Exact Analysis of
Bi-Periodic
Exact Analysis of
Bi-Periodic Structures
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Exact Analysis of

Bi-Periodic

Exact Analysis of

Bi-Periodic Structures

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Exact Analysis of

Bt-Periodic Structures CWCai Department of Mechanics, Zhongshan University, China

JKLiu Department of Mechanics, Zhongshan University, China

H C Chan Department of Civil Engineering, The University of Hong Kong, Hong Kong

V f e World Scientific w l

New Jersey London • Sine Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

EXACT ANALYSIS OF BI-PERIODIC STRUCTURES Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4928-4

Printed in Singapore.

PREFACE In the book "Exact Analysis of Structures with Periodicity using UTransformation" (World Scientific 1998), a comprehensive and systematic explanation has been given on the U-transformation method, its background, physical meaning and mathematical formulation. The book has demonstrated the application of the U-Transformation method in the analyses of many different kinds of periodic structures. As it has been rightly pointed out in the book, the method has a great potential for further development. With the research efforts by the authors and others in recent years, important advancement in the application of the U-transformation method has been made in the following areas: • The static and dynamic analyses of bi-periodic structures • Analysis of periodic systems with nonlinear disorder. The static and dynamic analyses of bi-periodic structures When the typical substructure in a periodic structure is itself a periodic structure, the original structure is classified as a bi-periodic structure: for example, a continuous truss supported on equidistant supports with multiple equal spans. As a singly periodic structure, the truss within each bay or span between two adjacent supports is a substructure. But there could be many degrees of freedom in such a substructure. If the U-transformation method is applied to analyze this structure as illustrated in the previous book, every uncoupled equation still contains many unknown variables, the number of which is equal to the number of degrees of freedom in each substructure. Therefore, it is not possible to obtain the explicit exact analytical solution yet. Though the substructure is periodic, it is not cyclic periodic. Hence, it is not possible to go any further to apply the same Utransformation technique directly to uncouple the equations. One of the main objectives for writing this new book is to show how to extend the U-transformation technique to uncouple the two sets of unknown variables in a bi-periodic structure to achieve an analytical exact solution. Through an example consisting of a system of masses and springs with bi-periodicity, this book presents a procedure on how to apply the U-transformation technique twice to uncouple the unknowns and get an analytical solution. The book also produces the static and dynamic analyses for certain engineering structures with bi-periodic properties. These include continuous truss with any number of spans, cable network and grillwork on supports with periodicity, and grillwork with periodic stiffening members or equidistant line supports. Explicit exact solutions are given for these examples. The availability of these exact solutions not only helps the checking of the convergence and accuracy of the numerical solutions for these structures, but also provides a basis for the v

Preface

vi

optimization design for these types of structures. It is envisaged that there may be a great prospect for the application of this technique in engineering. Analysis of periodic systems with nonlinear disorder The study on the force vibration and localized mode shape of periodic systems with nonlinear disorder is yet another research area that has attained considerable success by the application of the U-transformation method. The localization of the mode shape of nearly periodic systems has been a research topic attracting enormous attention and concern in the past decade. In the same way, localization problem also exists in periodic systems with nonlinear disorder. This book illustrates the analytical approach and procedure for these problems together with the results. It looks that there are big differences in the physical and mechanical meaning of the problems in the above-mentioned two areas. But as a matter of fact there are similarities in the approaches to their analyses. It is appropriate to present them all together in this book. They are both good examples of the amazing successful application of the U-transformation method. The advantage of applying the U-transformation method is to make it possible for the linear simultaneous equations, either algebraic or differential equations, with cyclic periodicity to uncouple. The first chapter in this book will provide a rigorous proof for this significant statement and give the form of the uncoupled equations. The result will be used in the procedure to obtain the solutions for the example problems in this book. Many achievements in this new book are new results that have just appeared in international journals for the first time together with some which have not been published before. This book can be treated as an extension of the previous book "Exact Analysis of Structures with Periodicity using U-Transformation" with the latest advancement and development in the subject. Nevertheless, sufficient details and explanations have been given in this book to make it a new reference book on its own. However, it will be helpful if readers of this book have obtained some ideas of the mathematical procedures and the applications of the U-transformation method from the previous book.

Prof. H.C. Chan Oct. 30, 2001

CONTENTS Preface Chapter 1 U Transformation and Uncoupling of Governing Equations for Systems with Cyclic Bi-periodicity 1.1 Dynamic Properties of Structures with Cyclic Periodicity 1.1.1 Governing Equation 1.1.2 U Matrix and Cyclic Matrix 1.1.3 U Transformation and Uncoupling of Simultaneous Equations with Cyclic Periodicity 1.1.4 Dynamic Properties of Cyclic Periodic Structures 1.2 Uncoupling of Simultaneous Equations with Cyclic Periodicity for Variables with Two Subscripts 1.2.1 Double U Transformation 1.2.2 Uncoupling of Simultaneous Equations with Cyclic Periodicity for Variables with Two Subscripts 1.3 Uncoupling of Simultaneous Equations with Cyclic Bi-periodicity 1.3.1 Cyclic Bi-periodic Equation 1.3.2 Uncoupling of Cyclic Bi-periodic Equations 1.3.3 Uncoupling of Simultaneous Equations with Cyclic Bi-periodicity for Variables with Two Subscripts Chapter 2 Bi-periodic Mass-Spring Systems 2.1 Cyclic Bi-periodic Mass-Spring System 2.1.1 Static Solution 2.1.1a Example 2.1.2 Natural Vibration 2.1.2a Example 2.1.3 Forced Vibration 2.1.3a Example 2.2 Linear Bi-periodic Mass-Spring Systems 2.2.1 Bi-periodic Mass-Spring System with Fixed Extreme Ends 2.2.1a Natural Vibration Example 2.2.1b Forced Vibration Example 2.2.2 Bi-periodic Mass-Spring System with Free Extreme Ends 2.2.2a Natural Vibration Example 2.2.2b Forced Vibration Example 2.2.3 Bi-periodic Mass-Spring System with One End Fixed

v

1 1 1 2 6 9 11 11 13 15 15 16 20 27 27 28 32 34 36 39 42 44 44 50 55 57 62 67

viii

Contents

and the Other Free 2.2.3a Natural Vibration Example

69 75

Chapter 3 Bi-periodic Structures 3.1 Continuous Truss with Equidistant Supports 3.1.1 Governing Equation 3.1.2 Static Solution 3.1.2a Example 3.1.3 Natural Vibration 3.1.3a Example 3.1.4 Forced Vibration 3.1.4a Example 3.2 Continuous Beam with Equidistant Roller and Spring Supports 3.2.1 Governing Equation and Static Solution 3.2.2 Example

81 81 81 90 94 99 102 107 112 115 116 121

Chapter 4 Structures with Bi-periodicity in Two Directions 4.1 Cable Networks with Periodic Supports 4.1.1 Static Solution 4.1.1a Example 4.1.2 Natural Vibration 4.1.2a Example 4.1.3 Forced Vibration 4.1.3a Example 4.2 Grillwork with Periodic Supports 4.2.1 Governing Equation 4.2.2 Static Solution 4.2.3 Example 4.3 Grillwork with Periodic Stiffened Beams 4.3.1 Governing Equation 4.3.2 Static Solution 4.3.3 Example

125 125 128 134 139 144 149 153 156 159 165 169 175 178 185 191

Chapter 5 Nearly Periodic Systems with Nonlinear Disorders 5.1 Periodic System with Nonlinear Disorders — Monocoupled System 5.1.1 Governing Equation 5.1.2 Localized Modes in the System with One Nonlinear Disorder 5.1.3 Localized Modes in the System with Two Nonlinear Disorders

203

5.2 Periodic System with One Nonlinear Disorder

203 204 210 212

Exact Analysis of Bi-periodic Structures

— Two-degree-coupling System 5.2.1 Governing Equation 5.2.2 Perturbation Solution 5.2.3 Localized Modes 5.3 Damped Periodic Systems with One Nonlinear Disorder 5.3.1 Forced Vibration Equation 5.3.2 Perturbation Solution 5.3.3 Localized Property of the Forced Vibration Mode

ix

219 220 224 229 239 239 245 255

References

263

Nomenclature

265

Index

267

Chapter 1 U TRANSFORMATION AND UNCOUPLING OF GOVERNING EQUATIONS FOR SYSTEMS WITH CYCLIC BI-PERIODICITY 1.1 1.1.1

Dynamic Properties of Structures with Cyclic Periodicity Governing Equation

In general, the discrete equation for cyclic periodic structures without damping may be expressed as MX + KX = F

(1.1.1)

where a superior dot denotes differentiation with respect to the time variable t, K and M are stiffness and mass matrices and X and F are displacement and loading vectors respectively. Generally they can be written as Jsr, , M = Km

K>

K>

M,

Mv

M2X

M22

Mm

MN2

(1.1.2a, b) MK

and

(1.1.3a, b)

where N represents the total number of substructures; the vector components x. and Fj (j = l,2,...,N)

denote displacement and loading vectors for the y'-th

substructure, respectively. The numbers of dimensions of submatrices Krs,

l

Mrs

2

Exact Analysis of Bi-periodic Structures

(r,s = 1,2,...,N) and vector components Xj and Fj (j = l,2,...,N) are the same as the degrees of freedom for a single substructure and let J denote the number of degrees of freedom of a substructure. The stiffness and mass matrices for the cyclic periodic structures possess cyclic periodicity as well as symmetry, namely Krs^Kl'

r,s = \,2,...,N

(1.1.4)

Kn = K22 = --- = Km K\,s = K2,s+i = •'• = KN-S+\,N =

KN-S+2,1

Mrs = Ml'

=

''' =

KN,S-\

(1.1.5a) '

s = 2,3,...,N

r,s = 1,2,. ..,N

MU = M22 = — = MNN Mij = M2lJ+1 = • • • = MN-,w

\ • • ) (1.1.6) (1.1.7a)

= MN_S+1X =••• = MNs_x • s = 2,3,...,N

(ll-7b)

where [ ] denotes the transposed matrix of [ ] . The simultaneous equation (1.1.1) with K, M having cyclic periodicity may be called a cyclic periodic equation. 1.1.2

U Matrix and Cyclic Matrix

Let U=[Ut

U2

... U„]

(1.1.8a)

with the submatrices

emvIj

U_ =

J2my, j

y[W ei(N-l)my/j

m = l,2,...,N

(1.1.8b)

U Transformation and Uncoupling of Governing Equations

3

in which y/ = 2n/N, / = V—1 and J y denotes the unit matrix of order J. It can be shown that VIV, = — (1 + ei{s-r)v + e , ' 2(l -' > + •• + N

i(N-l)(s-r)y,

e

)Ij

r=s 1 1 ' ., . [N \-e

(1.1.9)

I, = 0

r*s

That leads to UTU-

(1.1.10)

where the superior bar denotes complex conjugation. U satisfying Eq. (1.1.10) is referred to as unitary matrix or U matrix. Eq. (1.1.10) indicates that the column vectors of U are a set of normalized orthogonal basis in the unitary space with ./V • J dimensions. The columns of Um are made up of the basis of the m-th subspace with J dimensions. An arbitrary vector, say Umxm (x m is a J dimensional vector), in the m-th subspace possesses the cyclic periodicity. If xm x„e

X{t) = Umxmer =

4N

x„e

x„e

i(ajt+2my/)

(1.1.11)

i[0M+(N~\)my/]

represents a vibration mode for a cyclic periodic structure with N substructures, then this mode is a rotating one, namely the deflection of one substructure has the same amplitude as, and a constant phase difference my/{= 2mn/N) from, the deflection of the preceding substructure, y/ is referred to as the period of the cyclic periodic structure. All of the rotating modes, the phase difference between two adjacent substructures must be 2rnn/N (m = 1,2, ...,N) due to cyclic periodicity. As a result, all of the mode vectors lie in the TV subspaces respectively.

4

Exact Analysis of Bi-periodic Structures

A matrix with cyclic periodicity shown in Eq. (1.1.5) is referred to as cyclic matrix, such as the stifmess and mass matrices of structures with cyclic periodicity are cyclic matrices. The elementary cyclic matrices can be defined as 1 2 0 0

0

I

j = 0,1,2,...,N-\

e. =

(1.1.12)

j

J 1 2

0

j

••

where the empty elements are equal to zero, f„ is a unit matrix and each element of matrix

e. is a J dimensional square matrix. An arbitrary cyclic matrix can be

expressed as the series of the elementary cyclic matrix, such as

7-1

Kt

or

7=1

N-j+\

(1.1.13a)

7=1

and -•

N

Mu 7=1

where eN = eQ and

V>

or

M=Y,

-. Mfl

N-j+\

7=1

denotes the quasi-diagonal matrix, i.e.,

(1.1.13b)

U Transformation and Uncoupling of Governing Equations

x

5

0 X

(1.1.14)

Noting the cyclic periodicity of Um and t. l/m = e W [ / m

(1.1.15)

it can be verified that UTe.U=0j,

j = 0,1,2,...,N-]

(1.1.16a)

with e'""I, ni

*I,

0

(1.1.16b)

j

e'"'vIj

0

It is obvious that + mh

7.'

V •

k

9s\

".

H»>

=

•

(1.1.27)

fN.

where N

1

f -_L_V e -'0- 1 )'-icc-

(1.1.28)

Eq. (1.1.27) is made up of N independent equations, i.e., »tr9r+krqr=fr,

r=\,2,...,N

Noting the definitions of mr, kr and fr (1.1.28) respectively, it is obvious that mN_r =mr, and mN,

mNI2

shown in Eqs. (1.1.21), (1.1.18) and

kN_r - kr,

(if N is even), kN,

kN/2

(1.1.29)

fN_r = fr

(1.1.30)

(if N is even) are real symmetric

matrices, so qN_r = qr and qN, qN/2 (if N is even) are real vectors. N N +l We need only consider — + 1 (N is even) or (N is odd) equations, i.e., r = \,2,...,—,N

(Nis even) or r = 1,2,...,^^-,N

(Nis odd) in Eq. (1.1.29).

U Transformation and Uncoupling of Governing Equations

1.1.4

9

Dynamic Properties of Cyclic Periodic Structures

Consider now the natural vibration of rotationally periodic structures. The natural vibration equation can be expressed in terms of the generalized displacements as krqr=co2mrqr,

r = l,2,...,N

(1.1.31)

where a> denotes the natural frequency, qr represents the amplitude of the r-th generalized displacement and kr, mr denote generalized stiffness and mass matrices as shown in Eqs. (1.1.18) and (1.1.21) respectively. It is well known that the eigenvalues of the eigenvalue equation (1.1.31) with Hermiltian matrices are real numbers. The eigenvalues can be denoted as a)2{, co22,---,

G>2rJ (a>2s < co2s+i,s = l,2,...,J-I)

and the corresponding normalized

orthogonal eigenvectors may be written as qrX,

qr2, ..., qrJ.

They satisfy the

eigenvalue equation and the normalized orthogonal condition, i.e., krqrs=a>lmrqrs,

5 = 1,2,...,7; r = l,2,...,W

(1.1.32)

and £>,*,,, =1.

s=\,2,...,J;

r = l,2,...,N

(1.1.33)

leading to ^r,. = 9r,skr9r^ = r e a l number Noting kN_r = kr,

mN^r =ntr, it is obvious that ® L v = Ks.

and qNs,

qN

(1.1.34)

qN-r,s=qr,s,

s=\,2,...,J

(1.1.35)

(if Nis even) s = 1,2,..., J are real eigenvectors.

2

Let us consider the natural modes. Corresponding to the eigenvector ( s = 1,2,...,/), the natural mode can be expressed as

qNs

10

Exact Analysis of Bi-periodic Structures

X = UNq^

(1.1.36a)

*i =*2 = xi = — = xN =-J=

=

--

"

=

• * {M,k)(M~j+\,v)

j = l,2,-,M;

=

*

(U)(M-;+2,v)

k,v = l,2,...,N

(1.2.6a)

14

Exact Analysis ofBi-periodic Structures

and *t/,*X«.0

=

•** (y,t+l)(u,2) = ••• = &(j,N)(u,N-k+\)

= - = K(j2

M

;

v = l,2,...,tf

(1.3.32)

where M

If

"(.,)=EE^iH)"^"(,4,"^MW)

and

(1-3-33)

U Transformation and Uncoupling of Governing Equations

rO

m

f°

v

"'V) "

n

y

r - ' ( ' - ' ) p i " ^ r-'(i-i)/>iV»'i r

^-J^mJ

JM-JN

23

/ i -J - M U \

X

(l+(r-l)p,,l+(S-l),,2)

U-J-''tD>

From Eq. (1.3.32), g(u v) can be formally expressed as *(.,v, =, = 2;r/?n and (p2 = 2njn . Premultiplying both sides of Eq. (1.3.39) by the operator

-rr= Z S e"'0_1)""e_

Xs — xx+,sn

(2.1.9)

and 1

=

H

e'(s-u)mpv

&.« ^Z^T^

7' *>»=1>2>->"

(211°)

N ~ K + 2k{\ - cos m y/) Psu denotes the influence coefficient for the single periodic system. By using the U-transformation once, the equilibrium equation (2.1.1) with N (= pn) unknowns becomes Eq. (2.1.8) with n unknowns. Note that the simultaneous equations (2.1.8) possess the cyclic periodicity, i.e., fl,i = £ 2 , 2 =••• = &,»

P..X = A+1.2 = - = Pn,n-s+i = A,„-,+2 = ... = /?,_,,. , 5=2,3,...,«

(2.1.11a)

(2.1.11b)

Bi-periodic Mass-Spring Systems

31

One can now apply the U-transformation again to Eq. (2.1.8). Introducing Xs=-^Yeiis-^Qr,

s=l,2,...,n

(2.1.12a)

and Qr=-^Ye-*s-l)r*Xs,

r=\,2,...,n

(2.1.12b)

with (p = In/n = p y/, into Eq. (2.1.8) results in Qr=-AK^0sle-i{s-

(2.1.22b)

Substituting Eqs. (2.1.20a) and (2.1.22) into Eq. (2.1.7b), we have

1

(K + k)P K(K + 3k)

4

(2.1.23a)

kP x' = x' = x' = , ; = 2

3

5

6

(2.1.23b) K(K + 3k)

'

It can be verified thatx^. (/=1,2,...,6) is the exact displacement solution for the system with AA' = 0 subjected to the loads shown in Eq. (2.1.20b). Recalling the definition shown in Eq. (2.1.9) and/?=3 gives X[ = X'2 = (K + ^P 1 2 K(K + 3k)

(2.1.24)

Inserting Eqs. (2.1.24) and (2.1.20a) into Eq. (2.1.14) yields bi=0, 1

^ I K +W 2 K(K + 3k)

(2.1.25)

and then introducing Eqs. (2.1.25), (2.1.20a) and (2.1.17) into Eq. (2.1.16) results in

a = 0 1

,

e

2

= —

2

r 2 i K + k ) P

—

(2.1.26)

K(K + 3k) + M:(K + k)

From Eq. (2.1.19), f° (m=l,2,...,6) can be obtained as /„°=0

m = 1,3,5

(2.1.27a)

34

Exact Analysis of Bi-periodic Structures

+ P /: =- ^ W » 3 K(K + 3k) + AK(K + k)

m = 2,4,6

(2.1.27b)

Finally, substituting Eqs. (2.1.5b), (2.1.27) and (2.1.20a) into Eq. (2.1.7a) we have

1

4

K(K + 3k)[K(K + 3k) + AK(K + k)]

x l = , 3 ° = *s° = x l = 2

3

5

6

AKk(K

+

k)P

K(K + 3k)[K(K + 3k) + AK(K + k)]

and then inserting Eqs. (2.1.28) and (2.1.23) into Eq. (2.1.6) results in

Xi=x^ 1

The

displacement

4

£±*£

(2L29a)

K(K + 3k) + AK(K + k) kP K(K + 3k) + AK(K + k)

Xj (j = 1,2,... ,6)

shown

in

(2.1.29b)

Eq. (2.1.29) satisfies the

equilibrium equations (2.1.1) with the parameters shown in Eq. (2.1.20). 2.1.2

Natural Vibration

The natural vibration equation for Fig. 2.1.1 may be expressed as

the

cyclic bi-periodic system shown

in

(K + 2k- M(Dl)xj - k(xj+x + Xj_x) = -(AK - AMco2)Xj , j = \,\ + p,...,\ + (n-\)p (K + 2k-Ma)2)Xj

-jfc(*, + ,+*,-,) = 0,

j*l,l

(2.1.30a) + p,...,l + (n-l)p

(2.1.30b)

where co denotes the natural frequency, Xj denotes the amplitude of 7-th subsystem and the term - (AK -

AMQ)2)XJ

may be formally treated as the load.

Applying the U-transformation (2.1.2) to Eq. (2.1.30) results in

Bi-periodic Mass-Spring Systems {K + 2k-Mco1)qm-2kcosmVqm=fm,

m=l,2,-,N

35

(2.1.31)

where »

,2N

f.=-iAK-Ka}

!. Xj

'

N

^-fj-f

ei(j-\)mVe-i(u-\)mpV

K + 2k -Ma2

-2kcosmy/^H"~1)p

j=l,2,...,N Introducing the notation Xs=xlHs_l)P

(2.1.34)

and inserting j = l + (s-l)p

(s=l,2,...,ri)

in Eq. (2.1.34) we have n

X,=-(AAT-AM« 2 ) £ # „ * „ ,

i=l,2,...,«

(2.1.35)

where 1

N

B' =—Y su N ^-f K +

i(s-u)mplft

—; , s,u=l,2,...,n 2k-Mco2-2kcosmw

(2.1.36)

P*su denotes the harmonic influence coefficient for the considered system with AK = AM =0. Obviously J3SU also possesses cyclic periodicity. Applying the Utransformation (2.1.12) to Eq. (2.1.35) results in

36

Exact Analysis of Bi-periodic Structures

1 + (AK-AMa)2)£j Substituting Eq. (2.1.36) Eq. (2.1.15), we have 1

/?,>"' = n,

yr = —

(2.1.41b)

The frequency equation (2.1.39) becomes l + (K- Meo2)-^(K

+ 2k- Ma2 - 2kcos[r + 2{u -1)]—)"' = 0, r = l,2

(2.1.42)

The solution for a>2 of Eq. (2.1.42) can be found as 2

a

=

K + {2-42)k K + {2 + 42)k K x — '—, — '—,

(for r = l )

tniAi\ (2.1.43a)

and

^=ILt!L±2Lt

(for

r

= 2)

(2.1.43b)

These natural frequencies are corresponding to the modes with xl and x4 nonvanishing. Consider the other frequency equation (2.1.40), i.e., K+ 2k(l-cos m—)-Mco2

=0,

m = l,2

(2.1.44)

The square of frequency can be expressed as K +k , „ K + 3k w22 = ^ — - (m = 1), — - — (m = 2) M M

(2.1.45a,b)

xx and x4 are identically equal to zero in the corresponding modes. Consider now the natural modes. Corresponding to the natural frequencies

38

Exact Analysis ofBi-periodic Structures

shown in Eq. (2.1.43a), the modes can be expressed as 0*0,

Q2=0

(2.1.46)

Substituting Eqs. (2.1.46) and (2.1.41) into Eq. (2.1.12a), we have A ' 1 ( = J C 1 ) = 1,

X 2 ( = x 4 ) = -1

(2.1.47)

where an arbitrary constant factor is neglected. Introducing Eqs. (2.1.47), (2.1.41) and to2 = K+(2~^2)k M the natural mode can be found as

j ^

Eq

x1=l,x2=j2-l,x3=-(j2-l),x4=-l,x5=-(-j2-l),x6=<j2-l Substituting Eqs. (2.1.47), (2.1.41) and ^

=

K+ (2 + ^2)k ^ M

(2.1.34),

(2.1.48)

£ q

(22 34)

results in JC, =l,x2 = - ( l + V2),x 3 =l + ^,x4

=-l,x5

= \ + Jl,x6

= -(l + V2)

(2.1.49)

Similarly, corresponding to the natural frequencies shown in Eq. (2.1.43b), the modes in terms of the generalized displacements can be expressed as 02*O,

0 = 0

(2.1.50a)

xA=l

(2.1.50b)

That leads to x,=l,

Corresponding to co2 = K/M , the natural mode is Xj=l,

7=1,2,...,6

and corresponding to a2 = (K + 2k) JM, the natural mode is

(2.1.51)

Bi-periodic Mass-Spring Systems

x,=*4=l,

x2 =JC3 = x5 = x6 = -1

39

(2.1.52)

Consider the other kind of modes with xx = x4 = 0. The mode in terms of the generalized displacements can be expressed as u+i;

s = u,...,«

(2.1.6I)

where *,

s

*.«,-.,„,

x:=xlH,_l)p

(2.1.62)

^=TiTir^—rrT—r,

(2-1-63)

N^-f K + 2k-Mco -Ikcosmtf/ x^if^e'W'q:

(2.1.64)

Applying the U-transformation (2.1.12) to Eq. (2.1.61) yields ar(w)Qr=br

r = l,2,...,n

(2.1.65)

where 1

p

ar(co) = 1 + (AK- AMco2) — Y^(K + 2k-Ma2 PT:-.

-2kcos[r + (u-\)n\y/)'x

b^-j^e^-^X',

(2.1.66)

(2.1.67)

If ar(a>)*0, Qr=~^—

r = l,2,...,«

(2.1.68)

ar(ffl)

Recalling Eqs. (2.1.59a) and (2.1.12b), we have (AK-AMco2) /r+(B-i)n=-?= - 2 , r = l,2,...,«,

Bi-Periodic

Exact Analysis of

Bi-Periodic Structures

This page is intentionally left blank

Exact Analysis of

Bt-Periodic Structures CWCai Department of Mechanics, Zhongshan University, China

JKLiu Department of Mechanics, Zhongshan University, China

H C Chan Department of Civil Engineering, The University of Hong Kong, Hong Kong

V f e World Scientific w l

New Jersey London • Sine Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

EXACT ANALYSIS OF BI-PERIODIC STRUCTURES Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4928-4

Printed in Singapore.

PREFACE In the book "Exact Analysis of Structures with Periodicity using UTransformation" (World Scientific 1998), a comprehensive and systematic explanation has been given on the U-transformation method, its background, physical meaning and mathematical formulation. The book has demonstrated the application of the U-Transformation method in the analyses of many different kinds of periodic structures. As it has been rightly pointed out in the book, the method has a great potential for further development. With the research efforts by the authors and others in recent years, important advancement in the application of the U-transformation method has been made in the following areas: • The static and dynamic analyses of bi-periodic structures • Analysis of periodic systems with nonlinear disorder. The static and dynamic analyses of bi-periodic structures When the typical substructure in a periodic structure is itself a periodic structure, the original structure is classified as a bi-periodic structure: for example, a continuous truss supported on equidistant supports with multiple equal spans. As a singly periodic structure, the truss within each bay or span between two adjacent supports is a substructure. But there could be many degrees of freedom in such a substructure. If the U-transformation method is applied to analyze this structure as illustrated in the previous book, every uncoupled equation still contains many unknown variables, the number of which is equal to the number of degrees of freedom in each substructure. Therefore, it is not possible to obtain the explicit exact analytical solution yet. Though the substructure is periodic, it is not cyclic periodic. Hence, it is not possible to go any further to apply the same Utransformation technique directly to uncouple the equations. One of the main objectives for writing this new book is to show how to extend the U-transformation technique to uncouple the two sets of unknown variables in a bi-periodic structure to achieve an analytical exact solution. Through an example consisting of a system of masses and springs with bi-periodicity, this book presents a procedure on how to apply the U-transformation technique twice to uncouple the unknowns and get an analytical solution. The book also produces the static and dynamic analyses for certain engineering structures with bi-periodic properties. These include continuous truss with any number of spans, cable network and grillwork on supports with periodicity, and grillwork with periodic stiffening members or equidistant line supports. Explicit exact solutions are given for these examples. The availability of these exact solutions not only helps the checking of the convergence and accuracy of the numerical solutions for these structures, but also provides a basis for the v

Preface

vi

optimization design for these types of structures. It is envisaged that there may be a great prospect for the application of this technique in engineering. Analysis of periodic systems with nonlinear disorder The study on the force vibration and localized mode shape of periodic systems with nonlinear disorder is yet another research area that has attained considerable success by the application of the U-transformation method. The localization of the mode shape of nearly periodic systems has been a research topic attracting enormous attention and concern in the past decade. In the same way, localization problem also exists in periodic systems with nonlinear disorder. This book illustrates the analytical approach and procedure for these problems together with the results. It looks that there are big differences in the physical and mechanical meaning of the problems in the above-mentioned two areas. But as a matter of fact there are similarities in the approaches to their analyses. It is appropriate to present them all together in this book. They are both good examples of the amazing successful application of the U-transformation method. The advantage of applying the U-transformation method is to make it possible for the linear simultaneous equations, either algebraic or differential equations, with cyclic periodicity to uncouple. The first chapter in this book will provide a rigorous proof for this significant statement and give the form of the uncoupled equations. The result will be used in the procedure to obtain the solutions for the example problems in this book. Many achievements in this new book are new results that have just appeared in international journals for the first time together with some which have not been published before. This book can be treated as an extension of the previous book "Exact Analysis of Structures with Periodicity using U-Transformation" with the latest advancement and development in the subject. Nevertheless, sufficient details and explanations have been given in this book to make it a new reference book on its own. However, it will be helpful if readers of this book have obtained some ideas of the mathematical procedures and the applications of the U-transformation method from the previous book.

Prof. H.C. Chan Oct. 30, 2001

CONTENTS Preface Chapter 1 U Transformation and Uncoupling of Governing Equations for Systems with Cyclic Bi-periodicity 1.1 Dynamic Properties of Structures with Cyclic Periodicity 1.1.1 Governing Equation 1.1.2 U Matrix and Cyclic Matrix 1.1.3 U Transformation and Uncoupling of Simultaneous Equations with Cyclic Periodicity 1.1.4 Dynamic Properties of Cyclic Periodic Structures 1.2 Uncoupling of Simultaneous Equations with Cyclic Periodicity for Variables with Two Subscripts 1.2.1 Double U Transformation 1.2.2 Uncoupling of Simultaneous Equations with Cyclic Periodicity for Variables with Two Subscripts 1.3 Uncoupling of Simultaneous Equations with Cyclic Bi-periodicity 1.3.1 Cyclic Bi-periodic Equation 1.3.2 Uncoupling of Cyclic Bi-periodic Equations 1.3.3 Uncoupling of Simultaneous Equations with Cyclic Bi-periodicity for Variables with Two Subscripts Chapter 2 Bi-periodic Mass-Spring Systems 2.1 Cyclic Bi-periodic Mass-Spring System 2.1.1 Static Solution 2.1.1a Example 2.1.2 Natural Vibration 2.1.2a Example 2.1.3 Forced Vibration 2.1.3a Example 2.2 Linear Bi-periodic Mass-Spring Systems 2.2.1 Bi-periodic Mass-Spring System with Fixed Extreme Ends 2.2.1a Natural Vibration Example 2.2.1b Forced Vibration Example 2.2.2 Bi-periodic Mass-Spring System with Free Extreme Ends 2.2.2a Natural Vibration Example 2.2.2b Forced Vibration Example 2.2.3 Bi-periodic Mass-Spring System with One End Fixed

v

1 1 1 2 6 9 11 11 13 15 15 16 20 27 27 28 32 34 36 39 42 44 44 50 55 57 62 67

viii

Contents

and the Other Free 2.2.3a Natural Vibration Example

69 75

Chapter 3 Bi-periodic Structures 3.1 Continuous Truss with Equidistant Supports 3.1.1 Governing Equation 3.1.2 Static Solution 3.1.2a Example 3.1.3 Natural Vibration 3.1.3a Example 3.1.4 Forced Vibration 3.1.4a Example 3.2 Continuous Beam with Equidistant Roller and Spring Supports 3.2.1 Governing Equation and Static Solution 3.2.2 Example

81 81 81 90 94 99 102 107 112 115 116 121

Chapter 4 Structures with Bi-periodicity in Two Directions 4.1 Cable Networks with Periodic Supports 4.1.1 Static Solution 4.1.1a Example 4.1.2 Natural Vibration 4.1.2a Example 4.1.3 Forced Vibration 4.1.3a Example 4.2 Grillwork with Periodic Supports 4.2.1 Governing Equation 4.2.2 Static Solution 4.2.3 Example 4.3 Grillwork with Periodic Stiffened Beams 4.3.1 Governing Equation 4.3.2 Static Solution 4.3.3 Example

125 125 128 134 139 144 149 153 156 159 165 169 175 178 185 191

Chapter 5 Nearly Periodic Systems with Nonlinear Disorders 5.1 Periodic System with Nonlinear Disorders — Monocoupled System 5.1.1 Governing Equation 5.1.2 Localized Modes in the System with One Nonlinear Disorder 5.1.3 Localized Modes in the System with Two Nonlinear Disorders

203

5.2 Periodic System with One Nonlinear Disorder

203 204 210 212

Exact Analysis of Bi-periodic Structures

— Two-degree-coupling System 5.2.1 Governing Equation 5.2.2 Perturbation Solution 5.2.3 Localized Modes 5.3 Damped Periodic Systems with One Nonlinear Disorder 5.3.1 Forced Vibration Equation 5.3.2 Perturbation Solution 5.3.3 Localized Property of the Forced Vibration Mode

ix

219 220 224 229 239 239 245 255

References

263

Nomenclature

265

Index

267

Chapter 1 U TRANSFORMATION AND UNCOUPLING OF GOVERNING EQUATIONS FOR SYSTEMS WITH CYCLIC BI-PERIODICITY 1.1 1.1.1

Dynamic Properties of Structures with Cyclic Periodicity Governing Equation

In general, the discrete equation for cyclic periodic structures without damping may be expressed as MX + KX = F

(1.1.1)

where a superior dot denotes differentiation with respect to the time variable t, K and M are stiffness and mass matrices and X and F are displacement and loading vectors respectively. Generally they can be written as Jsr, , M = Km

K>

K>

M,

Mv

M2X

M22

Mm

MN2

(1.1.2a, b) MK

and

(1.1.3a, b)

where N represents the total number of substructures; the vector components x. and Fj (j = l,2,...,N)

denote displacement and loading vectors for the y'-th

substructure, respectively. The numbers of dimensions of submatrices Krs,

l

Mrs

2

Exact Analysis of Bi-periodic Structures

(r,s = 1,2,...,N) and vector components Xj and Fj (j = l,2,...,N) are the same as the degrees of freedom for a single substructure and let J denote the number of degrees of freedom of a substructure. The stiffness and mass matrices for the cyclic periodic structures possess cyclic periodicity as well as symmetry, namely Krs^Kl'

r,s = \,2,...,N

(1.1.4)

Kn = K22 = --- = Km K\,s = K2,s+i = •'• = KN-S+\,N =

KN-S+2,1

Mrs = Ml'

=

''' =

KN,S-\

(1.1.5a) '

s = 2,3,...,N

r,s = 1,2,. ..,N

MU = M22 = — = MNN Mij = M2lJ+1 = • • • = MN-,w

\ • • ) (1.1.6) (1.1.7a)

= MN_S+1X =••• = MNs_x • s = 2,3,...,N

(ll-7b)

where [ ] denotes the transposed matrix of [ ] . The simultaneous equation (1.1.1) with K, M having cyclic periodicity may be called a cyclic periodic equation. 1.1.2

U Matrix and Cyclic Matrix

Let U=[Ut

U2

... U„]

(1.1.8a)

with the submatrices

emvIj

U_ =

J2my, j

y[W ei(N-l)my/j

m = l,2,...,N

(1.1.8b)

U Transformation and Uncoupling of Governing Equations

3

in which y/ = 2n/N, / = V—1 and J y denotes the unit matrix of order J. It can be shown that VIV, = — (1 + ei{s-r)v + e , ' 2(l -' > + •• + N

i(N-l)(s-r)y,

e

)Ij

r=s 1 1 ' ., . [N \-e

(1.1.9)

I, = 0

r*s

That leads to UTU-

(1.1.10)

where the superior bar denotes complex conjugation. U satisfying Eq. (1.1.10) is referred to as unitary matrix or U matrix. Eq. (1.1.10) indicates that the column vectors of U are a set of normalized orthogonal basis in the unitary space with ./V • J dimensions. The columns of Um are made up of the basis of the m-th subspace with J dimensions. An arbitrary vector, say Umxm (x m is a J dimensional vector), in the m-th subspace possesses the cyclic periodicity. If xm x„e

X{t) = Umxmer =

4N

x„e

x„e

i(ajt+2my/)

(1.1.11)

i[0M+(N~\)my/]

represents a vibration mode for a cyclic periodic structure with N substructures, then this mode is a rotating one, namely the deflection of one substructure has the same amplitude as, and a constant phase difference my/{= 2mn/N) from, the deflection of the preceding substructure, y/ is referred to as the period of the cyclic periodic structure. All of the rotating modes, the phase difference between two adjacent substructures must be 2rnn/N (m = 1,2, ...,N) due to cyclic periodicity. As a result, all of the mode vectors lie in the TV subspaces respectively.

4

Exact Analysis of Bi-periodic Structures

A matrix with cyclic periodicity shown in Eq. (1.1.5) is referred to as cyclic matrix, such as the stifmess and mass matrices of structures with cyclic periodicity are cyclic matrices. The elementary cyclic matrices can be defined as 1 2 0 0

0

I

j = 0,1,2,...,N-\

e. =

(1.1.12)

j

J 1 2

0

j

••

where the empty elements are equal to zero, f„ is a unit matrix and each element of matrix

e. is a J dimensional square matrix. An arbitrary cyclic matrix can be

expressed as the series of the elementary cyclic matrix, such as

7-1

Kt

or

7=1

N-j+\

(1.1.13a)

7=1

and -•

N

Mu 7=1

where eN = eQ and

V>

or

M=Y,

-. Mfl

N-j+\

7=1

denotes the quasi-diagonal matrix, i.e.,

(1.1.13b)

U Transformation and Uncoupling of Governing Equations

x

5

0 X

(1.1.14)

Noting the cyclic periodicity of Um and t. l/m = e W [ / m

(1.1.15)

it can be verified that UTe.U=0j,

j = 0,1,2,...,N-]

(1.1.16a)

with e'""I, ni

*I,

0

(1.1.16b)

j

e'"'vIj

0

It is obvious that + mh

7.'

V •

k

9s\

".

H»>

=

•

(1.1.27)

fN.

where N

1

f -_L_V e -'0- 1 )'-icc-

(1.1.28)

Eq. (1.1.27) is made up of N independent equations, i.e., »tr9r+krqr=fr,

r=\,2,...,N

Noting the definitions of mr, kr and fr (1.1.28) respectively, it is obvious that mN_r =mr, and mN,

mNI2

shown in Eqs. (1.1.21), (1.1.18) and

kN_r - kr,

(if N is even), kN,

kN/2

(1.1.29)

fN_r = fr

(1.1.30)

(if N is even) are real symmetric

matrices, so qN_r = qr and qN, qN/2 (if N is even) are real vectors. N N +l We need only consider — + 1 (N is even) or (N is odd) equations, i.e., r = \,2,...,—,N

(Nis even) or r = 1,2,...,^^-,N

(Nis odd) in Eq. (1.1.29).

U Transformation and Uncoupling of Governing Equations

1.1.4

9

Dynamic Properties of Cyclic Periodic Structures

Consider now the natural vibration of rotationally periodic structures. The natural vibration equation can be expressed in terms of the generalized displacements as krqr=co2mrqr,

r = l,2,...,N

(1.1.31)

where a> denotes the natural frequency, qr represents the amplitude of the r-th generalized displacement and kr, mr denote generalized stiffness and mass matrices as shown in Eqs. (1.1.18) and (1.1.21) respectively. It is well known that the eigenvalues of the eigenvalue equation (1.1.31) with Hermiltian matrices are real numbers. The eigenvalues can be denoted as a)2{, co22,---,

G>2rJ (a>2s < co2s+i,s = l,2,...,J-I)

and the corresponding normalized

orthogonal eigenvectors may be written as qrX,

qr2, ..., qrJ.

They satisfy the

eigenvalue equation and the normalized orthogonal condition, i.e., krqrs=a>lmrqrs,

5 = 1,2,...,7; r = l,2,...,W

(1.1.32)

and £>,*,,, =1.

s=\,2,...,J;

r = l,2,...,N

(1.1.33)

leading to ^r,. = 9r,skr9r^ = r e a l number Noting kN_r = kr,

mN^r =ntr, it is obvious that ® L v = Ks.

and qNs,

qN

(1.1.34)

qN-r,s=qr,s,

s=\,2,...,J

(1.1.35)

(if Nis even) s = 1,2,..., J are real eigenvectors.

2

Let us consider the natural modes. Corresponding to the eigenvector ( s = 1,2,...,/), the natural mode can be expressed as

qNs

10

Exact Analysis of Bi-periodic Structures

X = UNq^

(1.1.36a)

*i =*2 = xi = — = xN =-J=

=

--

"

=

• * {M,k)(M~j+\,v)

j = l,2,-,M;

=

*

(U)(M-;+2,v)

k,v = l,2,...,N

(1.2.6a)

14

Exact Analysis ofBi-periodic Structures

and *t/,*X«.0

=

•** (y,t+l)(u,2) = ••• = &(j,N)(u,N-k+\)

= - = K(j2

M

;

v = l,2,...,tf

(1.3.32)

where M

If

"(.,)=EE^iH)"^"(,4,"^MW)

and

(1-3-33)

U Transformation and Uncoupling of Governing Equations

rO

m

f°

v

"'V) "

n

y

r - ' ( ' - ' ) p i " ^ r-'(i-i)/>iV»'i r

^-J^mJ

JM-JN

23

/ i -J - M U \

X

(l+(r-l)p,,l+(S-l),,2)

U-J-''tD>

From Eq. (1.3.32), g(u v) can be formally expressed as *(.,v, =, = 2;r/?n and (p2 = 2njn . Premultiplying both sides of Eq. (1.3.39) by the operator

-rr= Z S e"'0_1)""e_

Xs — xx+,sn

(2.1.9)

and 1

=

H

e'(s-u)mpv

&.« ^Z^T^

7' *>»=1>2>->"

(211°)

N ~ K + 2k{\ - cos m y/) Psu denotes the influence coefficient for the single periodic system. By using the U-transformation once, the equilibrium equation (2.1.1) with N (= pn) unknowns becomes Eq. (2.1.8) with n unknowns. Note that the simultaneous equations (2.1.8) possess the cyclic periodicity, i.e., fl,i = £ 2 , 2 =••• = &,»

P..X = A+1.2 = - = Pn,n-s+i = A,„-,+2 = ... = /?,_,,. , 5=2,3,...,«

(2.1.11a)

(2.1.11b)

Bi-periodic Mass-Spring Systems

31

One can now apply the U-transformation again to Eq. (2.1.8). Introducing Xs=-^Yeiis-^Qr,

s=l,2,...,n

(2.1.12a)

and Qr=-^Ye-*s-l)r*Xs,

r=\,2,...,n

(2.1.12b)

with (p = In/n = p y/, into Eq. (2.1.8) results in Qr=-AK^0sle-i{s-

(2.1.22b)

Substituting Eqs. (2.1.20a) and (2.1.22) into Eq. (2.1.7b), we have

1

(K + k)P K(K + 3k)

4

(2.1.23a)

kP x' = x' = x' = , ; = 2

3

5

6

(2.1.23b) K(K + 3k)

'

It can be verified thatx^. (/=1,2,...,6) is the exact displacement solution for the system with AA' = 0 subjected to the loads shown in Eq. (2.1.20b). Recalling the definition shown in Eq. (2.1.9) and/?=3 gives X[ = X'2 = (K + ^P 1 2 K(K + 3k)

(2.1.24)

Inserting Eqs. (2.1.24) and (2.1.20a) into Eq. (2.1.14) yields bi=0, 1

^ I K +W 2 K(K + 3k)

(2.1.25)

and then introducing Eqs. (2.1.25), (2.1.20a) and (2.1.17) into Eq. (2.1.16) results in

a = 0 1

,

e

2

= —

2

r 2 i K + k ) P

—

(2.1.26)

K(K + 3k) + M:(K + k)

From Eq. (2.1.19), f° (m=l,2,...,6) can be obtained as /„°=0

m = 1,3,5

(2.1.27a)

34

Exact Analysis of Bi-periodic Structures

+ P /: =- ^ W » 3 K(K + 3k) + AK(K + k)

m = 2,4,6

(2.1.27b)

Finally, substituting Eqs. (2.1.5b), (2.1.27) and (2.1.20a) into Eq. (2.1.7a) we have

1

4

K(K + 3k)[K(K + 3k) + AK(K + k)]

x l = , 3 ° = *s° = x l = 2

3

5

6

AKk(K

+

k)P

K(K + 3k)[K(K + 3k) + AK(K + k)]

and then inserting Eqs. (2.1.28) and (2.1.23) into Eq. (2.1.6) results in

Xi=x^ 1

The

displacement

4

£±*£

(2L29a)

K(K + 3k) + AK(K + k) kP K(K + 3k) + AK(K + k)

Xj (j = 1,2,... ,6)

shown

in

(2.1.29b)

Eq. (2.1.29) satisfies the

equilibrium equations (2.1.1) with the parameters shown in Eq. (2.1.20). 2.1.2

Natural Vibration

The natural vibration equation for Fig. 2.1.1 may be expressed as

the

cyclic bi-periodic system shown

in

(K + 2k- M(Dl)xj - k(xj+x + Xj_x) = -(AK - AMco2)Xj , j = \,\ + p,...,\ + (n-\)p (K + 2k-Ma)2)Xj

-jfc(*, + ,+*,-,) = 0,

j*l,l

(2.1.30a) + p,...,l + (n-l)p

(2.1.30b)

where co denotes the natural frequency, Xj denotes the amplitude of 7-th subsystem and the term - (AK -

AMQ)2)XJ

may be formally treated as the load.

Applying the U-transformation (2.1.2) to Eq. (2.1.30) results in

Bi-periodic Mass-Spring Systems {K + 2k-Mco1)qm-2kcosmVqm=fm,

m=l,2,-,N

35

(2.1.31)

where »

,2N

f.=-iAK-Ka}

!. Xj

'

N

^-fj-f

ei(j-\)mVe-i(u-\)mpV

K + 2k -Ma2

-2kcosmy/^H"~1)p

j=l,2,...,N Introducing the notation Xs=xlHs_l)P

(2.1.34)

and inserting j = l + (s-l)p

(s=l,2,...,ri)

in Eq. (2.1.34) we have n

X,=-(AAT-AM« 2 ) £ # „ * „ ,

i=l,2,...,«

(2.1.35)

where 1

N

B' =—Y su N ^-f K +

i(s-u)mplft

—; , s,u=l,2,...,n 2k-Mco2-2kcosmw

(2.1.36)

P*su denotes the harmonic influence coefficient for the considered system with AK = AM =0. Obviously J3SU also possesses cyclic periodicity. Applying the Utransformation (2.1.12) to Eq. (2.1.35) results in

36

Exact Analysis of Bi-periodic Structures

1 + (AK-AMa)2)£j Substituting Eq. (2.1.36) Eq. (2.1.15), we have 1

/?,>"' = n,

yr = —

(2.1.41b)

The frequency equation (2.1.39) becomes l + (K- Meo2)-^(K

+ 2k- Ma2 - 2kcos[r + 2{u -1)]—)"' = 0, r = l,2

(2.1.42)

The solution for a>2 of Eq. (2.1.42) can be found as 2

a

=

K + {2-42)k K + {2 + 42)k K x — '—, — '—,

(for r = l )

tniAi\ (2.1.43a)

and

^=ILt!L±2Lt

(for

r

= 2)

(2.1.43b)

These natural frequencies are corresponding to the modes with xl and x4 nonvanishing. Consider the other frequency equation (2.1.40), i.e., K+ 2k(l-cos m—)-Mco2

=0,

m = l,2

(2.1.44)

The square of frequency can be expressed as K +k , „ K + 3k w22 = ^ — - (m = 1), — - — (m = 2) M M

(2.1.45a,b)

xx and x4 are identically equal to zero in the corresponding modes. Consider now the natural modes. Corresponding to the natural frequencies

38

Exact Analysis ofBi-periodic Structures

shown in Eq. (2.1.43a), the modes can be expressed as 0*0,

Q2=0

(2.1.46)

Substituting Eqs. (2.1.46) and (2.1.41) into Eq. (2.1.12a), we have A ' 1 ( = J C 1 ) = 1,

X 2 ( = x 4 ) = -1

(2.1.47)

where an arbitrary constant factor is neglected. Introducing Eqs. (2.1.47), (2.1.41) and to2 = K+(2~^2)k M the natural mode can be found as

j ^

Eq

x1=l,x2=j2-l,x3=-(j2-l),x4=-l,x5=-(-j2-l),x6=<j2-l Substituting Eqs. (2.1.47), (2.1.41) and ^

=

K+ (2 + ^2)k ^ M

(2.1.34),

(2.1.48)

£ q

(22 34)

results in JC, =l,x2 = - ( l + V2),x 3 =l + ^,x4

=-l,x5

= \ + Jl,x6

= -(l + V2)

(2.1.49)

Similarly, corresponding to the natural frequencies shown in Eq. (2.1.43b), the modes in terms of the generalized displacements can be expressed as 02*O,

0 = 0

(2.1.50a)

xA=l

(2.1.50b)

That leads to x,=l,

Corresponding to co2 = K/M , the natural mode is Xj=l,

7=1,2,...,6

and corresponding to a2 = (K + 2k) JM, the natural mode is

(2.1.51)

Bi-periodic Mass-Spring Systems

x,=*4=l,

x2 =JC3 = x5 = x6 = -1

39

(2.1.52)

Consider the other kind of modes with xx = x4 = 0. The mode in terms of the generalized displacements can be expressed as u+i;

s = u,...,«

(2.1.6I)

where *,

s

*.«,-.,„,

x:=xlH,_l)p

(2.1.62)

^=TiTir^—rrT—r,

(2-1-63)

N^-f K + 2k-Mco -Ikcosmtf/ x^if^e'W'q:

(2.1.64)

Applying the U-transformation (2.1.12) to Eq. (2.1.61) yields ar(w)Qr=br

r = l,2,...,n

(2.1.65)

where 1

p

ar(co) = 1 + (AK- AMco2) — Y^(K + 2k-Ma2 PT:-.

-2kcos[r + (u-\)n\y/)'x

b^-j^e^-^X',

(2.1.66)

(2.1.67)

If ar(a>)*0, Qr=~^—

r = l,2,...,«

(2.1.68)

ar(ffl)

Recalling Eqs. (2.1.59a) and (2.1.12b), we have (AK-AMco2) /r+(B-i)n=-?= - 2 , r = l,2,...,«,

denotes the vibration frequency; \ + (m-\)p ( m = l,2,...,2n ) is the ordinal number of the subsystem with mass M + AM and the loads F, (j = \,2,...,2np) must satisfy the symmetric condition, i.e., F2M-J=-F]

J = 2X.,N

(2.2.2a)

Fj M A k s'x

- » Xj

M+AM

M+AM

k 2

p+1

l+(n-l)p

symmetric l i n e M M+AM

I

M+AM

-Fj

M M+AM

k

2

P

7

np np+1

Original system

Fj k

s,

Y/

l+2p

(a)

M+AM

M

k

M+AM

I

p+!

j

l+(n-l)p

np

i+np 4np

l+(n+l)p

N+2-j N=2np

(b) Figure 2.2.1

2np l+2np xi+2np=xi

Equivalent system

Bi-periodic mass spring system with fixed extreme ends

Bi-periodic Mass-Spring Systems

F,=Fw+1=0

47

(2.2.2b)

where N = np and Fj(j = 2,3,...,N) denote the real loads acting on the original system. The term AMo 2 x. acts as a kind of loading. The expressions on the left sides of Eqs. (2.2.1a) and (2.2.1b) possess cyclic periodicity. We can now apply the U-transformation to Eq. (2.2.1). Let xj=-^=Yei(J-^qm V2JV tTi

j=l,2,...,2N

(2.2..3a)

m = l,2,...,2N

(2.2.3b)

or 1 1

2JV

1 = -fL=ye-'u-l)m"x,.

V2JV^ where y/ = — , i = v - 1 and 2N denotes the total number of subsystems. 1 2N Premultiplying Eq. (2.2.1) by - = = Y e~^-\^w ^ r e s u l t s

m

V2iv^r {2k-Ma>2)qm-2kzosm¥qm=f°m+fm

m =l,2,...,2N

(2.2.4)

where

V2N

"

/ =

* -p = E e ~' 0 ~ 1 ) m r j p ; V2./V , =!

Introducing Eq. (2.2.2) into Eq. (2.2.5b) gives

(2 2 5b)

--

48

Exact Analysis of Bi-periodic Structures

2i N / ; = — ^ X s i n O - - \)m y,Fj \2N j=2

(2.2.5c)

As a result / ; = 0 and f;N=0

(2.2.5d)

From Eq. (2.2.4), gm can be expressed as qn=ql+q'a

(2-2.6a)

?• = f°/(2k-Mw2

-Ikcosmy/)

(2.2.6b)

9™ =fm/(2k-Mco2

-2kcosmy/)

(2.2.6c)

Substituting Eqs. (2.2.5), (2.2.6) and j = l + (s-l)p (2.2.3a), we have

into the U-transformation

In

X s =AMa> 2 £ & „ * „ + * ;

5 = l,2,...,2n

(2.2.7)

where Xs 1 ;0„ =

Xs =x\+u-\)p

=XI+(S-\)P>

(2.2.8)

2N

r l (j u) p,, 2 ^[e' - '" '/(2fc-M)

r = 1,2,...,«-1

(2.2.14a)

50

Exact Analysis of Bi-periodic Structures

Q^=Qr

and

Q„=Q2n=0

(2.2.14b)

where 1 ar(co) = I-AMeo2—Y

p

(2k-Mw2-2kcos[r

+ (u-1)2^1//)-'

(2.2.15)

PTt Making a comparison between Eqs. (2.2.5a) and (2.2.11b) gives AMo2 fr+(u-l)ln=—^Qr

T = 1,2

fp

2»

«=l,2,...,/7

(2.2.16)

Finally substituting Eq. (2.2.6) into the U-transformation (2.2.3a), the solution for x, can be expressed as XJ=XJ+X'J 2N

1

(2.2.17a)

r

i

x°j = -jL=^[eiU-i)m"f°/(2k-Mca2

-2kcosmy/)\

(2.2.17b)

and x'j is defined in Eqs. (2.2.5c), (2.2.6c) and (2.2.10). 2.2.1a

Natural Vibration Example

Letting Fj=0

(j = \,2,...,2N ) as a result x\, = 0 and br = 0 ( r = 1,2,...,2w ),

the independent frequency equation can be obtained from Eqs. (2.2.14) and (2.2.15) as 1

p

l-AMco2—Y(2K-Mco2-2kcos[r

+ (u-l)2n]y/yl

=0

PTt r = \,2,...,n-\

(2.2.18)

if Xs (s = 1,2,...,2ri) are not identically equal to zero. Consider the case of Xs(s = 1,2,...,2n) vanishing. As a result the terms on the

Bi-periodic Mass-Spring Systems

51

right side of Eq. (2.2.4) are equal to zero. Corresponding frequency equation can be obtained as my/)-Mo)2

2k(\-cos

=0

/n = «,2«,...,(/?-l)n

(2.2.19)

where m denotes the half wave number of the mode for the original system. Taking a specific example as shown in Fig. 2.2.2.

Pe

icot

->

kM k M ^1

2

2M 4

Figure 2.2.2

M

k

5

6

K/ 7^

Bi-periodic mass spring system with fixed ends, p=3 and n=2

The parameters are n = 2,

p =3

AM=M

(2.2.20a)

leading to N=6

Y=^ o

^

=

(2.2.20b)

T/

Substituting Eq. (2.2.20) into Eq. (2.2.18) yields 2

3

Mco 2 1 l_i^y(2A;-Mft> -2A:cos[r + 4( M -l)]-)=0, 3 6

Tt

A nondimensional frequency parameter may be defined as

r=\

(2.2.21)

52

Exact Analysis of Bi-periodic Structures

flo-^r1

(2 2 22)

- '

k The frequency equation (2.2.21) can be rewritten, in term of fi0 as Q^-5Q^+6Q0-1 = 0

(2.2.23a)

The roots of Eq. (2.2.23a) are Q 0 =0.198062264, 1.55495813,3.24697960

(2.2.23b)

Noting Eq. (2.2.20), the other frequency equation (2.2.19) becomes 2k(\ - cos w - ) - Mco1 = 0 6

m = 2,4

(2.2.24a)

That leads to Q 0 =2(1-cosm—) 6

/n = 2,4

(2.2.24b)

The two roots are Q 0 = 1 (m = 2 ) ,

3 (m = 4)

(2.2.24c)

There are five natural frequencies altogether. The total number of the natural frequencies is in agreement with the number of degrees of freedom for the original system. Consider now the natural modes. Corresponding to the frequency equation (2.2.21) with Q 0 shown in Eq. (2.2.23b) the modes can be expressed in terms of the generalized displacements, as 2, " 0 ,

03=

ft

(2.2.25)

with the other Qr vanishing. Introducing Eqs. (2.2.20) and (2.2.25) into (2.2.11a) and letting Xx = 0 yields

Bi-periodic Mass-Spring Systems

Qx = imaginary number = -i Xj = sinC/ -1) Y

Qi=Qx= i

j = 12,3,4

53

(2.2.26a) (2.2.26b)

neglecting an arbitrary constant factor in the expression of the mode. Inserting Eqs. (2.2.20a) and (2.2.26a) into Eq. (2.2.16) gives Mco2 1 f\ =/5 =h = J^

(2.2.27a)

Ma1 f?=fi=ti=Lil£-i

(2.2.27b)

fi=f:=--fn=0

(2.2.27c)

Substituting Eqs. (2.2.20) and (2.2.27) into Eq. (2.2.17b), the natural mode can be obtained as .A-i — ^ - i ~~ \J }

_ ..o _

" 7 — J\r*i — V7

o

* 2 ~ * 2 ~ ( 2 - a o ) [ ( 2 - Q 0 ) 2 - -3] 0

^ 0

(2-Q0)2-3

.0 — At —

(2.2.28b)

(2.2.28c)

(2.2.28d)

* 4 = X° = 1

X

Q 0 [(2-Q 0 ) 2 -l]

(2.2.37b)

2V3(Qo-5Qo+6Qo-1) /m° = 0

m = 2,4,...,12

(2.2.37c)

Bi-periodic Mass-Spring Systems

57

Finally inserting Eq. (2.2.37) (2.2.17b) (2.2.33) and (2.2.31a) into Eq. (2.2.17a), the frequency response functions for the amplitudes of displacements can be found as

*,=*, = 0 1

P X

6

x2

=

x

= *5

i

(2.2.38a)

* 4 = '

+ 6Q 0 -1

(2.2.38b)

2-Q 0 P Ik Ql--5Q20 + 6Q 0 -1

(2.2.38c)

2k

Q

3 0

•-5QJ;

P [(2-- " o ) 2 - -1] i t n 3 _

i

(2.2.38d)

When Q 0 approaches zero the solution Xj shown in Eq. (2.2.38) will approach the static displacement, namely x, = xn = 0,

x,=x5=

P —, k

P x2 =x6 = — Ik

(2.2.39a)

3P xA=— 2k

(2.2.39b)

Obviously the resonance frequencies are the roots of Eq. (2.2.23a). When Q 0 approaches each value shown in Eq. (2.2.23b), the amplitudes of displacement will approach infinity. It can be shown that, when each amplitude of displacement approaches a finite value at one natural frequency, the harmonic force is acting at a nodal point of the corresponding mode. 2.2.2

Bi-periodic Mass-Spring Systems with Free Extreme Ends

Consider a bi-periodic mass-spring system with free extreme ends and 2np (p=2d+l) particles altogether as shown in Fig. 2.2.3(a) where M, AM , k, xJ7 Fj and N have the same meanings as those in Fig. 2.2.1 .The bi-periodic system can also be regarded as single periodic. Each subsystem is made up of 2d particles with mass M and one particle having mass M + AM . The bi-periodic system shown in

„Fj M

M k

I

M+AM k

2

d

p=2d+l

Xj

M+AM

M

d+l+(n-l)p

np

k

d+1 d+2

J 3d+l d+l+p

(a) Original system

center line | p=2d+l

Fj

•*• X j

._ k

k _

M+AM _

k

M | M I

M+AM k

F2N+l-j=Fj M+AM —*"*J - , k |

(b) Equivalent system Figure 2.2.3 Bi-periodic mass spring system withfreeextreme ends

M

2N-2np s . a' M k k

Bi-periodic Mass-Spring Systems

59

Fig. 2.2.3(a) has n subsystems and np (p=2d+\) mass points altogether. In order to form an equivalent system having cyclic bi-periodicity, it is necessary to extend the original system by its symmetrical image and apply antisymmetric loading on the corresponding extended particles. In such an extended system the vibration displacements will be antisymmetric about the center line if the initial displacements and velocities are also antisymmetric. For each pair of symmetric particles, both longitudinal displacements are the same. As a result xnp = xnp+x and x2ap = xi which indicate that the connecting spring between both particles np and np+l dose not transmit any force and both extreme ends of the extended system can be imaginarily connected by the same spring as the other one, i.e., the 2n/?-th particle can be regarded as the preceding one of the first particle. Therefore the extended system shown in Fig. 2.2.3 (b) can be regarded as cyclic bi-periodic. The harmonic vibration equation takes the form as (2k - Ma)2)Xj - k(xJ+l + xH) = Fj+ AMco2Xj j = d + l,d + l + p,...,d +1 + (2n - \)p

(2.2.40a)

(2k - Mco2 )xj - k(xJ+1 + Xj_x) = Fj j*d

+ \,d + \ + p,...,d + \ + (2n-\)p

and j = l,2,...,2N(= 2np) (2.2.406)

where 2N denotes the total number of particles for the equivalent system, p=2d+l, Xj, Fj denote the amplitudes of the longitudinal displacement and loading for the 7-th particles and a denotes the frequency of the external excitations. For the equivalent system, the loads must satisfy the antisymmetric condition, i.e., F2N«.j=Fj,

y=l,2,...,«p

(2.2.41)

where 7V=«p and F. (j=l,2,...,np) are real loads acting on the original system . Applying the U transformation (2.2.3) to Eq. (2.2.40) we have (2k-Ma)2-2kcosmW)qm=f°+f:, where

m=\,2,...,2N

(2.2.42)

60

Exact Analysis of Bi-periodic Structures

>ly 4IN

tf 2W

1

r (y 1,m /:=-r=Z< ^ 2 W % '' ~

(2 2 44a)

--

with y/ = rv/N and /' = v - 1 . Introducing Eq. (2.2.41) into Eq. (2.2.44a) yields 7

N

i-m

1

1

f' = - = e "" V cost/ — ) » V Ft

(2.2.44b)

/; a 0

(2.2.44c)

That leads to

The solution for r'Qr+br

r=l,2,...,2n

(2.2.50)

where cp = K]YI and br=J=Ye-«°-l)r*X's

(2.2.51a)

V2« J=, X's must satisfy the antisymmetric condition, i.e., X'2n+l_s = X's (s=l,2,...,n). As a result Z>r =-^=e'2riPYcos(s V2« Tf

)r

b

2n-r=K

Introducing Eqs. (2.2.51c) and (2.2.48) into Eq. (2.2.50) gives Qr=-%— ar(d))

and

r = l,2,...,«-l,2«

(2.2.52a)

62

Exact Analysis of Bi-periodic Structures

2 „ = 0 , Q2„_r=Qr

r = l,2,...,»-l

(2.2.52b)

where ar(co) = 1 - AMa2 — V ( 2 k -Mco2 -2kcos[r + {u-l)2n]^)_1

(2.2.53)

Making a comparison between Eqs. (2.2.43) and (2.2.11b) and noting the definition of Xs shown in Eq. (2.2.47) yields

fX-mn^^-e-'^-^Qr y/P 2.2.2a

r=l,2,...,2n u=l,2,...,p

(2.2.54)

Natural Vibration Example

Let Fj = 0 (j=l,2,...,2N). That leads to x'j=0

{j=\,2,...,2N) and br = 0

(r=l,2,...,2«). If Xs (5=1,2,.,.2n) are not identically equal to zero, the frequency equation can be expressed as ar (cci) = 0, namely 2\{2k-Ma>2 PTt

1-AMOJ2 —

-2kcos[r + (u-\)2n]y/Yl

r = 1,2,..., n-1,2/1

=0 (2.2.55)

If Xs (5=1,2,...,2n) are identically equal to zero the frequency equation can be obtained from Eq. (2.2.42) with /ra° = f'm = 0 as 2k-M(02

-2kcosmy/

=0

m=n,3n,5n,...,(p-2)n

(2.2.56)

It is interesting to note that Eqs. (2.2.55) and (2.2.56) formally are the same as Eqs. (2.2.18) and (2.2.19) respectively, but the parameters r and m take different values in corresponding Eqs. (2.2.55) and (2.2.18) and Eqs. (2.2.56) and (2.2.19) respectively. Consider the system with the following parameters d=\,

« = 2,

AM=M

(2.2.57a)

Bi-periodic Mass-Spring Systems

63

That leads to p = 3,

JV = 6 ,

V =- ,

(2.2.57b)

M

k

1 Figure 2.2.4

^M 2

M

k

3

4

2|M 5

M

6

Bi-periodic mass spring system with free extreme ends, p=3 and n=2

Introducing Eq. (2.2.57) into Eq. (2.2.55) yields M

1

2

3

—Y(2it-MfiJ 2 -2A:cos[r + 4(M-l)]—)"* = 0

3

6

Tt

When r = 1, the roots for Q 0 (=

r = l,4

(2.2.58)

) of Eq. (2.2.58) are the same as those k

shown in Eq. (2.2.23b), i.e., n0 = 0.198062264,

1.55495813,

3.24697960

(2.2.59a)

When r = 4, the root for Q n is n0=2 Inserting Eq. (2.2.57) into Eq. (2.2.56) gives

(2.2.59b)

64

Exact Analysis of Bi-periodic Structures

2k - Ma2-2k

cosm-

=0

m=2

(2.2.60a)

That leads to Q0=l

(2.2.60b)

The corresponding mode possesses the property of x2 = x5 = 0. Let us now pay attention to the natural modes. Firstly consider the modes with x2 and xs non-vanishing. Corresponding to the natural frequencies shown in Eq. (2.2.59a), the mode, in terms of the generalized displacement, is e,*0,

03 = 6 i .

22=04=0

(2.2.61)

Because of the antisymmetry of displacement for the equivalent system, namely X2n+l_s = Xs (s = 1,2,...,n ), Eq. (2.2.11b) can be rewritten as

Qr = - r ^ Z «>s(* ~hr

(2.2.103)

The final results for JC. can be found by introducing Eqs. (2.2.103), (2.2.90) and (2.2.92) into Eq. (2.2.3a). 2.2.3a

Natural Vibration Example

There are two sets of frequency equations corresponding to Xs non-vanishing

75

76

Exact Analysis of Bi-periodic Structures

and vanishing respectively , namely ar(o}) = l-AMco2 — y^ilk-Ma2 PTt

-2kcos[r + (u-l)(4n

+ 2)]i//yl = 0

r = l,3,5,...,2n-l

(2.2.104)

and Ik - Mco2 - 2k cos m y/ = 0 m = 2n + l,3(2n + l),5(2n + l),...(p - 2)(2n +1)

(2.2.105)

with p = 2d + \ and y/ = n/(2n + \)p . When the specific parameters n, d and AM are given, the natural frequency can be found from Eqs. (2.2.104) and (2.2.105). Taking a specific example as shown in Fig. 2.2.6.

, M M , 2M , M k k k / A A ^ ) - > W 0 ^ V Q A A < )

/'l

Figure 2.2.6

2

3

Bi-periodic mass spring system with fixed and free ends d - n = 1

The parameters are given as d = \,

n=\,

AM=M

(2.2.106a)

That leads to p = 2,

N = 9,

y/ = — ,

N ^

f

*

(4,7)

>->"

{"("-)>

+ JC

i

"^(^)]

(3L29b)

where / ( ° m ) is dependent on the unknown supporting reactions which can be determined by the compatibility condition at supports, i.e., v (1 , (s - 1 ,^ 1) =0,

s = l,2,---,2n

Substituting Eqs. (3.1.21d), (3.1.29a) with j = (s-l)p equation, the restraint condition can be expressed as IX*n+^=0,

(3.1.30) + l into the above

* = l,2,-,2n

(3.1.31)

where

n Hs k

'

1 2N K J_y ei(s-k)pmv _n£_ 2NJ-1 A„ tn—\

1 V

2N

(3.1.32)

in

1

> = -f=ulle'{S~')PmWT-(^,„/(1.m,

-Kl2,mf(2J

(3.1.33)

Here Vs denotes the transverse displacement at the 5-th supported node caused by the external force for the equivalent system without supports. The compatibility equation (3.1.31) is linear simultaneous equations with unknown Pk (k=\,2,---,2n). The coefficients fisk {s,k = \,2,---,2n) of Eq. (3.1.31) possess cyclic periodicity, i.e.,

92

Exact Analysis of Bi-periodic Structures

A , . = A , 2 = - = A„,2„ A , I = A + I , 2 = - " = A . . 2 . - , + I = " - = A-I.2.>

(3.1.34a) 5 = 2,3,-,2«

(3.1.34b)

The independent coefficients are flkx (k =l,2,---,2«). Eq. (3.1.34) indicates the simultaneous equations have the cyclic periodicity. One can now apply the U-transformation to Eq. (3.1.31). Let (3.1.35a)

V2« Tt or 1

2

"

Qr=-f=Ye-iU-^Ps, V2n

r = l,2,.--,2«

(3.1.35b)

"

with cp = n/n = piy . Premultiplying Eq. (3.1.31) by the operator (l/V2n) ] T e'iU'])r,p results in 2n

S A j « * ^ e , + *>r = 0 ,

r = l,2,-,2«

(3.1.36)

where

^=-4=Je-' (s -'^K s

(3.1.37)

By using the U-transformation twice, the governing equation becomes a set of one degree of freedom equations as shown in Eq. (3.1.36). Obviously the solution for Qr ofEq. (3.1.36) is

Qr=-—„

"

,

r = l,2,---,2«

(3.1.38)

Bi-periodic Structures

93

where 1

a

K

J_y IN*-!

•

«t-i)-f _J^L A

m=\

(3.1.39)

m

Consider now the denominator on the right side of Eq. (3.1.38). Note that y

2t In,

eHk-l)(m-r)9 =\

*=1

m = r,r + 2n,---,r + (p-l)2n m *r,r + 2n,---,r + (p-l)2n

I"'

r = l,2,-,2n;

m = \,2,-,2N

(3.1.40)

Substituting Eq. (3.1.39) into Eq. (3.1.38) results in Qr=

-j-p1 V P

-1

"•

k=\

r = l,2,-,2»

(3.1.41)

22,r+(k-\)2n ^r+{k-\)2n

When the specific structure parameters and external loads are given, the generalized supporting reactions can be calculated from Eq. (3.1.41). Then the supporting reactions and the displacements for all nodes can be found from the related formulas given above. Recalling the definitions of both generalized supporting reactions / ( ° m ) and Qr shown in Eqs. (3.1.21d) and (3.1.35b) respectively, there is a simple relation, i.e., /(U*-.)2n)=^em,

m=l,2,-,2n,

k = l,2,-,p

(3.1.42)

Consequently, when we are only interested in the nodal displacements, it is not necessary to find the supporting reactions. In order to explain the procedure of the calculation and verify the exactness of the formulas given in the present section, we need to consider a specific truss with loading.

94

Exact Analysis of Bi-periodic Structures

3.1.2a Example Consider a Warren truss having six substructures and four supports subjected to a concentrated load of magnitude F at the center node as shown in Fig. 3.1.4.

K

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

Figure 3.1.4 Plane truss with six substructures and four supports subjected to a concentrated force of magnitude F at the center node The structural parameters are given as N = 6,

«=3, p=2, Kt=K2=K,

a = n/Z

(3.1.43a)

which lead to yr = * / 6 ,

F (2 ,.,=0 (

F

M=°>

J*A

j = 1,2,- -,6

(3.1.44a) (3.1.44b)

Introducing Eqs. (3.1.43) and (3.1.44) into Eq. (3.1.22), the generalized loads can be obtained as

Bi-periodic Structures

fM=-J-sm^y-F,

m = l,2,-,12

/(2, M )=0,

m=l,2,-,12

The stiffness coefficients of Eq. (3.1.25) Eq. (3.1.43) into Eq. (3.1.26) as Ku,m=K22tm

(3.1.45a)

(3.1.45b)

can be found by substituting

=—_(ii-7cos—-) oJJm

(3.1.46a)

O

Kn,m -K2hm = - ^ - ( 3 - 2 c o S - ^ ) ( l

+

e-""-)

^ 49 . m;r Dm = 2cos 16 6 Inserting Eqs. (3.1.45) and (3.1.43) into Eq. (3.1.33) gives

Vs =

95

(3.1.46b)

,„ . , , . (3.1.46c)

Ly sin^sinfc-!)^:^*. 3 M ^, 5 2 3 Ara

(3.1.47)

where A"22m and Am can be calculated from Eqs. (3.1.46) and (3.1.28) if m is given. Substituting Eqs. (3.1.46) and (3.1.28) into Eq. (3.1.47) results in Fj=0,

K2=13^,

K3=13^

K

VA=0,

(3.1.48a)

K

K5=-13^,

K6=-13^

(3.1.48b)

Introducing Eqs. (3.1.48) and (3.1.43) into Eq. (3.1.37) yields fc, = -I3V2/—, K

b2=0,

fe3=0

(3.1.49a)

96

Exact Analysis of Bi-periodic Structures

b4=0,

b5 = 13V2i — , K

b6 = 0

(3.1.49b)

Now the generalized supporting reaction can be found by substituting Eqs. (3.1.49), (3.1.46), (3.1.28) and (3.1.43) into Eq. (3.1.41), as Ql=^-iF,

02=O,

04=0,

Q5=-^^iF,

&=0

(3.1.50a)

Q6=0

(3.1.50b)

Inserting Eqs. (3.1.50) and (3.1.43) into the U-transformation (3.1.35a) results in

P=0, 1

39 p=-—F, 2 70

P=0, 4

39 p. = — F , ^ 5 70

39 P.=-—F 3 70

(3.1.51a)

39 p6=—F 6 70

(3.1.51b)

Since we consider the equivalent truss subjected to the antisymmetric loads instead of the original one, the supports at the symmetric line are not subjected to any loads, i.e., P^ = 0 and P„+1 = 0. The real supporting reactions at two extreme ends of the original trass can be found easily by solving the equilibrium equation for 2 the whole truss, i.e., P.=P.= —F . 4 35 Introducing Eq. (3.1.50) and p = 2 into Eq. (3.1.42) gives

/(u, = /«u) = f

tF.

/o, 5 , = / ; „ = ~iF

(3.1.52)

with the other components vanishing. Substituting Eqs. (3.1.45), (3.1.52), (3.1.43), (3.1.46) and (3.1.28) Eq. (3.1.29), the transverse displacements for all nodes can be found as n Vo..,=0,

8 F v(1,2)=-—-,

v ( 1 , 3 ) =0,

353 F v(M)=— _ ,

into

Bi-periodic Structures o V

v0,5)=0,

p

=

v

('.s> -^J7'

('. 7 ) =0

(3.1.53a)

7=1,2,-,6

__j_L

I_F_ v(2,„-

2lK>

v(2>2)-

i 5

v„ „ = v

l

5

K

-2LE-

v(2i3)-35^,

,

(2,5)

(3.1.53b)

-ILL

^,

v(2,4)-35/r

v„ M =

>

97

(3.1.53c)

(2,6)

(2,n-y) =- v (2,;)'

2

{

K

7=l>2,--,6

(3.1.53d)

The results show that the restraint condition Eq. (3.1.30) is satisfied. The longitudinal displacements for all nodes also can be obtained by inserting Eqs. (3.1.24), (3.1.27), (3.1.28), (3.1.46), (3.1.45), (3.1.52) and (3.1.43) into the first and third component equations in Eq. (3.1.9a), i.e., 1

_

_

1

"o.)=-7=Ee''0'"1)'"^o.»)' ^=l^Te'iM)m¥^

(3-1-54)

The results are summarized as follows

_S_F_ "(U) -

7o

A:

'

_V£iL

U

™ - 30 K '

19V3 F

U,,~=

°''

210 K

,

«^6^ = (

' >

M(u

19S F

> _ 210 K ' " ( M )

VJ F 30 A:

»

fi

M

(n7i

' '

=

'

F 70 AT

y = l,2,-,6

V^F_

llV3F

70 AT '

=

9>/3F 70

A:

"

.

in which Af26 = 0 and AT212 = 0 indicate the axial forces vanishing for two additional bars. It can be verified easily that the equilibrium equation for every node is satisfied and then the solution for displacement and axial force is an exact one for the truss shown in Fig. 3.1.4. 3.1.3

Natural Vibration [12]

Consider now the natural vibration of the continuous truss with equidistant supports. The natural vibration equation can be obtained easily from the equilibrium equation (3.1.25) by using the inertia force instead of the static loading. The masses of the bars are assumed to be lumped at the nodes. Two lumped masses denoted by M, and M2 are attached to each of the lower and upper nodes respectively as shown in Fig. 3.1.1. We also assume that the inertia forces in the longitudinal direction may be neglected. By using the co2Mxv(XJ) and = / ( U n > = ^ | iA

/o,2»-r) = /(i,4»-o = — 7 =

(3.1.82a)

(3-1 -82b)

Bi-periodic Structures

107

with the other / ( ° m ) vanishing. Substituting Eqs. (3.1.82), (3.1.75), (3.1.26) and (3.1.64) into Eq. (3.1.67) gives

"(U)

-=-

£

Sin[0-l)m^]—-

7

(3.1.83a)

2

IAl (m)-Qr]

•sJZnK „=r,r+2n

v(2 n - — ? = —

Kl(m)-Qr

-K2(m) K3{m)

sintO' — ) m y/] cos(— m y/)

m)-Qrf-K2(m) (3.1.83b)

where K{(m) and K2(m) have been defined in Eqs. (3.1.77a) and (3.1.77b) and 24(3 - 2 cos m y/) 49 - 32 cos my/

K3(m)-

(3.1.84)

with y/ = nfln . It can be proved that when r is odd, the mode shown in Eq. (3.1.83) is symmetric, i.e., v(12n+2_y) = v(1J) and v(2 2n+1_>) = v ( 2 ; ) ; when r is even, the mode is antisymmetric, i.e., v(12n+2_y) = -v ( M ) and v ( 2 2 „ + w ) = - v ( 2 ; ) . Also, the mode satisfies the constraint condition V

(l.2(,-!)+l) = 0 >

3.1.4

(3.1.85)

5 =1,2,-.11 + 1

Forced Vibration [12]

The continuous trass subjected to transverse harmonic loads acting at the nodes is considered. The forced vibration equation can be expressed as

K^-a2Mx K 2\,m

Klu, K

22,m-

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