Conference Proceedings of the Society for Experimental Mechanics Series
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Tom Proulx Editor
Advanced Aerospace Applications, Volume 1 Proceedings of the 29th IMAC, A Conference on Structural Dynamics, 2011
Editor Tom Proulx Society for Experimental Mechanics, Inc. 7 School Street Bethel, CT 06801-1405 USA
[email protected] ISBN 978-1-4419-9301-4 e-ISBN 978-1-4419-9302-1 DOI 10.1007/978-1-4419-9302-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011922492 © The Society for Experimental Mechanics, Inc. 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Advanced Aerospace Applications represents one of six clusters of technical papers presented at the 29th IMAC, A Conference and Exposition on Structural Dynamics, 2011, organized by the Society for Experimental Mechanics, and held in Jacksonville, Florida, January 31 - February 3, 2011. The full proceedings also include volumes on Linking Models and Experiments, Modal Analysis; Civil Engineering; Rotating Machinery, Structural Health Monitoring, and Shock and Vibration; and Sensors, Instrumentation, and Special Topics. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. The current volume on Advanced Aerospace Applications includes studies on Aeroelasticity, Ground Testing, and Dynamic Testing of Aerospace Structures. It could be said that many early developments in the field of structural dynamics were motivated by the needs of aviation, and later aerospace. By their very nature aerospace products are susceptible to vibration and they operate in high vibration environments. Structural dynamics plays a key role in aerospace design and testing, impacting flight safety, product durability, performance, and comfort. As in other industries, today’s aerospace products are pushing the limits of performance, pose increased demands for structural dynamic analysis and testing, and will benefit greatly from the recent developments in this field that are the topics of technical sessions at IMAC. The organizers would like to thank the authors, presenters, session organizers and session chairs for their participation in this track.
Bethel, Connecticut
Dr. Thomas Proulx Society for Experimental Mechanics, Inc
Contents
1
High Frequency Optimisation of an Aerospace Structure Through Sensitivity to SEA Parameters A. Culla, Università di Roma ’La Sapienza’; W. D’Ambrogio, Università dell’Aquila; A. Fregolent, Università di Roma ’La Sapienza’
1
2
Benefit of Acoustic Particle Velocity Based Reverberant Room Testing of Spacecraft E.H.G. Tijs, Microflown Technologies; J.J. Wijker, Dutch Space BV; A. Grillenbeck, IABG mbH
15
3
Ultrasonic Vibration Modal Analysis Technique (UMAT) for Defect Detection J.L. Rose, Pennsylvania State University/FBS, Inc.; F. Yan, FBS, Inc.; C. Borigo, Y. Liang, Pennsylvania State University
25
4
Acoustic Testing and Response Prediction of the CASSIOPE Spacecraft V. Wickramasinghe, A. Grewal, D. Zimcik, National Research Council Canada, Institute for Aerospace Research; A. Woronko, P. Le Rossignol, V-O. Philie, MDA Space Missions; M. O'Grady, R. Singhal, Canadian Space Agency
33
5
Force Limited Vibration Testing Applied to the JWST FGS OA Y. Soucy, Canadian Space Agency; P. Klimas, COM DEV Canada
45
6
On Force Limited Vibration for Testing Space Hardware Y. Soucy, Canadian Space Agency
63
7
Calculation of Rigid Body Mass Properties of Flexible Structures K. Napolitano, M. Schlosser, ATA Engineering, Inc.
73
8
Simulating Base-shake Environmental Testing J. Steedman, NAVCON Engineering Network; B. Schwarz, M. Richardson, Vibrant Technology, Inc.
95
9
Geometry-based Updating of 3D Solid Finite Element Models T. Lauwagie, E. Dascotte, Dynamic Design Solutions
103
10
A PZT-based Technique for SHM Using the Coherence Function J. Vieira Filho, F.G. Baptista, Sao Paulo State University; D.J. Inman, Virginia Polytechnic Institute and State University
111
11
The Best Force Design of Pure Modal Test Based Upon a Singular Value Decomposition Approach J.M. Liu, Tsinghua University/China Orient Institute of Noise & Vibration; Q.H. Lu, Tsinghua University; H.Q. Ying, China Orient Institute of Noise & Vibration
119
viii 12
Modal Identification and Model Updating of Pleiades F. Buffe, CNES; N. Roy, TOP MODAL; S. Cogan, FEMTO-ST Institute
131
13
Aircraft GVT Advances and Application – Gulfstream G650 R. Brillhart, K. Napolitano, ATA Engineering, Inc.; L. Morgan, R. LeBlanc, Gulfstream Aerospace Corporation
145
14
Aircraft Dynamics and Payload Interaction – SOFIA Telescope R. Brillhart, K. Napolitano, ATA Engineering, Inc., T. Duvall, L-3 Communications, Integrated Systems
159
15
Application of Modal Analysis for Evaluation of the Impact Resistance of Aerospace Sandwich Materials A. Shahdin, J. Morlier, G. Michon, L. Mezeix, C. Bouvet, Y. Gourinat, Université de Toulouse
171
16
An Integrated Procedure for Estimating Modal Parameters During Flight Testing W. Fladung, G. Hoople, ATA Engineering, Inc.
179
17
Multiple-site Damage Location Using Single-site Training Data R.J. Barthorpe, K. Worden, The University of Sheffield
195
18
Assessment of Nonlinear System Identification Methods Using the SmallSat Spacecraft Structure G. Kerschen, University of Lige; L. Soula, J.B. Vergniaud, Astrium Satellites; A. Newerla, European Space Agency (ESTEC)
19
Ground Vibration Testing Master Class: Modern Testing and Analysis Concepts Applied to an F-16 Aircraft J. Lau, B. Peeters, J. Debille, Q. Guzek, W. Flynn, LMS International; D.S. Lange, Air Force Flight Test Center; T. Kahlmann, AICON 3D Systems GmbH
20
Advanced Shaker Excitation Signals for Aerospace Testing B. Peeters, J. Lau, LMS International; A. Carrella, University of Bristol; M. Gatto, G. Coppotelli, Università di Roma “La Sapienza”
21
System and Method for Compensating Structural Vibrations of an Aircraft Caused by Outside Disturbances W. Luber, J. Becker, CASSIDIAN - Air Systems
203
221
229
243
22
Operational Modal Analysis on a Modified Helicopter E. Camargo, D. Strafacci, Centro Técnico Aeroespacial; N-J. Jacobsen, Brüel & Kjær Sound & Vibration Measurement A/S
265
23
Development of New Discrete Wavelet Families for Structural Dynamic Analysis J.R. Foley, J.C. Dodson, U.S. Air Force Research Laboratory; A.J. Dick, Q.M. Phan, P.D. Spanos, Rice University; J.C. Van Karsen, Michigan Technological University; G.L. Falbo, LMS Americas, Inc.
275
24
Model Updating With Neural Networks and Genetic Optimization M.E. Yumer, E. Cigeroglu, H.N. Özgüven, Middle East Technical University
285
25
A Piezoelectric Actuated Stabilization Mount for Payloads Onboard Small UAS K.J. Stuckel, W.H. Semke, University of North Dakota
295
ix 26
Extraction of Modal Parameters From Spacecraft Flight Data G.H. James, T.T. Cao, V.A. Fogt, R.L. Wilson, NASA Johnson Space Center; T.J. Bartkowicz, The Boeing Company
307
27
Dynamic Characterization of Satellite Components Through Non-invasive Methods D. Macknelly, Imperial College London; J. Mullins, Vanderbilt University; H. Wiest, Rose-Hulman Institute of Technology; D. Mascarenas, G. Park, Los Alamos National Laboratory
321
28
An Inertially Referenced Non-contact Sensor for Ground Vibration Tests B. Allen, Moog CSA Engineering; C. Harris, D. Lange, Edwards Flight Test Center
339
29
Reliability of Experimental Modal Data Determined on Large Spaceflight Structures A. Grillenbeck, S. Dillinger, IABG mbH
351
30
Operational Modal Analysis of a Spacecraft Vibration Test M. O’Grady, R. Singhal, Canadian Space Agency
363
High frequency optimisation of an aerospace structure through sensitivity to SEA parameters Antonio Culla, Walter D'Ambrogio and Annalisa Fregolent
Abstract Classical (FEM, BEM) structural optimisation techniques fail to solve medium high frequency dynamic problems because too many DoFs are involved and eigenvalues and eigenvectors loose the significance due to high modal density. Using a SEA model, the subsystem energies are controlled by (coupling) loss factors, under the same loading conditions. In turn, coupling loss factors (CLF) depend on physical parameters of the subsystems. The idea is to determine an approximate relation between CLF and physical parameters that can be modified in the structural optimisation process, for instance, by using Design of Experiment (DoE). Starting from this relation, an optimisation problem can be formulated in order to bring the subsystem energies under prescribed levels. A preliminary analysis of subsystem energy sensitivity to CLF can be performed to save time in looking for the approximate relationship between CLF’'s and physical parameters. The approach is applied on a typical aerospace structure.
1 Introduction In Statistical Energy Analysis, the studied systems belong to a random population of similar systems [9]. Systems are considered similar if their physical parameters are slightly different. SEA considers a structure as the union of several subsystems. Antonio Culla Dipartimento di Meccanica e Aeronautica, Università di Roma 'La ’ Sapienza', ’ Via Eudossiana 18, 00184 Roma, Italy, e-mail:
[email protected] Walter D’'Ambrogio Dipartimento di Ingegneria Meccanica, Energetica e Gestionale, Università dell'Aquila, ’ Piazza E. Pontieri 2, 67040 Roio Poggio (AQ), Italy e-mail:
[email protected] Annalisa Fregolent Dipartimento di Meccanica e Aeronautica, Università di Roma 'La ’ Sapienza', ’ Via Eudossiana 18, 00184 Roma, Italy, e-mail:
[email protected] T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_1, © The Society for Experimental Mechanics, Inc. 2011
1
2 Each of them is a modal group, i.e. a set of similar modes. For instance, considering two plates welded together, six modal groups can be identified, one set of flexural modes and two sets of in plane modes for each plate. SEA estimates the mean value of the energy stored in the modal groups constituting the studied system. The mean value provided by SEA equations is in principle the average response of a set of similar systems. However, SEA equations are represented by a linear system of equations for each frequency, or better for each frequency band. The solution of each linear system gives the energy of each subsystem in a given frequency band. No average operation is explicitly performed, but all the statistics is not visible to the user. In general, this is not a problem because many simple relationships used by physicists and engineers are the result of more complicated mathematical procedures. Unfortunately, in this case this simple model holds only under many strong hypotheses, listed in Section 2. The linear system results from some mathematical manipulations, that include also averages on frequency bands, on the classical equations of motion of multi degrees of freedom systems, and the observance of the strong hypotheses mentioned before. The coefficients of the linear system, named coupling loss factor (CLF) and internal loss factor (ILF), are the result of these average processes and account for the parameters of the native physical system. Therefore, SEA gives the energy of each modal group belonging to the studied system. This energy is the most representative sample of a statistical population of similar systems and on a frequency band. No information is given about the dispersion of the data around the result. In order to provide a true statistic solution it is necessary, at least, to know the variance of the result. All the methods to approach the problem to calculate this variance use either parametric models [11, 2, 5, 1] or nonparametric models [9, 8, 6, 7]. In the first case the physical properties of the system are considered to be uncertain. The uncertainty of the response is calculated by considering the propagation of the physical uncertainties through the model. If the system is very complex and random, its natural frequencies statistics can be assumed, that is the statistics of the response can be considered independent of the statistics of the physical quantities. This is the nonparametric case. In this paper a parametric approach is adopted. However, the study on the effect of SEA parameter variations on SEA results is an opportunity to perform an optimization problem in order to reduce the energy level of some critical subsystem of a complex structure. The effect of uncertainties is modeled by using a sensitivity approach [4, 3]. The sensitivity of the energies stored into the subsystems is calculated by considering uncertainties on ILF’'s and CLF’'s. The goal is to preliminarily understand how much the energies (SEA solution) depend on variations of CLF’'s and ILF’'s. After discarding CLF's ’ and ILF’'s that are less effective on subsystem energies, a model of the remaining SEA coefficients as function of selected physical parameters is developed. The selected physical parameters are that can be modified. Subsequently, a simple mathematical model of how they affect CLF’'s and ILF’'s can be obtained using Design of Experiments [10].
3 A constrained optimisation problem is formulated in order to bring the subsystem energies under prescribed levels. The variables of the problem are the relative deviation of the selected physical parameters from their nominal value. Upper and lower bounds of these design variables are defined. Finally, the energy of the subsystems are constrained to be lower than a prescribed value. In section 4 the procedure described before is performed to optimize the energy of three subsytems of an aerospace structure by modifying three physical parameters. First, a preliminary sensitivity analysis is performed in order to select the more significant CLF’'s to which the considered energies are more sensitive. Then, by using DoE an approximate relation between the more sensitive CLF’'s and the three selected physical parameters is obtained. Finally, the results of the optimisation gives the set of physical parameters values corresponding to the constrain on the energies.
2 SEA equations Under some particular hypotheses, it is possible to assume that the transmitted power between two subsystems is proportional to the difference of the energy stored in each subsystem. A list of these hypotheses is presented below: • all the modes of a subsystem must be similar (i.e. they must have almost the same energy, damping, coupling with the other subsystems and they must be almost excited by the same input power), • the coupling between the subsystems must be conservative, • the eigenfrequencies must be uniformly probable in the frequency range, • the force exciting the subsystems must be random and uncorrelated, • the interactions between the subsystems must be weak. Thus, the SEA equations of Nsub coupled subsystems can be written as follows: Nsub
Pi,in j = ω ηi Ei + ω
∑
(ηi j Ei − η ji E j )
(1)
j=1, j6=i
where i and j are indexes of the subsystems, ηi and ηi j are the internal loss factors (ILF) and the coupling loss factors (CLF), respectively, Pi,in j is the power injected into the subsystem i, E is the energy in a given subsystem and ω is the central frequency of the considered band. Equations (1) represents the energy balance of the subsystems. The power dissipated in the subsystem i is: Pi,d = ω ηi Ei
(2)
The power transmitted from subsystem i to the subsystem j is: Pi j = ω (ηi j Ei − η ji E j )
(3)
4 The solution of the linear system (1) provides the energy stored in each subsystem. The set of equations (1) can be rewritten in a more convenient way as follows: p = ω Ce
(4)
where the coefficients of matrix C are combinations of ILF’'s and CLF’'s as shown in the following equations: ( Ci j = −η ji i, j = 1 . . . Nsub , i 6= j (5) Nsub C j j = η j + ∑i=1, i6= j η ji If ni and n j are the modal densities of subsystems i and j, the following reciprocity relationship holds: ηi j ni = η ji n j (6) By enforcing reciprocity, under the assumption that only the η ji with j > i are known, it is: −η ji if j > i nj Ci j = if j < i −η ji (7) ni C j j = η j − Nsub Ci j ∑i=1, i6= j
3 Effect of SEA parameter variation on subsystem energies Here a study on the effect of SEA parameter variations on SEA results is performed. Therefore, the goal is to preliminarily understand how much the energies (SEA solution) depend on variations of CLF’'s and ILF’'s. The energy of each subsystem is calculated by solving equation (4) with the obvious implication that energies depend on the CLF’'s and the ILF’'s of the considered system. By defining a range of variability of CLF's ’ and ILF's, ’ a sensitivity approach is used to account for the dependence of the energy on the variations of SEA parameters.
3.1 Sensitivity to SEA parameter variation Sensitivity to loss factors is evaluated in correspondence to nominal values ηˆ of the CLF’'s and ILF’'s. To compare different sensitivity factors, it is assumed that changes ∆ ηkl in the coupling loss factors are a given fraction of nominal value or, with additional effort, they are calculated in agreement with prescribed variations of the physical parameters.
5 ¯ ∂ e ¯¯ ∆ ηkl ∂ ηkl ¯η =ηˆ
∆ ekl =
(8)
and similarly for ILF’'s. To find ∂ e/∂ ηkl , it is necessary to differentiate the solution of Eq. (4): e=
1 −1 C p ω
⇒
1 ∂ C−1 ∂e = p ∂ ηkl ω ∂ ηkl
(9)
and similarly if internal loss factor ηk are considered instead of ηkl . Here it is assumed that the injected power is not affected by changes in CLF’'s and ILF’'s. The derivative of C−1 can be easily obtained from the identity C C−1 = I
∂ C−1 ∂ C −1 = −C−1 C ∂ ηkl ∂ ηkl
(10)
where ∂ C/∂ ηkl can be computed from Eq. (7): 1 nk ∂ Ci j nl = −1 ∂ ηkl − nk nl 0
if i = j and i = k if i = l and j = l if i = l and j = k
(11)
if i = k and j = l else
and similarly for ∂ C/∂ ηk :
∂ Ci j = ∂ ηk
( 1 0
if i = j and i = k else
(12)
At the end of this stage, CLF’'s and ILF’'s that are less effective in changing the subsystem energies, i.e. for which ∆ ekl is lower, can be discarded.
3.2 CLF’s and ILF’s as functions of physical parameters After discarding CLF’'s and ILF’'s that are less effective on subsystem energies, a model of the remaining SEA coefficients as function of selected physical parameters must be developed. First of all, physical parameters that can be modified are selected; subsequently, a simple mathematical model of how they affect CLF’'s and ILF’'s can be obtained using Design of Experiments (see appendix). At the end of this stage, the most effective CLF’'s and ILF’'s are expressed as:
ηi j = ηi j (x)
(13)
6 where x contains the relative deviations of the selected physical parameters from their nominal values. Therefore, from Eq. (5) or Eq. (7) it is possible to express Ci j as: Ci j = Ci j (x)
⇒
C = C(x)
(14)
Finally, the subsystem energies can be obtained as function of x as: e(x) =
1 −1 C (x) p ω
(15)
3.3 Modification of physical parameters to reduce subsystems energies A constrained optimization problem can be defined in order to reduce the energy level of some critical subsystems by varying the selected physical parameters: Np
arg min f (x) = ∑ wi xi2 x
subject to
(
i=1
Eˆlk (x) ≤ Elk∗
l = 1, . . . , NE
xLi ≤ xi ≤ xUi
i = 1, . . . , N p
k = 1, . . . , N f
(16)
where x contains the relative deviations of the selected physical parameters from their nominal values, wi are weights associated to possible costs of modifications, N p is the total number of physical parameters that can be modified. NE is the number of subsystems for which it is required to reduce or control the energy level, N f is the number of frequency bands. Ekl (x) is the energy of subsystem l at the frequency band k that must be lower than a prescribed value, Elk∗ : it defines N f ×NE constraints. Moreover, upper and lower bounds of the design variable x are defined, xL and xU , so that the solution is always in the range xL ≤ x ≤ xU .
4 Results The studied structure is a launcher fairing. A sketch of the system is shown in figure 1. Subsystems 1, 2 and 3 are made of composite material: the core is aluminum honeycomb and the skins are carbon fiber layers. Subsystem 4 is made of ribbed aluminum shell. The scheme in Figure 2 shows the coupling between the SEA subsystems. The system is excited by an external acoustic diffuse field. The SEA solution is sought at 18 third octave bands between 25 Hz and 1250 Hz.
7
1
5 2
3
4 6
Fig. 1 Launcher fairing: 1 ogive, 2 cylinder, 3 payload adapter, 4 interstage, 5 top cavity, 6 bottom cavity.
!"#"$%&'("
)"#"*+,&-.(/"
0"#"&-1(/213%("
4"#"1$5"*3'&1+"
6"#"53+,$3." 3.351(/"
7"#"8$9$:" *3'&1+"
Fig. 2 SEA subsystems coupling
The considered problem concerns the optimization of the SEA model by imposing an upper bound on the SEA solution when the variability of some CLF’'s is taken into account. In particular, the analysis is focused on the energy of the subsystems 3, 4 and 5, represented by the flexural modes of the structural subsystems payload
8 adapter and interstage skirt and by the acoustic modes of the cavity 5, because they are considered the most critical system of the studied structure. A preliminary sensitivity analysis is performed in order to select the more significant CLF’'s to which the considered energies are more sensitive. A range of variability of ±10% around the nominal values is considered for some selected physical parameters that might be subjected to design modifications: the Young’'s modulus of the aluminum core, the thickness of the carbon fiber skins, the centroid offset of the ribbed shell. The reciprocity relationship (6) is used to get the ηi j and the ∆ ηi j with i > j. The modal densities in the reciprocity relationship are those corresponding to the nominal values of the physical parameters. The ILF’'s are varied of ±10% around their nominal values. Sensitivities of energies in the three critical subsystems, with respect to coupling loss factors and internal loss factors are evaluated according to the procedure outlined in section 3.1. In practice, each value represents the first order approximation of variation of the energy stored in a given subsystem due to a change ∆ ηi j (∆ ηi ) of a given CLF (ILF). Table 1 shows the norm of the sensitivity of each studied energy with respect to CLF’'s and ILF’'s, on all the frequency bands. Since the analysis if performed on the CLF’'s, the ILF’'s contribution is not considered also if the sensitivity values are very high. CLF’'s η21 , η32 , η42 and η43 are identified as the most important, because their correspondent sensitivity values are globally the higher over the three energies. Table 1 Norm of subsystem energies sensitivities with respect to coupling and internal loss factors Sensitivity to η21 η32 η42 η43 η51 η52 η53 η63 η64 η1 η2 η3 η4 η5 η6
kE3 k · 105 1258.6021 73.935694 29.4092 94.7500 12.6385 0.5420 0.0383 0.0002 0.0516 4191.6864 11765.0668 24006.9360 1340.5043 2.6775 0.0710
kE4 k · 105 802.1298 6.149145 899.4531 1080.4808 7.1617 0.3844 0.0072 0.0025 0.65275 2706.9862 7410.2918 13225.5494 985.2324 1.7235 0.0497
kE5 k · 105 161.5913 5.2900 3.1386 1.4237 14.6329 126.2108 1.3447 0.0012 0.0705 552.0421 1501.4571 2670.0953 174.0167 1.7052 0.0139
Figures 3 and 4 show the sensitivity of the studied energies with respect to the most significant CLF’'s, for all third octave bands. Consequently, the optimisation of energies E3 , E4 and E5 is performed by taking into account the variability of these CLF’'s.
9 −3
0
x 10
Sη21
−2 −4 −6 −8 25
50
100
200 [Hz]
400
800
100
200 [Hz]
400
800
−4
6
x 10
Sη32
4 2 0 −2 25
50
Fig. 3 Sensitivities of subsystem energies E3 (o), E4 (∗) and E5 (¦) to coupling loss factors η21 and η 32
By using Design of Experiment an approximate relation between CLF’'s and physical parameters is obtained. A set of numerical experiments is performed, by varying of ±10% around the nominal values, the previuosly selected physical parameters: the Young’'s modulus of the aluminum core, the thickness of the carbon fiber skins, the centroid offset of the ribbed shell. Starting from this relation, an optimisation problem is formulated by following the scheme described in section 3.3. The energies are constrained to assume values lower than the 90% of their nominal values. The physical parameters are bounded between ±10% of their nominal values, the bounds considered for the DoE.
10 −3
2
x 10
Sη42
0 −2 −4 −6 −8 25
50
100
200 [Hz]
400
800
100
200 [Hz]
400
800
−3
2
x 10
Sη43
0 −2 −4 −6 −8 25
50
Fig. 4 Sensitivities of subsystem energies E3 (o), E4 (∗) and E5 (¦) to coupling loss factors η42 and η 43
The result of the optimisation gives the following set of physical parameters values: Young’'s modulus of the aluminum core -10% of its nominal value, thickness of the carbon fiber skins +10% of its nominal value and the centroid offset of the ribbed shell +10% of its nominal value. Therefore, the results correspond to the bounds selected for the optimisation procedure, because not all energies satisfy the prescribed constraints as shown in figure 5.
11
Energy decrease [%]
25 20 15 10 5 0 25
50
100
200 [Hz]
400
800
Fig. 5 Percent energy decrease after design optimisation
5 Conclusions Since classical structural optimisation techniques fail to solve medium high frequency dynamic problems an optimisation problem can be formulated by using a SEA model in order to bring the subsystem energies under prescribed levels. In this paper a constrained optimisation problem is formulated on a typical aerospace structure. The variables of the problem are the relative deviation of the selected physical parameters from their nominal value. Upper and lower bounds of these design variables are defined. Finally, the energy of the subsystems are constrained to be lower than a prescribed value. A preliminary sensitivity analysis is performed in order to select the more significant CLF’'s. By using DoE an approximate relation between the more sensitive CLF’'s and the three selected physical parameters is obtained. Finally, the results of the optimisation correspond to the bounds selected for the optimisation procedure, because not all energies satisfy the prescribed constraints. Next activities will consider the study of more complicated optimisation problem by involving more energies and more physical parameters by considering also the junctions dynamics. Acknowledgements This research is supported by University of Rome La Sapienza grants.
12
Appendix: short background on Design of Experiment In Design of Experiments (DoE), the values of the variables that affect an output response are appropriately modified by a series of tests, to identify the reasons for changes in the response. This does not prevent from performing numerical tests whenever this may be convenient for a better understanding of the numerical problem under investigation. Since many experiments involve the study of the effects of two or more variables or factors, it is necessary to investigate all possible combinations of the levels of the factors. This is performed by factorial designs which are very efficient for this task. To account for possible non linear effects, Central Composite Design (CCD) can be used, that starts from factorial design (2 p observations for p factors) and augments it with the center point i.e. a single observation with all factors at intermediate level (denoted as 0), and axial runs where each factor is considered at two levels (the low level −1 and the high level +1) while the remaining factors are at the intermediate level, for a total of 2p observations (see Figure 6).
(-1,1,1) s
(1,1,1)
z
6
s
s(0,0,1)
(-1,-1,1)
(1,-1,1) s y
s
s
(0,1,0) s
s
(-1,0,0)
x -
s
(0,0,0) (1,0,0)
s s (0,-1,0) (-1,-1,1) s(0,0,-1) s
s
(-1,-1,-1)
(1,-1,-1)
s
(1,-1,1)
Fig. 6 Central Composite Design for p = 3
Overall, a central composite design for p factors requires n = 2 p + 2p + 1 observations. For p control factors, the experimental response can be expressed as a regression model representation of a 2 p full factorial experiment (involving 2 p terms), augmented with p quadratic terms:
13 p
p i−1
i=1 p i−1 m−1
i=1 j=1
f = α0 + ∑ αi xi + ∑ ∑ α ji x j xi + . . . + + ∑ ∑ ··· i=1 j=1
p
∑ αnm··· ji xn xm · · · x j xi + ∑
n=1
(17)
αii xi2 + ε
i=1
The expression contains 2 p + p parameters α , each one providing an estimate of the effect of a single factor (linear or quadratic) or of a combination of them. Note that Eq. (17) is linear in the parameters α , and it can be rewritten as: α0 £ ¤ α1 f = 1 x1 · · · x2p +ε (18) .. . α pp having arranged the parameters in a vector α . A different equation can be written for each observation by varying the factors (x1 , . . . , x p ) as indicated by CCD. By arranging the experimental responses in a vector f, a linear relationship between f and α can be expressed in matrix notation as: f = Xα + ε where X is a
(2 p + 2p + 1) × (2 p + p)
(19)
matrix. The least square estimate of α is:
αˆ = (XT X)−1 XT f
⇒
ˆf = Xαˆ
(20)
where fˆ is the fitted regression model. The difference between the actual observations vector f and the corresponding ˆ The residuals account both for the fitted model fˆ is the vector of residuals e = f − f. modelling error ε and for the fitting error due to the least square estimation.
References 1. Bussow, R., Petersson, B.: Path sensitivity and uncertainty propagation in SEA. Journal of Sound and Vibration 300(3-5), 479–489 (2007) 2. Culla, A., Carcaterra, A., Sestieri, A.: Energy flow uncertainties in vibrating systems: Definition of a statistical confidence factor. Mechanical Systems and Signal Processing 17(3), 635–663 (2003) 3. Culla, A., D’'Ambrogio, W., Fregolent, A.: Parametric approaches for uncertainty propagation in SEA. Mechanical Systems and Signal Processing (2010), doi:10.1016/j.ymssp.2010.05.001 4. D’'Ambrogio, W., Fregolent, A.: Reducing variability of a set of structures assembled from uncertain substructures. In: Proceeding of 26th IMAC. Orlando (U.S.A.) (2008) 5. de Langhe, R.: Statistical analysis of the power injection method. Journal of the Acoustical Society of America 100(1), 294–304 (1996) 6. Langley, R., Cotoni, V.: Response variance prediction in the statistical energy analysis of builtup systems. Journal of the Acoustical Society of America 115(2), 706–718 (2004)
14 7. Langley, R., Cotoni, V.: Response variance prediction for uncertain vibro-acoustic systems using a hybrid deterministic-statistical method. Journal of the Acoustical Society of America 122(6), 3445–3463 (2007) 8. Lyon, R.: Statistical analysis of power injection and response in structures and rooms. Journal of the Acoustical Society of America 45(3), 545–565 (1969) 9. Lyon, R., De Jong, R.: Theory and Applications of Statistical Energy Analysis. The MIT Press, Cambridge (U.S.A.) (1995) 10. Montgomery, D.: Design and Analysis of Experiments, 6th edn. Wiley, New York (2005) 11. Radcliffe, C.J., Huang, X.: Putting statistics into the statistical energy analysis of automotive vehicles. Journal of Vibration and Acoustics 119(4), 629–634 (1997) 12. Weaver, R.: Spectral statistics in elastodynamics. Journal of the Acoustical Society of America 85, 1005–1013 (1989)
Benefit of Acoustic Particle Velocity Based Reverberant Room Testing of Spacecraft Ing. E.H.G Tijs1, Ir. J.J. Wijker2, Dr. Ing. A. Grillenbeck3 1
Microflown Technologies, PO Box 2205, 6802 CE Arnhem, The Netherlands
2
Dutch Space BV, PO Box 32070, 2303 DB Leiden, The Netherlands
3
IABG mbH, Einsteinstrasse 20, 85521 Ottobrunn, Germany
Abstract
The vibro-acoustic load during launch takes a big toll on space structures. In order to simulate the dynamic loading as encountered during launch, both shaker facilities and high sound pressure reverberant rooms are used. Acoustic particle velocity sensors offer interesting new opportunities, for measuring both the applied noise field as well as the structural responses. Single particle velocity sensors in a so-called U probe can be used for very near field vibration measurements. When they are combined with a microphone in a PU probe the full sound vector can be measured. The novel perspectives of using PU probes for reverberant room testing comprise: The classical control of the noise field and the measurement of the sound pressure level, and of acoustic quantities like the reverberation time may be complemented by making reference to the total acoustic energy. The input power of the sound source can be measured directly using PU probes. Arrays of U probes can be used to measure contactless surface vibrations at multiple measurement points simultaneously as an alternative to accelerometers or laser vibrometers. PU probes can be used to measure local acoustic quantities near the structure like sound radiation, acoustic impedance and energy. 3D sound fields around structures can be visualized. The degree of diffusion in a reverberation room can be better characterized. In this paper, the first experience with practical implementation and the results of recent measurements using the PU probe will be presented.
Introduction Nowadays, particle velocity sensors are used for many applications in industry such as automotive and aerospace, complementing traditional sensor technology, or are even offering entirely novel measurement capabilities. The most important features for some of these applications are the small size of the sensors, their intrinsic directional and broad band behavior, and their low influence to background noise and reflections under certain conditions. Particle velocity sensors are sometimes used alone, to measure structural vibration, or in combination with a sound pressure microphone to measure the sound intensity, impedance and energy. Here, the perspective of measuring in reverberant chambers under high sound levels is being investigated. Today, typically only two types of sensors are used for the acoustic noise tests for spaceflight: accelerometers are used to measure the vibration; microphones are used to measure the sound pressure. Accelerometers are relatively cheap, but although they are available in various types their mass can influence the response of the lightweight object under test. Laser vibrometers can be used for non-contact measurements instead, but their application is difficult because their mountings are subjected to vibrations, the number of measurement points is restricted due to the limited test time, and more importantly, it is extremely complicated to reach all outer/inner surfaces or measurement points. Velocity sensors placed close to the surface could be used for non-contact measurements. In the
T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_2, © The Society for Experimental Mechanics, Inc. 2011
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16 (very) near field of a surface the structural vibration almost equals the particle velocity because the air layer can be considered to be almost incompressible [2-5]. Microphones of are used to measure the sound pressure level in the reverberant chamber. But the sound pressure level only represents the potential energy of the reverberant noise field. Using particle velocity sensors, also the kinetic energy could be measured complementing the measurement to record the total acoustic energy at any arbitrary point. Depending on the location in the reverberant chamber the standing waves result in different sound pressure readings from point to point. To reduce this effect, the mean sound pressure of microphone measurements on multiple locations is used. Sound pressure measured in the vicinity of a structure is also used to estimate the local acoustic load as input for simulation. However, some principles based on sound pressure measurements cannot be used here. E.g. P-P sound intensity probes cannot be used because of the reverberant conditions and because the relation between pressure and velocity cannot be assumed constant at these high sound levels. This paper will describe the applications of PU probes that are envisaged at the moment. To demonstrate some of these applications the first results of a test on the FRED/ATV solar array wing of Dutch Space [1] will be shown. Many accelerometers and sound pressure microphones were also installed on the panel during this test, and a comparison with these sensors will be made. These activities were made in the frame if the ESA funded project VAATLMDS conducted by Dutch Space and partners. The acoustic test and the measurements were carried out in the reverberant chamber of IABG [6].
1. Applications of PU probes in reverberant conditions at high sound levels Several applications of PU probes for reverberant room testing of space structures can be envisaged and will be discussed here. The applications involve enhanced or new methods to improve testing on test objects or the performance of the reverberant room.
1.1 Measurement of the full energy density By measuring the sound pressure only the potential energy is measured. The energy density is approximated with the potential energy alone via [7-10]:
E ≈ 2 *U = 2 *
p2 [W ] 2 ρc 2
(1)
With E being the sound energy, U the potential energy, p pressure, ρ air density and c speed of sound From the particle velocity vector u the kinetic energy K in one direction can be calculated:
r ρvr 2 K= [W ] 2
(2)
By measuring both the potential and kinetic energy in three directions the full energy density E can be calculated: 2
2
2
ux + u y + uz p2 E = U + K = U + Kx + Ky + Kz = +ρ [W ] 2 2 2 ρc
(3)
While the conventional sound pressure based energy measurements by using intensity probes are place dependent, actually the full energy density is a place independent quantity. Accordingly, sound energy measurements could be per-
17 formed in addition or could even replace all control microphones in a reverberant noise field (typically 5 to 10 control microphones are normally used). Also for measurements of the acoustic characteristics like the reverberation time and equivalent absorption area, full energy density measurements could be done with fewer probes than with microphones only.
1.2 Direct measurement of the input power A reverberant room for testing of space structures generally consists of several noise sources to cover a broad frequency range. The noise level inside the reverberation chamber is influenced by the presence of test structures and measurement equipment. The adjustment of the sound level is not so straightforward because the energy is measured indirectly via several control microphones. Because of the long reverberation time it can take a certain amount of time to obtain a stable sound pressure level after adjustment of the sound source level. A direct measurement of the input power with a PU probe could save time for such an adjustment. The input power could be measured with particle velocity alone at low frequencies, and with sound intensity probes at high frequencies. In the near field of a sound source almost all direct energy is kinetic energy. The potential energy near the source is much more influenced by the reflections. The assumption here is that the amount of energy being absorbed by the sound source is low. By measuring the kinetic energy with a particle velocity sensor the source can then be characterized at low frequencies [2-5]. This direct measurement of the sound source could also be used as a reference for calculation of the impulse response of all the other sensors, which could be used for measurement of the magnitude and delay of each reflection. At high frequencies the near field conditions do no not hold anymore. This means the sound intensity should be measured instead [2-5, 11]. However, sound intensity measurements with traditional P-P probes are impossible in these reverberant conditions because of the amount of reflections, or in other terms the p/I (pressure/intensity) index is too high. PU intensity probes are not affected by this p/I index problem.
1.3 Non contact vibration measurements Close to the surface, in the so called very near field, the particle velocity is almost equal to the surface velocity, due to the almost incompressible characteristic of air. For this relation to become valid two conditions have to be met: the distance rn to the surface should be much smaller its typical size L, and the wavelength λ should be larger than the vibration surface L [3]:
rn
