fundamental frequency maximization. Fitness function: After computing objective function f (s) from Eq. 1 or 2 and constraint violations from Eq 3–7, the individual modified objective function f¯(s)i can be derived: violi 2 f¯(s)i = f (s)i + pc (8) Note that the influence of viol on the above modified objective function is controlled by an input penalty multiple pc . The maximum and minimum modified objective function ( f¯(s)i )max and ( f¯(s)i )min in the population of individuals can then be used to calculate the fitness value of each individuals designs: ndv f (s)i ˆ fav = dv=1 (9) f iti = fi / fav nz where:

fî = f¯(s)max + f¯(s)min + f¯(s)i

(10)

3. Geometry definition and mesh generation Among the various types of the curves used to representing shape, cubic splines are the most popular. The cubic spline is the spline of lowest degree with C(2) continuity which meets the needs of most problems arising in practical applications. The detail treatment of cubic spline curves and varieties of cubic spline functions can be found in (Tays¸i, 2005). The automatic mesh generator is of prime importance in the automated structural optimization. In the present work, a mesh generator for arches has been developed based on approach of (Sienz, 1994). The mesh generator can generate meshed of two- three- and four noded elements on the natural line of the arch which is defined using parametric cubic splines. Moreover the thickness within the arch are also interpolated from the key points to the nodal points using cubic splines.

OPTIMIZATION OF VIBRATING ARCHES BASED ON GA

Figure 2.

479

Pinned-pinned arches of continuously varying cross-section

4. Example Problem definition: The following example deals with a continuously varying cross-section arch supported by pin joints at both ends as shown in Figure 2. The arch was originally analyzed (but not optimized) by (Gutierrez and Laura, 1989). The arches are optimized for following cases (a) maximization of the fundamental frequency with a constraint that the total material volume of the structure should remain constant, and (b) volume (or weight) minimization subject to the constraint that the fundamental frequency should remain constant. The location of the design variables and boundary condition for the arch is shown in Figure 2 where ratio of thickness of beginning section to end section tb /te = 0.43 and β = 60◦ . The geometry of the arch is modeled using five key points and one segment. Six design variables are considered. These are thicknesses of five key points and width of arch. The width of the arch at the key points is same. Discussion of results: Table 1 shows initial and optimum values of design variables and objective functions together with bounds on the design variables for cubic thickness variation. For case (a), the problem of fundamental frequency maximization — a 77% increase in the fundamental frequency from 5.58 to 9.69 is obtained. In case (b) involving volume minimization, a reduction of 53% from 104.61 to 49.09 is obtained. Large numbers of thickness design variables, apart from leading to impractical geometries, can sometimes lead to negative thicknesses between key points. Constraints on the bounds of the design variables are used to guard against negative or zero element thickness. As seen in Table 1 the optimum thicknesses of arch shows that, the symmetry varying of thickness gives better results and upper part of arch thickness is smaller than lower parts.


480 TABLE I.

Initial and optimum design values of continuously varying arch

DV

Minumum

Initial

Maximum

t1 t2 t3 t4 t5 w1

0.1 0.1 0.1 0.1 0.1 1.0

0.6 0.6 1.0 1.2 1.4 1.0

2.0 2.0 2.0 2.0 2.0 2.0

Optimum frequency Optimum weight

Optimum design variables Frequency maximizaiton Weight Minimization 1.2302 1.9472 1.4721 1.9853 1.5835 0.5586

1.6118 0.9711 0.8596 0.9209 1.6508 0.4436

9.6905 —

— 49.0884

5. Conclusions In the present work, computational tools have been developed for the geometric modeling, automatic mesh generation, analysis and GA optimization of arch structures. It is observed that the number of the design variables and the type of thickness variation used for thickness definition greatly affects the amount of improvement and smoothness of the final optimum shape. The optimum thickness variation may not be practical, however, we believe that the tools developed in the present work could be assisted engineering their quest for novel structural forms. References Arora, J.S., Huang, M.W. and Hsieh, C.C. (1999) Methods for Optimization of Nonlinear Problems with Discrete Variables: A Review, Structural Engineering and Mechanics 8, 465–476. Farin, G. (1996) Curves and Surfaces for CAGD, Academic Press, 5 edition. Gutierrez, R. and Laura, P., In-Plane Vibration of Non-Circular Arcs of Non-Uniform Cross Section, Journal of Sound and vibration 129. ¨ Hinton, E., Ozakc ¸ a, M. and Jantan, M.H. (1992) A Computational Tool For Determining Optimum Shapes Of Vibrating Arches, Struct. Eng. Review. 4, 162–174. Schittkowski, K., Zillober, C. and Zotemantel, R. (1994) Numerical Comparison of Nonlinear Programming Algorithms for Structural Optimization, Structural Optimization 7, 1–19. Sienz, J. (1994) Integrated Structural Modeling, Adaptive Analysis and Shape Optimization, Ph.D. thesis, University of Wales, Swansea. Tadjbakhsh, I. (1981) Stability and Optimum Design of Arch-type Structures, Int. J. of Solids and Structures 17, 565–574. Tays¸i, N. (2005) Analysis and Optimum Design of Structures Under Static and Dynamic Loads, Ph.D. thesis, University of Gaziantep, Gaziantep. ¨ Dalo˘glu, A. and Saka, M.P. (1999) Optimum Design Of Parabolic And Circular Arches Uzman, U., With Varying Cross-Section, Structural Engineering and Mechanics 8, 465–476.

PROPAGATION OF LONG EXTENSIONAL NONLINEAR WAVES IN A HYPER-ELASTIC LAYER Mevlüt Teymür ˙ ˙ Faculty of Science and Letters, Istanbul Technical University, 34390, Istanbul, Turkey

Abstract. Propagation of small but finite amplitude waves in a nonlinear hyper-elastic plate of uniform thickness is considered. By employing a perturbation expansion, extensional waves are examined under the long wave limit. It is shown that the asymptotic wave field is governed by a Korteweg-DeVries (K-dV) equation. Then the propagation characteristics of the asymptotic waves are discussed via the well known solutions of the K-dV equation. Key words: nonlinear waves, hyper-elastic, KDV, extensional waves

1. Introduction The propagation of linear elastic waves in wave guides such as rods, plates, layered half spaces, etc. have been studied extensively because of their important applications in certain areas such as geophysics, non-destructive testing of materials, electronic signal processing devices, etc. (Ewing et al., 1957, Farnell, 1978, Morgan, 1998). In these wave propagation problems because of repeated reflection processes which take place at the boundaries of wave guides, waves become dispersive, i.e. the phase velocities of waves depend on the wave number. In recent years the effect of the constitutional nonlinearity on the propagation characteristics of dispersive elastic waves has been the subject of many investigations for somewhat similar reasons mentioned above. By employing the asymptotic perturbation methods previously used in the fields of fluid mechanics, plasma physics, etc., to investigate the propagation of weakly nonlinear waves (Jeffrey and Kawahara, 1981, Johnson, 1997), several problems related with the propagation of nonlinear dispersive elastic waves are examined (Ahmetolan and Teymur, 2003, Fu, 1994, Mayer, 1995, Teymur, 1988, 1996). In these works as a balance between nonlinearity and dispersion, various scalar nonlinear evolution equations, which were obtained previously in the above mentioned fields, have been derived to describe the wave motions asymptotically. Then several aspects of problems under consideration, such as nonlinear instability of modulated waves, the existence of solitary waves, etc., were discussed on the basis of these equations. 481 .


¨ TEYMUR

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In the present work the propagation of small, but finite amplitude extensional waves in a nonlinear elastic plate of uniform thickness is considered. The perturbation analysis developed in (Teymur, 1996) is applied to examine the problem. The balance between the nonlinearity and dispersion in the analysis yields a KdV equation for the asymptotic wave field. Then the propagation characteristics of nonlinear extensional waves are discussed via the well known solutions of the K-dV equation. 2. Formulation of the problem Let (x1 , x2 , x3 ) and (X1 , X2 , X3 ) be the spatial and material coordinates of a point in space referred to the same rectangular Cartesian system of axes respectively. Consider a plate of uniform thickness 2h occupying the region; D = {(X1 , X2 , X3 )| − h ≤ X2 ≤ h, − ∞ < X1 < ∞, − ∞ < X3 < ∞}

(1)

in the reference frame XK and assume that the boundaries X2 = h and X2 = −h are free of traction. Then a wave motion described by the equations; x1 = X1 + u1 (X1 , X2 , t),

x2 = X2 + u2 (X1 , X2 , t),

x3 = X3

(2)

is supposed to propagate along the X1 -axis in the plate, where u1 and u2 are displacements in the (X1 , X2 )−plane in X1 and X2 directions respectively, and t is the time. In the absence of body forces, the equations of motion in the reference state take the following forms; T β1,β = ρ0 u¨ 1 , T β2,β = ρ0 u¨ 2

(3)

where T Kk is the first Piola-Kirchoff stress tensor. The subscripts preceded by a comma indicate partial differentiation with respect to coordinates X1 or X2 and an over-dot represents the partial differentiation with respect to t. The assumption of vanishing of traction on the free surfaces of the plate imposes the boundary conditions; T 21 = T 22 = 0 on X2 = ±h. (4) The constituent material is assumed to be homogenous, isotropic and compressible elastic. Since the propagation of small but finite amplitude waves are considered the stress constitutive relation is taken in the following quadratic nonlinear form; , > + λ u p,M u p,M + 1 (6l + 3m + n)(trE) > 2 − 1 (m + n)(trE >2 )-δKk T Kk = λ(trE) 2 2 2 , -> , >(5) + 2µ − (m + n)(tr> E) E KL δLk + λ(tr E) uk,K >KL uk,L + µu p,K u p,L δLk + nE >KN E >NL δLk +2µE

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> is the linear Lagrangian deformation tensor with the components E KL = where E (1/2)(uk,K δLk + uk,L δKk ) and λ, µ are the Làme (linear elastic) constants, l,m,n are Murnaghan (nonlinear elastic) constants. 3. Extensional waves We now investigate the propagation of small, but finite amplitude extensional (symmetric) waves by a perturbation method. In the analysis, the following nondimensional variables are defined x = X1 /L, y = X2 /h, τ = (c/L)t, u = u1 /L, v = u2 /h, δ = h/L

(6)

where c and L denote a characteristic velocity and a characteristic wavelength respectively. Here δ is the ratio of vertical to horizontal length scales. Note that, for extensional waves we have the following symmetry conditions to be satisfied by the displacement functions; u(x, y, τ) = u(x, −y, τ),

v(x, y, τ) = −v(x, −y, τ).

(7)

We consider the unidirectional waves propagating along the positive x−axis generated by a time dependent boundary loading impinged on x = 0. For this boundary value problem to develop a uniformly valid asymptotic solution, we use the asymptotic perturbation method given in (Teymur, 1996). As in (Teymur, 1996) we introduce new independent variables x0 = x, x1 = εx, τ = τ, y = y where x1 = εx characterizes the slow spatial variation along the x−axis, and assume that u and v are functions of x0 , x1 , τ, and y. We now expand u and v in the following asymptotic power series in a small parameter ε > 0 which measures the degree of nonlinearity; u=

∞ n=1

εn un (x0 , x1 , , y, τ) , v =

∞

εn vn (x0 , x1 , y, τ).

(8)

n=1

Here, we also make an assumption on δ, the ratio of vertical to horizontal length scales, that is δ2 = O(ε) as ε → 0, i.e.ε = δ2 = h2 /L2 . Hence employing the expansions (8) in the equations of motion (3) and the boundary conditions (4), after using the constitutive relation (5) in them, and then collecting the terms of like powers in ε we obtain a hierarchy of problems from which it is possible to determine un and vn , successively. The solutions of the first and second order problems are found to be; 5 6 ∂A y , ξ = τ − x0 u1 = A(x0 , x1 , τ) , v1 = 1 − 2c2T − c2L ∂ξ

(9)

5 6 ∂2 A y2 + J1 (x0 , x1 , τ) u2 = 1 − 2c2T − c2L ∂ξ2 2

(10)

¨ TEYMUR

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where J1 is an arbitrary function to be determined in higher order perturbation problems and A satisfies the wave equation; 2 5 c2 ∂2 A 2 26∂ A − 4 1 − c − c = 0. T L c2T ∂τ2 ∂x02

(11)

5 6 c2 /c2T = 4 1 − c2T − c2L

(12)

If we now choose c2 as the equation (11) reduces to

∂2 A ∂2 A − = 0. (13) ∂τ2 ∂x02 & A close examination of (12) reveals that c = E/(1 − ν2 )ρ0 . Hence, c is nothing but the longitudinal thin plate velocity. The solution of this equation for the waves propagating along the positive x−axis is of the form A = A(τ − x0 , x1 ). Note that, at this order the function A could not be determined completely. The result implies that A remains in a frame of reference moving with the unit velocity. The dependence of A on τ−x0 and x1 is determined proceeding to the next order. From the third order problem it is found that J1 satisfies the following inhomogeneous wave equation; ∂ 5 ∂A 62 ∂4 A ∂2 A ∂2 J1 ∂2 J1 − = 2 − Λ − Λ . 0 1 ∂ξ∂x1 ∂ξ ∂ξ ∂τ2 ∂ξ4 ∂x02

(14)

Here, while the constant Λ1 depends only on linear elastic constants, Λ0 also depends on nonlinear ones and it becomes zero when the nonlinear effects are neglected. Note that, the solution of this equation will exhibit a secular behavior. But this can be eliminated by equating the terms on the right-hand side of (14) to zero; ∂2 A Λ0 ∂ 5 ∂A 62 Λ1 ∂4 A − − = 0. (15) ∂ξ∂x1 2 ∂ξ ∂ξ 2 ∂ξ4 By doing this not only the uniformity of the asymptotic expansions (8) are maintained, but also a differential equation for the function A, which was left undetermined in the previous steps, is deduced. Under the condition (15) the solution of (14) for the waves propagating along the positive x−axis is of the form J1 = J1 (τ − x0 , x1 ). Note that, at this order it is only revealed that J1 remains constants in the frame of reference moving with the unit velocity. The function could not have been determined completely. The dependence of on τ − x0 and x1 can be determined in higher order problems. But, since this work is centered around the small but finite amplitude waves it is aimed to obtain just the uniformly valid first

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order solutions u1 and v1 . Therefore, to obtain this solution the determination of A as a solution of (15) will be sufficient. Now let us define a function w(ξ, x1 ) by ∂A/∂ξ = (1/|Λ0 |) w,

(16)

we then write the equation (15) as w x1 − αwwξ − βwξξξ = 0,

(17)

where α = sgnΛ0 = ±1, β = Λ1 /2. This new equation is the K-dV equation and once a solution for w is derived from (17) for a given initial value of the form w(0, ξ) = w0 (ξ), then the first order solutions u1 and v1 can be constructed via (16). The initial condition w0 (ξ) is related with an applied boundary traction creating a symmetric motion in the plate. Note that, if α = −1, then the substitution w → −w, ξ → −ξ, x1 → x1 in (17) reduces it to the following standard form; w x1 + wwξ + βwξξξ = 0.

(18)

If α = +1, the substitution w → w, ξ → −ξ, x1 → x1 yields this form. It is known that this equation has the solitary wave solution (Jeffrey and Kawahara, 1981); & w = w∞ + a sech2 ( a/12β ζ), ζ = ξ − (w∞ + a/3)x1 , (19) if w → w∞ as ζ → −∞ or +∞. Here w∞ represents the uniform state at infinity and a is the amplitude of the solitary wave. For w∞ = 0, we obtain the displacements as; & & u1 = ( 12aβ/|Λ0 |) tanhΘ , v1 = (1 − 2c2T − c2L )(a/|Λ0 |) sech2 Θ y, Θ = a/12β ζ. They respectively represent a shock wave and a solitary wave propagating along the positive x−axis with the same speed. References Ahmetolan S., Teymur M. (2003) it Nonlinear modulation of SH waves in a two layered plate and formation of surface SH waves, Int. J. Non-Linear Mechs. 38 1237-1250. Ewing W. M., Jardetsky W. S., Press F. (1957) Elastic Waves in Layered Media, McGraw Hill, New York. Farnell G. W. (1978) it Types and properties of surface waves, Acoustic Surface Waves, (Ed: Oliner A. A), 24 13, Springer, Berlin. Fu Y. B. (1994) On The propagation of nonlinear travelling waves in an incompressible elastic plate, Wave Motion 19 271-292. Jeffrey A., Kawahara T. (1981) Asymptotic Methods in Nonlinear Wave Theory, Pitman, Bostan. Johnson R. S. (1997) A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press. Mayer A. P. (1995) Surface acoustic waves in nonlinear elastic media, Physics Reports 256 (4&5).

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Morgan D. P. (1998) History of SAW devices, IEEE Intl. Frequency Control Symposium, 3450-3458. Teymur M. (1988) it Non-linear modulation of Love waves in a compressible hyperelastic layered half space, Int. J. Eng. Sci. 26 907-927. Teymur M. (1996) Small but finite amplitude waves in a two-layered incompressible elastic medium, Int. J. Eng. Sci. 34 227-241.

QUENCHING OF SELF-EXCITED VIBRATIONS IN A SYSTEM WITH TWO UNSTABLE VIBRATION MODES ∗ Aleˇs Tondl,1 Radoslav Nabergoj2 and Horst Ecker3 1 Zborovsk´ a 41, Prague 5, CZ-l5000 Czech Republic 2 University of Trieste, Department of Naval Architecture, Ocean and Environmental Engineering, I-34127 Trieste, Italy 3Vienna University of Technology, Institute of Mechanics and Mechatronics, A-1040 Vienna, Austria

Abstract. A self-excited chain system consisting of three-mass is investigated. The equilibrium position of the system is unstable in two vibration modes. Two of the masses are self-excited by a Van der Pol type self-excitation, the third mass is positively damped and parametrically excited due to a stiffness variation of its elastic mounting. The results from numerical simulations show that single mode vibrations prevail, either with the first or the third mode. These vibrations are locally stable and there exist two domains of attraction. Only one mode vibration can be partly or fully suppressed with single-harmonic variation of spring stiffness at parametric anti-resonance frequency. Key words: self-excitation, Van der Pol type, parametric excitation, stiffness variation

1. Introduction This study investigates a three-mass system with nonlinear self-excitation forces acting on two masses. The equilibrium position is unstable for two vibration modes, the first and the third mode, while the second mode is stable. From this system the following vibrations can be expected in principal: single-frequency vibrations with the first or the third mode, double-frequency vibrations or eventually chaotic vibrations. It is known from (Tondl, 1998) and other references that parametric excitation due to a periodic variation of a spring stiffness can be used for vibration quenching. The suppression effect to a certain vibration mode can be achieved at a Parametric Excitation (PE-)frequency given by the difference of the natural frequencies corresponding to the vibrating mode due to self-excitation and to the natural frequency of the stable vibration mode. Single-mode vibration can be even fully cancelled and therefore this phenomenon was called parametric ∗

This work was partly funded by the Austrian Science Fund under Grant P-16248 487

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anti-resonance. Several questions arise on what may happen when parametric excitation is applied to the considered system, where a multi-mode instability of the equilibrium position occurs. Special attention will be given to PE-frequencies close to Ω2 − Ω1 (for suppressing the first mode vibration) and to Ω3 − Ω2 (for suppressing the third mode vibration). 2. Equations of motion and basic considerations Let us consider a three-mass system as shown in Figure l. The masses are connected by springs of stiffness k j ( j = 1, 2, 3). The upper masses m1 , m2 are self-excited, e.g. by a constant flow, which can be described by a Van der Pol model. Mass m3 represents a foundation mass being elastically mounted and damped by linear viscous damping. The stiffness of the elastic mounting has constant and periodic components, e.g. k3 = k0 (1 + ε cos ωt), which represents the source of parametric excitation. The deflections from equilibrium positions are y j ( j = 1, 2, 3). Thus, the considered system is governed by the following Figure 1. Schematic representation of the system with parametric excitation at foundation equations m1 y¨ 1 − b1 − d1 y21 y˙ 1 + k1 (y1 − y2 ) = 0 , m2 y¨ 2 − b2 − d2 y22 y˙ 2 − k1 (y1 − y2 ) + k2 (y2 − y3 ) = 0 , (1) m3 y¨ 3 + b3 y˙ 3 − k2 (y2 − y3 ) + k3 y3 = 0 . For the sake of simplicity let us assume that k1 = k2 = k0 = k,√m1 = m2 = m, m3 = m/2. Using the time transformation ω0 t = τ, where ω0 = k/m, the following equations are obtained: 2 y 1 − β1 − δ1 y1 y1 + y1 − y2 = 0 , 1 2 (y1 + y3 ) = 0 , y (2) 2 − β2 − δ2 y2 y2 + y2 − 2 1 y 3 + κy3 − y2 + (1 + ε cos ντ) y3 = 0 , 2 where βk = bk /mk ω0 , δk = dk /mk ω0 , for k = 1, 2, and κ = b3 /m3 ω0 , ν = ω/ω0 .

QUENCHING OF SELF-EXCITED VIBRATIONS

489

It is useful to transform the sets of differential equations (1) and (2) into the quasi-normal form of xs

+

Ω2s x s

3 + Θ sk xk + Q sk xk cos ντ = 0 ,

(s = 1, 2, 3) .

(3)

k=1

For both alternatives this transformation reads: y1 = x1 + x2 + x3 , √ √ 3 3 x1 − x3 , y2 = (4) 2 2 1 1 y3 = x1 − x2 + x3 . 2 2 In the case when Θ j j is negative and Θ ss positive, then the vibration mode corresponding to the natural frequency Ω j can be stabilized in the interval, see (Tondl, 1998): $ $ $ $$ $Ω − Ω $$ − ∆ < ν < $$Ω − Ω $$ + ∆ , ( j, s = 1, 2, 3; s j) (5) j

s

js

where ∆ js =

j

s

js

1/2 Θ j j + Θ ss Q js Q s j + Θ Θ . & j j ss 2 |Θ j j Θ ss | 4Ω j Ω s

(6)

The following conditions must be met to achieve full vibration suppression: Θ j j + Θ ss > 0 , Q js Q s j + Θ j j Θ ss > 0 . 4Ω j Ω s For the considered system (2) the following relations hold:

(7) (8)

1, 1 − (2β1 + 3β2 ) + κ , Θ22 = (−β1 + 2κ) , (9) 6 3 ε ε , Q21 = Q23 = . (10) Q12 = Q32 = 12 6 From these conditions it is evident, that Θ22 would be the first to become positive when starting from zero and increasing the dimensionless damping parameter κ. Coefficients β1 , β2 and κ were chosen such that Θ11 + Θ22 and Θ33 + Θ22 will be of positive value. Θ11 = Θ33 =

490

TONDL, NABERGOJ, ECKER

Figure 2. Extreme values [x1 ], [x2 ] and [x3 ] of normal mode vibration amplitudes x1 , x2 and x3 versus applied parametric excitation frequency ν for κ = 0.15, β1 = 0.05, β2 = 0.05, δ1 = 0.50, δ2 = 0.50, and ε = 0.25. (Ω1 = 0.37, Ω2 = 1.0, Ω3 = 1.37)

3. Numerical results Numerical simulation results are presented in diagrams showing the extreme values of xk (k = 1, 2, 3) (denoted as [xk ]) in dependence of the parametric excitation frequency ν. For this ν was increased by small steps, starting from zero up to a maximum value. Also a second run was calculated by decreasing the PE-frequency to check the nonlinear behavior and to look for hysteresis effects. Figure 2 shows a result for a system with parameters as defined in the figure annotation. Note that normal mode deflections are plotted in this and the following diagram. Within a broad interval around ν = Ω3 − Ω2 the single-frequency vibration with the third mode is suppressed, but the first mode vibration is initiated. For

QUENCHING OF SELF-EXCITED VIBRATIONS

491

ν Ω2 − Ω1 the first mode vibration is suppressed, but the third mode vibration is initiated. Strong parametric resonances occur, especially for the first mode x1 : the parametric resonance of the first kind at ν ≈ 2Ω1 and the combination resonances at ν = Ω1 + Ω2 and ν = Ω1 + Ω3 can easily be seen. As mentioned earlier, a system with double component parametric excitation has been investigated also. This system is governed by equations: 2 y 1 − β1 − δ1 y1 y1 + y1 − y2 = 0 , 1 2 (y1 + y3 ) = 0 , (11) y 2 − β2 − δ2 y2 y2 + y2 − 2 1 y 3 + κy3 − y2 + {1 + ε [cos (Ω2 − Ω1 ) τ + cos (Ω3 − Ω2 ) τ]} y3 = 0 . 2 Normal mode deflections [x1 ], [x2 ], [x3 ] have been calculated for ε ranging from zero to (a rather high value of) 0.5. From Figure 3 one can see that with increasing ε also [x1 ] increases from zero value, but [x3 ] starts from a non-zero value and decreases. Close to ε = 0.2 [x3 ] reaches its minimum close to 0.1. Further increase of ε results in vibrations having higher extreme values than for ε = 0. Single-frequency vibrations with the third vibration mode turn into vibrations resulting from more than one vibrational mode. This means that full suppression cannot be achieved. From this it follows that the double-frequency parametric excitation using stiffness variation of one spring cannot fully suppress both modes of vibration. For a large amplitude value parametric excitation can even results in more pronounced vibrations than it is the case without parametric excitation. One possibility of quenching both modes of vibration with a single-frequency parametric excitation would be to tune the system so that the relation Ω2 − Ω1 = Ω3 − Ω2 is met, which may not be an easy matter. Another approach could be a combination of passive means together with parametric excitation such that only one mode would be necessary to quench and therefore the single-frequency parametric excitation would be sufficient. 4. Conclusions In the case when the equilibrium position is unstable in two vibration modes then using a single-frequency parametric excitation can fully suppress only one mode of self-excited vibrations, even when conditions (7) and (8) are met at certain frequency of parametric excitation. When just condition (7) is met, then only partial suppression can be expected. The application of a two-frequency component parametric excitation corresponding to different parametric anti-resonances was not successful with respect to full vibration cancelling and needs further investigations.

492 Vibration Amplitude , [x1]

TONDL, NABERGOJ, ECKER 2.0 Excitation Up

1.0 0.0 -1.0 Excitation Down

-2.0

Vibration Amplitude , [x2]

0.0

0.2

0.3

0.4

0.5

0.2

0.3

0.4

0.5

0.2

0.3

0.4

0.5

2.0 Excitation Up


-2.0 0.0

Vibration Amplitude , [x3]

0.1

0.1

2.0 Excitation Up


-2.0 0.0

0.1

Parametric Excitation Amplitude ,

ε

Figure 3. Extreme values [x1 ], [x2 ] and [x3 ] in dependence of the PE-amplitude ε for κ = 0.15, β1 = 0.05, β2 = 0.05, δ1 = 0.50, δ2 = 0.50, and ν1 = Ω2 − Ω1 , ν2 = Ω3 − Ω2

References Ecker H., Tondl A. (2000) Suppression of flow-induced vibrations by a dynamic absorber with parametric excitation, Proc. 7th Int. Conference on Flow-Induced Vibrations - FIV. Nabergoj R., Tondl A. (2001) Self-excited vibration quenching by means of parametric excitation, ˇ Acta Technica CSAV 46 107-211. ˇ Tondl A. (1998) To the problem of quenching self-excited vibrations, Acta Technica CSAV 43 109-116. ˇ Tondl A. (2002a) Two parametrically excited chain systems, Acta Technica CSAV 47 67-74. ˇ Tondl A. (2002b) Three-mass self-excited systems with parametric excitation, Acta Technica CSAV 47 165-176. Tondl A., Ecker H. (2003) On the problem of self-excited vibration quenching by means of parametric excitation, Archive of Applied Mechanics 72 923-932.

FREE VIBRATION ANALYSIS OF LAMINATED PLATES USING FIRST-ORDER SHEAR DEFORMATION THEORY ¨ Umut Topal and Umit Uzman Karadeniz Technical University, Department of Civil Engineering, 61080, Trabzon, Turkey

Abstract. This paper deals with free vibration analysis of simply supported laminated composite plates using first-order shear deformation theory (FSDT). The displacement field of a laminated composite plate is given for FSDT. The numerical studies are conducted to determine the effect of width-to-thickness ratio, degree of orthotropy, fiber orientation, aspect ratio on the nondimensionalized fundamental frequency for laminated composite plates. Also, the effect of shear deformation, rotatory inertia and shear correction coefficient on the nondimensionalized fundamental frequency is examined. A MATLAB code is written for free vibration of laminated plates. However, the problem is modeled using finite element package program ANSYS for different meshes. Finally, the results are given in graphical and tabular form and compared. Key words: laminated composite plate, first-order shear deformation theory, free vibration analysis, non-dimensional fundamental frequency

1. Introduction The application of laminated fiber reinforced composites as structural plate elements or members has been steadily increasing in civil, aerospace, mechanical and marine structures, etc. due to their advantages of high stiffness and strengthto-weight ratios (Guo et al., 2002). An understanding of the composite plates’ behaviour under dynamic loading is essential because the loading can cause severe damage in composite plates, such as, matrix cracking, fibre cracking and delaminating (Latheswary et al., 2004). Determining the free vibration characteristics of a structural system often appears to be the fundamental task in dynamic analysis. Many researchers have investigated the vibration of laminated plates in the past (Dai et al., 2004), (Leissa and Narita, 1989), (Afshari and Widera, 2000), (Ray, 2003),(Bert and Chen, 1978), (Dawe and Wang, 1993).

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2. First order displacement fields Based on the first-order shear deformation plate theory, the displacement field of a laminated plate is expressed as u(x, y, z) = uo (x, y) + zφ x (x, y)

(1)

v(x, y, z) = vo (x, y) + zφy (x, y)

(2)

w(x, y, z) = wo (x, y)

(3)

The strain-displacement and stress-strain relations for any ply are given by ε = LU, σ = Qε

(4)

where U T = [u v w φ x   ∂ x  0  L =  0  0  ∂y

0 ∂y 0 0 ∂x

φy ], εT = [ xx

yy

γyz

γ xz

γ xy ]

   Q11 Q12 0 0 Q16 0 z∂ x 0    0 Q26  Q12 Q22 0 0 0 z∂y    ∂y 0 1  , Q =  0 0 Q44 Q45 0   ∂ x 1 0   0 0 Q45 Q55 0  0 z∂y z∂ x Q16 Q26 0 0 Q66

       

(5)

(6)

in which σi j , εi j , Qi j are the stress, strain components and plane-stress reduced constitutive matrix, respectively. 3. Numerical results and discussions The numerical results are given for laminates having the following properties. E1 = 25x106 N/m2 , E2 = 1x106 N/m2 , ν12 = 0, 25 G12 = G13 = 0, 5x106 N/m2 , G23 = 0, 2x106 N/m2 , ρ = 1kg/m3 3.1. EFFECT OF WIDTH-TO-THICKNESS RATIO

Simply supported four-layer cross-ply and angle-ply laminates with symmetric and anti-symmetric arrangement and having different width-to-thickness ratios are analysed. The variation of non-dimensional fundamental frequency is shown in Fig. 1. As seen, the frequency increases abruptly as b/h increases in the thick plate, but the increase is small beyond b/h=20. The value of non-dimensional fundamental frequency becomes practically constant for very thin plates. Also,

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FREE VIBRATION ANALYSIS OF LAMINATED PLATES

symmetric lay-up has higher non-dimensional frequency as compared to antisymmetric lay-up in the case of cross-ply laminates whereas the reverse is the case with angle-ply laminates. Also in Fig. 1, the non-dimensional fundamental frequency obtained from ANSYS for (10x10 and 15x15 meshes) is shown for (0/90/90/0) laminate. 16

23 21

14

19

Z

17

12

15 13

10

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8

9 0

20

40

60

80

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0

120

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0/90/0/90 45/-45/45/-45

0/90/90/0 45/-45/-45/45

40

60

80

100

120

0/90/90/0,FSDT,RI=0 0/90/90/0,ANSYS,10X10 0/90/90/0,ANSYS,15x15

b/h Figure 1. Variation of non-dimensional fundamental frequency with b/h ratio

3.2. EFFECT OF MATERIAL ANISOTROPY

Three-layer cross-ply (0/90/0) and angle-ply (45/-45/45) laminates with b/h=10 and 100 are analysed for this case. An increase of E1 /E2 ratio, keeping same E2 , leads to an increase in the fundamental frequency in the case of both cross-ply and angle-ply laminates in thin and thick plates (Fig. 2). This is due to the increase in stiffness with increase in E1 . The rate of increase is more in the case of thin plates. Also in Fig. 2, the non-dimensional fundamental frequency obtained from ANSYS is shown for (0/90/0)laminate. 28

15

25 13

22

Z

19

11

16 13

9

10 7

7 0

10

20

30

40

50

0/90/0(b/h=10)

0/90/0(b/h=100)

45/-45/45(b/h=10)

45/-45/45(b/h=100)

0

10

20

30

0/90/0,FSDT,RI=0 0/90/0,ANSYS,10x10 0/90/0,ANSYS,15x15

E1 / E 2 Figure 2. Variation of fundamental frequency with material anisotropy

40

50

496

TOPAL, UZMAN

3.3. EFFECT OF FIBRE ORIENTATION

Symmetric (α/-α/-α/α) and antisymmetric (α/-α/α/-α) laminates with the fibre orientation varying from 0o to 45o with b/h=10 and 100 are analysed. A change in fibre orientation angle leads to an increase in the fundamental frequency in the (b/h=10) and (b/h=100) plates as shown in Fig. 3. The difference being more for higher values of α . The non-dimensional fundamental frequency obtained from ANSYS is shown for (α/-α/-α/α) laminate (b/h=10) in Fig. 3.

Z

22

17

20

16

18

15

16

14

14

13

12

12 0

15

30

45

0

15

Į/-Į/-Į/Į(b/h=10) Į/-Į/-Į/Į(b/h=100) Į/-Į/Į/-Į(b/h=10) Į/-Į/Į/-Į(b/h=100)

30

45

Į/-Į/-Į/Į,FSDT,RI=0 Į/-Į/-Į/Į,ANSYS,10x10 Į/-Į/-Į/Į,ANSYS,15x15

T Figure 3. Variation of fundamental frequency with fibre orientation angle

3.4. EFFECT OF ASPECT RATIO

Cross-ply (0/90/0) laminates with b/h=10 and 100 by varying a/b ratio, keeping the value of ‘b’ constant are analyzed. As seen, the fundamental frequency is found to decrease gradually with increase in aspect ratio (Fig. 4). This is due to the decrease in stiffness of the plate with increasing aspect ratio. In Fig. 4, the nondimensional fundamental frequency obtained from ANSYS is shown for (0/90/0) laminate (b/h=10).

Z

17

14

15

12

13

10

11

8

9

6

7

4

5

2 0,8

1

1,2 b/h=10

1,4

1,6

1,8

2

2,2

0,8

b/h=100

a/b

1

1,2

1,4

1,6

1,8

0/90/0,FSDT,RI=0 0/90/0,ANSYS,10x10 0/90/0,ANSYS,15x15

Figure 4. Variation of fundamental frequency with aspect ratio

2

2,2

FREE VIBRATION ANALYSIS OF LAMINATED PLATES

497

3.5. EFFECT OF SHEAR DEFORMATION, ROTATION INERTIA AND SHEAR CORRECTION COEFFICIENT

In Table 1, the effect of shear deformation, rotatory inertia (RI) and shear coefficient (K) on nondimensionalized frequencies of simply supported cross-ply (0/90/0) square plates for a/h=10 is given for different m and n modes. (*) corresponds to K=1 and the second line corresponds to K=5/6. As seen, nondimensionalized frequencies are higher for classical plate theory (CLPT) than those for first order shear deformation (FSDT) theory for both with RI and without RI cases. The effect of the K is to decrease the frequencies, i.e. the smaller K, the smaller is the frequency. The RI also has the effect of decreasing frequencies. TABLE I. Effect of shear deformation, rotatory inertia and shear coefficient on nondimensionalized natural frequencies of simply supported symmetric cross-ply (0/90/0) square plates for a/h=10 m

n

CLPT w/o RI

CLPT with RI

FSDT w/o RI

FSDT with RI

1

1

15,2278

15,1041

12,5931 12,2235

12,5269* 12,163

1

2

22,8772

22,4209

19,4402 18,9422

19,2032 18,7294

1

3

40,2992

38,7377

32,4963 31,4213

31,9219 30,9322

2

1

56,8855

55,7508

33,0978 31.1316

32,9316 30.9916

16 14

Z

12 10 8 0

20

40

60

80

100

0/90/0,FSDT,RI=0 0/90/0,CLPT,RI=0 0/90/0,FSDT,RI0 0/90/0,CLPT,RI0

a/h Figure 5. a/h

Effect of transverse shear deformation and rotary inertia on fundamental frequency for

498

TOPAL, UZMAN

In Fig. 5, the nondimensionalized fundamental frequency versus side-to thickness ratio (a/h) for simply supported symmetric cross-ply (0/90/0) laminates for K=5/6 is given. As seen, the effect of shear deformation decreases with increasing values of a/h. The effect of rotatory inertia is negligible in FSDT, whereas it is significant in CLPT only for very thick plates. 4. Conclusions In this paper, free vibration analysis of laminated plates based on first-order shear deformation theory is presented. The non-dimensional fundamental frequency of vibration is found to increase with increase in width-to-thickness ratio, material anisotropy and angle of fibre orientation. The fundamental frequency of vibration decreases gradually with increase in aspect ratio. The nondimensionalized natural frequencies are higher for classical plate theory (CLPT) than those first order shear deformation (FSDT) theory for both with RI and without RI cases. The effect of the K is to decrease the frequencies, i.e. the smaller K, the smaller is the frequency. The RI also has the effect of decreasing frequencies. The effect of rotatory inertia is negligible in FSDT, whereas it is significant in CLPT only for very thick plates. References Afshari P., Widera G. E. O. (2000) Free vibration analysis of composite plates Transactions of the ASME, 122 390-398. Bert C. W., Chen T. L. C. (1978) Effect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plates, Int. J. Solids Struct. 14 465-473. Dai K. Y., Liu G. R., Lim K. M., Chen X. L. (2004) A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates, Journal of Sound and Vibration, 269 633-652. Dawe D. J., Wang S. (1993) Free vibration of generally-laminated, shear-deformable, composite rectangular plates using a spline Rayleigh-Ritz method, Structural Mechanics 25 77-87. Guo M., Harik I. E., Ren W. X. (2002) Free vibration analysis of stiffened laminated plates using layered finite element method. Structural Engineering and Mechanics, 14 245-262. Latheswary S., Valsarajan K. V., Rao Y. V. K. S. (2004) Free vibration analysis of laminated plates using higher-order shear deformation theory, IE(I) Journal-AS, 85 18-24. Leissa A. W., Narita Y. (1989) Vibration studies for simply supported symmetrically laminated rectangular plates, Composite Structures, 12 113-132. Ray M. C. (2003) Zeroth-order shear deformation theory for laminated composite plates, Transactions of the ASME, 70 374-380.

THE GENERALIZED BOLOTIN METHOD AS AN ALTERNATIVE TOOL FOR COMPLETE DYNAMIC STABILITY ANALYSIS OF PARAMETRICALLY EXCITED SYSTEMS: APPLICATION EXAMPLES ¨ ur Turhan Ozg¨ ˙ Faculty of Mechanical Engineering, Istanbul Technical University, Gümüs¸uyu ˙ 34437, Istanbul, Turkey

Abstract. Stability analysis of the solutions of linear differential equations with periodic coefficients or Mathieu-Hill equations constitutes a topic of constant research interest. Many methods have been devised for this purpose, among which is the generalized Bolotin method developed by the present author. This study presents some application examples of that method, by comparing them with the results obtained via monodromy matrix method. Key words: Mathieu-Hill equations, stability, infinite determinant method

1. Introduction Many problems of mathematical physics lead to linear ordinary differential equations with periodic coefficients (or Mathieu-Hill equations), which can be written x¨ +

1 1 C(τ)˙x + 2 K(τ)x = 0; C(τ) = C(τ + 2π), K(τ) = K(τ + 2π) ω ω

(1)

where τ = ωt, ω = 2π/T , T=Period in t domain. Systems whose behaviour is governed by an equation of form (1) are referred to as parametrically excited systems and the theory of that class of equation is called Floquet theory (Cesari, 1959). Parametrically excited systems have special resonance conditions and the important problem of determining these conditions is called dynamic stability analysis. It is customary to differentiate between two kinds of resonance: Parametric resonances encountered in both single and multi-d.o.f systems and combination resonances peculiar to multi-d.o.f ones. The problem of devising comprehensive methods for dynamic stability analysis of Mathieu-Hill equations remained a challenge since more than one century. References to the existing methods may be found in (Nayfeh and Mook, 1979) and (Seyranian and Mailybaev, 2003). We also cite (Guttalu and Flashner, 1996) and (Pernot and Lamarque, 2001). 499 .


500

TURHAN

The generalized Bolotin method is a complete dynamic stability analysis method for non-canonical systems, proposed by the present author (Turhan, 1998). It consists in a generalization of the well-known Bolotin method (Bolotin, 1964), which is an infinite determinant boundary trace method valid for single-d.o.f systems (parametric resonances) only, so as to cover multi-d.o.f systems (combination resonances) too. Here, we present some application results of that method and compare them to the results of another, essentially numerical scatter plot method known as monodromy matrix method. 2. The method According to the Floquet theory, a fundamental solution of Eq. (1) can be written x(τ) = q(τ) · eρτ ; q(τ) = q(τ + 2π)

(2)

where the constant ρ is called a Floquet exponent. Inserting Eq. (2) into Eq. (1), introducing the complex Fourier series expansions ∞

x(τ) = eρτ

Dk eikτ , C(τ) =

m

C p eipτ , K(τ) =

p=−m

k=−∞

m

K p eipτ

(3)

p=−m

and requiring harmonic balance, one is led to an infinite set of linear homogeneous algebraic equations for the unknown Fourier coefficients’ vectors Dk , the solvability condition of which yields '

( 1 1 1 det ρ I + ρ(E0 + E1 ) + (F0 + F1 ) + 2 F2 ) = 0 ω ω ω 2

(4)

where Ei , Fi ’s are infinite dimensional hyper-matrices made up of nxn sub-matrices k,q k,q k,q k,q 2 Ek,q 0 = 2ikIδkq , E1 = C p F0 = −k Iδkq , F1 = ikC p , F2 = K p

(5)

where p=k-q, δkq is the Kronecker delta and the superscripts k and q refer to the hyper-row and column indices. Given the value of the Floquet exponent ρ on the stability boundaries, Eq. (5) can be used to calculate the corresponding ω values. It is known that on parametric resonance boundaries a certain s-th exponent takes the value ρ s = 0 (a harmonic resonance boundary) or ρ s = 2i (a sub-harmonic resonance boundary), while in a non-canonical system, on a combination resonance boundary a certain pair ρ s , ρt of Floquet exponents take values so that ρ s + ρt = 0 (Turhan, 1998). Whence, substituting ρ = 0 into Eq. (5), one has for harmonic resonance boundaries ' ( det F0 +

and substituting ρ =

i 2

1 1 F1 + 2 F2 = 0 ω ω

(6)

one has for sub-harmonic parametric resonance ones,

( '' ( 1* i 1 i + 1 det F0 + E0 − I + F1 + E1 + 2 F2 = 0. 2 4 ω 2 ω

(7)

GENERALIZED BOLOTIN METHOD AND APPLICATION EXAMPLES 501

As for the combination resonance boundaries, first linearize the matrix polynomial of Eq. (4) in ρ, to obtain the infinite Hill’s determinant of the problem '' ( ( 1 1 det U0 + U1 + 2 U2 − ρI = 0; U0 = ω ω

−E0 I

−F0 0

, U1 =

−E1 0

−F1 0

, U2 =

0 −F2 0 0

,

(8)

then introduce the bialternate sum matrices B(Ui ) of the matrices Ui , whose eigenvalues are the sums of the eigenvalues of the argument matrix taken in pairs (Fuller, 1968), and write ' ( 1 1 det B(U0 ) + B(U1 ) + 2 B(U2 ) = 0 ω ω

(9)

as a condition for ρ s + ρt = 0. ω values on the stability boundaries can be approximately calculated from finite dimensional portions of the determinants of Eqs. (6), (7) and (9) (obtained by truncating the first Fourier series of Eq. (3) at k=K), by solving an eigenvalue problem. Details of the method and some notes on its implementation can be found in (Turhan, 1998) and (Turhan and Bulut, 2005). Let one be contented here with noting that some of the ω values calculated through Eq. (9) must be eliminated as they correspond to unconverged ρ s,t values. 3. Application examples Here we present some application examples of the generalized Bolotin method and compare its results to those of the monodromy matrix method. The former method is implemented by means of a special FORTRAN code and the latter is implemented by using the MATLAB package. As a first example, we consider a system of two coupled damped Mathieu equations depending on a parameter λ, '

( '' ( ' ( ( 0.1 0 0.5 0 0.4 λ x¨ + x˙ + + cos τ x = 0, 0 0.1 0 1.5 λ 0.4

(10)

first studied by (Szemplinska-Stupnicka, 1978). That problem has already been considered by the author (Turhan, 1998), but the developments in computer capacities made it possible finer approximations to be obtained today. Accordingly, the generalized Bolotin method is implemented by truncating the Fourier series of Eq. (3) at K=30 when determining the parametric resonance boundaries through Eqs. (6) and (7), and at K=6 when determining the combination resonance boundaries through Eq. (9). The result is presented in Fig. 1a as a stability chart constructed on the ω − λ parameter plane, where different shadings are applied for different kinds of instability zones. Fig. 1b depicts the same chart as obtained via the monodromy matrix method. It can be seen that the two charts are in perfect accordance. Notice in Fig. 1a the existance of three summation type combination resonance regions

502

TURHAN

√ located at the vicinites of (ω +ω ), (ω +ω )/2 and (ω +ω )/3 where ω = 0.5 1 2 1 2 1 2 1 √ and ω2 = 1.5 are the natural frequencies of the system. As a second example we consider the dynamic stability of the viscoelastic coupler of an otherwise rigid, offset slider-crank mechanism with a crank of radius r2 rotating at constant rate ω2 , and a slider with mass m4 . The coupler is a uniform rod with cross section A, length , flexural rigidity EJ and mass m3 made of a Kelvin-Voigt material so that the stress (σ)-strain (ε) relationship reads σ = E(ε + ηε). ˙ The system is characterized by the dimensionless parameters ω = ω2 /ω∗ , & ζ = ηω∗ , λ = m4 /m3 , µ2 = r2 /, µ = e/ where ω∗ = EJ/m3 3 and e is the offset of the mechanism. When discretisized through Galerkin’s method, the system dynamics can be shown to be governed by a set of coupled, damped Hill’s equation whose homogeneous part has the form ' ( ζ 1 g¨ + E˙g + 2 E + P(λ, µ2 , µ, ; τ) g = 0 (11) ω ω where the matrix P is 2π-periodic in τ = ωt. See (Turhan, 1996) for the details of the formulation. Here, two Galerkin terms are used in discretization so that the system is reduced to a 2-d.o.f one. Figure 2a and b show the stability analysis results on the crank speed-offset plane for a mechanism with = 0.3 m, µ2 = 0.3, λ = 0.5 and ζ = 0.01. Eight harmonics are considered in expanding the matrix P, and the problem is truncated at K=16. One sees that the results of the two methods agree well. No combination resonance is encountered in this problem, but interesting overlapping of different resonance regions exist.

Figure 1. The Szemplinska-Stupnicka problem (a) Generalized Bolotin method (b) Monodromy matrix method

GENERALIZED BOLOTIN METHOD AND APPLICATION EXAMPLES 503

Figure 2. Stability of the coupler of a slider-crank mechanism (a) Generalized Bolotin method (\\\ : harmonic, /// : subharmonic resonance region) (b) Monodromy matrix method

A number of further examples can be found in (Turhan and Bulut, 2005) where the set of coupled, damped Hill’s equations + ζ 1 * δ g¨ + Λ4 g˙ + 2 Λ4 − β20 (1 + sinτ)2 · (αA + B + I) g = 0 (12) ω 2 ω is considered. Figs. 3a,b are reproduced from Fig. 8 of that work. In Fig. 3a, a 5-d.o.f model and K=10 are considered for parametric stability analyses of Eqs. (6) and (7), while a 3-d.o.f model and K=3 are taken in the determination of combination resonance regions through Eq. (9). The monodromy matrix method

Figure 3. Stability of a rotating beam (a) Generalized Bolotin method (\\\ : harmonic, /// : subharmonic, ≡ : combination resonance region) (b) Monodromy matrix method

504

TURHAN

is applied to the 3-d.o.f model. Two strong combination resonance regions, which can be shown to correspond to ω1 + ω2 and ω1 + ω3 , are visible on Fig. 3a, and the results of the two methods are seen to be in perfect harmony. 4. Conclusions The Generalized Bolotin method is applied to 2, 3 and partially 5-d.o.f system examples. It is seen that its results check well with those of the monodromy matrix method. The method, possess the obvious advantages of being a boundary tracing method: It is less time consuming and less exposed to the risk of missing narrow regions of a stability chart as compared to scatter plot methods. Also, it provides direct information on the bifurcation values of the considered parameters, whose accurate knowledge is important in most applications. As it treats different types of resonance regions separately, the method has also the advantage of pre-classifying the instability regions; a feature which facilitates getting insight into the problems. The limitations of the method come from the high dimensionality of the involved matrices and of the elimination procedure mentioned in Sec. 2. These make of the combination resonance boundaries calculation problem of Eq. (9) a time-consuming one, although not comparable with the excessive time consumption (and sometimes failure) of the monodromy matrix method. References Bolotin, V. V. (1964) The Dynamic Stability Of Elastic Systems, Holden-Day, San Francisco. Cesari, L. (1959) Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer-Verlag, Berlin. Fuller, A. T. (1968) Conditions for a matrix to have only characteristic roots with negative real parts, Journal of Mathematical Analysis and Applications 23, 71–98. Guttalu, R. S. and Flashner, H. (1996) Stability study of a periodic system by a period-to-period mapping, Applied Mathematics and Computation 78, 123–135. Nayfeh, A. H. and Mook, D. T. (1979) Nonlinear Oscillations, Wiley, New York. Pernot, S. and Lamarque, C. (2001) Journal of Sound and Vibration, A wavelet-Galerkin procedure to investigate time-periodic systems: Transient vibration and stability analysis 245, 845–875. Seyranian, A. P. and Mailybaev, A. A. (2003) Multiparameter Stability Theory With Mechanical Applications, World Scientific, New Jersey. Szemplinska-Stupnicka, W. (1978) The generalized harmonic balance method for determining the combination resonance in the parametric dynamic systems, Journal of Sound and Vibration 79, 581–588. ¨ (1996) Dynamic stability of four-bar and slider-crank mechanisms with viscoelastic Turhan, O. (Kelvin-Voigt model) coupler, Mechanism and Machine Theory 31, 77–89. ¨ (1998) A generalized Bolotin’s method for stability limit determination of parametrically Turhan, O. excited systems, Journal of Sound and Vibration 216, 851–863. ¨ and Bulut, G. (2005) Dynamic stability of rotating blades (beams) eccentrically clamped Turhan, O. to a shaft with fluctuating speed, Journal of Sound and Vibration 280, 945–964.

COUPLING EFFECTS BETWEEN SHAFT-TORSION AND BLADE-BENDING VIBRATIONS IN ROTOR-BLADE SYSTEMS ¨ ur Turhan and Gökhan Bulut Ozg¨ ˙ Faculty of Mechanical Engineering, Istanbul Technical University, Gümüs¸suyu ˙ 34437, Istanbul, Turkey

Abstract. Coupling effects between shaft-torsion and blade-bending vibrations in rotor-blade systems are studied. Certain features of the related eigen-analysis problem are underlined. Eigenanalyses are performed for multi-stage rotor-blade system examples. Eigen value-loci-veering phenomena are shown to occur, causing substantial departure of the eigen-characteristics from those obtained via an uncoupled analysis. Key words: rotor-blade systems, turbomachinery, coupled vibrations

1. Introduction Due to their very important applications, accurate prediction of the vibration characteristics of rotor-blade systems is of crucial importance. The common practice is to consider the vibrations of the shaft and those of the blading separately through uncoupled models. Some exceptions are (Loewy and Khader, 1984), (Crawley et al., 1986), (Chun and Lee, 1996), (Okabe et al., 1991), (Huang and Ho, 1996) and (Chatelet et al., 2005). All these studies imply that coupling between shaft and blade vibrations is of substantial consequences on the vibration characteristics of rotor-blade systems. This study focuses on the coupling between shaft-torsional and blade-(in plane) bending vibrations through an idealized model consisting in a torsionally elastic shaft carrying a number of rigid discs, which in their turn, carry a number of identical blades modelled as Euler-Bernouilli beams (Fig. 1a,b). The equations of motion of this multi-continuous-body system is obtained via a mixed finite element-Galerkin method. The method is introduced in (Turhan and Bulut, 2005) and will briefly be outlined below. The related eigen-analysis problem reveals that the system has two different kinds of normal mode motions. These are referred to as “coupled shaft-torsion-blade-bending modes” and “rigid shaft modes” by the authors. Numerical examples show that the eigen-frequencies pertaining to the coupled shaft-torsion-blade-bending modes are subject to eigenvalue-loci-veering 505 .


506

TURHAN, BULUT

Figure 1.

The system

phenomena, which causes the vibration characteristics of the system to depart considerably from those obtained via an uncoupled analysis. 2. Modelling and analysis Consider Fig. 1, let the shaft carry p rigid disks, the i-th of which carry, in its turn, ki identical blades undergoing bending vibrations in the rotation plane. Let the left end of the shaft rotate at constant rate Ω0 . The equation of the torsional motion of the shaft acted upon by the reaction torque T i of the i-th disk, can be written via finite element method as d2 θ GI p + · K · θ = e si ·T i , 0 dt2 i=1 p

ρ0 I p 0 · M ·

(1)

where ρ0 =mass density, G=shear modulus, I p =polar area moment of inertia, 0 = length of the shaft, θ is the 2n-dimensional nodal coordinates vector, e j is the j-th 2n-dimensional unit vector, si is the attachment station of the i-th disk, and M and K are the numerical parts of the 2nx2n, global mass and stiffness matrices. Next consider the i-th rigid disk acted upon by the reaction torque −T i of the shaft and the reaction forces −Fi j and torques −Mi j ; j = 1, 2, . . . , ki of the ki blades it carries. Its equation of motion reads i d2 θ ) = −T − (ri Fi j + Mi j ), i dt2 j=1

k

Ji · (eTsi ·

(2)

where ri is the radius and Ji the mass moment of inertia of the disk. Finally consider the j-th blade (beam) of the i-th disk. It can be shown that a linearized and non-dimensionalized equation of motion of its in-plane, transverse vibrations

507

COUPLING EFFECTS IN ROTOR-BLADE SYSTEMS

relative to the frame B; xi j yi j zi j attached to the i-th rigid disk can be given as 2 2 T ¨ v¨ i j + η2i ·viv i j − β {[αi (1 − ui j ) + 0.5(1 − ui j )]vi j − (αi + ui j )vi j + vi j } + (αi + ui j ) · e si · θ = 0 (3)

where ρi = mass density, (EI)i = flexural rigidity, Ai = cross section area and i = x y ∗ 0 length of the beam and ui j = iij , vi j = iij , τ = ω∗1 t, αi = rii , β = Ω ω∗1 , ωi = % ω∗ (EI)i , ηi = ω∗i with overdots and primes representing differentiation w.r.t. τ ρi Ai i4 1 and ui j , respectively. Equation (3) may be approximated by a finite set of ordinary differential equations by means of Galerkin’s method. To this end, inm troduce vi j (ui j , τ) = gi jk (τ) · ϕk (ui j ), use the eigen-functions of a stationary k=1

cantilever as the set of comparison functions ϕk (ui j ), follow the usual procedures of Galerkin’s method to obtain in matrix-vector form ¨ = 0, g¨ i j (τ) + [η2i · Λ4 − β2 · (αi · A + B + I)] · gij (τ) + (αi · c + d) · eTsi · θ(τ)

(4)

where Λ, A and B are mxm matrices and c and d are m-dimensional

1

1 vectors whose elements are defined as Ars = 0 [(1−ui j )ϕr −ϕr ]·ϕ s ·dui j , Brs = 0 [0.5(1−u2i j )ϕr −

1

1 ui j ϕr ] · ϕ s ·dui j , Λrs = λr δrs , cr = 0 ϕr ·dui j , dr = 0 ui j ϕr ·dui j , where λr ’s are the dimensionless eigenfrequencies of a cantilever, δrs is the Kronecker delta, and gi j (τ) is the m-dimensional Galerkin coordinates vector of the related blade. Let one also note that the shearing force Fi j and bending moment Mi j exerted to the j-th beam by the i-th disk can be calculated as Fi j (τ) = −2

(EI)i T (EI)i T e gi j (τ), Mi j (τ) = −2 f gi j (τ), i i2

(5)

where e and f are m-dimensional vectors with elements er = κr λ3r , fr = λ2r with κk = (cosh λk + cos λk )/(sinh λk + sin λk ). To synthesize the equation of motion of the whole system, eliminate first Fi j , Mi j and T i between Eqs. (1), (2) and (5) to obtain, after amening to the time scale τ and non-dimensionalizing   k   p p i      ¨ + µ2 Kθ(τ) − M + γi · e si eTsi  · θ(τ) 2δi ·e si αi ·eT + f T ·  gi j (τ) = 0 i=1

where γi =

i=1

Ji ρ0 I p 0 ,

δi =

ρi Ai i3 ρ0 I p 0 ,

µ=

ω∗0 ω∗1 ,

ω∗0 =

(6)

j=1

%

G . ρ0 02

Combining now Eqs. (4) and

(6) into a single hyper-matrix-vector equation one obtains finally   M00  M  10  M20   ..  . M p0

 ¨ θ   0     ¨ g   1  0       g¨ 2   0   .  .. ..      .     g¨ p   0 0 ··· I   0 I 0 .. .

0 0 I .. .

··· ··· ··· .. .

                   +                 

K00 δ1 K01 δ2 K02 0 K11 0 0 0 K22 .. .. .. . . . 0 0 0

· · · δ p K0p ··· 0 ··· 0 .. .. . . · · · K pp

 θ       g 1         g  2   ..       .      g   p   

       0                0             0  =         ..              .             0 

(7)

508

TURHAN, BULUT

where gi = {gTi1 gTi2 · · · gTiki }T , Kii = blockdiag[Kii ], Mi0 = [MTi0 MTi0 · · ·

MTi0 ]T , K0 j = [K0 j K0 j · · · K0 j ], with

p M00 = M + γi · e si eTsi , Mi0 = (αi · c + d) · eTsi , K00 = µ2 K, i=1 K0 j = −2e s j α j · eT + f T , Kii = η2i · Λ4 − β2 · (αi · A + B + I) .

(8)

It can be shown that Eq. (7) leads to the factorized frequency equation     det   

∆1 B01 ∆2 B02 ··· ∆ p B0p A00 C10 ∆1 D11 + K11 ∆2 B12 ··· ∆ p B1p C20 ∆1 D21 ∆2 D22 + K22 · · · ∆ p B2p .. .. .. .. .. . . . . . ∆1 D p1 ∆2 D p2 · · · ∆ p D pp + K pp C p0 p < 3ki −1 2 =0 det Kii − σ2 I ·

         2   I − σ      

(9)

i=1

−1 where ∆i = ki δi and A00 = M−1 00 K00 , B0 j = M00 K0 j , Ci0 = −Mi0 A00 , Di j = −Mi0 B0 j . Eq. (9) shows that the eigen-analysis problem splits into p+1 independent sub-problems corresponding to two different class of normal mode motions of the system. These are referred to as coupled shaft-torsion-blade-bending modes (first determinant in Eq. (9)) and rigid shaft modes (the remaining p determinants) by the authors. It can be shown that, in the rigid shaft modes, the entire shaft and the blades of all the disks, except one, are at rest and the blades of the remaining disk vibrate in such a manner that their total effect on the disk vanishes.

3. Numerical examples Let first a rotor-blade system with four identical and equally spaced stages be considered where µ = 5, αi = 0.5, γi = 0.1, ∆i = 0.4, ηi = 1; i = 1, 2, . . . , 4. Fig. 2a shows the variation of the eigenfrequencies with the dimensionless rotation speed β as obtained from an uncoupled analysis, while Fig. 2b depicting the results of the coupled analysis outlined above. The horizontal lines in Fig. 2a are the eigen-frequencies of the shaft and the rising lines are those of the blades. Comparing Fig. 2a and b one notes that the uncoupled blade eigen-frequencies also subsist in the coupled case. These frequencies correspond to the rigid shaft modes and are p(ki − 1)-fold, as implied by Eq. (9). The other frequencies on Fig. 2b are those of the coupled shaft-torsion-blade-bending modes. It can be seen that these are subject to eigenvalue-loci-veering phenomena, which makes the coupled frequencies to depart considerably from the uncoupled ones. Next consider a rotor-blade system with four different,equally spaced stages where µ = 5, α1 = 0.4762, α2 = 0.5, α3 = 0.4545, α4 = 0.4348, ∆1 = 0.4,

COUPLING EFFECTS IN ROTOR-BLADE SYSTEMS

509

Figure 2. Eigen-frequencies versus rotation speed:Four identical stages (a, uncoupled; b, coupled)

∆2 = 0.4631, ∆3 = 0.5324, ∆4 = 0.6083, η1 = 1, η2 = 0.9070, η3 = 0.8264, η4 = 0.7561, γ1 = γ2 = γ3 = γ4 = 0.1. The variation of the eigen-frequencies with β is given in Fig. 3a and b. It can be seen that the figures are similar to Figs. 2a and b except there are now four different blade frequencies at each mode, this being a result of the fact that the blades of different stages are now differing. Finally consider an example with four identical stages where µ = 5, β = 10, γi = 0.1, ∆i = 0.4, ηi = 1; i = 1, 2, . . . , 4 and study the variation of the eigen-frequencies with the dimensionless disk radii α. The results are presented in Figs. 4a and b where negative ranges of α, which correspond to systems where the blades are oriented towards the centre of rotation, are also considered. It can be seen that the blade frequencies vanish at a certain negative value of α. This means buckling of the blade, due to compressive effect of the centrifugal forces. Furthermore, the shaft and blade frequencies are again seen to be involved in lociveering phenomena, which makes of the coupled frequency loci something very different than the uncoupled ones.

Figure 3. Eigen-frequencies versus rotation speed:Four different stages (a, uncoupled; b, coupled)

510

Figure 4.

TURHAN, BULUT

Eigen-frequencies versus disk radius:Four identical stages (a, uncoupled; b, coupled)

4. Conclusions Coupling effects between shaft-torsional and blade-bending vibrations of multistage rotor-blade systems are studied. It is shown that the system has two different kind of normal mode motions. These are referred to as rigid shaft modes and coupled shaft-torsion-blade-bending modes by the authors. The coupled shafttorsion-blade-bending modes are shown to be subject to eigenvalue-loci-veering phenomena. It is concluded that coupling must be taken into account if accuracy is required in the calculations. References Chatelet, E., D’Ambrosio, F., and Jacquet-Richardet, G. (2005) Toward global modelling approaches for dynamic analyses of rotating assemblies of turbomachines, Journal of Sound and Vibration 282, 163–178. Chun, S. B. and Lee, C. W. (1996) Vibration analysis of shaft-bladed disk system by using substructure synthesis and assumed modes method, Journal of Sound and Vibration 189, 587–608. Crawley, E. F., Ducharme, E. H., and Mokadam, D. R. (1986) Analytical and experimental investigation of the coupled bladed disk/shaft whirl of a cantilevered turbofan, Journal of Engineering for Gas Turbines and Power 108, 567–576. Huang, S. C. and Ho, K. B. (1996) Coupled shaft-torsion and blade-bending vibrations of a rotating shaft-disk-blade unit, Journal of Engineering for Gas Turbines and Power 118, 100–106. Loewy, R. G. and Khader, N. (1984) Structural Dynamics of Rotating Bladed-Disk Assemblies Coupled with Flexible Shaft Motions, AIAA Journal 22, 1319–1327. Okabe, A., Otawara, Y., Kaneko, R., Matsushita, O., and Namura, K. (1991) An equivalent reduced modelling method and its application to shaft-blade coupled torsional vibration analysis of a turbine-generator set, Journal of Power and Energy 205, 173–181. ¨ and Bulut, G. (2005) Linearly coupled shaft-torsional and blade-bending vibrations in Turhan, O. multi-stage rotor-blade systems, Journal of Sound and Vibration (Submitted).

FORCED VIBRATION OF THE PRE-STRETCHED SIMPLY SUPPORTED STRIP CONTAINING TWO NEIGHBOURING CIRCULAR HOLES Nazmiye Yahnio˘glu and Surkay D. Akbarov Yıldız Technical University, Department of Mathematical Engineering, Davutpa¸sa ˙ Campus, Topkapı, Istanbul, Turkey

Abstract. Within the framework of the Three-Dimensional Linearized Theory of the Elastic Waves in Initially Stressed Bodies under plane-strain state the influence of the initial stretching of the simply supported plate strip containing two circular holes on the stress concentration around the holes caused by the action of the additional uniformly distributed normal dynamical (time-harmonic) forces on the plane of its upper face is studied. For the solution of the problem the FEM is employed. The numerical results on the stress concentration and the influence of the statical initial stresses and the frequency of the additional dynamical external forces frequency to this concentration are presented. Key words: dynamical (time-harmonic) stress field, initial stresses, plate-strip, forced vibration

1. Introduction Investigation on the stress concentration around holes has a wide range of applications in almost all branches of modern industry. There are many monographs, such as Savin (1951) and Savin (1968), that contain the results of these investigations. A review of the above-mentioned research has been given in references Kosmodumianskii (2002), Maksimyuk et al. (2003), Lei et al. (2001) and Hyde et al. (2003). It should be noted that these investigations are being studied continuously by many researches at present. It follows from the analysis of the above-mentioned investigations that among those there are no studies on the influence of the initial stresses arising as a result of an initial stretching or another type of initial loading, on the stress concentration caused by an additional loading in the case in which the superposition principle is not applicable. Here under non-applicability of the superposition principle it is understood that the stress field caused by additional loading depends significantly on the initial loading. The theoretical investigation of the phenomenon corresponding to these cases requires the use of complicated geometrical nonlinear equations of mechanics of the deformable body. However, according to well-known mechanical considerations for the cases where the magnitude of the 511 .


512

˘ YAHN˙IOGLU, AKBAROV

Figure 1.

The geometry of the considered strip.

initial loading is greater than that of the additional loading, these investigations can be carried out within the framework of the Three-Dimensional Linearized Theory (TDLT) (Biot (1965) and Guz (1999)) of the deformable body. In the paper Akbarov et al. (2005), an attempt was made in this field and the influence of the initial tension of a simply supported strip containing a rectangular hole on the stress concentration around the hole caused by bending of the strip under the action of the uniformly distributed normal forces on the plane of its upper face. However, in Akbarov et al. (2005) it was assumed that the initial and additional loading are the statical ones. In the present paper the investigation Akbarov et al. (2005) is developed for the case where the additional loading is dynamical (time-harmonic) one and the strip contains two neigbouring circular holes. Throughout the investigations, repeated indices are summed over their ranges.

2. Formulation of the problem Consider a plate strip containing two neighboring circular holes. The geometry of the plate strip is shown in Fig. 1. The Cartesian coordinate system Ox1 x2 x3 is associated with the strip so as to give Lagrange coordinates in the initial state. Assume that the plate strip occupies the region, {0 ≤ x1 ≤ , 0 ≤ x2 ≤ h, −∞ < x3 < +∞}, and the axis Ox3 is directed normal to the plane of Fig. 1. Suppose that the material of the strip is orthotropic with principal axes Ox1 , Ox2 and Ox3 . Moreover, assume that the strip is simply supported at the ends and in the initial state the uniformly distributed normal stretching forces with intensity q act on these ends. The additional uniformly distributed dynamical (time-harmonic) normal forces with amplitude p ( q) act on the plane of the strip’s upper face. The influence of the initial stretching of the strip the stress concentration around the holes caused by the additional dynamical forces with amplitude p is investigated below. Henceforth all the quantities referred the initial state will be labelled by the superscript (0).

FORCED VIBRATION OF THE PRE-STRETCHED STRIP

513

According to the above, the initial stress state can be determined by the solution of the boundary value problem ∂σ(0) ij ∂x j

= 0,

(0) (0) σ(0) 11 = A11 ε11 + A12 ε22 ,

(0) (0) (0) σ(0) σ(0) 22 = A12 ε11 + A22 ε22 , 12 = 2A66 ε12 ,    ∂u(0) ∂u(0)  $ 1 j  i +  , u(0) $$$ = ε(0) ij 2 x =0; = 0, 2  ∂x j ∂xi  1 $$ $$ $$ (0) $ $ $ σ(0) σ(0) 11 $ x =0 = σ11 $ x = = q, i1 $ x =0;h = 0, 1 1 2 $ (0) $$ σi j n j $ = 0, i; j = 1.2; k = 1, 2 Lk 3 2 L1 = (x1 , x2 )| (x1 − (E + R))2 + (x2 − (HA + R))2 = R2 3 2 L2 = (x1 , x2 )| (x1 − ( − (E + R)))2 + (x2 − (HA + R))2 = R2

(1)

In equation (1) the conventional notation is used. Moreover, in equation (1) L1 and L2 denote the contour of the holes, and n j are the component of the unite normal vector to the contours Lk . To determine the stress state caused by additional dynamical (time-harmonic) loading the following boundary value problem must be solved: ∂ui ∂2 ui ∂ σ ji + σ(0) = ρ0 2 in ∂x j ∂xn ∂t σ11 = A11 ε11 + A12 ε22 , σ22 = A12 ε22 + A22 ε22 , σ12 = 2A66 ε12 , 1 ∂ui ∂u j + , u2 | x1 =0; = 0, εi j = 2 ∂x j ∂xi $$ (0) ∂u1 $$ = 0, σi2 | x2 =h = p eiωt δ2i , σi2 | x2 =0 = 0, σ11 + σ1n $ ∂xn $ x1 =0; $$ (0) ∂ui $$ 2 2 δ1 = 0, δ2 = 1, σ ji + σin $ n j = 0, i; j; n = 1.2; k = 1, 2 (2) ∂xn $Lk In (2) the corresponding equations and relations of the Three-Dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies (TLTEWISB) are written.


514 3. FEM modelling

For the FEM modelling of the boundary value problem (1) the functional # h # h $ $$ 1 (0) (0) (0) $$ (0) $ σi j εi j dx1 dx2 − q u1 $ dx2 − q u(0) Π = 1 $ x1 = dx2 x1 =0 2 0 0 Ω− ΩL1 ∪ ΩL2

(3) is used, where Ω = {0 x1 , 0 x2 h} and ΩLk is the region occupied by the holes. Using the virtual work principle and employing the well-known Ritz technique, FEM modelling of the problem (1) is obtained from the equation 5 6 δΠ(0) = 0. In this case the region Ω − ΩL1 ∪ ΩL2 is divided into certain number rectangular Lagrange family quadratic elements. For around of the holes, the curvilinear triangular finite elements with six nodes are used. The selection of NDOF values follows from the requirements that the boundary conditions should be satisfied with very high accuracy and the numerical results obtained for various NDOFs should converge. For the FEM modelling of the problem (2) the variational principle of TDLT of elasticity given in Akbarov et al. (2005) is used. In this case, first, the sought values are presented as g(x1 , x2 , t) = g(x1 , x2 ) eiωt and for computed values g(x1 , x2 ) the functional # ∂u j 1 2 + ρ0 ω u j u j dx1 dx2 − p u2 | x2 =h dx1 (4) Π= Ti j 2 ∂xi 0 Ω− ΩL1 ∪ ΩL2

is used where

∂ui (5) ∂xn With the same finite elements and with the same arrangements as used for the FEM modelling of the problem (1), the FEM modelling for the problem (2) is formulated from the equation δΠ = 0 . The validity of the algorithm and PC programs are tested by the corresponding known results. T i j = σi j + σ(0) in

4. Numerical results and discussion Consider the case where the geometry of the strip has a symmetry with respect to x1 = /2 and x2 = h/2 . Assume that the material of the strip is a composite consisting of a large number of alternating layers of two materials. Suppose that the material of each layer is isotropic and these layers are located on the planes x2 = costant. In the following, the values related to the matrix and to the reinforcing material will be indicated by the superscripts (1) and (2) respectively.E (k) are

515

FORCED VIBRATION OF THE PRE-STRETCHED STRIP TABLE I. The values of the fundamental frequency Ω2 = E (2) /E (1)

ω2 ρ 2 A22

for various and E (2) /E (1)

0.000

0.0001

0.001

q/E (1) 0.002 0.003

1

0.0367

0.0375

0.0442

0.0516

0.0590

0.0739

0.0888

0.1111

5

0.0653

0.0657

0.0698

0.0742

0.0786

0.0875

0.0964

0.1098

10

0.1077

0.1082

0.1118

0.1159

0.1200

0.1281

0.1362

0.1484

20

0.1893

0.1897

0.1932

0.1970

0.2009

0.2087

0.2164

0.2281

Figure 2.

0.005

0.007

0.010

The influence of the initial stretching to the stress concentration

Figure 3.

The influence of the frequency on the stress concentration

516


Young moduli, are ν(k) are Poisson’s ratios and η(k) are the concentrations of the components in the representative pack. It is known that in the considered case the material of the representative pack (or the material of the strip) within the framework of the continuum approach can be modelled as homogeneous transversal isotropic and its effective mechanical constants Ai j are calculated according, for example, to references Akbarov et al. (2000) and Cristensen (1979). Assume that ν(1) = ν(2) = 0.3, η(1) = η(2) = 0.5, h/ = 0.075, R/ = 0.00625. Introduce the 2 2 dimensionless frequency Ω2 = ωAρ22 and consider the fundamental values of this frequency obtained for various E (2) /E (1) and q/E (1) . Note that the values of q/E (1) characterize the magnitude of the initial stretching. The values of the fundamental frequency are given in Table I and these values does not depend on the distance between the holes, i.e. on the c/R (Fig. 1). According to Table I, the values of the fundamental frequency (denote it by Ω∗ ) increase with q/E (1) and E (2) /E (1) . Below we will assume that Ω < Ω∗ and will analyse the concentration of the stress σθθ (in the corresponding Orθx3 cylindrical coordinate system (Fig. 1)). Figure 2 shows the influence of the initial stretching, i.e. of the q/E (1) to the values of σθθ /p at r = R for the case where c/R = 2, E (2) /E (1) = 5, Ω2 = 0.008 . It follows from the graphs that the absolute maximum values of σθθ /p decrease with q/E (1) . Figure 3 shows the influence of the frequency Ω to the values of σθθ /p which increase monotonically with Ω. The numerical results given in Table II shows the influence of the interaction between the holes to the values of σθθ /p. According to these results, it can be concluded that the influence of the initial stretching of the strip to the values of the σθθ /p becomes more significantly with decreasing distance between the holes. 5. Conclusion In the present paper within the framework of the TLTEWISB under plane-strain state the influence of the initial stretching of the simply supported plate strip containing two circular holes on the stress-concentration around the holes caused by the action of the additional uniformly distributed normal dynamical (timeharmonic) forces on the plane of its upper face has been studied. For the solution of the problem the FEM was employed. As a result of the numerical investigations the following were established: 1. The absolute maximum values of the σθθ /p decrease with initial stretching. 2. An increase in the values of the frequency of the external forces causes to increase in the values of the |σθθ |/p.

517

FORCED VIBRATION OF THE PRE-STRETCHED STRIP

3. The influence of the pre-stretching of the stripto the dynamical(time-harmonic) stress concentration σθθ /p becomes more significantly with decreasing of the distance between the holes. TABLE II. The values of σθθ /p obtained under E (2) /E (1) = 5, Ω2 = 0.008 for various c/R, q/E (1) and θ θ c/R q/E (1) 0.00 18 0

π/8

π/4

3π/8

π/2

5π/8

3π/4

7π/8

π

-3.0404 21.8993 24.6854 -62.1360 -94.3030 -82.4453 -7.5757 5.2369 -3.0433

0.005 -3.0793 14.8341 16.6586 -44.8140 -67.1678 -58.4656 -5.0526 3.0875 -1.8390 12 0.000 -3.0194 19.4773 19.8129 -65.8340 -95.0808 -79.9433 -2.5784 7.9169 -3.0311 0.005 -3.0443 13.0943 13.3289 -47.2814 -67.6756 -56.7577 -1.7103 4.9548 -2.1469 6

0.000 -3.0014 16.3132 13.9976 -69.3742 -95.1827 -76.6646 2.8040 10.7257 -3.0183 0.005 -3.013 10.8343 9.34482 -49.6476 -67.7292 -54.5314 1.8823 6.9045 -2.4642

4

0.000 -3.0014 16.3132 13.9976 -69.3742 -95.1827 -76.6646 2.8040 10.7257 -3.0183 0.005 -3.0130 10.8343 9.34482 -49.6476 -67.7292 -54.5314 1.8823 6.9045 -2.4642

2

0.000 -3.2190 9.4705 7.5990 -68.8237 -92.2121 -72.0739 7.2436 12.8491 -3.1550 0.005 -3.2209 5.8890 4.8470 -49.1649 -65.6003 -51.3561 4.8437 8.3373 -2.8182

1

0.000 -3.7280 4.1388 6.6393 -65.1413 -89.8233 -70.2884 8.1412 13.2205 -3.3056 0.005 -3.7218 1.9806 4.1412 -46.5000 -63.8640 -50.0935 5.4089 8.5467 -3.0206

References Akbarov, S.D., Guz, A.N. Mechanics of Curved Composites, (Kluwer, Dordrecht, The Netherlands 2000). Akbarov, S.D., Yahnioglu, N., Yucel, A.M. (2005) On the influence of the initial tension of a strip with a rectangular hole on the stress concentration caused by additional loading. J. Strain Analysis, 39(6), 615-624.

518


Biot, M.A. Mechanics of Incremental Deformations, (John Wiley, New York 1965). Cristensen, R.M. Mechanics of Composite Materials, (John Wiley, New York 1979). Guz, A.N. Fundamentals of the Three-dimensional Theory of Stability of Deformable Bodies, (Springer-Verlag, Berlin, Germany 1999). Hyde, T.H., Notarangelo, G., Pappalettero, C., Sun, W. (2003) Stress concentration factors in square and circular cross-sectioned bars with axial slots. J. Strain Analysis, 38(3), 247-259. Kosmodumianskii, A.S. (2002) Accumulation of internal energy in multiply connected bodies. Int. Appl. Mechanics, 38(4), 399-422. Maksimyuk, V., Mulyar, V.P., Chernyshenko, I.S. (2003) Stress state of thin spherical shells with an off-centre elliptic hole. Int. Appl. Mechanics, 39(5), 595-598. Lei, G.H., Cherles, W.W. Ng., Rigby, D.B. (2001) Stress and displacement around an elastic artificial rectangular hole. J. Engng. Mechanics, 127(9), 880-890. Savin, G.N., Stress Concentration Around Holes, (Pergomon, Oxford 1951). Savin, G.N. Stress Distribution Around Holes (in Russian), (Naukova Dumka, Kiev, Ukraine 1968).

FREE VIBRATION OF CURVED LAYERED COMPOSITE BEAMS Mustafa Yavuz and M. Ertaç Ergüven ˙ ˙ Faculty of Civil Engineering, Istanbul Technical University, Istanbul, Turkey

Abstract. In practice, fibrous and layered composite beams have periodically and locally curved layers because of the design considerations and manufacturing processes. In this study, the effect of these curvatures and composite material properties to the natural frequencies of the beams is investigated. The periodically curved layered composite material of the considered beam is modelled with the use of the continuum theory proposed by Akbarov and Guz. The free vibration problems are solved by employing the finite element method. Obtained natural frequencies of the beams are presented for the different parameters of the curvature, modulus of elasticity and support condition of the beams. For the case that the ratio of the modulus of elasticity of the layers equals to one and the parameter of the curvature equals to zero, the results converge to natural frequencies of a classical Euler-Bernoulli beam. Results are in good agreement with the literature. Key words: composite beam, finite element method, natural frequency, periodically curved layers

1. Introduction Free vibration analysis of the periodically curved layered unidirectional fibrous composite beams is performed. Model of the imperfection of the layers is based on the continuum theory presented in (Akbarov and Guz, 1991). In literature, Akbarov determined normalized nonlinear mechanical properties of composite materials with periodically curved layers (Akbarov, 1995). Kutug studied free vibrations and buckling loads of composite plates with curved layers (Kutug, 1997). Free vibration analysis and buckling loads of cracked beams resting on an elastic foundation is presented by (Kocer, 1997). Yokoyama studied parametric instability of Timoshenko beams resting on an elastic foundation (Yokoyama, 1988). 2.

Mechanical properties of composite materials with curved layers

The normalized constants are as follows: A011 = µ1 η1 +µ2 η2 +(µ1 +λ1 )η1 +(µ2 +λ2 )η2 −η1 η2

(λ1 − λ2 )2 (1) (λ1 + 2µ1 )η2 + (λ2 + 2µ2 )η1

519 .


¨ YAVUZ, ERGUVEN

520

A012 = λ1 η1 + λ2 η2 − (λ1 − λ2 )η1 η2

[(λ1 + 2µ1 ) − (λ2 + 2µ2 )] [(λ1 + 2µ1 )η2 + (λ2 + 2µ2 )η1 ]

A022 = (λ1 + 2µ1 )η1 + (λ2 + 2µ2 )η2 − η1 η2 A066 =

[(λ1 + 2µ1 ) − (λ2 + 2µ2 )]2 [(λ1 + 2µ1 )η2 + (λ2 + 2µ2 )η1 ]

µ1 µ2 µ1 η2 + µ2 η1

(2)

(3)

(4)

The main constants are A11 (x) = A011 Φ41 + 2(A012 + 2A066 )(Φ1 Φ2 )2 + A022 Φ42

(5)

A12 (x) = (A011 + A022 − 4A066 )(Φ1 Φ2 )2 + A012 (Φ41 + Φ42 )

(6)

A22 (x) = A011 Φ42 + 2(A012 + 2A066 )(Φ1 Φ2 )2 + A022 Φ41

(7)

Functions representing periodically curved layers are given below. 1 Φ1 (x) = & 1 + [a(x)]2

(8)

Φ2 (x) = Φ1 (x1 )a(x)

(9)

a(x) =

∂ f (x) ∂x

f (x) = sin(γπx + δ) π γ = 10 δ = ν = 0.2 2

(10) (11) (12)

The stress-strain relationship for transversely isotropic material (Razaqpur, 1991) is σ  A 0   11   11   11 A12    σ22  =  A21 A22 0   22  (13)    σ12 0 0 2A66 12 E(x) = A11 (x) − where E(x) is moduli of elasticity.

A211 (x) A22 (x)

(14)

521

FREE VIBRATION OF COMPOSITE BEAMS

3.

Finite element formulation

Euler-Bernoulli beam theory was used. Cubic displacement functions are selected. Five elements used during the calculations. Element rigidity matrix and element mass matrix respectively are as follows:  12 6l −12 6l     4l2 −6l 2l2  e 3  [K] = E(x)(I/l )  (15)  12 −6l   2 sym. 4l  13/35   e [M] = ρAl   sym.

9/70 −13l/420   13l/420 −l2 /140   13/35 −11l/210  2 l /105

11l/210 l2 /105

(16)

Well-known equation for free vibration is ([K] − w2 [M])q = 0

(17)

where w represents natural frequency. TABLE I.

Classical Euler-Bernoulli beam theory

m (mode)

w

1

0.395

2

1.582

TABLE II. Simply supported beam with small curvatures (mode 1)

w E2 /E1

0

0.01

0.02

0.03

0.04

0.05

0.06

1

0.403

0.403

0.403

0.403

0.403

0.403

0.403

10

0.945

0.943

0.939

0.931

0.921

0.908

0.893

20

1.306

1.303

1.295

1.282

1.263

1.240

1.212

50

2.035

2.030

2.016

1.990

1.953

1.902

1.841

100

2.864

2.857

2.834

2.792

2.725

2.632

2.520

¨ YAVUZ, ERGUVEN

522 TABLE III.

Simply supported beam with small curvatures (mode 2)

w E2 /E1

0

0.01

0.02

0.03

0.04

0.05

0.06

1

1.614

1.614

1.614

1.614

1.614

1.614

1.614

10

3.786

3.780

3.762

3.732

3.691

3.640

3.582

20

5.231

5.221

5.190

5.137

5.064

4.971

4.861

50

8.152

8.134

8.077

7.976

7.827

7.629

7.388

100

11.470

11.440

11.360

11.190

10.920

10.560

10.120

TABLE IV.

Simply supported beam with large curvatures (mode 1)

w

4.

E2 /E1

0

0.1

0.2

0.3

0.4

0.5

0.6

1

0.403

0.403

0.403

0.403

0.403

0.403

0.403

10

0.945

0.824

0.697

0.644

0.619

0.605

0.596

20

1.306

1.080

0.848

0.755

0.709

0.681

0.663

50

2.035

1.560

1.140

0.974

0.887

0.832

0.795

100

2.864

2.051

1.452

1.216

1.088

1.005

0.947

Numerical results

Natural frequencies for modes I and II of classical Euler-Bernoulli beam are presented in Table 1. It is given that natural frequencies for modes I and II of simply supported periodically curved layered composite beam with small and large curvatures, respectively, in Table 2 to 5. 5.

Conclusions

In this study, natural frequencies of the periodically curved layered unidirectional fibrous composite beams are obtained. Periodically curved structure of the composite material cause the loss of rigidity, therefore frequencies become smaller. In the case that = 0 and E2 /E1 = 1, the results converge to natural frequencies of classical Euler-Bernoulli beam.

523

FREE VIBRATION OF COMPOSITE BEAMS TABLE V.

Simply supported beam with large curvatures (mode 2)

w E2 /E1

0

0.1

0.2

0.3

0.4

0.5

0.6

1

1.614

1.614

1.614

1.614

1.614

1.614

1.614

10

3.786

3.308

2.801

2.590

2.490

2.434

2.397

20

5.231

4.338

3.417

3.042

2.855

2.744

2.670

50

8.152

6.275

4.602

3.935

3.583

3.363

3.211

100

11.470

8.259

5.870

4.922

4.403

4.069

3.833

It is shown that Akbarov and Guz Continuum Theory works well even with five elements via FEM. References Akbarov S. D., Guz A. N. (1991) On the continual theory in mechanics of composite materials with curved layers, Prikl. Mech. (in Russian), 20 3-9. Akbarov S. D. (1995) it On the determination of normalized nonlinear mechanical properties of composite materials with periodically curved layers, Int. J. Solids and Structures 32 3129-3143. Kocer M. (1997) Free vibration analysis and buckling loads of cracked beams resting on an elastic foundation, (in Turkish) Ph.D. Thesis, Istanbul Technical University, Istanbul, Turkey. Kutug Z. (1997) Free vibrations and stability of composite plates with curved layers (in Turkish), Ph. D. Thesis, Yildiz Technical University, Istanbul, Turkey. Razaqpur A. G. (1991) Method of analysis for advanced composite structures, Specialty Conference on Advanced Composites Materials in Civil Engineering Structures, Las Vegas, Nevada, USA Yokoyama T. (1988) Parametric instability of Timoshenko beams resting on an elastic foundation, Computers & Structures 28 207-216.

VIBRATIONS OF BEAM CONSTRUCTIONS SUBMERGED INTO A LIQUID Vladimir V. Yeliseyev and Tatiana V. Zinovieva St. Petersburg State Polytechnic University, Russia

Abstract. Coupled hydroelasticity problems for beam systems in compressible viscosity liquid are considered in connection with designing of offshore drilling rigs. The use of singular perturbation method to take effects of stretch, shear, initial stress and damping into account is discussed. The main term of asymptotics is found by the solvability requirements for the small correction terms on the following steps. Non-stationary coupled hydroelasticity problem of cantilever beam resonance oscillations in a liquid is considered. The resistance force is determined by solving the two-dimensional problem of hydromechanics for a cylinder in boundless liquid medium. The Navier-Stokes equations are corrected for a case of compressibility. Key words: rods, perturbation method, viscosity liquid, coupled problem, resonance oscillations

1. The equations of linear theory of rods The one-dimensional model of Cosserat rod represents the material line consisting of elementary solid bodies. An angular position s at reference configuration is attached to each particle. The position of each particle is given by the position vector r . The particles displacements are defined by the vector of small displacement u(s, t) and the vector of small rotational displacement ϑ(s, t). The element of material line ds has a mass ρ(s)ds, eccentricity ε(s) (sets the center of mass position to a pole) and tensor of inertia I ds. External force q ds and moment mds act on it. Internal interactions are defined by force Q(s, t) and moment M (s, t). The linear balance equations of forces and moments are given by: Q + q = ρ u¨ + ϑ¨ × ε , M + r × Q + m = ρε × u¨ + I · ϑ¨ , (1) see (Yeliseyev, 2003), elastic stress-strain relations are as follows

M = a · ϑ + c · γ ,

γ ≡ u − ϑ × r ,

Q = b · γ + ϑ · c.

(2)

All coefficients here are functions of angular position, they are prescribed by initial state. Rigidity of the rod is characterized by the second-rank tensors: a — 525 .


526

YELISEYEV, ZINOVIEVA

tensor of bending and torsion rigidity, b — tensor of tension and shear rigidity, c — tensor of cross relations. The reciprocity theorem of works is valid in a linear statics of rods #l

$l (q 1 · u2 + m1 · ϑ2 )ds + (Q1 · u2 + M 1 · ϑ2 )$$0 ≡ A12 = A21 ,

(3)

0

where two different states of a rod are denotes by subscripts 1 and 2, q and m are external distributed force and moment, Q and M — loads on the ends of rod. 2. Perturbation method The asymptotic methods are very effective at presence of many factors influencing the solution. Usually formulation of the problem is reduced to dimensionless form for introduction a small parameter λ → 0. Then a small dimensionless combination is denoted as λ. But another way exists: if it is known, that a some parameter p changes the solution slightly, then it is possible to rename it λp and to begin the asymptotic analysis at λ → 0. Let us consider the problem of natural frequencies and natural modes definition of the linear elastic discrete system with the small perturbations of masses and rigidities: C − ω2 A U = 0, A = A0 + λA1 , C = C0 + λC1 , λ → 0, (4) here A — matrix of masses, C — matrix of rigidities. Let us search the solution in the form (5) ω = ω0 + λω1 + . . . , U = U0 + λU1 + . . . We receive a sequence of problems, substituting (5) in (4) and equating the coefficients at identical powers of λ: C0 − ω20 A0 U0 = 0, (6) C0 − ω20 A0 U1 = −C1 + ω20 A1 + 2ω0 ω1 A0 U0 , . . . On the first of these steps we find the natural frequencies ω0 and the modes U0 of unperturbed system. On the second step we have non-uniform system with a matrix whose determinant equals zero; it is solvable only in a case of orthogonality of its right member to the solutions of conjugate uniform system. Since C0 and A0 are symmetric, conjugate system coincides with initial. Therefore the requirement of solvability takes the following form U0T C1 − ω20 A1 U0 . U0T −C1 + ω20 A1 + 2ω0 ω1 A0 U0 = 0 ⇒ ω1 = 2ω0 U0T A0 U0

BEAM CONSTRUCTIONS SUBMERGED INTO A LIQUID

527

Similarly, on the following steps we obtain the allowances ω2 , ω3 etc. The requirement of solvability is expressed by the reciprocity theorem of works (Yeliseyev, 2003) in the rod systems. This approach is used in (Yeliseyev and Zinovieva, 2004) for taking the effects of tension, shear and initial perturbations in rod models into account. Nextly it is used to solve the problem of beam resonance oscillations in a liquid. 3.

Resonance oscillations of an elastic beam in a compressible viscosity liquid

The calculations of beam systems oscillations in a liquid are necessary for an estimation of rigidity of offshore drilling rigs constructions. The hydrodynamic forces change inertial and damping properties of system. They can cause cross oscillations of body when viscosity liquid flows about it. 3.1. LINEAR VIBRATIONS OF A LIQUID

Usually it is assumed that a liquid is ideal and incompressible when the motion and the deformation of a body in the liquid are considered. However a preanalysis has shown the necessity of taking viscosity and compressibility into account. The stress tensor in a compressible viscosity liquid is given by 1 τ = −pE + 2η ∇v S − ∇ · vE , (7) 3 where p — pressure, v — velocity vector, E — unit tensor, η — viscosity. In a compressible liquid at constant temperature p = p (ρ) — density function. The balance equations of forces and mass should be satisfied: 1 (8) ρ (v˙ + v · ∇v ) = ∇ · τ = −∇p + η ∆v + ∇∇ · v , 3 ρ˙ + ∇ · (ρv ) = 0

(9)

((. . .)· — local derivative with respect to time t). Further the perturbations of pressure p˜ = p − p0 , density ρ˜ = ρ − ρ0 and velocity v are considered as small quantities. The linearized set of equations is as follows ∇ · τ˜ = ρ0 v˙ , (10) ρ˙˜ + ρ0 ∇ · v = 0, p˜ = c2 ρ, ˜

$ d p $$ $ . c2 ≡ dρ $0

(11) (12)

528


Let us introduce a displacement field u (r , t) (r — position vector of location): ˙ ˙ + v · ∇u = v ⇒ v = u du/dt ≡ u (13) after a linearization. Then from (11) at zero initial conditions ρ˜ = −ρ0 ∇ · u. Varying (7) and taking account of (12), we obtain 2 2 τ˜ = c ρ0 − η∂t ∇ · uE + 2η∂t ∇uS . (14) 3 It looks as a relation of the classical theory of elasticity

τ = λ∇ · uE + 2µ∇uS , only Lame coefficients became the operators. At harmonic oscillations u (r , t) = u (r ) eiωt etc. We obtain for complex amplitudes (λ + µ) ∇∇ · u + µ∆u + ω2 ρ0 u = 0, 2 λ = c2 ρ0 − i ωη, µ = iωη. 3 Usually the scalar and the vector potentials are introduced to solve this equation (Slepyan, 1972): u = ∇ϕ + ∇ × ψ , ∇ · ψ = 0; c21 ∆ϕ + ω2 ϕ = 0,

c22 ∆ψ + ω2 ψ = 0,

c21 = (λ + 2µ)/ρ0 ,

c22 = µ/ρ0 .

(15)

Let us search a liquid reaction on the solid cylinder of radius R, which oscillates in a plane under the law

uˆ (t) = weiωt i. Using relations (15), we obtain the expression of force per unit of cylinder length in Hankel functions: #2π

F =R

er · τ (R, θ) dθ = −F i, 0

β1 α2 − 2β1 β2 + β2 α1 , (16) α1 α2 − β1 β2 ωR , βk ≡ H1(2) (γk ) , k = 1, 2. αk ≡ H˙ 1(2) (γk ) γk , γk ≡ ck The calculations by formula (16) shows that the liquid gives an additional inertia (associated mass) and a damping. F = πR2 ρ0 wω2

BEAM CONSTRUCTIONS SUBMERGED INTO A LIQUID

529

3.2. RESONANCE OSCILLATIONS OF BEAM

The resonance oscillations are very dangerous conditions, the dynamic rigidity of a construction becomes minimal. The resistance forces restrict a vibration amplitude on a resonance. Let us find a resonance amplitude and phase of bending oscillations of cantilever beam simulated by the Bernoulli-Euler’s beam. Let us presume small the inducing and damping forces and use the asymptotic method. The frequency ω of an external force coincides with one of natural frequencies of a “dry” beam. The statement of problem for the beam deflection u (x, t) is as follows auIV + λ f (˙u) + ρ¨ ˆ u = λq (x) sin ωt, u (0) = u (0) = u (l) = u (l) = 0.

(17)

Here λ → 0 — formal small parameter, a — bending rigidity, ρˆ — beam mass per unit of length. The natural modes and frequencies of “dry” beam are found from the equation aU IV − ω2 ρU ˆ =0

(18)

with boundary conditions (17). The modes are orthonormal: 4

#l ρU ˆ n Uk dx = δnk , 0

δnk =

1, n = k 0, n k

(19)

The periodic solution of problem (17) we search as Poincare expansion (Nayfe, 1976) u (x, t) = u0 (x, t) + λu1 (x, t) + . . . We have for a main term au0IV + ρ¨ ˆ u0 = 0,

u0 = ΘU sin (ωt − α) ,

(20)

as the frequency ω = ωi — coincides with one of natural one. Amplitude factor Θ and phase shift α are found from an equation for u1 au1IV + ρ¨ ˆ u1 = q (x) sin ωt − f (˙u0 ) ≡ β (x, t)

(21)

with boundary conditions (17). The solution is searched as expansion in terms of natural modes Θk (t) Uk (x). u1 =

530


Considering (18) and (21) as two problems of statics and applying the theorem (Yeliseyev, 2003), we can determine the principal coordinates Θk #l

ω2n ρU ˆ n u1 dx

#l =

0

(β − ρ¨ ˆ u1 ) Un dx.

(22)

0

Whence, allowing for a normality condition (19), it is follows ¨k + Θ

ω2k Θk

#l =

βUk dx.

(23)

0

The right side of (23) has a period T = 2π/ω. The absence of the first harmonics in the right side of the equation for a resonating mode U (corresponding ωi = ω) is necessary for periodicity of u1 . So for Θ and α we obtain a set of equations #l #T #l 2 qUdx − sin ωt f (˙u0 )Udxdt = 0, T 0

0

#T

0

#l cos ωt

0

f (˙u0 )Udxdt = 0.

(24)

0

The used asymptotic method reminds a principle of balance of works. References Nayfe A. (1976) Perturbation methods, M. Mir. Slepyan L. I. (1972) Non-steady elastic waves, L. Sudostroenie. Yeliseyev V. V. (2003) Mechanics of elastic bodies, St.-Petersburg State Polytech. Univ. Yeliseyev V. V., Zinovieva T. V. (2004) Perturbation method in problems of oscillations of beam systems, Mechanics of materials and strength of constructions, SPbSPU papers, St. Petersburg, 489 200-209.

VIBRATION BEHAVIOR OF COMPOSITE BEAMS WITH RECTANGULAR SECTIONS CONSIDERING THE DIFFERENT SHEAR CORRECTION FACTORS Vebil Yıldırım Department of Mechanical Engineering, University of Çukurova, 01330 Adana, Turkey

Abstract. As is well known, there are the first and higher order shear deformation theories that involve the shear correction factor (k- factor), which appears as a coefficient in the expression for the transverse shear stress resultant, to consider the shear deformation effects with a good approximation as a result of non-uniform distribution of the shear stresses over the cross-section of the beam. Timoshenko’s beam theory (TBT) accounts both the shear and rotatory inertia effects based upon the first order shear deformation theory which offers the simple and acceptable solutions. The numerical value of the k- factor which was originally proposed by Timoshenko depends upon generally both the Poisson’s ratio of the material and the shape of the cross-section. Recently, especially the numerical value of the k-factor for rectangular sections is examined by both theoretical and experimental manners. Although there are no large numerical differences among the most of the theories, a few of them says that the k-factor varies obviously with the aspect ratio of rectangular sections while Timoshenko’s k-factor is applicable for small aspect ratios. In this study, the effect of the different k-factors developed by Timoshenko, Cowper and Hutchinson on the in-plane free vibration of the orthotropic beams with different boundary conditions and different aspect ratios are studied numerically based on the transfer matrix method. For the first six frequencies, the relative differences of among the theories are presented by charts. Key words: shear correction factor, Timoshenko’s theory, vibration, rectangular section, first-order

1. Introduction The forth-order differential equation for the flexural motion of bars in y − z plane based on the TBT by two coefficients is given by (Craig, 1981) (out-of-plane bending) (EI x )

∂4 v ∂4 v ∂2 v E 1 ∂4 v 2 ) ) + (ρA) − ρI (1 + + ρ I ( =0 x x k1G ∂z2 ∂t2 k2G ∂t4 ∂z4 ∂t2

(1)

where k1 and k2 are the shear correction factors (k−factors), t is the time, v is the transverse displacement, ρ is the density, A is the cross-sectional area, I x is 531 .


532

YILDIRIM

the moment of inertia, E and G are Young’s and shear modulus, respectively. The roots of the frequency equation, which are obtained from the solution of equation (1) with the boundary conditions, give the natural frequencies. As seen in eqn. (1) when k → ∞ the effect of the shear deformation is negligible, just the rotational inertia effect remains. If the rotatory inertia is also neglected (h/L → 0), the following well-known Bernoulli-Euler equation is obtained. EI

∂4 v ∂2 v + (ρA) =0 ∂z4 ∂t2

(2)

In most cases, the effects of the shearing deformations for isotropic materials, k = k1 = k2 = 5/6 = 0.8333 and υ = 0.3 are observed as remarkable significant than the rotatory inertia effects. In general, as the mode number increases the effects of both the rotatory inertia and shearing deformations increase. Those effects decrease with the increasing slenderness ratio. Apart from these, those effects are affected from the boundary conditions. On the contrary to the isotropic beams, the effect of shear deformations becomes very important for even slender composite beams. The E/G ratio varies in the range of 20 and 50 for modern composites while that ratio is 2, 5 ÷ 3 for isotropic materials. For sandwich structures those effects take also a great deal significance. The range of the validity of the first-order shear deformation theory is strongly dependent on the shear correction factors used, which exactly depend upon both the geometry of the cross-section and material type of the beam. In this respect, the main problem in solving the problems by the first order shear deformation theory is that the decision to the best k− factor. Timoshenko (1921, 1922) offers the k-factor dependent on just Poisson’s ratio as follows by solving 2-D elasticity bending problem 5 + 5υ kTimoshenko (υ) = k1 = k2 = (3) 6 + 5υ (Cowper, 1966) recommended the following k-factor which depends upon Poisson’s ratio for the 3-D integral solution of the elasticity problem 10 + 10υ kCowper (υ) = k1 = k2 = (4) 12 + 11υ In both beam and plate analyses, the most used factor is given as k = k1 = k2 = 5/6 ≈ 0.833333 (Reissner, 1947; Timoshenko, 1991; Stephen, 2002, Renton, 1991) which is obtained by parabolic shear distribution assumption. The corresponding factor in plate dynamics is taken as k = k1 = k2 = π2 /12 ≈ 0.822467 (Mindlin et al., 1956). The other k-factor used in the plate dynamics was offered by (Uflyand, 1948) as k = 2/3 = 0.666667. (Kaneko, 1975) reviewed the developments on k-factor up to that time. (Stephen and Levinson, 1979) proposed the beam theory with two coefficients k1 = kTimoshenko (υ),

k2 = kCowper (υ)

(5)

VIBRATION BEHAVIOR OF COMPOSITE BEAMS WITH . . .

533

The earliest study about the relationship between k-factor and the aspect ratio was performed by (Stephen, 1980). For this relation, (Hutchinson and Zilmer, 1986) offered a serial solution includes the aspect ratio. (Stephen, 1997) outlined the k−factors used in plate dynamics. For isotropic beams, (Pai and Schulz, 1999) suggested a k−factor formulation called energy-consistent without solving the elasticity equations. kPai/Schulz =

36

h4 (1 + υ)2 h4 (1+υ)2 30

+

υ2 b 4 180

−

υ2 b 5 2π5 h

∞ n=1

tanh πh b n5

(6)

The above yields k = 5/6 for h/b → 0 and/or υ → 0 and verifies the third order theory. (Hutchinson, 2001) formulated the k−factor consists of both υ and (b/h) effects based on the basic dynamic beam theory. b kHutchinson (υ, ) = k1 = k2 = − h C4 =

2(1 + υ) 9 C 4h5 b 4

+ υ(1 −

b2 ) h2

∞ 4 3 16υ2 b5 [nπh − b tanh(nπh)] h b(−12h2 − 15h2 υ + 5b2 υ) + 45 n5 π5 (1 + υ) n=1

(7)

(Stephen, 2001) showed that, the results in his previous study (Stephen, 1980), which was obtained by using the different method, are the same as (Hutchinson, 2001). (Renton, 2001) studied the elasto-dynamic Timoshenko’s solution of plane stress problem. (Stephen, 2002) expanded the Renton’s work. For orthotropic Bickford beam and plates, (Soldatos and Sophocleous, 2001), proposed the k−factor as k = k1 = k2 = 14/17 ≈ 0.823529 based on the parabolic shear distribution. For both two-skin as well as multi-skin sandwich structures, (Birman, 2002) states that the k−factor should be taken as unity. (Puchegger et al., 2003) have performed several experiments with the ”ideal” computer-generated material specimens with free-free ends. They show that Hutchinson’s k−factor is the only one to describe the clear dependence on the aspect ratio correctly. While Hutchinson’s k−factor is used for large aspect ratios (b/h) and Poisson’ ratios, Timoshenko’s k−factor is valid for just small aspect ratios. (Puchegger et al., 2003) also expressed that the effect of the slenderness ratio (L/h) on the formulation of k-factor is negligible except for very short and very long beams. As seen from the above, there are not enough experimental studies to verify the true value of k−factor in the literature except the particular work of (Puchegger et al., 2003). Numerical investigation of the effect of k−factors frequently used on the free vibration of beams will be very useful to the readers.

534

YILDIRIM

Figure 1.

The variation of the shear coefficient reciprocal wrt the aspect ratio

Figure 2.

The geometry of the beam and coordinate axes

2. Formulation and examples The in-plane (axial and flexural) linear free vibration equations of the orthotropic beams with rectangular sections are given as follows (Yıldırım et al., 1999a-b) dUz dz dT z dz

= A11 T z , =

−ρAω2 U

dU x dz z,

= Ωy + A22 kT x , dT x dz

=

−ρAω2 U

x,

dΩy dz

= D33 My ,

dMy dz

= −ρIy

ω2 Ω

(8) y

− Tx

Where ω is the natural frequency, T z is the axial force, T x is the shear force, My is the bending moment, Ωy is the rotation, U x and Uz are displacements. The other terms are computed with the help of the elements of the stiffness and compliance matrices, Ci j and S i j (Yıldırım et al., 1999a-b). Equation (8) corresponds to a set of six linear ordinary differential equations. In order to obtain its exact solution, the transfer matrix method together with an effective numerical algorithm developed by the author to obtain the element transfer matrix is used in this work (Yıldırım et al., 1999a-b). Two boundary conditions are considered with two aspect ratios in this study for the slenderness ratio, L/h = 10. Figure 3

VIBRATION BEHAVIOR OF COMPOSITE BEAMS WITH . . .

535

Figure 3. Variation of the six first natural frequencies%with the boundary conditions, k−factors and aspect ratios (Dimensionless frequency= ω ¯ = ωL2 E ρh2 ; E1 = 144.8GPa, E2 = 9.65GPa, 1

G12 = G13 = 4.14GPa, G23 = 3.45GPa, ρ = 1389.23kg/m3 , ν = 0.3)

shows the variation of the first six natural frequencies with the boundary conditions, k-factors and aspect ratios. As seen from Figure (3a), for higher frequencies, Cowper’s k−factor gives a little bit higher frequencies than k = 5/6. In general, the same results are obtained with both Timoshenko’s and Hutchinson’s k−factors for higher frequencies of square sections (b/h = 1). For higher aspect ratios, e.g. b/h = 5, it is obviously seen that the relative differences between Hutchinson’s k−factor and the other k−factors becomes very significant. It may be over 100% for some frequencies, e.g. for the second frequency of fixed-fixed ends and b/h = 5. This ratio may be greater for the other high aspect ratios. 3. Conclusions The effect of the k−factors offered by different researchers on the free vibration of orthotropic beams is investigated numerically. It is observed that the results are very close to each other except Hutchinson’s results. To get rid of the expected confusion in choosing the k-factors to be used in the analysis, some experiments related the k−factors should be performed.

536

YILDIRIM

References Birman, V. (2002) On the choice of shear correction factor in sandwich structures, Journal of Sandwich Structures & Materials 4, 83-95. Cowper, G. R. (1966) The shear coefficient in Timeshenko’s beam theory, ASME JAM 33, 335-340. Craig, R. R. (1981) Structural Dynamics, John Wiley & Sons. Hutchinson, J. R., Zilmer, S. D. (1986) On the transverse vibration of beams of rectangular crosssection, ASME JAM 53, 39-44. Hutchinson, J. R. (2001) Shear coefficient for Timeshenko beam theory, ASME JAM 68, 87-92. Kaneko, T. (1975) On Timoshenko’s correction for shear in vibrating beams, Journal of Physics D 8, 1927-1936. Mindlin, R. D., Schacknow, A., Deresiewicz, H. (1956) Flexural vibrations of rectangular plates, ASME JAM 23, 430-436. Pai, P. F., Schulz, M. J. (1999) Shear correction factors and an energy-consistent beam theory, Int. J. Sol. & Str. 36, 1523-1540. Puchegger, S., Bauer, S., Loidl, D., Kromp, K., Peterlik, H. (2003) Experimental validation of the shear correction factor, JSV 261, 177-184. Reissner, E. (1947) On bending of elastic plates, Quarterly of Applied Mathematics 5, 55-68. Renton, J. D. (1991) Generalized beam theory applied to shear stiffness, Int. J. Solid. Struct. 27, 1955-1967. Renton, J. D. (2001) A check on the accuracy of Timoshenko’s beam theory, JSV 245, 559-561. Soldatos, K. P., Sophocleous, C. (2001) On shear deformable beam theories: The frequency and normal mode equations of the homogeneous orthotropic Bickford beam, JSV 242, 215-245. Stephen, N. G., Levinson, M. (1979) A second order beam theory, JSV 67, 293-305. Stephen, N. G. (1980) Timoshenko’s shear coefficient from a beam subjected to gravity loading, ASME JAM 47, 121-127. Stephen, N. G. (1997) Mindlin plate theory: Best shear coefficient and higher spectra validity, JSV 202, 539-553. Stephen, N. G. (2001) On “Shear coefficient for Timoshenko beam theory”, ASME, JAM 68, 959960. Stephen, N. G. (2002) On “A check on the accuracy of Timoshenko’s beam theory”, JSV 257, 809-812. Timoshenko, S. P. (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine 41, 744-746. Timoshenko, S. P. (1922) On the transverse vibrations of bars of uniform cross-section, Philosophical Magazine 43, 125-131. Timoshenko, S. P. (1991) Mechanics of Materials, Chapman & Hall. Uflyand, Y. S. (1948) The propagation of waves in the transverse vibrations of bars and plates, Prikladnaia Mathematika Mekhanica 12, 287-300. Yıldırım, V., Sancaktar, E., Kiral, E. (1999a) Free vibration analysis of symmetric cross-ply laminated composite beams with the help of the transfer matrix approach, Com. Num. Meth. Eng. 15, 651-660. Yıldırım, V., Sancaktar, E., Kiral, E. (1999b) Comparison of the in-plane natural frequencies of symmetric cross-ply laminated beams based on the Bernoulli-Euler and Timoshenko beam theories, ASME, JAM 66, 410-417.

AIR BLAST-INDUCED VIBRATION OF A LAMINATED SPHERICAL SHELL Hüseyin Murat Yüksel and Halit S. Türkmen ˙ Faculty of Aeronautics and Astronautics, Istanbul Technical University, Maslak, ˙ Istanbul 34469, Turkey

Abstract. The scope of this study is to investigate the dynamic behavior of a laminated spherical shell subjected to air blast load. The shell structure considered here is a hemisphere in shape and made of a glass/epoxy laminated composite material. The blast experiments are performed on the spherical shell. The strain-time history of the center of the spherical shell panel is obtained experimentally. The blast loaded spherical shell is also modeled and analyzed using ANSYS finite element software. The static analysis is performed to characterize the material. The dynamic response of the spherical shell panel obtained numerically is compared to the experimental results. It is observed that the response frequency corresponds to the higher vibration modes of the panel. The qualitative agreement is found between the numerical and experimental results. Key words: air blast, vibration, spherical shell, laminated composite

1. Introduction The dynamic response of laminated composite structures to the blast load is an extremely important subject in engineering. The suddenly applied loading such as a blast load is known as a critical loading condition. They may cause the fiber brekage, delamination and finally failure of the structure. Theoretical studies on this area give many useful information about the response of laminated structures to the blast load. However, it is necessary to support all these studies with experiments. It is difficult to perform real blast tests on the structures. In this study it is aimed to make a simple experimental setup and perform the blast tests. There are many studies reported on the response of laminated structures to the blast load. Many of them are related to the laminated plates ((Librescu,1990), (Wiernicki, 1990), (Turkmen, 1999), (Turkmen, 1999)). Some of them are related to the laminated cylindrical shell panels subjected to the blast load ((Mukhopadhyay, 1996), (Zhu, 1997), (Turkmen, 2002), (Stein, 1986)). There are very few studies reported on the spherical shell panels subjected to the blast load. The spherical, or in general doubly curved, shell panels have been found many application areas particularly in the aerospace industry. The radom, external storage tank, leading 537 .


¨ ¨ YUKSEL AND TURKMEN

538

edge of a missile, wheel pant can be given as examples for the use of doubly curved shell panels. These panels usually are made of laminated composites to save weight. Therefore, a spherical shell panel made of laminated composites is investigated under the effect of the blast load, experimentally and theoretically. The strain at the center of the shell is measured during the effect of the blast load. The displacement, strain and stress are calculated using the finite element method. The vibration frequencies are obtained from both experimental and numerical data. 2. Experimental procedure The static and blast tests are carried out to investigate the dynamic behavior of the laminated spherical shell. The shell structure considered in this study has a hemispherical shape with a radius of 101.5 mm. The number of layers are 3 and each layer is made of a glass/epoxy laminated composite material with 0.16 mm thickness. The material properties are given in Table 1. TABLE I. Mechanical properties of the glass/epoxy laminate E x = Ey (GPa)

Ez (GPa)

G xy (GPa)

Gyz = G xz (GPa)

ν xy

νyz = ν xz

ρ(kg/m3 )

24.14

3.0

3.79

1.076

0.11

0.4

1800

The experimental setup is composed of a pressure tube, a main frame, a compressor, a digital scope, a dynamic strain meter, a multimeter, and a computer (Yuksel, 2005). A schematic of the experimental setup is shown in Figure 1. The static tests are carried out to characterize the material. The metal objects with known weights are put on the top face at the center of the clamped panel and the strain on the bottom face at the center of the panel is measured using a dynamic strainmeter and a multimeter. The static test results is shown in Table 2. The measured strains are proportional to the loads linearly. This indicates that the material response stays in the elastic regime for the loads applied in the tests. The air blast load is obtained as a result of a sudden charge from a compressor. The procedure to obtain this sudden charge is as follows: 1) The air is fed into a steel tube from one end and the other end of the tube is closed using a thin membrane. 2) A manometer mounted on the tube is used to read the pressure value inside the tube. 3) The membrane is torn when the air pressure reaches a critical value for the membrane and the sudden discharge occurs. The straintime history on the bottom face at the center of the panel are measured using strain gauges and a dynamic strain meter. The signals obtained from the strain gauges and the dynamic strainmeter are digitized using a digital scope and transferred

AIR BLAST-INDUCED VIBRATION

Figure 1. TABLE II.

539

The schematic of the experimental setup Static test and analysis results (microstrain)

Weight(gr)

Test

Analysis

56.69 134.08 190.77 236.39 366.03

73.0 131.5 184.5 229.0 374.5

77.8 184.0 262.0 325.0 503.0

to a personal computer using an RS232C serial interface. The tests are repeated to decrease the experimental uncertainties. The strain-time history obtained by four different tests with a time duration of 20 ms are shown in Figure 2. In these tests, the pressure read on manometer is 1.7, 1.6, 1.7, 1.8 bar, respectively. The observations in these tests are as follows: 1) The first peak is negative in each test. 2) Damping becomes effective after 2 ms. 3) The response frequency is about 4500 Hz (18 cycles per 4 ms). 4) The highest peak value is negative in each test. Although the tube pressure is almost same in each test, there are also some differences observed in the strain-time history. The highest peak strain values are -1269, -2187, -1542, -1542, respectively. 3. Finite element model of the blast loaded shell structure On the numerical side of the study, the static and blast tests are modeled using a finite element software (ANSYS). The mesh structure used to model the static tests is shown in Figure 3a. In this model, the mesh is dense at the center which the load is applied and the stress is concentrated (Figure 3b). Because of the concentrated load at this point it is needed finer mesh. The strain is calculated on the bottom face at the center of the panel for each weight. The results of the


540 1200

1200 Test 3 800

400

400 Microstrain

Microstrain

Test 1 800

0 -400 -800 -1200

-800 -1200

-1600

-1600 0

0.5

1

1.5

1200

2 2.5 Time (ms)

3

3.5

4

0

0.5

1

1.5

2 2.5 Time (ms)

3

3.5

4

3

3.5

4

1200

Test 2

800

Test 4 800

400

400

0

Microstrain

Microstrain

0 -400

-400 -800 -1200 -1600

0 -400 -800 -1200

-2000 -2400

-1600 0

0.5

Figure 2.

1

1.5

2 2.5 Time (ms)

3

3.5

4

0

0.5

1

1.5

2 2.5 Time (ms)

The experimental strain-time history on the bottom face at the center of the panel

Figure 3. a) The finite element model used in the static analysis b) The detail of the refined mesh at the center of the sphere c) The finite element model of the hemispherical shell panel used in the transient analysis

static analyses are shown in Table 2. The model used in the static analysis is not used in the dynamic analysis, because of the distributed nature of the blast load. The laminated spherical shell subjected to air blast load is modeled using the finite element method. For this purpose a finite element software (ANSYS) is used. The modal analysis is achieved to obtain the free vibration frequencies. The free vibration frequencies of the first twenty modes ranged from 2373 Hz to 4216 Hz. The effect of the mesh density on the free vibration frequencies is investigated to decide the number of elements of the final model. The increase in the mesh density slightly effected the free vibration frequencies. This effect is most pronounced for the higher vibration modes. It is decided that using a moderate number of elements is enough for the accuracy during the transient

541

AIR BLAST-INDUCED VIBRATION 60000 (a)

0.1

2

Pm=57812 N/m

(b) 0.05 Displacement (mm)

Pressure (N/m2)

40000 20000 0 -20000 -40000 -60000

0 -0.05 -0.1 -0.15 -0.2

0

5

10

15

20 25 Time (ms)

30

35

40

0

15

0.5

1

1.5

2 2.5 Time (ms)

3

3.5

4

3

3.5

4

400 (c)

10

(d) 200

0

Microstrain

Stress (MPa)

5

-5 -10

0 -200 -400

-15 -600

-20 -25

-800 0

0.5

1

1.5

2 2.5 Time (ms)

3

3.5

4

0

0.5

1

1.5

2 2.5 Time (ms)

Figure 4. a) The variation of air blast pressure b) The displacement-time history during the blast loading c) The stress-time history during the blast loading d) The strain-time history during the blast loading

analysis. The model which has 1200 elements is chosen for the transient analysis (Figure 3c). The transient analysis is achieved using the final model to obtain the strain-time histories. The air blast load is represented by using a Friedlander decay function shown in Figure 4a. This load is applied to the circular area around the center of the shell panel. The displacements in the perpendicular direction to the shell panel, stresses, and strains on the bottom face at the center are obtained numerically using the finite element method. These are shown in Figure 4b,c,d, respectively. The results of the analysis are listed below: 1) The peak displacement is -0.18 mm. This is in the linear elastic range. 2) The vibration follows the blast load. The vibration amplitude takes its minimum value when the load is zero. 3) The vibration frequency is about 4500 Hz. 4) The peak stress is -21.5 Mpa. 5) The peak strain is -794.8 microstrain. 4. Comparisons of the experimental and numerical results The strains and vibration frequencies obtained experimentally and numerically are compared. A qualitative agreement is found between the results. The response frequency is predicted very well using the finite element method. The peak strain obtained numerically differs from the experimental ones. In this study the air pressure on the panel is not measured. The air pressure is predicted from the pressure read on the manometer and the decrease in the pressure with the distance from the open end of the tube (Turkmen, 1999). Therefore, the predicted air blast pressure

542


may differ from the exact pressure value on the panel. This mainly effects the peak strains. 5. Conclusion The dynamic behavior of a hemispherical laminated composite panel is investigated under the air blast load. The experiments indicate the higher vibration modes are stimulated under the air blast load. The response frequency obtained numerically is in an agreement with the experimentally obtained response frequencies. There is a qualitative agreement between the numerical and experimental results. However the amplitudes are not predicted correctly because the air blast pressure is not measured. The numerical errors, experimental errors are the other reasons for the discrepancies. This study presents an experimental method to perform the blast experiments and to investigate the dynamic behavior of panels. The panels made of different materials can be tested using this method. So the structural optimization can be achieved for the blast loaded panels. The experimental setup can be modified to investigate the impact of a subject on the panels. These will be the subject of future studies. References Librescu L., Nosier A. (1990) Response of Laminated Composite Flat Panels to Sonic Boom and Explosive Blast Loadings, AIAA Journal 28 345-352. Mukhopadhyay M., Goswami S. (1996) Transient finite element dynamic response of laminated composite stiffened shell, Aeronautical Journal 100 223-233. Stein M. (1986) Nonlinear Theory for Plates and Shells Including the Effects of Transverse Shearing, AIAA Journal 24 1537-1544. Turkmen H. S., Mecitoglu Z. (1999) Dynamic response of a stiffened laminated composite plate subjected to blast load, Journal of Sound and Vibration 3 371-389. Turkmen H. S., Mecitoglu Z. (1999) Nonlinear structural response of laminated composite plates subjected to blast loading, AIAA Journal 12 1639-1647. Turkmen H. S. (2002) Structural response of laminated composite shells subjected to blast loading: Comparison of experimental and theoretical methods, Journal of Sound and Vibration 249 663678. Wiernicki C. J., Liem, F., Woods, G. D., Furio, A. J. (1990) Structural Analysis Methods for Lightweight Metallic Corrugated Core Sandwich Panels Subjected to Blast Loads, Naval Engineers Journal 28 192-203. Yuksel H. M. (2005) Dynamic Behavior of a Laminated Hemispherical Shell Under the Blast Loading, M. Sc. Thesis, Istanbul Technical University, Istanbul, Turkey. Zhu W. H., Xue H. L., Zhou G. Q., Schleyer G. K. (1997) Dynamic response of cylindrical explosive chambers to internal blast loading produced by a concentrated charge, International Journal of Impact Engineering 19 831-845.

AUTHOR INDEX (*: not personally presented)

Adanur S. Akbarov S. D.

Akkas¸ N.

RC

1

Bikçe M.∗

RC

15

RC

9

Birlik G.

RC

85

RC

225

Biswas P.

RC

265

RC

469

RC

271

RC

243

RC

91

RC

243

M.∗

RC

511

Bonadies

ST

7

Bonifasi-Lista C.∗ N.∗

Akköse M.

RC

9

Bontcheva

Akso˘gan O.

RC

15

Boyacı H.

RC

97

RC

21

Bulut G.

RC

505

RC

371

RC

27

Celep Z.∗

RC

231

RC

33

Chakrabarti B. K.

GL

103

Akyüz U. ¨ ∗ Aldemir U. Alıs¸veris¸çi F. Apaydın

39

Chatterjee

RC

389

Cherkaev A.

GL

111

181

E.∗

RC

91

RC

377

RC

45

As¸çı N.

RC

51

Aydın E.

RC

33

Aydo˘gdu M.

RC

57

Aslanyan A.

Cherkaev

317

RC G.∗

103

RC N.∗

Askes H.∗ ¨ Aslan O.

A.∗

111 Choo B.

S.∗

RC

15

Çelebi M. Çırak ˙I.

GL

123

RC

137

Demir F.∗

RC

137

Baltacı A.

RC

63

Demiray H.

GL

143

Banerjee M. M.

GL

69

Deniz A.

RC

437

RC

79

Do˘gan V.

RC

151

RC

1

RC

157

RC

45

Banks S.

P.∗

Bayraktar A.∗

RC

9

Dronka 543

J.∗

544

AUTHOR INDEX

Dumano˘glu A. A.∗

RC

1

RC

9

Kahali R. Kan C.

Ecker H. ˙I∗

GL

163

RC

487

RC

469

RC

15

¨ Erdem A. U.

RC

175

Erdik M.

GL

181

Ergüven M. E.∗

RC

519

Emiro˘glu Emsen

E.∗

Ertepınar

A.∗

RC

27

Kaplunov S. Karanji

M.∗

S.∗

Karmakar B. Kaya M. O.∗ Kırca

M.∗

Kırıs¸ A. Fedorov V.

A.∗

Fesenko T.

N.∗

RC

195

¨ Korfalı O. P.∗

RC

265

RC

271

RC

249

RC

195

RC

255

RC

265

RC

271

RC

265

RC

271

RC

279

RC

157

RC

285

RC

291

RC

213

RC

205

Kouzov D.

Filippenko G. V.

RC

213

Kumbasar N.

RC

297

V.∗

RC

195

RC

205

Levin V. M.∗

RC

317

RC

219

Makhutov N. A.∗

RC

255

RC

475

Mat˘ejec R.∗

RC

455

Güler C.

RC

225

Mehta A.

GL

305

Güler K.

RC

231

Michelitsch T. M.

RC

317

Güney D.

RC

39

Mishuris G. S.

RC

45

V.∗

RC

195

Mondal S.

RC

323

Movchan A. B.

GL

327

Movchan N. V.

RC

327

Mukherjee A.

RC

337

Fomenkov A. Foursov V.

N.∗

Gögüs¸ M. T.

Mizirin A. Haq

S.∗

Ivanova J. ˙Inan E. ˙Itik M.

327 RC

243

RC

285

RC

383

Nabergoj R.∗

RC

487

RC

237

Nath Y.

GL

415

Nordmann R.

GL

345

545

AUTHOR INDEX

Okrouhlik M. ¨ Onbas¸lı U. Oskouei A. V. ¨ Ozakc ¸ a M.

GL

357

RC

377

RC RC RC

¨ ¨ Ozdemir O. ¨ Ozdemir Z. P.

RC

371

Singh S.∗ Siv˘ca´ k M.∗ Skarolek ˘ Skliba J.

A.∗

219 475

Slepyan

RC

377

Sofiyev

A.∗

Sofiyev A. H.

383

RC

389

¨ Ozeren M. S.∗

RC

395

Pastrone Patel B.

RC RC

P.∗

RC

429

RC

429

RC

455

I.∗

RC

195

RC

21

RC

21

RC

437

RC

443

RC

255

291

Soyluk

K.∗

RC

1

243

¨ Sönmez U.

RC

449

415

Svoboda R.

RC

455

S¸enocak E.

RC

249

RC

463

Solonin V. F.∗

429

111 M.∗

Smolskiy S.

RC

RC

L.∗

279

¨ ¨ Ozer A. O. ¨ Ozer M.

Parlak B.

415

Pe˘sek L.

RC

403

Postacıo˘glu N.

RC

395

Pu st L.

RC

403

Ramachandran J.

GL

409

Tanrıöver H.

RC

463

Res¸ato˘glu N.

RC

15

Tas¸çı F.

RC

469

RC

57

RC

219

RC

475

◦

Tas¸kın Salamcı M. U. Samadhiya Sarıkanat Schnack ¨ Selsil O.

R.∗

M.∗

E.∗

79

RC

237

RC

337

Tekeli H.

RC

137

RC

63

Teymür M.

RC

481

RC

443

Tondl A.

RC

487

Topal U. ¨ Turhan O.

RC

493

RC

499

RC

505

RC

537

RC

137

RC

45

RC

85

Shariy N. V.∗

RC

255

¨ Sezgin O. Sharma

A.∗

Shtykov V. Shukla K.

415 V.∗

K.∗

N.∗

RC

C.∗

RC

195 415

Tays¸i

V.∗

Türkmen H. S. Türkmen

M.∗

546

AUTHOR INDEX

Urey H.∗

RC

249

Yavuz M.

H.∗

RC

51

Yeliseyev V.

¨ Uzman U.

RC

51

Yıldırım V.

Uysal

RC

493

Yıldız

H.∗

Yüksel H. ¨ Ulker F.

D.∗

RC

V.∗

M.∗

Wang

Yahnio˘glu N.

519

RC

525

RC

531

RC

63

RC

537

RC

525

RC

395

237 Zinovieva T. V.∗

J.∗

RC

RC

317

RC

511

Zora

B.∗

LIST OF PARTICIPANTS (with mailing addresses)

S¨ uleyman ADANUR

Surkay D. AKBAROV

Karadeniz Technical University Department of Civil Engineering 61080 Trabzon Turkey e-mail: [email protected]

Yıldız Technical University Faculty of Chemistry and Metallurgy Department of Mathematical Engineering Davutpasa Campus No:127 . 34010 Topkapı Istanbul Turkey e-mail: [email protected]

Nuri AKKAS ¸

¨ Mehmet AKKOSE

. ¨ TUBITAK Ankara, Turkey e-mail: [email protected]


˘ Orhan AKSOGAN

¨ U˘gurhan AKYUZ

C ¸ ukurova University Department of Civil Engineering 01330 Adana Turkey e-mail: [email protected]

Middle East Technical University Department of Civil Engineering 06531 Ankara Turkey e-mail: [email protected]

¨ ˙ Unal ALDEMIR

˙ ¸C G. F¨ usun ALIS ¸ VERIS ¸ I˙

. Istanbul Technical University Department of Civil Engineering Division of Mechanics . 34469 Maslak Istanbul Turkey e-mail: [email protected]

Yıldız Technical University Department of Mechanical Engineering Division of Mechanics . 80750 Yıldız Istanbul Turkey e-mail: [email protected]

Nurcan AS ¸C ¸I Nurdan APAYDIN

Karadeniz Technical University G¨ um¨ u¸shane Engineering Faculty Department of Civil Engineering 29000 G¨ um¨ u¸shane Turkey e-mail: [email protected]

General Directory of State Highways . Istanbul Turkey

547

548

LIST OF PARTICIPANTS Harm ASKES

A. G. ASLANYAN

The University of Sheffield Department of Civil and Structural Engineering Sir Frederick Mappin Building Mappin Street Sheffield S1 3JD UK

Druzhba St. 5 kv. 355 Khimki Moskow Region 141400 Russia

¨ Ozden ASLAN

Ersin AYDIN

Yıldız Technical University Faculty of Science and Art Department of Physics . Davutpasa 34010 Istanbul Turkey e-mail: [email protected]

. Istanbul Technical University Department of Civil Engineering Division of Mechanics . 34469 Maslak Istanbul Turkey e-mail: [email protected]

˘ Metin AYDOGDU

Aysun BALTACI

Trakya University Department of Mechanical Engineering 22030 Edirne Turkey e-mail: [email protected]

Ege University Department . of Mechanical Engineering Bornova Izmir Turkey e-mail: [email protected]

M. Muralimohan BANERJEE

Stephan P. BANKS

Retired Reader Department of Mathematics A. C. College Jalpaiguri 735101 W.B India e-mail: dgp [email protected]

The University of Sheffield Department of Automatic Control and System Engineering Sheffield UK

Alemdar BAYRAKTAR

˙ ¸E M. BIKC


University of Mustafa Kemal Department of Civil Engineering 31024 Hatay Turkey

˙ IK ˙ G¨ ulin BIRL

Paritosh BISWAS

Middle East Technical Univeristy Department of Engineering Sciences Ankara Turkey e-mail: [email protected]

Executive Secretary Von Karman Society for Advanced Study and Research& Founder Member ICOVP (India) old Police Line Jalpaiguri-735101 Bengal India e-mail: biswas [email protected]

M. BONADIES

C. BONIFASI-LISTA

University of Torino Department of Mathematics v.C. Alberto 10 10123 Torino Italy

University of Utah Department of Mathematics Salt Lake City UT 84112 USA

LIST OF PARTICIPANTS

549

N. BONTCHEVA

Hakan BOYACI

Bulgarian Academy of Sciences Institute of Mechanics Acad. G. Bonchev Str. bl. 4 113 Sofia Bulgaria

Celal Bayar University Department of Mechanical Engineering 45140 Muradiye Manisa Turkey e-mail: [email protected]

Gökhan BULUT

Zekai CELEP

Bikas K. CHAKRABARTI

A. CHATTERJEE

Saha Institute of Nuclear Physics Bidhan Nagar Kolkata 700064 India e-mail: [email protected]

Saha Institute of Nuclear Physics Bidhan Nagar Kolkata 700064 India

Andrej CHERKAEV

Elena CHERKAEV

University of Utah Utah Salt Lake City Utah USA e-mail: [email protected]

University of Utah Utah Salt Lake City Utah USA e-mail: [email protected]

B. S. CHOO

Mehmet C ¸ ELEBI˙

29 Shirley Road Nottingham NG35 DA UK

USGS (MS977) 345 Middlefield Rd. Menlo Park CA. 94025 e-mail: [email protected]

˙ Iffet C ¸ IRAK

˙ Fuat DEMIR

Suleyman Demirel University Department of Civil Engineering Isparta Turkey e-mail: iff[email protected]

S¨ uleyman Demiral University Department of Civil Engineering Isparta Turkey e-mail: [email protected]

˙ Hilmi DEMIRAY

˙ Ali DENIZ

I¸sık University Faculty .of Art & Sciences Maslak Istanbul Turkey e-mail: [email protected]

Karlsruhe University Institute of Solid Mechanics Karlsruhe Germany e-mail: [email protected]

˘ Vedat DOGAN

J. DRONKA

. Istanbul Technical University Faculty of Mechanical . Engineering 34439 G¨ um¨ u¸ssuyu Istanbul Turkey e-mail: [email protected]

. Istanbul Technical University Department of Aeronautical Engineering . Maslak Istanbul Turkey e-mail: [email protected]

. Istanbul Technical University Faculty of Civil . Engineering 34469 Maslak Istanbul Turkey e-mail: [email protected]

Rzeszow University of Technology Department of Mathematics W. Pola 2. 35-959 Rzeszow Poland

550


˘ A. Aydın DUMANOGLU

Horst ECKER

Grand National Assembly of Turkey Ankara Turkey

Vienna University of Technology Institute of Mech. and Mechatronics Wiedner Hauptstrasse 8-10/E325/A3 A-1040 Vienna Austria e-mail: [email protected]

˙ ˙ ˘ Ibrahim EMIRO GLU

E. EMSEN

Yıldız Technical University Faculty of Chemistry and Metallurgy Department of Mathematical Engineering Davutpa¸sa Campus No:127 . 34210 Topkapı Istanbul Turkey

C ¸ ukurova University Department of Civil Engineering 01330 Adana Turkey

¨ Ali Unal ERDEM

˙ Mustafa ERDIK

Gazi University Architecture of Engineering Faculty Ankara Turkey e-mail: [email protected]

Bogazici . University Bebek Istanbul Turkey e-mail: [email protected]

¨ M. Erta¸c ERGUVEN

Aybar ERTEPINAR

. Istanbul Technical University Faculty of Civil. Engineering 34469 Maslak Istanbul Turkey e-mail: [email protected]

Middle East Technical University Department of Civil Engineering 06531 Ankara Turkey e-mail: [email protected]

V. A. FEDOROV

Tatiana N. FESENKO

Moscow Power Engineering Institute (Technical University) Moscow Russia

Mechanical Engineering Research Institute of Russian Academy of Sciences Moskow Russia

George V. FILIPPENKO

A. V. FOMENKOV

Institute of Mechanical Engineering Vasilievsky Ostrov Bolshoy Prospect 61 St. Petersburg 199178 Russia e-mail: [email protected]


Valeriv N. FOURSOV

¨ US ¨¸ M. Tolga GOG

Mechanical Engineering Research Institute of Russian Academy of Sciences Moskow Russia

University of Gaziantep Department of Civil Engineering 27310 Gaziantep Turkey e-mail: [email protected]

LIST OF PARTICIPANTS ¨ Co¸skun GULER

S. KARANJAI

Yıldız Technical University Faculty of Chemistry and Metallurgy Department of Mathematical Engineering Davutpasa Campus No: 127 . 34210 Topkapi Istanbul Turkey e-mail: [email protected]

University of North Bengal Department of Mathematics North Bengal India

¨ Deniz GUNEY

. Istanbul Technical University Department of. Engineering 34469 Maslak Istanbul Turkey e-mail: [email protected]

˙ Esin INAN I¸sık University Faculty of Arts and Sciences Department of Mathematics Kumbaba Mevkii Sile 34980 . Istanbul Turkey e-mail: [email protected]

Jordanka IVANOVA Bulgarian Academy of Sciences Institute of Mechanics Acad. G. Bonchev Str. bl. 4 113 Sofia Bulgaria e-mail: [email protected]

¨ Kadir GULER

. Istanbul Technical University Faculty of Civil . Engineering 34469 Maslak Istanbul Turkey e-mail: [email protected]

S. HAQ University of Liverpool Department of Mathematical Sciences Liverpool UK

R. KAHALI Department of Physics P. D. Women’s College Jalpaiguri 735101 Bengal India

S. M. KAPLUNOV Imash ran MGTU by N. Bauman OKB GIDROPRESS e-mail: [email protected]

˙ IK ˙ Mehmet IT

Bapi KARMAKAR

Erciyes University Engineering and Architecture Faculty Mechanical Engineering Department Yozgat Turkey e-mail: [email protected]

Executive Secretary Von Karman Society for Advanced Study and Research& Founder Member ICOVP(India)Old Police Line Jalpaiguri-735101West Bengal India

Cihan KAN

. Istanbul Technical University Department of Mechanical Engineering . Maslak Istanbul Turkey e-mail: [email protected]

551

552


Metin Orhan KAYA

Mesut KIRCA

. Istanbul Technical University Faculty of Aeronautics and Astronautics . 34469 Maslak Istanbul Turkey e-mail: [email protected]

. Istanbul Technical University Department of Aeronautical Engineering . Maslak Istanbul Turkey e-mail: [email protected]

Ahmet KIRIS ¸

¨ Onder KORFALI

. Istanbul Technical University Faculty of Science and Letters Department of. Engineering Science Maslak 34469 Istanbul Turkey e-mail: [email protected]

Danill P. KOUZOV Institute of Mechanical Engineering Vasilievsky Ostrov Bolshoy Prospect 61 St. Petersburg 199178 Russia

Valery M. LEVIN Istituto Mexicano del Petroleo Eje Central Lazaro Cardenas No.152 Col. San Bartolo Atepehuacan C. P. 07730 D. F. Mexico

Galatasaray University Faculty of. Engineering and Tecnology Ortakoy Istanbul Turkey

Nahit KUMBASAR . Istanbul Technical University Faculty of Science and Letters . 34469 Maslak Istanbul Turkey e-mail: [email protected]

N. A. MAKHUTOV IMASH RAN MGTU by N. Bauman OKB GIDROPRESS Russian Federation

Anita MEHTA ˘ Radek MATEJEC Technical University of Lberec H´ alkova 6 461 17 Liberec Czech Republic

2H Cornfield Road 3rd Floor Calcutta 700 019 India e-mail: [email protected]

G. S. MISHURIS Thomas M. MICHELITSCH The University of Sheffield Department of Civil and Structural Engineering Sir Frederick Mappin Building Mappin Street Sheffield S1 3JD UK e-mail: t.michelitsch@sheffield.ac.uk

A. V. MIZIRIN Moscos Power Engineering Institute (Technical University) Moscow Russia

Rzeszow University of Technology Department of Mathematics W. Pola 2. 35-959 Rzeszow Poland

Subrata MONDAL Faculty Member of Mechanical Engineering Department Jalpaiguri Govt. Engg College Jalpaiguri West Bengal India e-mail: [email protected]


553

Alexander B. MOVCHAN

Natasha V. MOVCHAN

University of Liverpool Department of Mathematical Sciences Liverpool England UK e-mail: [email protected]

University of Liverpool Department of Mathematical Sciences Liverpool England UK e-mail: [email protected]

Abhijit MUKHERJEE

Radoslav NABERGOJ

Department of Civil Engineering IIT Bombay 400076 India e-mail: [email protected]

University of Trieste Via Valerio Department of Naval Architecture Ocean and Environmental Engineering 10 34127 Trieste Italy

Y. NATH

Ranier NORDMANN

Indian Institute of Technology Delhi Applied Mechanics Department Hauz Khas New Delhi 110 016 India e-mail: [email protected]

Arbeitsgruppe Mechatronik-TU Darmstadt Petersenstr. 30 D-64287 Lichtwiese Germany e-mail: [email protected]

Miloslav OKROUHLIK

¨ Ulker ONBAS ¸ LI

Institute of Thermomechanics Acad. Sci. Czech Rep. Dolejskova 5 CZ-182 00 Czech Republic e-mail: [email protected]

Yıldız Technical University Faculty of Science and Art Department of Physics . Davutpasa 34010 Istanbul Turkey e-mail: [email protected]

A. Vatani OSKOUEI

¨ Mustafa OZAKC ¸A

Shahid Rajaee University Buildings and Housing Research Center P. O. Box 13145-1696 Tehran IRAN e-mail: [email protected]

University of Gaziantep Department of Civil Engineering 27310 Gaziantep Turkey e-mail: [email protected]

¨ ¨ ˙ Ozge OZDEM IR

¨ ˙ Zeynep G¨ uven OZDEM IR

. Istanbul Technical University Faculty of Aeronautics and Astronautics . 34469 Maslak Istanbul Turkey e-mail: [email protected]

¨ Mutlu OZER 205 Commodore Dr. California USA e-mail: [email protected]

Yıldız Technical University Faculty of Science and Art Department of Physics . Davutpa¸sa 34010 Istanbul Turkey e-mail: [email protected]

¨ ¨ Ahmet Ozkan OZER

. Istanbul Technical University Faculty of Science and Letters . Maslak 34469 Istanbul Turkey e-mail: [email protected]

554


¨ M. Sinan OZEREN

. Istanbul Technical University Faculty of Mines . 34469 Maslak Istanbul Turkey e-mail: [email protected]

Burak PARLAK Galatasaray University Faculty of. Engineering and Tecnology Ortakoy Istanbul Turkey e-mail: [email protected]

F. PASTRONE

˘ Lud˘ek PESEK

University of Torino Department of Mathematics v. C. Alberto 10 10123 Torino Italy

Institute of Thermomechanics Dolej˘skova 5 Prague 8 CZ 182 00 Czech Republic e-mail: [email protected]

˘ Nazmi POSTACIOGLU

Ladislav PU ST

. Istanbul Technical University Faculty of Mines Main Campus . 80630 Maslak Istanbul Turkey

Institute of Thermomechanics Dolej˘skova 5 Prague 8 CZ 182 00 Czech Republic e-mail: [email protected]

J. RAMACHANDRAN

˘ Rıfat RES ¸ ATOGLU

Department of Applied Mechanics Indian Institute of Technology Chennai 600 036 Tamil Nadu India e-mail: [email protected]


Metin U. SALAMCI

Ritesh SAMADHIYA

Gazi University Engineering and Architecture Faculty Mechanical Engineering Department Maltepe Ankara Turkey e-mail: [email protected]

Department of Civil Engineering IIT Bombay 400076 India e-mail: [email protected]

Mehmet SARIKANAT

ECKART SCHNACK


Karlsrube University Institute of Solid Mechanics Karlsrube Germany e-mail: [email protected]

¨ ur SELSIL ˙ Ozg¨

Erol S ¸ ENOCAK

The University of Liverpool Department of Mathematical Sciences L69 7ZL Liverpool UK e-mail: [email protected]

. Istanbul Technical University Department of Mechanical Engineering . 34437 G¨ um¨ u¸ssuyu Istanbul Turkey e-mail: [email protected]

◦

LIST OF PARTICIPANTS ¨ ˙ Onder Cem SEZGIN

V. I. SOLONIN

Middle East Technical University Healt and Counselling Center Ankara Turkey e-mail: [email protected]

IMASH RAN MGTU by N. Bauman OKB GIDROPRESS Russian Federation

N. V. SHARIY

A. SHARMA

IMASH RAN MGTU by N. Bauman OKB GIDROPRESS Russian Federation

Thapar Institute of Engineering and Technology Patiala 1470004 India

V. V. SHTYKOV

K. K. SHUKLA


Motilal Nehru National Institute of Technology Allahabad 211 004 India

S. SINGH

˘ AK ` Michal SIVC

Applied Mechanics Department I.I.T. Delhi New Delhi 110 016 India

Antonin SKAROLEK

Technical University of Liberec Department of Mechanics and Stres Analysis Halkova 6 CZ-460 17 Liberec 1 Czech Republic

˘ Jan SKLIBA

Technical University of Liberec Department of Mechanics and Stres Analysis Halkova 6 CZ-460 17 Liberec 1 Czech Republic

Technical University of Liberec Department of Mechanics and Stres Analysis Halkova 6 CZ-460 17 Liberec 1 Czech Republic e-mail: [email protected]

Lenoid SLEPYAN

S. M. SMOLSKIY

Tel Aviv University Depatment of Solid Mechanics Materials and Structures Ramat Aviv 69978 Israel

Moscow Power Engineering Institute (Technical University) Moscow Russia e-mail: [email protected]

Abdullah H. SOFIYEV

Ali SOFIYEV

S¨ uleyman Demirel University Department of Civil Engineering Isparta Turkey e-mail: asofi[email protected]


555

556


Kurtulu¸s SOYLUK

Rudolf SVOBODA

Gazi University Department of Civil Engineering 06570 Maltepe Ankara Turkey e-mail: [email protected]

Techlab Ltd Sokolovsk 207 190 00 Praha 9 Czech Republic e-mail: [email protected]

¨ ¨ Umit SONMEZ

¨ Hakan TANRIOVER

. Istanbul Technical University Department of Mechanical Engineering . 80191 G¨ um¨ u¸ssuyu Istanbul Turkey e-mail: [email protected]

. Istanbul Technical University Department of Mechanical Engineering . G¨ um¨ u¸ssuyu 34439 Istanbul Turkey e-mail: [email protected]

Fatih TAS ¸C ¸I

Vedat TAS ¸ KIN

Yıldız Technical University Faculty of Chemistry and Metallurgy Department of Mathematical Engineering Davutpasa Campus No:127 . 34210 Topkapi Istanbul Turkey e-mail: [email protected]

Trakya University Department of Mechanical Engineering 22030 Edirne Turkey

N. TAYS ¸ I˙ University of Gaziantep Department of Civil Engineering 27310 Gaziantep Turkey e-mail: [email protected]

¨ Mevl¨ ut TEYMUR . Istanbul Technical University Department of Mathematics Faculty of Sciences and Letters . 80626 Maslak Istanbul Turkey e-mail: [email protected]

Umut TOPAL Karadeniz Technical University Department of Civil Engineering 61080 Trabzon Turkey e-mail: [email protected]

¨ Halit S. TURKMEN . Istanbul Technical University Faculty of Aeronautics and Astronautics . 34469 Maslak Istanbul Turkey e-mail: [email protected]

Hamide TEKELI˙ Suleyman Demirel University Department of Civil Engineering Isparta Turkey e-mail: [email protected]

Ales TONDL Zborovska 41 Prague 5 CZ-150 00 CZECH REPUBLIC e-mail: [email protected]

¨ ur TURHAN Ozg¨ . Istanbul Technical University Faculty of Mechanical . Engineering 34439 G¨ um¨ u¸ssuyu Istanbul Turkey e-mail: [email protected]

¨ F. Demet ULKER Middle East Technical University Aeronautical Engineering Department Ankara Turkey

LIST OF PARTICIPANTS Hakan UREY

¨ Umit UZMAN

Ko¸c University Department of Electrical Engineering . Istanbul Turkey

Karadeniz Technical University G¨ um¨ u¸shane Engineering Faculty Department of Civil Engineering 29000 G¨ um¨ u¸shane Turkey e-mail: [email protected]

557

Habib UYSAL Atat¨ urk University Engineering Faculty Department of Civil Engineering Erzurum Turkey e-mail: [email protected]

Jizeng WANG

˙ GLU ˘ Nazmiye YAHNIO

Mustafa YAVUZ

Yıldız Technical University Department of Mathematical Engineering Faculty of Chemistry and Metallurgy Davutpasa . Campus 34210 Topkapi Istanbul Turkey e-mail: [email protected]

Max Planck Institute For Metals Research Heisenbergstrasse 3 D-70569 Stuttgart Germany

. Istanbul Technical University Faculty of Civil. Engineering 34469 Maslak Istanbul Turkey e-mail: [email protected]

Vladimir V. YELISEYEV

Vebil YILDIRIM

St. Petersburg State Polytechnical University Computer Technologies in Engineering Department Polytechnicheskaya st. 29 St. Petersburg 195251 Russia

C ¸ ukurova University Faculty of Engineering and Architecture Department of Mechanical Engineering 01330-Balcalı Adana Turkey e-mail: [email protected]

Hasan YILDIZ

¨ H¨ useyin Murat YUKSEL


. Istanbul Technical University Faculty of Aeronautics and Astronautics . 34469 Maslak Istanbul Turkey

Tatiana V. ZINOVIEVA

Bahar ZORA

St. Petersburg State Polytechnical University Computer Technologies in Engineering Department Polytechnicheskaya st. 29 St. Petersburg 195251 Russia e-mail: [email protected]

. Istanbul Technical University Faculty of Mines . 34469 Maslak Istanbul Turkey

springer proceedings in physics 64 Superconducting Devices and Their Applications Editors: H. Koch and H. Lübbing

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81 Materials and Measurements in Molecular Electronics Editors: K. Kajimura and S. Kuroda 82 Computer Simulation Studies in Condensed-Matter Physics IX Editors: D.P. Landau, K.K. Mon, and H.-B. Schüttler 83 Computer Simulation Studies in Condensed-Matter Physics X Editors: D.P. Landau, K.K. Mon, and H.-B. Schüttler 84 Computer Simulation Studies in Condensed-Matter Physics XI Editors: D.P. Landau and H.-B. Schüttler 85 Computer Simulation Studies in Condensed-Matter Physics XII Editors: D.P. Landau, S.P. Lewis, and H.-B. Schüttler 86 Computer Simulation Studies in Condensed-Matter Physics XIII Editors: D.P. Landau, S.P. Lewis, and H.-B. Schüttler 87 Proceedings of the 25th International Conference on the Physics of Semiconductors Editors: N. Miura and T. Ando 88 Starburst Galaxies Near and Far Editors: L. Tacconi and D. Lutz 89 Computer Simulation Studies in Condensed-Matter Physics XIV Editors: D.P. Landau, S.P. Lewis, and H.-B. Schüttler

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