Exact Analysis of Biperiodic Structures

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Exact Analysis of

dic Cai -

K Liu

Exact Analysis of

6'-Periodic Structures

This page is intentionally left blank

E x a c t Analysis of

k

Periodic tructures C W Cai Department of Mechanics, Zhongshan University, China

J K Liu Department of Mechanics, Zhongshan University, China

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

vg b

World Scientific

New Jersey. London .Singapore Hong Kong

Published by

World Scientific Publishing Co. Re. Ltd. P 0 Box 128, Farrer Road, Singapore 912805 USA once: Suite lB, 1060 Main Street, River Edge, NJ 07661 UK once: 57 Shelton Street, Covent Garden, London WC2H 9HE

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

EXACT ANALYSIS OF BI-PERIODIC STRUCTURES

Copyright O 2002 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom 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

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. l b 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

...

MII

Contents

and the Other Free 2.2.3a Natural Vibration Example

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 Chapter 4 Structures with Bi-periodicity in Two Directions 4.1 Cable Networks with Periodic Supports 4.1.1 Static Solution 4.1.la 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 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 5.2 Periodic System with One Nonlinear Disorder

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 References Nomenclature Index

ix

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 In general, the discrete equation for cyclic periodic structures without damping may be expressed as

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

and

where N represents the total number of substructures; the vector components xi and Fj ( j = 42, ...,N ) denote displacement and loading vectors for the j-th substructure, respectively. The numbers of dimensions of submatrices K , , M ,

2

Exact Analysis of Bi-periodic Structures

( r ,s = 42,. ..,N ) and vector components x, and Fj ( j = 1,2,.

..,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

IT

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

with the submatrices

U Transformation and Uncoupling of Governing Equations

3

in which y = 2 ; l r / ~ , i = f i and I, denotes the unit matrix of order J. It can be shown that

That leads to

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 N .J dimensions. The columns of U , are made up of the basis of the m-th subspace with J dimensions. An arbitrary vector, say U,xm ( x , is a J dimensional vector), in the m-th subspace possesses the cyclic periodicity. If

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 m y (= 2mnlN) 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 2 m / N ( m = 1,2,. .., N ) due to cyclic periodicity. As a result, all of the mode vectors lie in the N 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 stiffness and mass matrices of structures with cyclic periodicity are cyclic matrices. The elementary cyclic matrices can be defined as

where the empty elements are equal to zero, Eo is a unit matrix and each element of matrix cj is a J dimensional square matrix. An arbitrary cyclic matrix can be expressed as the series of the elementary cyclic matrix, such as

and

where

E~

=

c0 and

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

U Transformation and Uncoupling of Governing Equations

Noting the cyclic periodicity of Urnand

it can be verified that

with

It is obvious that O j ( j

= 0,1,2,. .., N- 1) is

a diagonal matrix. One can now apply Eqs. (1.1.13) and (1.1.16) to derive the important formula:

5

Exact Analysis of Bi-periodic Structures

6

The matrix k, is a Herrniltian one, i.e.,

k-T, = k , In the same way, we have

and

in, is also a Hermiltian matrix.

1.1.3

U Transformation and Uncoupling of Simultaneous Equations with Cyclic Periodicity

The U transformation can be defined as

U Transformation and Uncoupling of Governing Equations

7

where X , U are defined as Eqs. (1.1.3a) and (1.1.8) respectively and

and q , ( m = 1,2,. ..,N ) are vectors of dimension J. Because the coefficient matrix in Eq. (1.1.22) is an unitary matrix satisfying Eq. (1.1.10), the complex linear transformation (1.1.22) is referred to as U transformation. Recalling Eq. (1.1.1O), premultiplying both sides of Eq. (1.1.22) by ET,the inverse U transformation can be obtained as

The component forms of the U and inverse U transformations are

and

with y/=2n/N and i=n. Usually, the original variables x, ,x,, ,x, are real vectors representing the displacement vectors of substructures for a cyclic periodic structure and q, ,q2,-..,qN are a set of generalized displacement vectors. Noting Eq. (1.1.25b) and x j being real vector, it can be shown that

8

Exact Analysis ofBi-periodic Structures

and qN, qNIZ(if N is even) = real vector

(1.1.26b)

Applying the U transformation (1.1.22) to Eq. (1.1. I), namely substituting Eq. (1.1.22) into Eq. (1.1.1) and premultiplying both sides of Eq. (1.1.1) by U T, we have

where

Eq. (1.1.27) is made up of N independent equations, i.e., mrq, + krqr = f, , r

= 1,2,..., N

(1.1.29)

Noting the definitions of m,, k, and f, shown in Eqs. (1.1.21), (1.1.18) and (1.1.28) respectively, it is obvious that

and m,, m,,, (if N is even), k,, k,,, (if N is even) are real symmetric matrices, so 9,-, = ijr and q , , q,,, (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., 2 2 N N-1 r=1,2 ,...,-,N (Niseven)or r=1,2,...,-,N (N is odd) in Eq. (1.1.29). 2 2

U Transformation and Uncoupling of Governing Equations

9

1.1.4 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

where w denotes the natural frequency, q, represents the amplitude of the r-th generalized displacement and k , , m , 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 (1.1.3 1) with Hermiltian matrices are real numbers. The eigenvalues can be denoted as w,?,, 2 w : , ~ , , w : , ~ ( w,,, Ia:,+,,s = 1,2,. ..,J - 1) and the corresponding normalized orthogonal eigenvectors may be written as q,,, , q r , 2 , ..., ( I , , ~They . satisfy the

eigenvalue equation and the normalized orthogonal condition, i.e.,

and

leading to w:,, = ijrTskrq,,, = real number

Noting kN-,

-

= k, ,

and qN,S, q -,s

(1.1.34)

-

mN_, = m , , it is obvious that

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

Let us consider the natural modes. Corresponding to the eigenvector qNSs ( s = 42,. ..,J ), the natural mode can be expressed as

10

Exact Analysis of Bi-periodic Structures

X = U~4iv.s

leading to

The vibrating displacements of all substructures possess the same amplitude vector and vibrating phase. Corresponding to the eigenvector q N (if N is even), the natural mode is -,s

which leads to

so the displacement vectors for any two adjacent substructures are equal and opposite. Such a mode doesn't occur when N is odd. ..

N

For other natural frequencies w r , ( r # N,-;

2

repeated frequencies: w , , = u,-,,~ (r

N + N , -; 2

s = 1,2,. ..,J ), which are

s = 42,. ..,J ), the corresponding

modes take the form as

This is a rotating mode, the deflection of any substructure has a constant phase difference 2 m / N from that for the preceding substructure, i.e., the mode lies in the r-th mode subspace. The real and imaginary parts of each rotating mode are two independent standing modes corresponding to the same natural frequency.

U Transformation and Uncoupling of Governing Equations

11

The rotating mode corresponding to o,-,,, is

The phase difference between two adjacent substructures is -2mlN. A pair of rotating modes shown in Eqs. (1.1.38) and (1.1.39) corresponding to the same natural frequency are complex conjugate modes and their rotating directions are opposite. Generally, the natural frequencies for periodic structures are densely distributed in pass bands. The number of pass bands is in agreement with that of the degrees of freedom for a single substructure. For the case under consideration, there are J pass bands altogether. In fact, w,,( r = 1,2,...,N ) lie in the s-th pass band. There are N natural frequencies lying in a pass band. If N is a large number, the natural frequencies are densely distributed in every pass band. It is of interest to note that the natural frequencies obtained from Eq. (1.1.31) with a given r are dispersed, namely they lie in different pass bands. The upper and lower bounds of the pass bandscanbe foundby solving Eq. (1.1.31)with r = N ( r y l = 2 r ) a n d r = N l 2 ( r ry = r ), respectively. 1.2

Uncoupling of Simultaneous Equations with Cyclic Periodicity for Variables with Two Subscripts

1.2.1 Double U Transformation If a structure considered possesses cyclic periodicity in two directions, say x and y directions, the governing equation can be uncoupled by using U transformation

two times (in x and y directions respectively). Let x(,,,) denote the displacement vector of substructure ( j , k) ( j = 1,2,...,M ;

k = 1,2,...,N ) and the subscriptsj and k denote the ordinal numbers of substructure along x and y directions respectively. Firstly the substructure with cyclic periodicity in x and y directions can be regarded as cyclic periodic in x direction. The corresponding U transformation can be expressed as

12

Exact Analysis of Bi-periodic Structures

where M and N denote the total numbers of substructures in x and y directions respectively and y , = 2 z / M denoting the period of the structure in x direction. The inverse U transformation is

The governing equation in terms of Q(,,,, ( r = 1,2,.

..,M

; k = 1,2,.

..,N ) also

possesses cyclic periodicity in y direction for the second subscript k. Secondly we can introduce again the U transformation into the governing equation with the unknown variables Q,,( ( r = 1,2,...,M ; k = 1,2,. ..,N ). The corresponding U and inverse U transformations can be expressed as

and

U Transformationand Uncoupling of Governing Equations

13

with y, = 2n/N denoting the period of the structure in y direction. Substituting Eq. (1.2.3a) into Eq. (1.2.lb) yields

with y, = 2n/M and y, = 2 n l N . Eq. (1.2.4a) is referred to as double U transformation, which may be regarded as an extension of the U transformation in two dimensional problems. Its inverse transformation can be obtained by introducing Eq. (1.2.2b) into Eq. (1.2.3b) as

1.2.2

Uncoupling of Simultaneous Equations with Cyclic Periodicity for Variables with Two Subscripts

Consider now the simultaneous equations having cyclic periodicity for variables with two subscripts. Such equations can be expressed as

possesses cyclic periodicity for subscriptsj, u and k, v respectively, where K(j,k,(u,v, namely

14

Exact Analysis of Bi-periodic Structures

and

Usually the equilibrium equation of structures with cyclic periodicity in two directions takes the form as Eqs. (1.2.5) and (1.2.6) where x ( ~ , and ~ ) F(j,k) represent the displacement and loading vectors of substructure (j,k) and K(j,k)(u,v) denotes the stiffness coefficient matrix. satisfying Eq. (1.2.6) can be uncoupled by using the Eq. (1.2.5) with K(j,k)(u,v) double U transformation shown in Eqs. (1.2.4a) and (1.2.4b). Premultiplying both sides of Eq. (1.2.5) by the operator 1 e - i ( j - l ) r v-i(k-1)s" el gives

fiJN

7, j=l

t=l

and

Noting the cyclic periodicity of K(j,,,(,,v,shown in Eq. (1.2.6), we have K(j,k)(u.v)

- K(j',k')(l,O

(1.2.9a)

with j f =j - u + 1 ,

kf=k-v+l

Introducing Eqs. (1.2.9) and (1.2.4b) in Eq. (1.2.7) results in

(1.2.9b)

U Transformation and Uncoupling of Governing Equations

15

Thus the simultaneous equations (1.2.5) have been uncoupled into M times N independent equations.

1.3 Uncoupling of Simultaneous Equations with Cyclic Bi-periodicity*

1.3.1 Cyclic Bi-periodic Equation

If the coefficient matrix of simultaneous equations is a superposition of two matrices with cyclic periodicity and different periods, the simultaneous equations are referred to as cyclic bi-periodic. The simple form of cyclic bi-periodic equation to be considered can be expressed as

where K,., ( r, s = 1,2,...,N ) possess cyclic periodicity, i.e.,

and

with the other K ; , vanishing and N = np . In physics,

KO

represents the

* See L. Gao and J.K. Liu, Uncoupling of governing equations for cyclic bi-periodic structures, Advances in Structural Engineering, An International Journal, Vol. 4, No. 3, 137-146 (2001).

16

Exact Analysis of Bi-periodic Structures

difference between the stiffness matrices of the [1 + (k - l)p] -th ( k = 1,2,. ..,n ) substructure and the other substructures. The j-th component equation of Eq. (1.3.1) is

Considering Eq. (1.3.2), Eq. (1.3.3) can be rewritten as

where x-,

,

= xN-, ( s = 0,1,2,. ..,N - 1) and the term ( - K Ox ) is treated as load as

well as F, . The coefficient matrices K,,, ( u = 42,. ..,N ) are the same in everyone of Eq. (1.3.4) ( j = 1,2,. ..,N ). It is a characteristic of cyclic periodic equations.

1.3.2 Uncoupling of Cyclic Bi-periodic Equations Applying the U transformation

and

U Transformation and Uncoupling of Governing Equations

17

with ly = 2n/N to Eq. (1.3.4), i.e., premultiplying both sides of Eq. (1.3.4) by the operator

L

JN

N

,and considering

e-i(j-l)m v j=,

r = +l,f2,...,fN

we have amqm= fm +f O , 3

m =1,2, ...,N

where

Eq. (1.3.7) leads to 4, = a i l ( f m+ fmO) Inserting Eq. (1.3.10) in the U transformation (1.3.5a), we obtain x j = x :I + x O ] ,

where

j = 1 , 2,..., N

18

Exact Analysis of Bi-periodic Structures

Here x; denotes the solution of Eq. (1.3.1) with

KO

vanishing and x9 denotes

the influence of K O on the solution x j . Note that x; is dependent on x,+(,-,,~ ( u = 1,2,...,n ). Introducing the notations

and inserting j = 1+ (s - l)p into Eqs. (1.3.11) and (1.3.12b) gives

where

The coupled terms in simultaneous equations (1.3.14) possess cyclic periodicity, i.e.,

By using the U transformation once, the cyclic bi-periodic equation (1.3.1) with N unknown vectors becomes the cyclic mono-periodic equation (1.3.14) with n ( = N/p ) unknown vectors. In order to uncouple Eq. (1.3.14), the U transformation needs to be used again. For the present case, the corresponding U and inverse U transformation can be given as

U Transformation and Uncoupling of Governing Equations

19

and Q

1 =-

J;I

" ii(s-l)rc X s , r =172,...,n

(1.3.17b)

s=l

with q = 2 z / n = p y / . 1 " Premultiplying both sides of Eq. (1.3.14) by the operator - e-i's-')rcand & ,=I considering Eq. (1.3.16), we have

where

Noting that

and substituting Eq. (1.3.20) into Eq. (1.3.18) we obtain ArQr=br,

r = l , 2,...,n

Exact Analysis of Bi-periodic Structures

20

and I denotes unit matrix. By means of U transformation twice the cyclic bi-periodic equation (1.3.1) has been uncoupled into a set of independent equations shown in Eq. (1.3.22) where everyone includes only one unknown vector. The number of unknowns in Eq. (1.3.22) with given r is in agreement with that of degrees of freedom for a single substructure. Recalling Eq. (1.3.13) and rp = p iy and making a comparison between Eqs. (1.3.9b) and (1.3.17b) gives

When the specific parameters in Eq. (1.3.1) are given, the solution for xi ( j = 1,2,...,N ) can be found by using the relevant formulas derived above.

1.3.3

Uncoupling of Simultaneous Equations with Cyclic Bi-periodicity for Variables with Two Subscripts

The simultaneous equations to be considered take the general form as

U Transformation and Uncoupling of Governing Equations

21

and K (j,k)(u,v) ( j , u = 1,2,. ..,M ; k, v = 1,2,. ..,N ) possess cyclic periodicity for two pairs of subscripts:j, u and k, v, respectively, namely

and K(j,k)(u,l)

- K(j,k+l)(u,Z)

- ". = K(j , N ) ( u , ~ - k + l ) - K(j,l)(u,N-k+2)

j,u=l,2. M ;

=

k=1,2,...,N

"

'

= K(j,k-l)(u,~)

(1.3.27b)

Introducing Eq. (1.3.27) and the notations X(-u,v)

-

= X(~-u,v)

-

,

=N - v )

u=0,1,2 ,..., M ;

X(-u,-v)

-

= X(M-~,N-~)

v=0,1,2,..., N

into Eq. (1.3.25), it can be rewritten as

j # 1 , 1 + p l , l + 2 p ,,...,l+(m-l)p, or k + l , l + ~ ~ , l ,..., + 2l +~( ~ n-l)p2 =

2. M ;

Applying the double U transformation

k=1,2,...,N

(1.3.29b)

22

Exact Analysis of Bi-periodic Structures

u=1,2 ,...,M ;

v = 1 , 2 ,...,N

(1.3.30b)

with ry, = 2 z l M and ry, = 2 z / N to Eq. (1.3.29),i.e., premultiplying Eq. (1.3.29) by the operator

e-i(j-l)uK

fifi

e -i(k-I),,

and noting that

j=l

we have a(,,,)q(,,,)= f(u,v) + f ( : , v ) , where

u = 1,2,...,M ;

v = 1,2,...,N

(1.3.32)

U Transformation and Uncoupling of Governing Equations

23

From Eq. (1.3.32), q(,,,, can be formally expressed as

f(3

-1

(1.3.35)

9(u,v) = a(u,v)(f(u,v) +

Substituting Eq. (1.3.35) into the double U transformation (1.3.30a) yields

+

0

x ( ~ ,=~x, ; ~ , ~x)( , , ~ ),

j = 1,2,..., M ; k = 1,2,...,N

(1.3.36)

where

and

Here x ; , , ~ ,denotes the solution of Eq. (1.3.25) with represents the influence of

KO

KO

vanishing and xpjSk,

on the solution x ( ~ , ,., xpj,,, is dependent on

x ( ~ + ( ~ - ~ ) ~ (~r , = ~ 1,2,. + ( ~ ..,m - ~; ) ~s ~ = )1,2,. ..,n)

Introducing the notations

in Eqs. (1.3.36) and (1.3.37b), we obtain

to be determined.

24

Exact Analysis of Bi-periodic Structures

in which

with 9, = p l y l = 2z/m and 9, = p 2 y 2 = 2z/n. It is obvious that $ j , k ) ( r , s ) ( j , r = 1,2,...,m ; k, s = 1,2,...,n ) possess cyclic periodicity for the two pairs of subscripts 0, r and k, s), namely

and

The term Xij,,, on the right side of Eq. (1.3.39) can be found by means of the formulas (1.3.33), (1.3.34a), (1.3.37a) and (1.3.38). The set of simultaneous equations (1.3.39) possess cyclic periodicity for two subscripts. It takes the same form as that of Eq. (1.2.5). As a result Eq. (1.3.39) can be uncoupled by means of the double U transformation (see section 1.2.2). Let

U Transformation and Uncoupling of Governing Equations

with 9,= 2 r / m and 9,= 2 n l n . Premultiplying both sides of Eq. (1.3.39) by the operator 1 " " e-i(i-l)une - i ( k - l ) n results in

rJ m n

j=l

k=l

where

and

Substituting Eq. (1.3.46) into Eq. (1.3.44) and noting that ( 1 j =u,u+m;..,u +(p, -1)m and

25

26

Exact Analysis of Bi-periodic Structures

results in

By applying the U transformation twice, the cyclic bi-periodic equation (1.3.25) with M x N unknown vectors is uncoupled into a set of independent equations shown in Eq. (1.3.43) where each includes only one unknown vector. Making a comparison between Eqs. (1.3.34b) and (1.3.42b), we have

where Q(,,+,,( u = 1,2,..., m ; v = 1,2,...,n ) can be found from Eq. (1.3.43). When the specific equation (1.3.25) is given, the solution for x ( ~ , ~ ) ( j= 1,2,...,M ; k = 1,2,...,N ) can be obtained from Eqs. (1.3.49), (1.3.34a), (1.3.35) and (1.3.30a).

Chapter 2 BI-PERIODIC MASS-SPRING SYSTEMS The physical meaning and mathematical formulation of the U-transformation have been described in reference [I]. Essentially the U-transformation method is a mode subspace method [2] for a cyclic periodic structure. In mathematics the Utransformation is an orthogonal linear transformation with complex coefficients. The linear simultaneous equations (algebraic or differential equations) with cyclic periodicity can be uncoupled by the U-transformation technique. The proof is given in the first Chapter. It is obvious that the governing equation of a cyclic periodic structure possesses the cyclic periodic property. Therefore the U-transformation can be applied to analyze cyclic periodic structures [3,4]. Furthermore the application of the U-transformation may be extended to analyze linear periodic structures [5-91, if its equivalent system with cyclic periodicity can be formed. What is referred to as a bi-periodic structure in this book is a structure consisting of two different sets of periodic properties. Certainly a bi-periodic structure may be regarded as a single periodic one but its substructure also possesses periodicity and many degrees of freedom. In order to utilize fully the property of biperiodicity, the proposed analysis method in this book requires the application of the U-transformation twice. As a result, the governing equation for a system with cyclic bi-periodicity may be fully uncoupled. That leads to the explicit analytical solution, which plays an important role in the optimal design or sensitivity analysis. 2.1

Cyclic Bi-periodic Mass-Spring System*

To illustrate the proposed method a simple model [lo] with cyclic bi-periodicity is analyzed. A general model, cyclic bi-periodic mass-spring system, is illustrated in Fig. 2.1.1 where all the coupling springs are of the same stiffness k ; K and K + AK denote the stiffness for two kinds of cantilever beams; M and M +AM denote two kinds of the lumped masses; and F, , x , denote the load, displacement for the r-th subsystem.

See L. Gao and J.K. Liu, Exact analytical solutions for static and dynamic analyses of cyclic biperiodic structures, Advances in Structural Engineering, An International Journal, Vol. 4, No. 3, 1471.58 (2001).

28

Exact Analysis of Bi-periodic Structures

\ \ - ' p + l JP' \ M

k

Figure 2.1.1 Rotationally bi-periodic mass spring system 2.1.1 Static Solution

The equilibrium equation can be expressed as

j

+ 11+ p . 1+ (n - 1

and

j =1,2,. ..A

(2.1.1b)

where x,+, = x, ; X , = X, and 1+ (s - l)p (s=1,2,. ..,n) indicate the ordinal on the right numbers of the subsystems with stiffness K + AK . The term side of Eq. (2.1.la) may be treated as the load as well as Fj . One can now apply the U-transformation [3,5] to Eq. (2.1.1). The U- and inverse U-transformation may be defined as

Bi-periodic Mass-Spring Systems

29

and

in which y/ = 2z/N, i = f i and N = total number of subsystems, for present case N =pn. The equilibrium equation (2.1.l) may be expressed in terms of the generalized displacements q m (m= 1,2,...,N) as

where

The generalized displacement q m in Eq. (2.1.3) may be formally expressed as

Substituting Eqs. (2.1.5) and (2.1.4) into Eq. (2.1.2a) yields

30

Exact Analysis of Bi-periodic Structures

1 x"-Cei(~-')m*q2

fi

M=I

=

AK ei(j-l)mye-i(u-l)mpy -NCF, n

N

U=I

M=I

K + 2k(l- cos m v)X1*'u-l'p

(2.1.7a)

where x; (j=1,2, ...,N) represents the solution for the perfect periodic system (i.e., AK = 0 ) subjected to the same loading as that acting on the bi-periodic system. When the specific loading condition is given, x; (j=1,2,. ..jV)can be obtained from Eqs. (2.1.4b), (2.1.5~)and(2.1.7b) Inserting j = 1+ (s - 1)p (s= 1,2,. ..,n) in Eqs. (2.1.6) and (2.1.7) gives

where

and

P,, 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.,

31

Bi-periodic Mass-Spring Systems

One can now apply the U-transformation again to Eq. (2.1.a). Introducing

and

with y, = 2n/n = py/ ,into Eq. (2.1.8)results in n

Q, = -MZps,le-i(s-l)rqQ, + b , ,

r=1,2 ,...,n

(2.1.13)

where

The governing equation (2.1.1) has been uncoupled and becomes a set of one degree of freedom equations by using the U-transformation twice. Substituting Eq. (2.1.10) into Eq. (2.1.13)and noting the identical relation

m = 1,2,...,N ( = pn) ;

we have

and

r

= 1,2

,..., n

(2.1.15)

Exnct Analysis of Bi-periodic Structures

32

Inserting Eqs. (2.1.16) and (2.1.17) into the later U-transformation (2.1.12a) yields

in which ty = 27r/N, p = 2n/n and N = pn . Making a comparison between Eqs. (2.1.4a) and (2.1.12b), we have

Finally the solution for xj of Eqs. (2.1.la) and (2.1 .lb) can be found by substituting Eqs. (2.1.9, (2.1.7), (2.1.19) and (2.1.4b) into Eq. (2.1.6). In order to explain the computational procedure and to verify the exactness of the formulas derived above, we need to consider a specific system and loading as an example. 2.1.la

Example

The systemic parameters and loads are given as n p=3, n=2 (as a result N=6, q~= n and y = - ) 3

and

F, = F, = P,

Fj = 0, j

Inserting Eq. (2.1.20) into (2.1.4b) yields

+ 1,4

(2.1.20a)

Bi-periodic Mass-Spring Systems

33

and then introducing Eqs. (2.1.21) and (2.1.20a) into Eq. (2.1.5~)gives 2P q:, = &K

1

+ 2k(l- cos rn n/3)

m = 2,4,6

(2.1.22a)

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

It can be verified thatx; (j=1,2,...,6) is the exact displacement solution for the system with AK = 0 subjected to the loads shown in Eq. (2.1.20b). Recalling the definition shown in Eq. (2.1.9) andp=3 gives

Inserting Eqs. (2.1.24) and (2.1.20a) into Eq. (2.1.14) yields

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

From Eq. (2.1.19),

fi (m=1,2,.. .,6) can be obtained as

Exact Analysis of Bi-periodic Structures

34

Finally, substituting Eqs. (2.1.5b), (2.1.27) and (2.1.20a) into Eq. (2.1.7a) we have 0

XI

0

X2

0

= X 40 = -

0

AK(K + k ) 2 ~ K(K + 3k)[K(K + 3k) + AK(K + k)]

0

= X 3 = X5 = X 6 = -

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

(2.1.28a)

(2.1.28b)

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

The displacement x j ( j = 1,2,. ..,6 ) shown in Eq. (2.1.29) satisfies the equilibrium equations (2.1.l) with the parameters shown in Eq. (2.1.20). 2.1.2

Natural Vibration

The natural vibration equation for the cyclic bi-periodic system shown in Fig. 2.1.1 may be expressed as

where w denotes the natural frequency, x j denotes the amplitude of j-th subsystem and the term - (AK - A M w 2 ) x j 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

( ~ + 2 k - ~ u ~ ) ~ ~ - 2 k c o s m l ym=1,2, q ~ =..., fN ~ ,

35

(2.1.31)

where

and then

Substituting Eqs. (2.1.32) and (2.1.33) into Eq. (2.1.2a) yields

x. =-

(AK - MU') N

Introducing the notation X , in Eq. (2.1.34) we have

ei(j-l)mve-i(u-l)mpy

2U=I2 K m =+~2 k - M u 2

-2kcosmy

= x,+(,-,,, and inserting j

XI+(~-~)~

= 1+ (s - l)p (s=1,2,.

..,n)

where

p,'

denotes the harmonic influence coefficient for the considered system with

AK=AM=o.

Obviously also possesses cyclic periodicity. Applying the Utransformation (2.1.12) to Eq. (2.1.35) results in

Exacr Analysis ofBi-periodic Structures

36

Substituting Eq. (2.1.36) Eq. (2.1.15), we have

and p y/ = p

into Eq. (2.1.37) and recalling

When Q, is non-vanishing, the frequency equation can be expressed as

,-,,,

.

= 0 (s=1,2,. .,n), When Q, (r=1,2,. ..,n) are identically equal to zero, i.e., x,+( the corresponding frequency equation can be obtained from Eqs. (2.1.31) and (2.1.32) as

2m =kn (niseven), k =1,2,..., p-1, 2m = 2kn (n is old), k = 1,2,..., (k < p 12)

(2.1.40)

where 2m is in agreement with the half wave number of the mode. In order that all mass points having mass M + Ah4 lie in the nodal points of the mode, the half wave number 2m must be equal to an integer times n and less than N. 2.1.2a

Example

The parameters are given as

Bi-periodic Mass-Spring Systems

p=3,

n=2,

AM=M,

AK=K

37

(2.1.41a)

That leads to

The frequency equation (2.1.39) becomes

r = 1,2 The solution for w of Eq. (2.1.42) can be found as

w

=

K+(2-fi)k M

7

~+(2+fi)k , (for r = 1 ) M

and K K + 2 k , (for r = 2 ) M' M

w =-

These natural frequencies are corresponding to the modes with x, and x, nonvanishing. Consider the other frequency equation (2.1.40), i.e.,

The square of frequency can be expressed as

x, and x, 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

Substituting Eqs. (2.1.46) and (2.1.41) into Eq. (2.1.12a), we have

where an arbitrary constant factor is neglected. Introducing Eqs. (2.1.47), (2.1.41) and w

=

~+(2-fi)k into Eq. (2.1.34), M

the natural mode can be found as

Substituting Eqs. (2.1.471, (2.1.41) and

02=

+(2 +fi)k M

into Eq. (2.1.34)

results in

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

That leads to XI

= 1,

X4

=1

Corresponding to w 2 = KIM ,the natural mode is

,

x . = l,

and corresponding to

02= (K

j=1,2 ,...,6

+ 2k)lM ,the natural mode is

Bi-periodic Mass-Spring Systems

39

Consider the other kind of modes with x, = x, = 0.The mode in terms of the generalized displacements can be expressed as

with the other qs vanishing. There are two independent modes corresponding to the same natural frequency for the cyclic periodic system. We are only interested in the mode with x , and x, vanishing. By substituting Eqs. (2.1.53) and (2.1.41) into Eq. (2.1.2a) and letting X, = 0,results in q, = imaginary number and

where an arbitrary constant factor is also neglected. K+k K+3k , the natural modes can be found Corresponding to w = - and M M by introducing m=l and 2 into Eq. (2.1.54) respectively as

and

2.1.3

Forced Vibration

The forced vibration equation for the system shown in Fig. 2.1.1 subjected to harmonic forces may be expressed as

40

Exact Analysis of Bi-periodic Structures

with x, = x,, x, = x, due to cyclic periodicity. In Eq. (2.1.57), Fj and xj denote the amplitudes of the loading and displacement for the j-th subsystem and denotes the frequency of the harmonic loads. Applying the U-transformation (2.1.2) to Eq. (2.1.57), yields

where

From Eq. (2.1.58), q , can be formally expressed as

9; = f i / ( K + 2 k - M w 2 -2kcosrny)

(2.1.60~)

Substituting Eqs. (2.1.59), (2.1.60) and j = 1 + (s - 1)p into Eq. (2.1.2a), results in

Bi-periodic Mass-Spring Systems

41

where

Applying the U-transformation (2.1.12) to Eq. (2.1.61) yields a,(u)Q,=b,

r=1,2 ,..., n

(2.1.65)

where

Recalling Eqs. (2.1.59a) and (2.1.12b), we have

fP+(u-l)n

=-

(AK - AMu2) Q, r

fi

=,

. . . ,n

u = 1,2,..., p

(2.1.69)

Exact Analysis of Bi-periodic Structures

42

Finally substituting Eqs. (2.1.60), (2.1.69) into Eq. (2.1.2a), the exact solution can be expressed as

0 qr+(u-1)n

-

--

(AK - AMw2)

Qr

K + 2k - M

W ~ - 2k cos[r

+ (u - l)n]y

and x; can be found from Eqs. (2.1.64), (2.1.60~)and (2.1.59b) if the loading is given. 2.1.3a

Example

The same system shown in Fig. 2.1.1 is considered. The structural and loading parameters are given as

and F, = F, = P,

F2 = F, = F,

= F, = 0

Inserting Eq. (2.1.71) into (2.1.59b) gives

and then substituting Eqs. (2.1.60c), (2.1.72) and (2.1.71a) into Eq. (2.1.64) results in

Bi-periodic Mass-Spring Systems

43

leading to

Inserting Eqs. (2.1.74) and (2.1.71a) into Eq. (2.1.67) gives

Substituting Eqs. (2.1.66), (2.1.71a) and (2.1.75) into (2.1.68) yields

That leads to

Finally substituting Eqs. (2.1.60b), (2.1.77) and (2.1.7la) into Eq. (2.1.70b), we have

44

Exact Analysis of Bi-periodic Structures

and then superposing this solution on x; ( j = 1,2,...,6) shown in Eq. (2.1.73), results in

This solution represents the steady state response. When w approaches zero the solution shown in Eq. (2.1.79) approaches the static one shown in Eq. (2.1.29) with K K+2k AK = K . When w 2 approaches - -, xj ( j = 1,2,...,6) will approach M' M K K+2k infinity. Corresponding to w = - and -, two natural modes possess the

M

M

property of x, = x4 # 0 . However, there are six natural frequencies for the considered system. It can be shown that, when x, approaches a finite value at one natural frequency, the work done by the external forces due to the displacement of the corresponding natural vibration is identically equal to zero, namely x, = -x4 including x, = x4 = 0 .

2.2 Linear Bi-periodic Mass-Spring Systems The U-transformation method is applicable to static and dynamic analyses of cyclic mono-periodic and bi-periodic systems. We can not directly apply the Utransformation method to analyze linear periodic systems. If the equivalent system with cyclic periodicity can be formed, the U-transformation method can be applied to the analysis of the equivalent system instead of the original linear periodic one. The following sections will illustrate how this can be done.

2.2.1 Bi-periodic Mass-Spring Systems with Fixed Extreme Ends Consider a bi-periodic mass spring system with fixed extreme ends and np - 1 mass points as shown in Fig. 2.2.1(a), where M and M + AM denote the masses of

Bi-periodic Moss-Spring Systems

45

two kinds of periodic particles respectively and k denotes the stiffness constant of all coupling springs. Assuming that the mass points can move only along the longitudinal direction. x, and F, denote the longitudinal displacement and load for mass point j. The fixed extreme end condition can be expressed as x, = 0 and x,+, = 0. The equivalent system with cyclic bi-periodicity must satisfy these restrained conditions. Such an equivalent system can be achieved by the following procedures. First, the mass-spring system is extended by its symmetrical image and the symmetric loading is applied on the corresponding extended part as shown in Fig. 2.2.l(b). In order to form a cyclic bi-periodic system, it is necessary that two fixed ends of the original system can be replaced by the particles with mass M + AM without any restriction and two extreme ends of the extended system may be imaginarily regarded as the same particle with mass M +AM , namely the first mass point is next to the 2np-th one. Because of the symmetry of the extended system and corresponding loading, the displacements of the symmetric centers must be equal to zero, i.e., x, = 0 and x,, = 0 . As a result the fixed end conditions of the original system can be satisfied automatically in its extended system. Therefore both systems shown in Fig. 2.2.l(a) and (b) are equivalent. Such an extended system with cyclic bi-periodicity is called equivalent system which can be analyzed by the U-transformation method. Consider now the harmonic vibration for the equivalent system shown in Fig. 2.2.l(b). The governing equation can be expressed as

j

+ 1,1+p,...,1+(2n -1)p

and j = 1,2,...,2np

(2.2.lb)

where xj and F, denote the amplitudes of the longitudinal displacement and load for the j-th mass point; w denotes the vibration frequency; 1+ (m - l)p ( m = 1,2, ...,2n ) is the ordinal number of the subsystem with mass M +AM and the loads F, ( j = 1,2,...,2np) must satisfy the symmetric condition, i.e.,

(a) Original system

symmetric line

,.-.+

FJ

I

-Fj

--l

(b) Equivalent system Figure 2.2.1 Bi-periodic mass spring system with fixed extreme ends

Bi-periodic Mass-Spring Systems

47

where N = np and F i ( j = 2,3,..., N) denote the real loads acting on the original system. The term AMw2xj acts as a kind of loading. The expressions on the left sides of Eqs. (2.2.la) and (2.2.lb) possess cyclic periodicity. We can now apply the U-transformation to Eq. (2.2.1). Let

7T

where y = -, i = f i and 2N denotes the total number of subsystems. N 1 ZN Prernultiplying Eq. (2.2.1) by -- e*"-')"* ,results in

Juv

j=l

( 2 k - ~ u ~ ) q-2kcosmyq, , = f; + f;

m = 1,2,...,2N

where f-" -

AMw2 - 2np

J2N

- " m p *

.=I

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

xl+(U-~)~

(2.2.4)

48

Exact Analysis of Bi-periodic Structures

As a result

f; = O and

fiN= O

From Eq. (2.2.4), q, can be expressed as 0

I

4, = 4 , + 4 ,

(2.2.6a)

q: = f : / ( 2 k - ~ w ~ -2kcosmy)

(2.2.6b)

q: = fi/(2k - M

(2.2.6~)

W ~- 2k cos m y )

Substituting Eqs. (2.2.5), (2.2.6) and j = 1+ (s - l)p into the U-transformation (2.2.3a), we have

where

x;(j = 1,2,...,2N) represents the solution of the perfect periodic system with AM vanishing. It can be expressed as

Because

p,,

possesses cyclic periodicity, as

shown in Eq. (2.1.11), the

Bi-periodic Mass-Spring Systems

49

simultaneous equations (2.2.7) can be uncoupled by using the U-transformation. Let

and

with

(D

= z/n and i =

fi. Because of

,,-,= 2 .

X, being real value, Q

1 2n Premultiplying two sides of equation (2.2.7) by -Ce-"'-""

J2n

results in

=.I

where

X j must satisfy the symmetric condition, i.e., X;,+,-, = -X: and X,' = X:,, = 0 . As a result

leading to b,, = b2,,= 0 . Substituting Eq. (2.2.9) with u = 1 into Eq. (2.2.12) yields

Exact Analysis of Bi-periodic Structures

50

Q2n-r= Q,

and

Q,

= Q2,

=0

where

Malung a comparison between Eqs. (2.2.5a) and (2.2.1 1b) gives

Finally substituting Eq. (2.2.6) into the U-transformation (2.2.3a), the solution for x j can be expressed as

and xi is defined in Eqs. (2.2.5c), (2.2.6~)and (2.2.10). 2.2.la

Natural Vibration Example

Letting F j =O ( j= 1,2,...,2 N ) as a result xl = O and b, =O ( r = 1,2,...,2n), the independent frequency equation can be obtained from Eqs. (2.2.14) and (2.2.15) as

if X , (s = 1,2,...,2n) are not identically equal to zero. Consider the case of X , (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

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

Figure 2.2.2 Bi-periodic mass spring system with fixed ends, p=3 and n=2

The parameters are

leading to

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

A nondirnensional frequency parameter may be defined as

52

Exact Analysis of Bi-periodic Structures

The frequency equation (2.2.21) can be rewritten, in term of Ro as

The roots of Eq. (2.2.23a) are

Ro = 0.198062264, 1.55495813,3.24697960

(2.2.23b)

Noting Eq. (2.2.20), the other frequency equation (2.2.19) becomes

That leads to

The two roots are

R, = l ( m = 2 ) ,

3(m=4)

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 SZ, shown in Eq. (2.2.23b) the modes can be expressed in terms of the generalized displacements, as

with the other Q, vanishing. Introducing Eqs. (2.2.20) and (2.2.25) into (2.2.11a) and letting X, = 0 yields

Bi-periodic Mass-Spring Systems

Ql = imaginary number = -i

Q, = Ql = i

53

(2.2.26a)

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

Substituting Eqs. (2.2.20) and (2.2.27) into Eq. (2.2.17b), the natural mode can be obtained as

54

Exact Analysis of Bi-periodic Structures

Obviously the modes depend on the nondimensional frequency parameter R, . For three values of R, shown in Eq. (2.2.23b), the numerical results of modes are given in Table 2.2.1.

x,lT Table 2.2.1 Natural modes [ x , x , x 3 for the system shown in Fig. 2.2.2 a - -

Mode X1

0

0

0

0

0

Corresponding to the frequency equation (2.2.24a) with R, shown in Eq. (2.2.24c), the natural modes can be expressed as

Introducing Eqs. (2.2.20b) and (2.2.29) into Eq. (2.2.3a), and letting x , results in

q , = imaginary number

=0

(2.2.30a)

Bi-periodic Mass-Spring Systems

55

where an arbitrary constant factor is neglected. The results for the modes shown in Eq. (2.2.30b) are also given in Table 2.2.1. All modes shown in Table 2.2.1 satisfy the fixed end conditions (x, = x, = 0) of the original system.

2.2.lb Forced Vibration Example Consider now the system shown in Fig. 2.2.2, subjected to a harmonic force Pei" acting at the center mass point. The parameters of the system and loading are given as z 7r n = 2 , p = 3 , A M = M , N = 6 , ty=- q = i 6'

(2.2.3 la)

and F, = P , I;I. = 0

j= 2,3,5,6

(2.2.31b)

Introducing Eq. (2.2.31) into Eq. (2.2.5~)yields

and then substituting Eqs. (2.2.6~)and (2.2.32) into Eq. (2.2.10) results in xi = X; = 0

(2.2.33a)

56

Exact Analysis of Bi-periodic Structures

j = 2,3,...,6

x ; ~= - ~-x;.

with Ro = ~ w ~ /That k .leads to

XI3 -- 7 -0,

X ~ = X ~ ~ = - X ~

(2.2.34b)

Inserting Eqs. (2.2.31a) and (2.2.34a) into Eq. (2.2.13b) gives

Substituting Eqs. (2.2.15), (2.2.31a) and (2.2.35) into Eq. (2.2.14) we have

Q,=a,Q 2 = Q 4 = 0 From Eqs. (2.2.16), (2.2.3la) and (2.2.36),

fi can be obtained as

(2.2.36b)

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 hnctions for the amplitudes of displacements can be found as

When R, approaches zero the solution xj shown in Eq. (2.2.38) will approach the static displacement, namely

Obviously the resonance frequencies are the roots of Eq. (2.2.23a). When R, approaches each value shown in Eq. (2.2.23b), the amplitudes of displacement will approach infmity. 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 @=2d+l) particles altogether as shown in Fig. 2.2.3(a) where M, AM ,k, xj , 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

(a) Original system

2N=2np

center line

!

p=2d+ I

FZN+I-j=Fj M

k

(b) Equivalent system Figure 2.2.3 Bi-periodic mass spring system with free extreme ends

a-

i

Bi-periodic Mass-Spring Systems

59

Fig. 2.2.3(a) has n subsystems and np @=2d+l) 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 x, = x,, and x,, = x, 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 2np-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

j i t d +1,d + l + p,..., d +1+(2n-1)p and j =1,2,...,2N(= 2np) (2.2.40b) where 2N denotes the total number of particles for the equivalent system, p=2d+l, xi, Fj denote the amplitudes of the longitudinal displacement and loading for the j-th particles and w denotes the frequency of the external excitations. For the equivalent system, the loads must satisfy the antisymmetric condition, i.e.,

where N=np and Fj (j=1,2,. ..,np) are real loads acting on the original system . Applying the U transformation (2.2.3) to Eq. (2.2.40) we have

60

Exact Analysis of Bi-periodic Structures

with i y = n / N and i = f i . Introducing Eq. (2.2.41) into Eq. (2.2.44a) yields

That leads to

f; G O The solution for q m of Eq. (2.2.42) can be formally expressed as

Substituting Eqs. (2.2.43), (2.2.45) and j U-transformation (2.2.3a), we have

where

=d

+ 1 + (s - l)p

(s=1,2,.. .,2n) into the

Bi-periodic Mass-Spring Systems

61

1 e"s-u)pmv/(2k- M C B-~2k cos rn ~y)] a,. =-C[ 2N 2N

,=I

and

Eq. (2.2.46) is similar to Eq. (2.2.7). Applying the U-transformation (2.2.11) to Eq. (2.2.46) results in

where y, = n/n and (2.2.5 la)

Xj must satisfy the antisymmetric condition, i.e., X;,+,-, = X: (s=1,2,. ..,n). As a result

that leads to

-

bn = O , b2,-, =b,

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

Introducing Eqs. (2.2.51~)and (2.2.48) into Eq. (2.2.50) gives

(2.2.5 1c)

Exact Analysis of Bi-periodic Structures

62

Q,

E

0,

-

Q2,-,

= Q,

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

where

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

2.2.2a Natural Vibration Example Let Fj = 0 (j=1,2,. ..,2N). That leads to x; = 0 (j=1,2,...,2N) and b, = 0 (r=1,2,. ..,2n). If X , (s=1,2,...2n) are not identically equal to zero, the frequency equation can be expressed as a, ( w ) = 0 , namely

If X, (s=1,2,. ..,2n) are identically equal to zero the frequency equation can be obtained from Eq. (2.2.42) with f : = fi = 0 as 2k - ~

a

- 2k) cos~ m y

=0

m=n,3n,5n,. ..,@-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

Bi-periodic Mass-Spring Systems

63

That leads to

The system considered is shown in Fig. 2.2.4 .

pe iot

pe iot

Figure 2.2.4 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 Mw2

1--z(2k 3 "=I

- MU'

-2kcos[r +4(u -1)lE)-' = 0 6

MU When r = 1 , the roots for Ro(= -) k shown in Eq. (2.2.23b), i.e.,

Ro = 0.198062264,

r = 1,4

(2.2.58)

of Eq. (2.2.58) are the same as those

1.55495813, 3.24697960

When r = 4 , the root for R, is

Inserting Eq. (2.2.57) into Eq. (2.2.56) gives

(2.2.59a)

64

Exact Analysis of Bi-periodic Structures

That leads to

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

Because of the antisymmetry of displacement for the equivalent system, namely X,,+,-, = X, (s =1,2,..., n),Eq.(2.2.11b)canberewrittenas

In order to make the mode antisymmetric. Ql must have a complex constant factor I.Z -

e

for the present case. Without loss of generality, let .?I

Q, = & ' a = l + i ,

e3= I - i ,

Q, =Q, = O

(2.2.63)

Substituting Eqs. (2.2.63) and (2.2.57) into Eq. (2.2.1 la) results in XI = X , = 1 ,

X, = X 3 =-1

Introducing Eqs. (2.2.57) and (2.2.63) into Eq. (2.2.54) gives

(2.2.64)

Bi-periodic Mass-Spring Systems

f:=O

miseven

65

(2.2.65~)

Inserting Eqs. (2.2.65), (2.2.45), (2.2.57) and f; = 0 into Eq. (2.2.3a) we have

which includes three modes corresponding to three values of 52, shown in Eq. (2.2.59a). The numerical results are given in Table 2.2.2. Table 2.2.2 Natural frequencies noand modes [ x, x, x, x, x , x, ] for the system shown in Fig. 2.2.4

no

0

0.198062264

1

1.55495813

2

3.24697960

mode

Corresponding to Ro=2 (see Eq. (2.2.59b)), the natural mode can be expressed as

66

Exact Analysis of Bi-periodic Structures

In view of Eq. (2.2.62), Q4 does not include any complex factor. Substituting Eqs. (2.2.67a) and (2.2.57) into Eq. (2.2.11a) and letting X, = 1 , results in

Substituting Eqs. (2.2.67) and (2.2.57) into Eq. (2.2.54) gives

Finally substituting Eqs. (2.2.69), (2.2.57) and (2.2.45) with Eq. (2.2.3a) results in

q: = 0

into

with Q, = 2 . That leads to

Because the system considered is not subjected to any constraint, there is a mode of rigid body motion which corresponding to zero frequency. Introducing o = 0 and F j ( j= 1,2,...,2N) = 0 into Eq. (2.2.42), the nontrivial solution can be expressed as

Bi-periodic Mass-Spring Systems

67

Substituting Eqs. (2.2.72) and (2.2.57b) into Eq. (2.2.3a) and letting x, = 1 (normalization) yields

fi

corresponding to q,, = . The rigid body mode can also be obtained from Eq. (2.2.70) with a, vanishing. Moreover, let us consider the mode with x, and x, vanishing. Corresponding to the natural frequency no= 1 (see Eq. (2.2.60b)), the nontrivial solution for generalized displacement can be expressed as

In order to make the mode antisymrnrtic for equivalent system, i.e., 1

x N + = x (j= 1 2 , .N) , q ,

must have a complex factor elTv . Letting

.Z

q, = 2e1%nd substituting Eqs. (2.2.74) and (2.2.57b) into Eq. (2.2.3a) we have

The results for all natural frequencies and modes are summarized in Table 2.2.2.

2.2.2b Forced Vibration Example Consider the system having the same parameters shown in Eq. (2.2.57) and subjected to two harmonic forces at the second and fifth mass points as shown in Fig. 2.2.4, where P and w denote the amplitude and frequency of the external excitation. The loading condition can be expressed as

Introducing Eqs. (2.2.76) and (2.2.57) into Eq. (2.2.44b) yields

68

Exact Analysis of Bi-periodic Structures

with the other f: vanishing. Substituting Eqs. (2.2.77), (2.2.45~)and (2.2.57b) into Eq. (2.2.49) results in

P = X; =-

(Qo-1) k Qo(3-Qo)

= X;

=x;

= X = ~

--P

(2.2.78a)

1

(2.2.78b)

k Qo(3-Qo)

~ l = XJ ~ - j ~= 172?...,6

(2.2.78~)

That leads to

x,' = x2 l--xr -x: 3 -

p =-

(Qo -1) k Qo(3-Qo)

(2.2.79)

Introducing Eqs. (2.2.79) and (2.2.57) into Eq. (2.2.5 1) results in

P 2(R, -1)

b4 = -

b,

k Qo(3-Qo)

=0

r = 1,2,3

Inserting Eqs. (2.2.80), (2.2.53) and (2.2.57) into Eq. (2.2.52) yields

Q4

p (Qo-1) Q, = o k ~~(2-a,)'

=-

r = 1,2,3

(2.2.81)

Substituting Eqs. (2.2.81) and (2.2.57) into Eq. (2.2.1la), we have

x

"

That leads to

(Qo-1) k 2Q0(2- a o )

=-p

s = 1,2,3,4

(2.2.82a)

Bi-periodic Mass-Spring Systems

69

This function represents the frequency response for displacement of the loaded mass point. Introducing Eqs. (2.2.81) and (2.2.57) into Eq. (2.2.54) gives f,"=-

p (Qo - 1)

J7(2-00)

-i:x

e

Y

p (00-1) f,"=j,", f l02 --- & (2 - 0 0 )

(2.2.83a)

Finally, substituting Eqs. (2.2.83), (2.2.77), (2.2.45) and (2.2.57) into Eq. (2.2.3a) we have

2.2.3

Bi-periodic Mass-Spring Systems with One End Fixed and the Other Free

Consider a bi-periodic mass spring system with fixed and free extreme ends and (2n+l)d+n particles as shown in Fig. 2.2.5(a) where n denotes the number of the particles with mass M +AM and 2d denotes the number of particles with mass M between two adjacent particles having mass M + AM . The symbols Fj and x j denote the amplitudes of the longitudinal load and displacement for the j-th particle and k denotes the stiffness of the coupling spring. In order to apply the U-transformation approach, we have to form a cyclic biperiodic system that is equivalent to the original one. It can be achieved by the following procedures. Firstly, the system is extended at the free end by its symmetrical image and an antisymmetric loading is applied on the corresponding extended part as shown in Fig. 2.2.5(b). Secondly, the new system can be treated as a bi-periodic mass spring system with fixed extreme ends as stated in section 2.2.1. The equivalent system with cyclic bi-periodicity and (4n + 2)p ( p = 2d + 1)

(a)

Original system

syaunetrical

p=2d+l N=(2n+ l )p

(b) Transitional system

M t

M+AM

pj

1

I

Fxtz-j=Fj

symmetrical line 2

XN+Z-j=X j

i

F

I

O - o . . . ~ . - . o ~ . - ~ . O O ~ - ~ . - u ~ O . . . C ) ? r r C ) . . ~ ~ I 21t2d l+p 1t2p j l+np i lyn+l)~N+Z-j 1+2n~ N+j

First particle =(1+2N)-th particle,

x, r x

,+,

(c) Equivalent system with cyclic bi-periodicity Figure 2.2.5 Bi-periodic mass spring system with fixed and free ends

72

Exact Analysis of Bi-periodic Structures

degrees of freedom is shown in Fig. 2.2.5(c), where the first and last particles are imaginarily regarded as the same particle. The loads and displacements of the equivalent system must satisfy the following relations:

and

where N = (2n + l ) p , p = 2d + 1 and n, d are the parameters of the original system. The harmonic vibration equation for the equivalent system with 2N particles can be expressed as

where w denotes the vibration frequency and 1 + ( m - l ) p ( m = 1,2,...,4 n + 2) is the ordinal number of the particle having mass M +AM . Applying the U transformation (2.2.3) to Eq. (2.2.87) results in

where

77

+ IY

=-

N

and

Bi-periodic Mass-Spring Systems

73

Substituting Eq. (2.2.85) into Eq. (2.2.89b) yields n -m F

f; =- - 4i i

Jzni

m is odd

j=2

fi = 0

m is even

Inserting Eq. (2.2.86) into Eq. (2.2.3b) gives

qN

= q Z N = 0,

qm= O

m is even

that indicates q m must be an imaginary number, From Eq. (2.2.88), q m can be expressed as

= f;/(2k

- M U *- 2k cos m y/)

Substituting Eqs. (2.2.89), (2.2.92) and j (2.2.3a) we have

=1

+ (s - 1)p

(2.2.92~)

into the U-transformation

Exact Analysis of Bi-periodic Structures

74

where

xl.(j = 1,2,...,2 N ) denotes the solution for displacement of the mono-periodic system (i.e., AM = 0 ) under the same loading as that acting on the considered system, namely XI

'

=

-c JWV 1

2N

ei"-"mY 4,'

=,,

Let

where p = ~ / ( 2 n + l ) = ~ yand / i=&. We can now apply the U-transformation (2.2.97) to Eq. (2.2.93). The uncoupling equation can be obtained as

where

Bi-periodic Mass-Spring Systems

75

Because X: ( s = 1,2,...,4n + 2 ) possess the property shown in Eq. (2.2.86), namely

XZ+,-, = Xi,+,+, = -X;,,-,

= -XJ

j = 2,3,..., n + 1

(2.2.100a) (2.2.100b)

X ( = Xi,,, = 0 Eq. (2.2.99b) becomes

b, =- - 4i

2

J4n+2 ,=2

sin(s - 1)rp X:

bzn+,= b4n+2= 0 , b, = 0

r is odd

r is even

That leads to

Q2n+l

= Q4n+2 = 0 ,

Q, = 0

Q4n+2-r = Qr

r is even

Making a comparison between Eqs. (2.2.89a) and (2.2.97b) yields

The final results for x j 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 X, non-vanishing

76

Exact Analysis of Bi-periodic Structures

and vanishing respectively , namely

and

m = 2n + 1,3(2n+ 1),5(2n+ 1), ...(p- 2)(2n + 1)

(2.2.105)

with p = 2d + 1 and y/ = 7r/(2n+ l ) 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.

Figure 2.2.6 Bi-periodic mass spring system with fixed and free ends d = n = 1

The parameters are given as d=l,

n=l,

AM=M

That leads to

Introducing Eq. (2.2.106)into the frequency equation (2.2.104)yields

(2.2.106a)

Bi-periodic Mass-Spring Systems

77

Applying the relations

to Eq. (2.2.107),the frequency equation becomes - 2Ri

+ 1oR;- 12Q0 + 1 = 0

MU '

with R, = -. The roots for R, of Eq. (2.2.109)are

k

0,= 0.08995531, 1.77031853, 3.13972616

(2.2.110)

Inserting Eq. (2.2.106)into the later frequency equation (2.2.105)gives

That leads to

Consider now the natural modes. Corresponding to the frequency equation

(2.2.109), the natural mode, in terms of the generalized displacement Q, ( F1,2,. ..,6), can be expressed as

78

Exact Analysis of Bi-periodic Structures

with the other Q, vanishing. From Eqs. (2.2.101a) and (2.2.102a), Q, must be an imaginary number. Introducing Eqs. (2.2.106) and (2.2.112a) into Eq. (2.2.97a) and letting x, = X 2 = 1 ,yields

Inserting Eqs. (2.2.106) and (2.2.112b) into Eq. (2.2.103) results in

&

f o1 = f o7 = f o1 3 -- - - M &

3

with the other f: vanishing. Substituting Eqs. (2.2.106), (2.2.114) and (2.2.92) with Eq. (2.2.3a), we have

That leads to

(2.2.114a)

qk = 0

into

Bi-periodic Mass-Spring Systems

79

Because the natural frequency R, is a root of Eq. (2.2.109), Eq. (2.2.117a) may be rewritten as

Inserting Eq. (2.2.117b) into Eq. (2.2.116) the natural modes can be expressed as

where Ro may be an arbitrary root of Eq. (2.2.109). Therefore Eq. (2.2.118) represents three natural modes. Their numerical results are given in Table 2.2.3 . Table 2.2.3 Natural frequency R, and mode [ x , x , ... x , ] for the system shown in Fig. 2.2.6 n o

Mode

0.0899553 1

1.77031853

3.13972616

1

80

Exact Analysis of Bi-periodic Structures

Corresponding to the frequency equation (2.2.11 la), the mode can be expressed, in terms of q, ,as

In view of Eq. (2.2.91) q, must be an imaginary number. Substituting Eqs. (2.2.106) and (2.2.119) into Eq. (2.2.3a), the natural mode can be obtained as

3fi i and SZ, = 1 . The result is also given in Table 2.2.3. corresponding to q, = - 2 Obviously the particle with mass 2M lies at the nodal point of the mode, i.e., x, = 0.

In conclusion of this chapter we must show clearly that the key to the settlement of the question lies in forming a cyclic bi-periodic system which is equivalent to the considered one. The necessary conditions are as follows: Firstly the system extended by the symmetric image must possess the cyclic biperiodicity and secondly the boundary condition of the original system must be satisfied automatically in its extended system by means of applying the symmetric or antisymmetric loading on the corresponding extended part.

Chapter 3 BI-PERIODIC STRUCTURES 3.1

Continuous Trusses with Equidistant Supports

The plane truss to be considered is the Warren truss. The transverse vibration of the Warren truss with two simply supported ends was investigated by using the Utransformation technique [7], where the truss is regarded as a mono-periodic structure. Recently the static and dynamic analyses of the continuous Warren truss with equidistant roller supports was performed by Cai et a1 [11,12] where the truss is treated as a bi-periodic structure and the U-transformation is also used.

3.1.1 Governing Equation

Figure 3.1.1 Warren truss with equidistant roller supports Consider the continuous truss resting on equidistant roller supports as shown in Fig. 3.1.1. The truss is subjected to transverse loads acting at the nodes. The truss is made up of four sets of bars pin-jointed at the nodes so that only axial forces but no bending moments and shear forces act on the cross-sections of the bars. The bars in the longitudinal direction have modulus of elasticity E, , cross-sectional area A,

82

Exact Analysis of Bi-periodic Structures

and length L, . The inclined bars have modulus of elasticity E,, cross-sectional area A, and length L, . In Fig. 3.1.1 N and n denote the total numbers of substructures and spans, respectively and p denotes the number of substructures between two adjacent supports. A typical substructure is shown in Fig. 3.1.2(a). Each substructure consists of four nodes and four bars. In order to avoid repetition, we consider only the nodal loads of two nodes on the left of every substructure. The serial number of both the node and bar is made up of two integer numbers in which the first one is the ordinal number of the node or bar in the substructure and the second one indicates the ordinal number of the substructure. In order to avoid ambiguity the serial numbers of the nodes are given in round brackets.

Bar 4,

(a) Serial numbers of nodes and bars

@)

Displacements and internal forces

Figure 3.1.2 Substructure At the outset, we must create a cyclic bi-periodic system which is equivalent to the original one. The considered truss is extended by its symmetrical image and apply the antisymmetric loading on the corresponding extended part as shown in Fig. 3.1.3 where two bars denoted by dotted lines are additional ones. Such an extended system may be regarded as a cyclic bi-periodic, when two pairs extreme nodes a , a' and b , b' are imaginarily jointed by hinges respectively, i.e., the first substructure is next to the last (2Nth) one. Two additional bars are not subjected to any load for antisymmetric deformation. Consequently the antisymmetrical deformation for the extended truss is not affected by such additional bars which are necessary in order to form the cyclic periodic system. If and only if the displacements

Symmetric line

Figure 3.1.3 Equivalent system with cyclic bi-periodicity subjected to antisymmetric loads

84

Exact Analysis of Bi-periodic Structures

of the extended truss possess antisymmetry, the extended truss with cyclic periodicity is equivalent to the original one. When the supports are replaced by the supporting reactions, the equivalent truss may be regarded as a cyclic mono-periodic structure which can be analyzed by using the U-transformation technique [7]. The total potential energy of the equivalent truss considered may be expressed as

where J3 denotes the potential energy of the j-th substructure and 2N denotes the total number of substructure for the equivalent structure. In general, the potential energy may be defined as

,

where [K],,, denotes the stiffness matrix of the substructure; { S ) ,, { F ) denote the displacement and loading vectors for the j-th substructure, respectively and superior bar indicates complex conjugation. It is necessary to have the superior bar in Eq. (3.1.2) at deriving the variational equation with complex variables. For the present case, the displacement vector (61, may be defined as

and

with its components shown in Fig. 3.1.2(b) where u and v denote the longitudinal and transverse displacement components, respectively. By using the conventional

assembly process the stiffness matrix for every substructure can be obtained from the stiffness matrices of four bar elements as

where

I

K I + ~ , c o s 2 a K2sinacosa K2sinacosa K , sin2a [Kill = -K2 c o s 2 a -K2sinacosa -K2sinacosa -K2sin2a

- ~ , c o s ~ a -K2sinacosa - K , sina cosa - K2sin2a K, + 2 K 2 c o s 2 a 0 0 2K2 sin2a

1

in which K, = E,A, /L, , K2 = E2A, /L2 and a denotes the inclination as shown inFig. 3.1.1. The loading vector should include both the external load and supporting reaction. Noting the longitudinal load vanishing, the loading vector takes the form as

86

Exact Analysis of Bi-periodic Structures

where p indicates the number of substructures between two adjacent supports; Pk indicates the supporting reaction at k-th support and F(l,j),F(,,,)denote the external loads acting at the nodes (1, j ) and (2, j ) respectively, which must satisfy the antisymmetric conditions, i.e.,

qaj)( j = 1,2; -,N ) are real loads acting on the original truss. The continuity condition between two adjacent substructures may be expressed as

in which F(,,,,( j = 2,3, - ..,N ) and

where (6,),,+, = {S,), due to cyclic periodicity. Eq. (3.1.8) shows the coupling of the energy of substructures. Obviously Eq. (3.1.8) possesses cyclic periodicity. One can now apply the U-transformation to Eqs. (3.1. I), (3.1.2) and (3.1.8). The U- and inverse U-transformations may be expressed as

and

Bi-periodic Structures

. The generalized displacement vector

with y = z / N and i = defined as

87

{q),,, may be

and

By means of the generalized displacement, the potential energy shown in Eqs. (3.1.1) and (3.1.2) can be written as

where

and the continuity condition shown in Eq. (3.1.8) becomes {q ) ~m

=e'"'W

{q~)m,

m = 1,2...,2N

which may be rewritten as

with

in which I is the unit matrix of fourth order. In order to eliminate the non-independent variables {qR},(rn = 1,2-..,2N ) in

Emct Analysis of Bi-periodic Structures

88

Eq. (3.1.1 I), substituting Eq. (3.1.14) into Eq. (3.1.11) yields

where

In Eq. (3.1.16), the real and imaginary parts of {q,), are independent variables. =I 0 , the Substituting Eq. (3.1.16) into the first order of variation equation & equilibrium equation can be obtained as

By inserting Eqs. (3.1.4), (3.1.5) and (3.1.15) into Eq. (3.1.17), results in

- K , c o ~ ~ a ( l + e - " ~ ) -K2sinacosa(l-e-'mv) - K2 sinacosa(1- e-imv) - K2sin2'a(l+ e-"v) 2Kl (1 - cos m y ) + 2K2 cos2a 0 0 2K, sin2a Eq. (3.1.19) is equivalent to the nodal equilibrium equation. Introducing Eq. (3.1.6) into Eq. (3.1.12) yields

in which &,,,,,

&,,,,

1

(3.1.20)

and f&, denote the generalized loads corresponding to the

Bi-periodic Structures

89

external loading and supporting reaction respectively, namely

Introducing the anti-symmetric condition for nodal loads shown in Eq. (3.1.7) into Eqs. (3.1.21b) and (3.1.21c) results in

Substituting Eqs. (3.1.15) and (3.1.21a) into Eq. (3.1.18) yields

Recalling the definition of {q, J, Eq. (3.1.19) for q (,,,, and q(,,,, gives

(1 - e-imY)[P(l - cosm p) + cos2 alsin a c o s a

-isinm ~ y s i n a c o a s~ 9(1,m)

=

Dm

q(2.m)

shown in Eq. (3.1.lob) and solving

+

Dm

q(4.m)

90

Exact Analysis of Bi-periodic Structures

q(3,m)=

(1 - eimv)[P(l- cos my) + cos2a ] sin a cos a Dm

q(2,m)

+ i sin my sin a cos3a q(4,m) Dm

where Dm= (1 - cosm y ) [ ~ / 3 ~ ( 1cosm y ) + 4flcos2 a + cos4a]

(3.1.24~)

Substituting Eqs. (3.1.24a) and (3.1.24b) into the second and fourth component equations in Eq. (3.1.19) results in

where

-

K I Z ,=~K Z I ,=~3.1.2

2Kl sin2a ( l + e-imv)[2cos2a + P(1- cos m ry)] (3.1.26b) 2~2(1-cosm~)+4~cos2a+cos4a

Static Solution [11]

Consider the governing equation (3.1.25). The solution for q(,,,, and q(,,m, of Eq. (3.1.25) may be expressed as

and

Bi-periodic Structures

91

(3.1.28) Substituting Eq. (3.1.27) into the U-transformation (3.1.9a) yields

where f(;,,,is dependent on the unknown supporting reactions which can be determined by the compatibility condition at supports, i.e.,

Substituting Eqs. (3.1.21d), (3.1.29a) with j = (s - l)p + 1 into the above equation, the restraint condition can be expressed as

where

Here V, denotes the transverse displacement at the s-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 = 1,2,...,2n ). The coefficients P,,, ( s, k = 1,2, - ,2n ) of Eq. (3.1.31) possess cyclic periodicity, i.e.,

Exact Analysis of Bi-periodic Structures

92

The independent coefficients are Pk,, ( k = 1,2;..,2n ). 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

with q

=z/n =p y

. 2n

Premultiplying Eq. (3.1.31) by the operator (I/&)

e-"'l"*

results in

s=l

where

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 Q, of Eq. (3.1.36) is

Bi-periodic Structures

93

where

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

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

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 f&, and

Q, shown in Eqs. (3.1.21d) and (3.1.35b) respectively, there is a simple relation, i.e.,

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.

Esnct Analysis of Bi-periodic Srrlrcftrres

94

3.1.2a

Example

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

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 K, = K, = K ,

N = 6 , n=3, p=2,

a =n/3

(3.1.43a)

which lead to w=z/6,

p=lr/3,

p=l

(3.1.43b)

The nodal loads can be expressed as =F

51.4)

62.j)

9

=O *

6l.j)= O J

7

j

= 1,2;..,6

#

4

(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

f(l,rn)

1 - --

6

rnz sin-. F, 2

95

rn = 1,2,...,12

The stiffness coefficients of Eq. (3.1.25) can be found by substituting Eq. (3.1.43) into Eq. (3.1.26) as

3K rnz K12,, = K ~ I , = , -(3 - 2 cos -)(I 40, 6 49 Dm =-16

+ e-imY)

rnz 2 cos 6

Inserting Eqs. (3.1.45) and (3.1.43) into Eq.(3.1.33) gives

where K,,,, and A, 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

Introducing Eqs. (3.1.48) and (3.1.43) into Eq. (3.1.37) yields

96

Exact Analysis of Bi-periodic Structures

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. (3.1.4I), as

Inserting Eqs. (3.1SO) and (3.1.43) into the U-transformation (3.1.35a) results in

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., 4 = 0 and P,, = 0 . The real supporting reactions at two extreme ends of the original truss can be found easily by solving the equilibrium equation for 2 the whole truss, i.e., P, = P4 = -F . 35 Introducing Eq. (3.1.50) and p = 2 into Eq. (3.1.42) gives

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

Bi-periodic Structures

97

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.,

The results are summarized as follows

98

Exact Analysis of Bi-periodic Structures

Eqs. (3.1.53b), (3.1.53d), (3.1.55b) and (3.1.55d) indicate that the nodal displacements possess antisymmetry. The axial force on the cross section of every bar can be found by using the Hook's law that may be specifically expressed as

N 3 j = K , ( U ( ~ ,cOSa ~ + ~ -) v ( , , ~ + sina , ) - u ( , , ~cosa ) + v ( ~ sina) ,~) N 4 j = K , ( U ( , , ~C)O

+

S ~v

( ~sina , ~ )- u ( , , ) cosa - v(,,,,sina)

(3.1.56~) (3.1.56d)

,

with j = 1,2, ,2N , where N,, on the left side of Eq. (3.1.56) denotes the axial force and its two subscripts denote the serial numbers of the bar and substructure respectively, as shown in Fig. 3.1.2. Introducing Eqs. (3.1.53), (3.1.55) and (3.1.43a) into Eq. (3.1.56) results in 2J;; N,, =-F, 105

N,, =-- l g A F , 210

N,, =--F,4 4 7 105

9J;; NZ3= -F, 35

447 N,, =-F 105

N3, = --FJ;;

3

, N,, =--F4 J 5

(3.1.57a)

105

J5

, N

= -F 43

3

(3.1.57~)

Bi-periodic Structures

99

in which N,, = 0 and N,,,, = 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 natu~alvibration 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 I and M 2 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 w 2 ~ , v ( , , and j ) w~M,v(,,~) instead of F(,,,) and F(,,,) ,namely

and substituting Eq. (3.1.59) into Eq. (3.1.25), the natural vibration equation can be expressed as

100

Exact Analysis of Bi-periodic Structures

and f(:,,, have been defined as shown in where K ,,,, , K,,, , K12,,, K,,,, Eqs. (3.1.26) and (3.1.21d) respectively; w denotes the natural frequency; q(,,,, , q(,,,, represent the amplitudes of the transverse generalized displacements. Noting Eq. (3.1.2 1d) and the antisymmetry of P, ,the generalized force has the property

That leads to

The solution for q(,,,, and q(,,,, of Eq. (3.1.60) can be expressed as

rn = 1,2,...,2N and rn

#

N,2N

(3.1.63)

where Am(@)= (K11.m-w2M1)(K22,m - w ~ M ~ ) - K I ~ , ~ K ~ (3.1.64) I,~ For the original truss with simply supported ends and without the internal supports (as a result f(:,,, = 0 ), the frequency equation can be expressed as

If A, ( 0 ) z 0 , substituting Eq. (3.1.63) into the U-transformation (3.1.9a), i.e.,

Bi-periodic Structures

101

we have

where A;,,, is a function of P, which can be determined by the constraint condition at supports shown in Eq. (3.1.30). Substituting Eqs. (3.1.67a) and (3.1.21d) into Eq. (3.1.30) ,we have

where

Eq. (3.1.68) is the linear simultaneous equations with unknown Pk (k = 1,2,...,2n) . The coefficients pSxkpossesses cyclic periodicity shown in Eq. (3.1.34). As a result, Eq. (3.1.68) can be uncoupled by means of the Utransformation. One can now apply the U-transformation (3.1.35) to Eq. (3.1.68). The uncoupling equation can be expressed as

where y, = ls/n and

102

Exact Analysis ofBi-periodic Structures

In view of Eq. (3.1.35b) and antisymmetry of Pk,it can be shown that

Recalling Eq. (3.1.40) and introducing Eq. (3.1.70b) into Eq. (3.1.70a) we have

and

The frequency equation can be expressed as

corresponding to P, non-vanishing. When w is a root of Eq. (3.1.73) the nontrivial solution for Qr of Eq. (3.1.72a) exists. If all supported nodes are located at the nodal points of the natural mode, the supporting reactions are identically equal to zero. The corresponding frequency equation can be obtained from Eq. (3.1.65) with rn = n,2n, ...,pn ,namely

where rn is in agreement with the half wave number of the natural mode for the original system. 3.1.3a Example

Consider a continuos Warren truss having 2n substructures and n spans. The

Bi-periodic Structures

103

structural parameters are

and

where n is an arbitrary positive integer. Introducing Eqs. (3.1.79, (3.1.26) and (3.1.64) into the frequency equation (3.1.73), it yields

where

K2 ( m ) =

288(1+ cos m y ) ( 3 - 2 cos m y ) (49-32c0smy)~

and Q denotes the non-dimensional frequency parameter. Eq. (3.1.76) depends on the positive integer n, i.e., the total number of spans of the continuous truss. For several values of n, the natural frequencies are calculated from Eq. (3.1.76)by using numerical method. The results are shown in Table 3.1.1. There are 3(n-1) natural frequencies altogether. One can now consider the other frequency equation corresponding to the case of Q, ( r = 42;-.,2n ) vanishing. Substituting Eqs. (3.1.75)and (3.1.26)into Eq. (3.1.74),one can show that

104

Exact Analysis of Bi-periodic Structures

Table 3.1.1 Natural frequency w of the continuous truss with p = 2 and n spans

JG

Multiplier: a k denotes the ordinal number of the pass bands [ W L ,WU )represents W L < W < WU .

Bi-periodic Structures

105

The analytical solution for Q in Eq. (3.1.78) is

Introducing Eq. (3.1.77) into Eq. (3.1.79a), it yields

The corresponding frequencies are

These natural frequencies are independent of n and in agreement with those for the sectional truss between two adjacent supports, i.e., the case n = 1. The numerical results are also shown in Table 3.1.1. They represent the lower limits of the three pass bands. The total number of the natural frequencies is equal to 3n, which is in agreement with the number of the degrees of freedom for the truss considered. The results in Table 3.1.1 show that all natural frequencies lie in the three pass bands. If n approaches inflnity, each pass band is full of natural frequencies. The upper and lower limits of the pass bands can be obtained by solving Eq. (3.1.76) with r = 0, n . The explicit solution for Q can be expressed as Q = L ( 7 1 - a ) , 51 and

It can be shown readily that

4

3

1 -(71+m) 51

for r = O

(3.1.80a)

106

Exact Analysis of Bi-periodic Structures

where w, and w,, ( k = 1,2,3) denote the lower and upper limits of the k-th pass band, respectively. In Table 3.1.1, the numerical results show that the natural frequencies of the continuous truss with arbitrary number of spans do not include any upper limit of the three pass bands, but they can approach every upper bound of the pass bands as a limit when n approaches infinity. When n is a large number, the natural frequencies are densely distributed in each pass band. It is not easy to find the dense natural frequencies by numerical methods. In the present approach, the frequency equation of the continuous truss with n spans is uncoupled to form n frequency equations. The natural frequencies obtained from each one are dispersed, namely, they lie in different pass bands. Consequently, accurate natural frequencies can be easily found no matter how large n is. Consider now the natural modes. If R, denotes a root for R of the r-th equation in Eq. (3.1.76) for a given r, the non-trivial solution for R m ( m = 1,2,. . ,2n ) can be expressed as

with the other Qm vanishing where A indicates an arbitrary real constant with the unit of force. Because the supporting reactions have antisymmetry for the extended truss, i.e., P2n+2-r =-Ps ( ~ = 2 , 3 , . . . , n )and P, = P,,, = 0 , according to Eq. (3.1.35b), Q, must be an imaginary number similar to in Eq. (3.1.22a). Substituting Eq. (3.1.8 1) into Eq. (3.1.42) with p = 2 , one can show that

A,,,)

Bi-periodic Structures

with the other

f(:,,,

107

vanishing.

Substituting Eqs. (3.1.82), (3.1.75), (3.1.26) and (3.1.64) into Eq. (3.1.67) gives

(3.1.83b) where K, (m) and K , (m) have been defined in Eqs. (3.1.77a) and (3.1.77b) and 24(3 - 2 cos m y ) K3(m)= 49-32cosmy with y/ = 7r/2n. It can be proved that when r is odd, the mode shown in Eq. (3.1.83) is symmetric, i.e., v ( , , , ~ + , - j ) - v(,,,) and v ( ~ , ~ , + , - - v ( , , ) ; when r is even, the mode is

,,

-

antisymmetric, i.e., v ( , , ~ , , + ~ - j ) satisfies the constraint condition

and

'(2,2n+l-j)

= -'(2,j)

. Also, the mode

3.1.4 Forced Vibration [l2]

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

108

Exact Analysis ofBi-periodic Structures

amplitudes; w denotes the excitating frequency and K,,,, , K 2 , K 2 , K22,m are the same as those shown in Eq. (3.1.26). Recalling Eq. (3.1.22), the generalized force has the property

In view of Eqs. (3.1.87a) and (3.1.61), the solution for q(,,,,, and q(,,,,, in Eq. (3.1.86) with m = 2N will vanish; namely

For the case of m = 2N (my/ = 2 n ) , all of the substructures having the same amplitude vector and phase, there is no antisymmetric mode for the equivalent system. Introducing Eqs. (3.1.87b), (3.1.61) and K,,,, = K,,,, = 0 into Eq. (3.1.86) with m = N ,we have

For m = N ( my/ = n ), two adjacent substructures have the same amplitude vector and opposite phase, i.e., the nodes (1, m) (m = 1,2-..,2N) are located at nodal points of the corresponding mode. Consequently, the generalized displacement q(,,,, corresponding to v(,,])( j = 1,2- - - ,2N ) is equal to zero. In general, the solution for q(,,,, and q(,,,, in Eq. (3.1.86) can be expressed as

( i f A,(@) where

#

0),

m = 1,2..-,2N

Bi-periodic Structures

109

One can show that

Substituting Eq. (3.1.89) into the U-transformation (3.1.66) results in

where v ( , , ~and ) v ( , , ~denote ) the amplitudes of the transverse displacements for

; is the function of P, which can be nodes (1, j) and (2, j), respectively, and )4 determined by the constraint condition at supports, i.e., Eq. (3.1.30). Substituting Eqs. (3.1.21d) and (3.1.92a) into Eq. (3.1.30), we have

in which

Here Vs represents the transverse amplitude of the s-th supported node caused by the external harmonic excitation for the equivalent system without supports. It can

Exact Analysis of Bi-periodic Structures

110

be verified that Vs ( s = 1,2,...,2n ) possess antisymrnetry, i.e.,

One can now apply the U-transformation (3.1.35) to Eq. (3.1.93). To this end, 2n

premultiplying Eq. (3.1.93) by the operator (I/&)

e-i(s-')r'

,we have

s=l

where

Therefore, Eq. (3.1.93) is uncoupled into a set of single degree of freedom equations as shown in Eq. (3.1.96) by using the U-transformation. Introducing Eq. (3.1.95) into Eq. (3.1.97b) yields

and

b2,-,

= b,

,

b, = b2, = 0

Considering Eq. (3.1.98b), Eq. (3.1.96) can be rewritten as ar(w)Qr +br = 0 ,

r =1,2;-.,n-l

Bi-periodic Structures

111

where

Recalling Eq. (3.1.40) and substituting Eq. (3.1.97a) into Eq. (3.1.1OO), one can show readily

If a r ( o ) # O (r=1,2;-.,n-I), expressed as

the solution for Qr in Eq. (3.1.99a) can be

When the specific structure and loading parameters are given, Qr can be calculated from Eqs. (3.1.101), (3.1.102), (3.1.94b) and (3.1.98). Then the supporting reactions and the nodal displacements can be found from the related formulas given above. Recalling the definitions of both generalized supporting reactions h;,,, and Qr shown in Eqs. (3.1.21d) and (3.1.35b), respectively, a simple relation between

A:,,,

and Q, can be established

Consequently, when we are only interested in the nodal displacements, it is not necessary to find the supporting reactions Pk. It will save considerable computing effort.

3.1.4a

Example

Consider now a Warren truss having six substructures and four roller supports subjected to a harmonic loading Fei" acting at the center node as shown in Fig. 3.1.5.

Figure 3.1.5

Plane truss with six substructures and four supports subjected to a harmonic force at the center node

The structural parameters are given as

and

The amplitudes of the nodal loads can be expressed as

51,.4) =F <2,j,

9

=0,

'(1.j)

=O

j#4

j = 1,2,...,6

Introducing Eqs. (3.1.105) and (3.1.104) into Eq. (3.1.22), we have

(3.1.105a) (3.1.105b)

Bi-periodic Structures

1 13

Substituting the relevant formulas, i.e., Eqs. (3.1.104), (3.1.106), (3.1.26), (3.1.94b), (3.1.97b) and (3.1.101)-(3.1.103), into Eq. (3.1.92a) with j = 4 , the amplitude response function for the loaded node can be found in explicit form as

where 4 (Kl(9 - Q)(K1(7) H (R) = 3 (K, (1) - Q)[(K, (7) - QI2 - K2(711 + (K, (7) - W K , (1)

- QI2 - K2(01

and K, (m) , K,(m) and Q have the same meanings as those in Eq. (3.1.77). The frequency response curve, v(,,,, versus R , is plotted in Fig. 3.1.6. From Fig. 3.1.6, one can easily find that there are five resonance frequencies. When R is close to these points, H(R) will approach infinity. The equations of the resonance frequencies can be found by setting the two denominators on the right side of Eq. (3.1.107b) to be zero, that is

(K, (3) -

-

K , (3) = 0

(3.1.108b)

Recalling Eqs. (3.1.76), (3.1.78) and n=3, and inserting r=l and m=3 into Eqs. (3.1.76) and (3.1.78), respectively, we can have Eqs. (3.1.108a) and (3.1.108b). 6 The roots for R in Eq. (3.1.108b) are -(1 1T 6&) and the roots in 49

Resonance frequencies:

0=0.3079247,0.3619367, 1.338599,2.385953,2.388870.

R

Figure 3.1.6

Frequency response curve.

R

H =v

,,,,FK .

Q = 0'

M K

Bi-periodic Structures

115

Eq. (3.1.108a) are 0.3619367, 1.338599, and 2.388870. These five resonance frequencies correspond to the symmetric modes of the considered truss. However, there are nine natural frequencies for the truss with n=3. From Eq. (3.1.107b), it can be shown that, when H(R) approaches a finite value at one natural frequency, the harmonic force is acting at a nodal point of the corresponding mode. In the present case, the loaded node is located at the nodal point of the antisymmetric modes. F When R approaches zero, v(,,,, approaches H(0) -,where K

This result is in agreement with the exact static displacement shown in Eq. (3.1.53a). It is well known that when R approaches infinity, v(,,,, approaches zero. 3.2

Continuous Beam with Equidistant Roller and Spring Supports [lo]

The static and dynamic analyses of the continuous beam with equidistant roller supports were investigated by using the U-transformation method [6],[ 5 ] . The static problem of the continuous beam having equidistant roller and spring supports was studied [lo] where the structure is regarded as bi-periodic and U-transformation is applied twice. Consider a beam with uniform flexural rigidity EI running over N + 1 number of roller supports and n elastic supports as shown in Fig. 3.2.l(a), where K denotes the stiffness of the elastic supports and 1 denotes the span length between any two adjacent roller supports. The distance between any two adjacent elastic supports is pl. It is assumed that each elastic support is located at midspan and N = pn , and a symmetric plane of the continuous beam exists, i.e., p must be an odd number. To form an equivalent system with cyclic bi-periodicity for the beam considered, it is necessary to extend the original beam by its symmetrical image and apply the antisymmetric loading on the corresponding extended part as shown in Fig. 3.2.l(b). Such an equivalent system can be regarded as a cyclic bi-periodic, because the slopes and moments at both extreme ends are the same, namely the first span can be imaginarily next to the last one (2N-th see Fig. 3.2.l(b)). The simply supported boundary conditions at both extreme ends of the original beam can be satisfied automatically in its equivalent system.

116

Exact Analysis of Bi-periodic Structures

N = np , jo =

e, p is odd number 2

(a) Original System

symmetric line I

(b) Equivalent System Figure 3.2.1 Continuous Beam with Bi-Periodic Supports

3.2.1 Governing Equation and Static Solution In each span a local coordinate system oxy with the origin o at its midspan, and x axis along the beam, is established. The equilibrium equations for the equivalent system can be expressed as

Bi-periodic Structures

1 17

where j, = ( p + 1)I 2 and j, denotes the ordinal number of the span with the first elastic support; wj(x) and Fj(x) denote deflection and loading functions for the j-th span; 6(x) denotes the Dirac delta function; 2N is the total number of spans of the equivalent system. The loading functions must satisfy the antisymmetric condition, i.e.,

in which Fj(x)( j= 1,2, ...,N) is the real loading acting on the original beam. The constraint and continuity across the roller supports require the following conditions to be satisfied:

where a prime denotes differentiation with respect to x and w,,,, cyclic periodicity. Introducing the U-transformation

with

I+V

=r

/ N into Eqs. (3.2.1) and (3.2.3) results in

s w,

due to the

118

Exact Analysis ofBi-periodic Structures

and

where

If the loading condition is given, the generalized load fA(x) can be found from Eq. (3.2.8). The formal solution for qm(x) of Eq. (3.2.5) subject to the boundary condition (3.2.6) can be expressed as

where

Bi-periodic Structures

1 19

and qk (x) represents the generalized displacement for the cyclic periodic system with K = 0 subjected to the same loading as that acting on the equivalent system. Substituting Eqs. (3.2.9) 43.2.1 1) into the U-transformation (3.2.4a) yields wj (x) = w; (x) + w; (x)

(3.2.12)

in which

and

w;(x) represents the deflection function of the j-th span for the equivalent system with K vanishing under the same loading, while wg(x) account for effect of the elastic supports. The latter depends on the deflections at the elastic supports. Inserting j = j, + (s - l)p and x = 0 into Eqs. (3.2.12) and (3.2.13a) gives

where

120

Exact Analysis of Bi-periodic Structures

It is obvious that P,,, (s,k = 42,...,2n) possesses the cyclic periodicity. The simultaneous equations (3.2.14) with unknown W ( j = 1 2 2 n ) can be uncoupled by using the U-transformation. Let

with y , = a / n = p y / . Eq. (3.2.14) can be expressed in terms of

Qr

( r = $2, ...,2n) as

in which

and

Introducing Eq. (3.2.19)into Eq. (3.2.18)results in

Bi-periodic Structures

12 1

where c , + ( ~ - , has ) ~ ~been , ~ defined in Eq. (3.2.11a), namely

The solution for Q, of Eq. (3.2.21) can be obtained as

in which b, shown in Eq. (3.2.20) depends on the loading condition. When the specific load is given, b, can be found without difficulty. 3.2.2 Example Consider a concentrated load of magnitude P acting at the midpoint of the middle span, say k, -th span [i.e., k, = (N + 1)/2 and N is odd number], as shown in Fig. 3.2.l(a) where j = k,. For the equivalent system, equal and opposite concentrated loads must be applied to the k, -th and (2N - k, + 1) -th spans with all other spans unloaded. The loading condition can be expressed as

where k, = ( N + 1)/2 and N is odd number. Inserting Eq. (3.2.24) into Eq. (3.2.8) yields

with y / = x / N .

Exact Analysis of Bi-periodic Structures

122

Noting that q k ( x ) represents the solution for q m ( x ) of Eq. (3.2.5) with f ; ( x ) vanishing subject to the boundary condition (3.2.6) and fL(x) is shown in Eq. (3.2.25), qk ( x ) can be found as

-

where cmo cm3have the same definition as those shown in Eq. (3.2.1 1). Introducing Eq. (3.2.26) into Eq. (3.2.13b) yields

'2

p ei(j-ko)m* w;.( x ) = EZN m=1,3,5

Recalling j = jo

W

= w o + s l (p 0 ,

+ ( s - 1)p

ko = ( N + 1 ) / 2 , and

3

1

3

12

j o = ( P + 1 ) / 2 , inserting

and x = 0 into Eq. (3.2.27) gives

and then substituting Eq. (3.2.28) into Eq. (3.2.20) we have

Since N, P are odd numbers, therefore, n(= N / p ) is also odd. This property has been used in deriving Eq. (3.2.29a). Afterward inserting Eqs. (3.2.29) and (3.2.22) into Eq. (3.2.23) results in

where

and

KO is a nondimensional parameter of the stiffness for the elastic support. Substituting Eq. (3.2.30) into Eq. (3.2.17a) yields

Now the deflection function for each span can be found by introducing Eqs. (3.2.27), (3.2.13a) and (3.2.34) into Eq. (3.2.12). The maximum deflection occurs at the midpoint of the loaded span. The maximum deflection can be obtained by inserting s = (n + 1)/2 in Eq. (3.2.34) as

in which H, shown in Eq. (3.2.31) depends on p and n or N. Noting N = np , Eq. (3.2.35) includes two independent parameters p and n or N besides K O .Some

124

Exact Analysis ojBi-periodic Structures

numerical results for Eq. (3.2.35) are given in Table 3.2.1 where p = 3 and total number of spans N and the nondimensional stiffness K O take several values, respectively. Consider the particular case of K O = 0 . By substituting K O = 0 and Eq. (3.2.31) into Eq. (3.2.35), the maximum deflection can be expressed as

The preceding result is in agreement with that obtained by Cheung et a1 [6]. Table 3.2.1 Maximum Deflections of Continuous Beams with Bi-periodic Supports Subjected to Concentrated Load P at Midpoint (p = 3)

Multiplier

~1 3 / 3 8 4 ~ ~

Chapter 4 STRUCTURES WITH BI-PERIODICITY IN TWO DIRECTIONS Cable and beam networks can be regarded as typical structures with periodicity in two directions. The static analyses of rectangular single and double layer grids were investigated by Chan et a1 [13][14] using the double U-transformation technique. The natural vibration and dynamic response analyses of cable and beam networks were studied by Cheung et a1 [15][16] employing the same method. A rectangular plate with uniform finite element meshes can also be regarded as a bi-directional periodic structure. Such a structure may be analyzed by means of the double U-transformation. The exact solution of the finite element equation for simply supported square plates was derived by Chan et a1 [17]. Recently the double U-transformation was successfully applied to the analysis of bi-directional bi-periodic structures. The static and vibration analyses of rectangular beam and cable networks with periodically distributed supports along two directions were performed by Chan et a1 [18][19] using the double Utransformation twice. The static analysis of rectangular grids with periodic stiffening beams is investigated in this book applying the same technique. 4.1

Cable Networks with Periodic Supports

The network considered is made up of two sets of pretensioned straight cables orthogonal to each other with fixed ends, meeting at spot-welded nodes and supported by periodically distributed posts. For generality, consider an n,xn, rectangular network with fixed ends at four edges as shown in Fig. 4.1.1 where the solid circles denote the nodes supported by posts. There are (rn, - 1) x ( m , - 1) internal supports. The equivalent network with cyclic periodicity in x- and y-directions can be produced by using image method [15][16]. At the outset, consider the extended network with 2n,x2n2 mesh shown in Fig. 4.1.2 where the loading pattern is antisymmetric with respect to two symmetric planes of the extended network. Moreover we regard the extended network as one having cyclic bi-periodicity in x- and ydirections, i.e., each pair of nodes (0, k) and (2n,, k) (k=O,1,2,...,2n, ) and (j,O ) and ( j, 2n2 ) (j=0,1,2,.. .,2n, ) may be imaginarily put together and treated as one point in mathematics. The boundary conditions of the original system can be satisfied automatically in its equivalent system where the additional supports located at boundary and symmetric lines are necessary in order to form the cyclic bi-periodic

126

Exact Analysis ofli-periodic Structures

system, but their supporting reactions are identically equal to zero.

n,=m,p,

Figure 4.1.1

n,=mzp,

n, x n, network with (m, - 1) x (m, - 1 ) supports

Sfntcfureswith Bi-periodicity in Tivo Directions

nl=m, p , . n,=m:p2

ti. O)e(j,2 n l ) (0. k)e(2n1, k)

j=0.1.2, .-,2n, k=O,1.2, ..-,Zn,

Figure 4.1.2 Equivalent network with 2n, x 2n, mesh and cyclic periodicity in x- and y-directions

127

128

Exact Analysis of Bi-periodic Structures

4.1.1

Static Solution

In order to use the periodicity, it is necessary that the supports should be replaced by the supporting reactions. The equilibrium equations for all nodes in the equivalent system can be expressed as

(j, k) = (jgl, kg2), jl=1,2 ,..., 2ml, k1=1,2,...,2m2

(4.1.1a)

where wGYk, and FGXk, denote the transverse displacement and loading of node (j, k) respectively; (jg,, kg,), j1=1,2,...,2ml, kl=l ,2,..., 2m2represent the nodal numbers for represents the supporting the nodes supported by posts (see Fig. 4.1.2); 4j,,kl, reaction at node (jg,, kg,);TI, T2denote the pretensions of the cables in the x and y directions respectively and a,b denote the spacing ofy- and x-cables. , ~Eq. ) (4.1.l) must be subjected to the restrained condition The solution for w ~ of at supports, i.e.,

In Eq. (4.1.1) the loading must be anti-symmetric and the supporting reaction can be determined by using Eq. (4.1.3). Eqs. (4.1.l) and (4.1.3) possess cyclic periodicity but the two periods are different. The cyclic periodic equation (4.1.1) subject to the cyclic periodic condition (4.1.3) is also cyclic bi-periodic. The solution for w(j,k, can be derived by applying the double U-transformation twice. Introducing the double U-transformation

Structures with Bi-periodicity in Two Directions

129

and its inverse transformation

with

in Eq. (4.1 .I), the equilibrium equation (4.1 .l) can be expressed in terms of the generalized displacement q(,, as

where

in which

h:,s, and h:,s, represent the generalized load and supporting reaction

respectively. Noting the anti-symmetric properties for the loading, the supporting reaction is also anti-symmetric about two symmetric planes of the extended network. As a result

Exact Analysis of Bi-periodic Structures

130

The solution for q(,, of Eq. (4.1.6) may be formally expressed as q(rj)

= qPr,,) + q ( r , s ) 7

~ 1 ,,..., 2 2nl; s=1,2,...,2n,

where 0

-

f(9.S)

q'r's' - 2Kl(l-cosrvl)+2K2(1-cosstyl)

7(' S) 2 (2n17 2n2)

q;r,s)

=0 7

(r7 s) = (2% 2n2 )

4;r,s)

=O 7

(7.7S) = (2n~,2n2 )

and

(4.1.10b)

Substituting Eqs. (4.1.9), (4.1.10) and (4.1.7) into Eq. (4.1.4a) results in w(i,k)

and

.

= ~ ; j , k )+ w(i.k)

(4.1.11)

Structures with Bi-periodicity in Two Directions

13 1

w:,,~, and wij,,, represent the displacements caused by the supporting reaction and loading for the equivalent network without supports. Noting that the loading pattern needs to satisfy the anti-symmetric condition, i.e.,

the Eq. (4.1.12a) becomes

~~~~~ nl-1 n,-1 nl-1

W"'k'

=

n2-1

.

sm jryr, sinksly, sin j l r y l sinklsyr2 2K1(1- cosrv,) + 2K2(1- cossyr,) F(~l'k"

(4.1.15)

Obviously, w;,, shown in Eq. (4.1.15) satisfies the anti-symmetric condition and boundary condition of the original network, i.e.,

and

Introducing the notations

and substituting Eqs. (4.1.1 I), (4.1.12) into Eq. (4.1.3) yields

132

Exact Analysis of Bi-periodic Structures

where Zm, 2m2

W(?l) =

P ( j . k ) ( j 1.*I)

P(11.h) .

and

with pl=.rrlml, q 2 = . r r / m 2 . Eq. (4.1.19) is the simultaneous equations with unknown Po,,. In order to determine the unknown supporting reaction Po.,,),Eqs. (4.1.19) need to be solved. ( j , j l = 1,2,...,2m1, k, kl = 1,2,...,2m2 ) of Note that the coefficients P(j,k)(j,,k,) Eq. (4.1.19) have the cyclic periodicity, i.e.,

In Eq. (4.1.19), the independent coefficien are &,,(,,,, (j=1,2 ,...,2m,, k=1,2,...,2m,). Eq. (4.1.22) indicates that Eq. (4.1.19) possesses cyclic periodicity and it can be uncoupled by using the double U-transformation. Let

Structures with Bi-periodicity in Two Directions

Premultiplying Eq. (4.1.19) by the operator

1

J2m,4 %

and noting Eq. (4.1.22), we have

where

and

P ( , , , , ( ,,,,

can be obtained from Eq. (4.1.21) as

Substituting the above equation into Eq. (4.1.24) results in

where

133

~%e-i(j-o,e-i~k-vs, j=l

t=l

134

Exact Analysis of Bi-periodic Structures

Obviously, A(,, is dependent on the structural parameters only and b(,,s, is dependent on the loading condition besides structural parameters. If the specific loads and structural parameters are given, the generalized reaction Q(,J, can be calculated and the supporting reaction Po&can be found by substituting Q(,, into Eq. (4.1.23a). Finally the transverse displacement for each node can be obtained from Eqs. (4.1.1 I), (4.1.12b) and (4.1.15) in which Po,, can be found as indicated above. The following example will show how this can be done.

4.1.la

Example

Consider a uniform square network with 6x6 mesh and 2x2 supports subjected to a concentrated load of magnitude P acting at the center node as shown in Fig. 4.1.3.

Figure 4.1.3

6 x 6 network with 2 x 2 supports

Structures with Bi-periodicity in 7ivo Directions

135

For the considered network, the specific parameters and loading may be written down as:

and

F(3,3)=P, F,kl=O,

O', k)*(3,3),

j , k = 1,2,...,5

(4.1.30)

Inserting Eqs. (4.1.29) and (4.1.30) into Eq. (4.1.15) yields R

5

=

-7 3

5

3

R

R

sin jr-sinks-sinr-sins-P 6

R

, j , h 1 , 2,...,12

(4.1.31)

%? %? 2 K ( 2 - cos - cos s -) 6 6

The nodal displacements caused by external loading without supporting reaction can be calculated from Eq. (4.1.31). The results are as follows:

w ; , ~ )= 0, j=6,12 (or 0 )

or k= 6,12 (or 0)

(4.1.32f)

136

Exact Analysis of Bi-periodic Structures

and

According to the definition shown in Eq. (4.1.18), the following can be written down directly

and

Noting that w(;,,,has anti-symmetricproperty as shown in Eq. (4.1.16), Eq. (4.1.25) may be rewritten as

7,

rn I m - l

b(,,,) = 4eir~leisC2

J2m,&

1 2 ,. 2

1-1

sin ks9,

w(;,,,

t=i

s=1,2,...,2m2

Substituting Eqs. (4.1.29) and (4.1.33) into the above equation results in

(4.1.34)

Structures with Bi-periodicity in Two Directions

137

For the case under consideration as given in Eq. (4.1.29), then Eq. (4.1.28) becomes

r, s=1,2,...,6 The useful results are

Now the generalized reaction Q(,, can be obtained by substituting Eqs. (4.1.35) and (4.1.37) into Eq. (4.1.27) as

Q(,,=O,

e 1 , 5 or s$1,5

(4.1.38~)

For the present case, the double U-transformation (4.1.23a) becomes

The supporting reaction Po.,,, can be found by inserting Eq. (4.1.38) into Eq. (4.1.39) as

138

Exact Analysis of Bi-periodic Structures

with the other PC,,)vanishing. The supporting reactions Po,,) ( j , k=1,2) are actual ones for the network under consideration shown in Fig. 4.1.3. The nodal displacements wpjc, caused by the supporting reactions for the system without any supports can be calculated by substituting Eqs. (4.1.29), (4.1.40) and (4.1.13) into Eq. (4.1.12b). The results are summarized as follows:

w;,,, = 0 ,

j=6,12 (or 0) or k=6,12 (or 0)

(4.1.41f)

and

Finally the real displacements w G ,for the original system can be easily obtained by superimposing wPjSk)on w ; ~ ,, giving ~,

Structures with Bi-periodicity in Two Directions

139

and the displacements for the nodes on the boundary of the original network are equal to zero. Obviously the displacement solution shown in Eq. (4.1.42) satisfies the restrained condition shown in Eq. (4.1.3) besides the boundary condition. It can be verified that the displacements shown in Eq. (4.1.42) and supporting reactions shown in Eq. (4.1.40) satisfy the nodal equilibrium equations. The results also demonstrate the exactness of the formulas in this section. 4.1.2 Natural Vibration [19]

The rectangular network shown in Fig. 4.1.1 with lumped mass M at each node is considered. The natural vibration equation can be obtained from Eq. (4.1.1) where the nodal load F(j,k, should be replaced by the inertial force MW w ( , , ~,i.e., )

140

Exact Analysis of Bi-periodic Structures

where w denotes the vibrational frequency; w ( ~ , , , and ~ j l , k l , denote the amplitude of transverse displacement and the support reaction respectively. Applying the double U-transformation (4.1.4) to Eq. (4.1.43), we have

where f(:,$, is given in Eq. (4.1.7b). q(,,s)

can be expressed as q(r9s'-

(4.1.45)

J (r,s)

ZK, +2K2 - ~ w -2Klcosryl ' -2K2 cossy,

Substituting Eqs. (4.1.45) and (4.1.7b) into the double U-transformation (4.1.4a) we have

in which

P;i.t)(~,v)

--

1

41' 2'

2n r=l

2n -I

s=l

ei(j-u)ry,

e

i(k-v)sy,

2Kl + 2K2 - Mw2 -2Kl cosry, - 2K2 coss y,

Introducing Eq. (4.1.46) into the restrained condition (4.1.3) gives

Structures with Bi-periodicity in Two Directions

141

with q, = p l y l = z / m , and q2= p 2 y 2 = z / m 2 . The simultaneous equations (4.1.48) are cyclic periodic for unknowns with two subscripts. They can be uncoupled by means of the double U-transformation (4.1.23).Applying the double U-transformation (4.1.23)to Eq. (4.1.48),we have

where

and 1 P(u.v)(l,l) E '"1'2

r=l

r=l

ei(u-l)rpl

ei ( v - 1 ) s ~ ~ 2 K l + 2 K , - M u 2 - 2 K l cos r y , - 2 K , toss (u2

CC

2n1-1 2n2-1

Inserting Eq. (4.1.52)into Eq. (4.1.51) results in A,,,, =

-xx 1

P'

Pz

PlP2

,=I

k=l

{2K1+ 2K2 -MU' - 2K1C O S+[(~j - 1 ) 2 ]- 2K2cos[s + ( k - 1)2m2]y2}-I

Eq. (4.1.50) is made up of single degree of freedom equations. Let us consider the property of Q(r,s,(r = 1,2,...,2ml ; s = 1,2,...,2 m 2 ). For the equivalent system, the supporting reactions

qj,k)( j= 42,...,2ml ;k = 1,2,...,2 m 2 ) must

symmetric condition, i.e.,

satisfy the anti-

142

Exact Analysis of Bi-periodic Structures

and

Substituting Eq. (4.1.54) into Eq. (4.1.23b) yields

That leads to

and the other Q ( , , , having the complex factor eirp'ekp2 When the generalized supporting reactions Q(,,, are not identically equal to zero, the independent frequency equation is

and A(,, is the function of o as shown in Eq. (4.1.53). Consider now the natural mode. From the definition of

shown in

= A ( r , 2 m z - s ) - A(2rn1-r,2rn2-s) .

Eq. (4.1.53), it can be verified that A ( , , s ) A ( 2 m , - r , s ) Corresponding to each natural frequency satisfying A(,)=O, four generalized supporting reactions Q ( r , s , Q ( 2 m l - r , s ) Q(r,2m2-s) Q(2ml-r,2m2-s) Can be equal to different constants for the extended network. We need to find the anti-symmetric mode. In view of Eq. (4.1.59, we must let 7

3

Structures with Bi-periodiciw in Two Directions

143

with the other Q,,, vanishing, in which c denotes an arbitrary real constant. Substituting Eq. (4.1.58) into Eq. (4.1.23a) yields

Fij,k)= csin jrp, sin ksp,

j = 1,2,...,2m,, k = 1,2,...,2m2

(4.1.59)

where r and s represent the half wave numbers in x and y directions for the original network, respectively. The corresponding mode can be found by substituting Eqs. (4.1.59), (4.1.47) and the value of w into Eq. (4.1.46). When the supporting reactions are identically equal to zero, i.e., all of the supported nodes lie in the nodal lines of the mode for the network without the internal supports, the frequency equation can be obtained from Eq. (4.1.44) with 0 f(r,s) = 0 , i.e., 2K, + 2K2 - ~ w -'2 ~cosr , V,

- 2K2coss y2= 0

Note that when r orland s is replaced by 2n1- r orland 2n2 - s in Eq. (4.1.60), the frequency equation has no change. Therefore corresponding to one natural frequency satisfying Eq. (4.1.60), there are four generalized displacements, q(,,,) q(2n1-r,s) q(r~4-s)and q(2nl-r,2n2-s)which are non-vanishing. In order to obtain the anti-symmetric mode, the corresponding generalized displacements should take the same form as that shown in Eq. (4.1.58), namely 7

with the other q(r,s,vanishing. The corresponding mode can be obtained by introducing Eq. (4.1.61) into Eq. (4.1.4a) as w ( ~ , =sin ~ , jriy, sin ksyl,

j = 1,2,...,2n,;

in which an arbitrary constant factor has been omitted.

k = 1,2,...,2n2

(4.1.62)

144

4.1.2a

Exact Analysis of Bi-periodic Structures

Example

Consider a uniform square network with 6 x 6 mesh and 2 x 2 internal supports as shown in Fig. 4.1.3. The specific structural parameters can be written down as

When the supporting reactions are not identically equal to zero, the frequency equation is Eq. (4.1.57). Substituting Eqs. (4.1.63) and (4.1.53) into Eq. (4.1.57) gives

The roots for w of the above frequency equation are summarized in Table 4.1.1.

Table 4.1.1 Natural frequency (1) (~9s)

w

Multiplier

(1,l)

(192)

(271)

(2,2)

4

4 2

4

4-J;S

4 2

4-JZ

4+&

6

6

4+&

KIM

When the supporting reactions are identically equal to zero, the corresponding frequency equation can be obtained from Eq. (4.1.60), where one of r and s must be equal to 3 for the present case, i.e.,

Structures with Bi-periodicity in Two Directions

r

= 3 , s = 1,2,-..,5

and s = 3 , r = 1,2;..,5

145

(4.1.65)

Because r and s on the left hand side of Eq. (4.1.65) are in agreement with the numbers of the half wave in x- and y-directions for the original network. When r or s is equal to 3, all of the supported nodes must necessarily lie on the nodal lines of the corresponding mode, i.e., the supporting reactions are identically equal to zero. The corresponding frequencies can be expressed as

r = 3 , ~ = 1 , 2 ; . . , 5 and s = 3 , r = 1 , 2 ; . . , 5

The result is summarized in Table 4.1.2. Table 4.1.2 Natural frequency ( 2 ) (r,s)

w2

Multiplier

(3,l) 4-&

(32)

(3,3) 4

(394)

(3,s) 4+&

(1,3) 4-&

(2,3)

(4,3)

(5,3) 4+&

KIM

The total number of natural frequencies is equal to 21 which is in agreement with the number of degrees of freedom for the original network. Next, consider the natural modes corresponding to the supporting reactions that do not vanish. The mode can be obtained by inserting the values of r , s, w and the supporting reactions shown in Eq. (4.1.59) into Eq. (4.1.46).Consider now the basic mode corresponding to the lowest natural frequency. The parameters r , s , w can be obtained from Table 4.1.1 as

146

Exact Analysis of Bi-periodic Structures

Substituting Eqs. (4.1.63) and (4.1.67a) into Eq. (4.1.59) gives

vanished. Inserting Eqs. (4.1.47), (4.1.63), (4.1.67b) and (4.1.68) while the other Po;,) into Eq. (4.1.46), the mode can be found as shown in Table 4.1.3, in which the mode has been normalized according to the maximum amplitude taken as 1. Obviously, the mode shown in Table 4.1.3 satisfies the boundary condition and restrained condition at supported nodes. It can be verified that the free vibration equation shown in Eq. (4.1.43) is satisfied at the fiee nodes. Noting Tables 4.1.1 and 4.1.2, it is interesting that there are five independent modes corresponding to one natural frequency, i.e., w 2 = 4 K I M , where four modes are corresponding to qj$) f 0. They can be found by using the same procedure described in the above. The results are shown in Tables 4.1.4 - 4.1.7. Table 4.1.3

K

Natural mode w(iPq corresponding to m = (4 - &)M

Structures with Bi-periodicity in Two Directions

147

Table 4.1.4 Natural mode 1 , wV,,) corresponding to w 2= 4K I M and ( r , s ) = ( l , l )

* if K , + K 2,the supporting reactions are non-zero.

Table 4.1.5 Natural mode 2,

corresponding to w = 4K I M and (r,s)=(1,2)

148

Exact Analysis of Bi-periodic Structures

corresponding to w 2 = 4K / M and (r,s)=(2,1) Table 4.1.6 Natural mode 3, wUSk)

Table 4.1.7 Natural mode 4 , wo.,,, corresponding to w Z = 4 K I M and (r,s)=(2,2)

* if

K,

#

K 2 ,the supporting reactions are non-zero.

Structures with Bi-periodicity in Two Directions

149

Table 4.1.8 Natural mode 5, w ~ corresponding ,~) to w = 4K I M

The another mode where all the supported nodes lie in its nodal lines can be obtained by substituting Eq. (4.1.63) and r=s=3 into Eq. (4.1.62) as

W(

.z . z = sin J -sin k -

2

2

(4.1.69)

The result is as shown in Table 4.1.8. In the same way as the above, the other modes can also be found without any difficulty. 4.1.3

Forced Vibration [l9]

Consider the same network shown in Fig. 4.1.1 with lumped mass M at each node and subjected to harmonic load qj,,,ei" at node(j,k). The harmonic vibration equation for node ( j , k) of equivalent system takes the form as

Exact Analysis of Bi-periodic Structures

150

where F( denotes the amplitude of harmonic force acting at node ( j , k) and the other notations have the same meaning as those in Eq. (4.1.43). In Eq. (4.1.70), F ( j , k , must satisfy the anti-symmetric condition as shown in Eq. (4.1.14). Applying the double U transformation (4.1.4) to Eq. (4.1.70), we have

where

&:,s,

q(,,,

and

f(:,s,

are as shown in Eqs. (4.1.7a) and (4.1.7b) respectively.

in Eq. (4.1.7 1) can be written as

where 0

q(r3s)

J(r,s) - 2K, +2K, - M o 2 -2K, cosry, -2K2 cossy,

(r,s) + (2n1,2n2 )

and qrr~s)

hr,s)

- ZK, +2K2 -MU,-2K, cosrp, -2K, cossy,

*

(r,s) (2n192n2 (4.1.73~)

Structures with Bi-periodicity in Two Directions

15 1

Substituting Eqs. (4.1.72), (4.1.73) and (4.1.7) into Eq. (4.1.4a), we have

and

where

fi(;,k,(j,,k,,has the same definition as that shown in Eq. (4.1.47), i.e., -- 1 -

B"2k""'1'

2n -I

2n -I

- 4n1n2

,i(i-il)rn ei(k-k~)sv2 ZK, + 2K2 - ~ m -'2K, cos r y, - 2K2 coss y2

and wpj+, represent the amplitudes of the displacements caused by the harmonic loading and the supporting reactions acting on the equivalent network without the internal supports, respectively. Substituting Eqs. (4.1.76) and (4.1.14) into Eq. (4.1.75b), results in w;,,

n -1 n

-I "1-1n

-1

sin jry1sinksy/, sin j l r y l s i n k l s y 2 m2 -2Kl cosry, -2K2 cossy2 F(jlAI )

W ; ' 9 k ' = 1 1 1 1 ) 1 ~ ~ ~ ~2KI +2K2 - M

It can be verified that w;,,,, shown in Eq. (4.1.77) satisfies the anti-symmetric condition and boundary condition of the original network with n, x n2 mesh, i.e., ~;2n~-,,2n~-k) - ~ ; , k ) ; ~ ; 2 n ~ - j , k ) - ~;j,2n~-k) - -~;j,k),

I

k=1,2,.'.,n2 and

152

Exact Analysis of Bi-periodic Structures

w ; , ~ ,= 0 ,j=O,n, or k-0,n,.

The displacements at supported nodes must be equal to zero, i.e.,

Introducing the notation

and Eq. (4.1.75a) into the restrained condition (4.1.78) yields

where

with

w(;,,, has been determined in Eqs. (4.1.79) and (4.1.77). of the simultaneous equations It can be shown that the coefficients fl(j,kNjl,kl, (4.1.80) have the cyclic periodicity, i.e.,

Structures with Bi-periodicity in Two Directions

153

Eq. (4.1.80) takes the same form as that shown in Eq. (4.1.19), therefore it can be uncoupled by applying the double U-transformation. Applying the double U-transformation (4.1.23) to Eq. (4.1.80), we have

where

and 1 P ( u * v ~ ( l ~ l '=

ei(u-l)rql i(v-l)sq,

e 2 K , + 2 K 2 -MU' - 2 K , cos r y, - 2 K 2 coss y2

2n1-1 2n2-1

(4.1.86)

Substituting Eq. (4.1.86) into Eq. (4.1.84) results in

where

If the specific load and structural parameters are given, the amplitudes of the nodal displacement and supporting reaction can be calculated from the relevant formulas given above. 4.1.3a

Example

Consider the same network shown in Fig. 4.1.3 subjected to the harmonic loading peim acting at its center, i.e.,

154

Exact Analysis of Bi-periodic Structures

Noting that the definition of qf,,,shown in Eq. (4.1.79) and substituting Eqs. (4.1.63), (4.1.89) and (4.1.76) into Eq. (4.1.75b), results in

where R denotes the nondirnensional parameter of frequency as shown in the following equation

Introducing Eqs. (4.1.63) and (4.1.90) into Eq. (4.1 3 5 ) yields

while the other b(,,s,vanished. Inserting Eq. (4.1.63) into Eq. (4.1 38) gives

Substituting Eqs. (4.1.92) and (4.1.93) into Eq. (4.1.87), we have

Structures with Bi-periodicity in Two Directions

155

Substituting Eqs. (4.1.63) and (4.1.94) into Eq. (4.1.23a)' the amplitudes of supporting reaction can be obtained as

All of the nodal displacements can be found by substituting Eqs. (4.1.63), (4.1.89), (4.1.95) and (4.1.76) into Eqs. (4.1.75) and (4.1.74). Consider now the displacement of the loaded node, i.e., w(,,,,. The final result can be expressed as

where

H (R) =

(R2 - 8R)' + 27(R2 - 8R) + 178 (4 - R)(l0 - 8R+ RZ)(13- 8Q + Q2)

(4.1.97)

where R has been defined as shown in Eq. (4.1.91). From Eq. (4.1.97), it can be shown that when H(R) approaches a finite value at a resonance frequency, the force is acting at a nodal pointlline of the corresponding mode. The frequency response curve governed by Eqs. (4.1.96) and (4.1.97), w(,,,, versus R ,is plotted in Fig. 4.1.4.

156

Exact Analysis of Bi-periodic Structures

Figure 4.1.4 Frequency response curve, w(,,~, versus R

4.2 Grillwork with Periodic Supports [18] The grillwork considered is made up of two orthogonal sets of beams, say x- and y-beams, which are connected at the nodes so that no moments are transmitted from one set to the other (i.e., the torsional rigidity of the beams is neglected). For generality, consider now an M x N rectangular grillwork with simply supported ends at four edges as shown in Fig. 4.2.1, where EI, and EI, denote the flexural rigidity of the beams in the x- and y-directions respectively; a and b denote the spacing of y- and x-beams; the solid circles denote the nodes supported by the posts. There are (m - 1) x (n - 1) internal supports.

Structures with Bi-periodicity in Two Directions

Figure 4.2.1

157

Grillwork with MxN mesh and (m- l)x(n- 1) internal supports

The equivalent grillwork with cyclic periodicity in x- and y-directions can be produced by using the image method as explained in Section 4.1. At the outset, consider an extended grillwork with 2M x 2N mesh as shown in Fig. 4.2.2 where the loading pattern is anti-symmetric with respect to two symmetric planes of the extended grillwork and the nodal loads acting at the nodes (r, s) ( r = 2,3, M ; a * . ,

s=2,3;.- N ) are the real loads acting on the original grillwork. Moreover we regard the extended grillwork as having cyclic bi-periodicity in x- and y-directions [i.e., each pair of nodes (1, k) and (2M+1, k) ( k = 1,2,---,2N+ 1) and 0, 1) and 0, 2N+1) ( j = 1,2,- -.,2M + 1) may be imaginarily regarded as the same node mathematically].

M=mp,, N-np,

, I

)2

1

( I , kF(2M+1, k)

Figure 4.2.2

j=1,2,

.-.,2M+I

k=1,2, -..,2N+ I

Equivalent grillwork with 2Mx2N mesh and cyclic periodicity in x- and y-directions

Structures with Bi-periodicity in Two Directions

159

Such an extended grillwork is equivalent to the original one, because the simply supported boundary condition of the original grillwork can be satisfied automatically in its equivalent one with antisymmetric displacements. In the equivalent grillwork (see Fig. 4.2.2) the additional supports located at boundary and symmetric lines are necessary in order to form the cyclic bi-periodic system, but their supporting reactions are identically equal to zero. 4.2.1 Governing Equation

The principle of minimal potential energy will be applied to establish the equilibrium equation. The potential energy of the equivalent grillwork with 2M x 2N mesh may be expressed as

where ITjk denotes the potential energy of substructure (j, k) and j, k denote the serial numbers of the substructure in the x- and y- directions, respectively. In general, the potential energy for the substructure can be written as

in which [ K ] , , denotes the stiffness matrix of the substructure; { d } ( j , k{)F, } ( j , k ) denote the displacement and loading vectors for the substructure (j, k), respectively and the overbar denotes complex conjugation. Here the loading vectors should include both the external load and supporting reaction. For the present case, the substructure is made up of two orthogonal segments of x- and y-beams as shown in Fig. 4.2.3. Each substructure includes three nodes. The substructural displacement vector is made up of three nodal displacement vectors; that is

where the nodal displacement vector can be expressed as

160

Exact Analysis of Bi-periodic Structures

in which w, ex,0, denote the transverse displacement and two angular rotations (19,E at)/ a,0, = at)/ @ ). It is stipulated that w ~ ( = ~w , ~( ~) , ~Bxl(j,k) ), 3 Bx(j,k), 19yl(j,k)

19y(i.k) '

Figure 4.2.3 Substructure and the ordinal number of nodes

By using the conventional stiffness superposition method, the stiffness matrix of the substructure can be obtained from two beam element stiffness matrices as

Structures with Bi-periodicity in Two Directions

161

where the torsional rigidity of the beams is neglected. The continuity across the nodes requires the following conditions to be satisfied

where {6~ ) ( 2 ~ + 1 , k ) (61 1 ( ~ , k,) {611 ( j , 2 N + 1 ) = {Bl due to the cyclic periodicity. The restrained condition at supports can be written as

where p , , p , and m, n are the geometric parameters of the grillwork (see Fig. 4.2.1). There are simple relations between the geometric parameters; that is

For the present case, the double U-transformation and its inverse can be given as

Exact Analysis of Bi-periodic Structures

162

and

inwhich

yl,

=z/M,yl,=*IN, i = f i and

where {qrtrjr is a vector while q(r,s,is a scalar. Applying the double U-transformation (4.2.11) to Eqs. (4.2.1), (4.2.2) and (4.2.8) results in

with

and

The continuity condition (4.2.15) can be rewritten

Structures with Bi-periodicity in Two Directions

163

where

and I,,, represents a unit matrix of order three. Substituting Eq. (4.2.16) into Eq. (4.2.13), we have

in which

and {ql)(r,s,(r = 1,2,.--,2M;s = 1,2;..,2N) are the independent variables. The necessary condition to make the potential energy minimum can be expressed as

which is equivalent to the nodal equilibrium equation. Each equation in Eq. (4.2.21) with given r and s includes only three variables. Introducing Eqs. (4.2.5), (4.2.6) and (4.2.17) into Eq. (4.2.19) gives

24K1(1- cosryl,) + 24K2(1-cossyl,) - 12Kliasinr yl, -12K2ibsinsy2

1

12Kliasinryll 12K,ibsinsyl, 4~,a~(2+cosryl,) 0 0 4K,b2(2 + cossV2)

which is the Hennitian matrix. Without loss of generality, it is stipulated that the loading vector for substructure

164

Exact Analysis of Bi-periodic Structures

only includes the load acting at the first node of substructure. The loading vector can be expressed as

where F(j,k, denotes the external load acting at the node (j, k) (i.e., the first node of substructure (j, k) ) and

q,,,,

denotes the supporting reaction acting at the

supported node [I + (u - l ) p , , 1 + ( v - l ) p 2 ]. The external loads of the equivalent grillwork must satisfy the anti-symmetric condition; that is

Introducing Eqs. (4.2.17), (4.2.14)and (4.2.23)into Eq. (4.2.20)results in

in which

Structures with Bi-periodicity in Two Directions

165

Noting the anti-symmetric condition (4.2.24), Eq. (4.2.26a) may be rewritten as

That leads to

In the right side of Eq. (4.2.21) includes the unknown support reactions besides external loads which can be determined by using the restrained condition (4.2.9).

4.2.2

Static Solution

At the outset we need to derive the restrained condition in terms of the support reactions. Next we solve this equation by using the double U-transformation for the second time. Inserting Eqs. (4.2.22), (4.2.25) and (4.2.12) into Eq. (4.2.21), the solution of Eq. (4.2.21) can be expressed as (2 + cosry,)(2 + cossy,)

(hrs) + f(:,,,),r # 2M, s # 2N

(4.2.29a)

- 3i sin rty, (2 + cos s y 2 ) 4 ~ ( , ,-~ ) (S,,,,)+S,;,,,), r * 2 M , s z 2 N

(4.2.30a)

4(,,,) =

D,r,s)

aD,,,

166

Exact Analysis of Bi-periodic Structures

Substituting Eqs. (4.2.29a) and (4.2.26) into the first component equation of the double U-transformation (4.2.1la), we have

where

and

Here w;,,, and wpj,,, denote the transverse displacements of the node (j, k) caused by the anti-symmetric external loads and supporting reactions, respectively. It is presupposed that F(i,k, satisfies the anti-symmetric condition shown in Eqs. (4.2.24), otherwise Eq. (4.2.34) is not true. The term w;,,, is dependent on the unknown supporting reaction Piu,,, which can be determined by means of the restrained condition (4.2.9). Inserting Eqs. (4.2.33) - (4.2.36) into Eq. (4.2.9) yields

Structures with Bi-periodicity in Two Directions

167

Eq. (4.2.37) is the set of simultaneous linear algebraic equations with 4mn numbers of variables. Noting that the coenicients fl(j,k)(u,v)( j, u = 1,2,--.,2m;

k, v = 1,2, ,2n)shown in Eq. (4.2.38) possess cyclic periodicity; that is

the deferent coefficients are

P(j,k)(l,l) ( j = 1,2,...,2m; k = 1,2,..-,2n). This property

indicates that the simultaneous equation (4.2.37) can be uncoupled by using the double U-transformation. For the present case, the double U- and inverse double Utransformations can be given as

with p, = n / m = p , y / , and p 2 = n 1 n = p 2 y / 2 . Premultiplying Eq. (4.2.37) by the operator gives

1

&J2n

7 2m

2n

j=1 k=1

e-i(j-l)me-i(k-l)sfi

168

Exact Analysis of Bi-periodic Structures

where

= 0 leading to Q(2m,2n, = 0 . Eq. (4.2.42) is made up It can be proved that b(,,,,,, of 4mn numbers of single degree of freedom equations. The solution is

where

When the specific structure parameters and external loads are given, the generalized supporting reaction Q(r,s, can be calculated from Eqs. (4.2.45), (4.2.46), (4.2.43), (4.2.39) and (4.2.34). Then the supporting reactions and all of the nodal displacements can be found from the related formulas derived above. Recalling the definitions of both generalized supporting reactions f(:,s, ( r = 1,2;..,2M; s = 1,2;..,2N)

and

Q(,,s,

( r = 1,2,...,2m; s = 1,2,...,2n) as

shown in Eqs. (4.2.26b) and (4.2.41b) respectively, there is a simple relation between both f(:,s, and Q(r,s, ;that is

Structures with Bi-periodicity in Two Directions

169

Consequently, if we are only interested in the nodal displacements, it is not necessary to find the supporting reactions. 4.2.3

Example

To explain the calculation procedure and to verify the exactness of the formulas derived in the above, a specific grillwork and loading is worked out as a numerical example. Consider a uniform square grillwork with 6 x 6 mesh simply supported along the four sides and by 2 x 2 internal supports subjected to a concentrated load with magnitude P acting at the centre node as shown in Fig. 4.2.4. The structural parameters are given as

That leads to

The anti-symmetric nodal loads for the equivalent grillwork can be expressed as q4,4)

= F(~o,~o)= lJ

with the other nodal loads vanishing. Substituting Eqs. (4.2.48) and (4.2.49) into Eq. (4.2.27) yields

Introducing Eqs. (4.2.48), (4.2.49) and (4.2.36) into Eq. (4.2.34) results in

(4.2.49a)

170

Exact Analysis of Bi-periodic Structures

Figure 4.2.4

Grillwork with 6x6 square mesh subjected to a concentrated load acting on the center node

It

P

w ( , , ~=, -

r-I

S-I

z .

K

~(-l)~(-l)isin(j-l)r-sm(k-l)s6 6

r=1,3,5 r=1,3,5

(2 + cos r -)(2 6 '(r,s,

+ cos s -)K 6

Structures with Bi-periodicity in Two Directions

171

where 7T

7T

D(r,s,= 1 2 ~ [ ( 1 -cos r ~ ) ~+(cos 2 s-) 6 6

7T 7T + (1 - cos s-)*(2 + cos r-)]

6

6

(4.2.52)

Obviously w;,,,) satisfies the anti-symmetric condition shown in Eqs. (4.2.24) with

w;,,,) instead of F(j,k, and M = N = 6 . Introducing Eq. (4.2.51) into Eq. (4.2.39) gives

and

. which indicates the anti-symmetry for w(:.,,, Substituting Eqs. (4.2.53) and (4.2.48) into Eq. (4.2.43) yields

ZI(,,~, = 0 r # 1,5 or s # 1,5

(4.2.54~)

Inserting Eqs. (4.2.48) and (4.2.52) into Eq. (4.2.46) results in

By introducing Eqs. (4.2.54) and (4.2.55) into Eq. (4.2.45a), the generalized

Exact Analysis of Bi-periodic Structures

172

supporting reactions can be obtained as

Inserting Eqs. (4.2.48a) and (4.2.56) into Eq. (4.2.47) yields

with the other components of f(:,,, vanishing. Substituting Eqs. (4.2.56) into the double U-transformation (4.2.41a) gives

and

Eqs. (4.2.58b) and (4.2.58~)indicate that the supporting reactions of the additional supports are equal to zero and the anti-symmetry of supporting reaction for the equivalent grillwork. The real supporting reaction for the original grillwork is given in Eq. (4.2.58a). Now all of the nodal displacements ( w, Ox, By ) can be calculated by introducing Eqs. (4.2.48), (4.2.50), (4.2.57) and (4.2.29) - (4.2.31) into the first three component equations in the vector equation (4.2.1la); that is

Structures with Bi-periodicity in Two Directions

173

The exact results for the original 6 x 6 grillwork are summarized in Tables 4.2.1 - 4.2.3. It can be proved that the results shown in Tables 4.2.1 - 4.2.3 satisfy all of the nodal equilibrium equations, the support conditions and the boundary conditions exactly.

Table 4.2.1

Multiplier

w(~,~)

Pl(75811 K)

174

Exact Analysis of Bi-periodic Structures

Multiplier

Pl(75811 a K )

Structures with Bi-periodicity in Two Directions

Table 4.2.3

Multiplier

175

ey(j,k,

Pl(758 11a K )

4.3 Grillwork with Periodic Stiffened Beams The grid to be considered is made up of two orthogonal sets of beams, namely the x- and y-beams, which are connected at the nodes. Under the assumption that the torsional rigidity of the beams can be neglected, no moments are transmitted from one set to the other. For generality, consider now an M x N rectangular grid with simply supported ends and equidistant stiffening girders in one direction as shown in Fig. 4.3.1. The flexural rigidities of the regular beams and the stiffening girders in the x-direction are EI, and (1 + y)EI, respectively, whereas the flexural rigidity of the regular beams in the y-direction is EI, . The spacing of the y- and x-beams are denoted by a and b respectively. Along the y-direction, the (n-1) internal stiffening girders divide the grid into n repetitive patterns, while there are p repetitive substructures

within each pattern. In order to apply the U-transformation technique, we must define a cyclic bi-periodic system that is equivalent to the original one. Firstly, the grid is extended in the x- and y-directions by its mirror images about two adjacent edges. Then the extended grid has a 2M x 2N mesh and its loading pattern must be

176

Exact Analysis of Bi-periodic Structures

Figure 4.3.1 A grid with an MxNmesh and n-1 interior stiffening girders

anti-symmetric with respect to the two axes of symmetry as shown in Fig. 4.3.2. The nodal loads acting at the nodes (j, k) ( j = 2,3, M; k = 2,3; .-N) are real loads acting on the original grid while the other nodal loads can be determined by using the properties of anti-symmetry. Herej and k denote the ordinal numbers of the node (j, k) in the x- and y-directions, respectively. Secondly, such an extended system may be considered as cyclic bi-periodic in x- and y-directions, when nodes (1, k) and a ,

Structures with Bi-periodicity in Two Directions

177

(2M+ 1, k) ( k = 1,2,- ,2N + 1) are hypothetically regarded as the same node mathematically, and nodes (j, 1) and (j, 2N+1) ( j= 1,2,--.,2M+ 1 ) are likewise regarded as the same node mathematically. The extended system with cyclic biperiodicity and anti-symmetric loading pattern is equivalent to the original one, as the simply supported boundary condition of the original grid can be satisfied automatically in its equivalent one having anti-symmetric displacements. Axis of symmetry

! !

(i,1) = (i,2N+1) ( I , k) 2 (2M+l, k)

j=1,2, ...,2 M t 1 k=1,2, .-, 2N+1

Figure 4.3.2 An equivalent grid with a 2MxUVrnesh, 2n stiffening girders and cyclic periodicity in the x- and y-directions

178

Exact Analysis of Bi-periodicStructures

4.3.1 Governing Equation

Figure 4.3.3 A typical substructure showing the ordinal numbers of nodes Fig. 4.3.3 shows a typical substructure (j, k) where j and k denote the ordinal numbers of the substructure in the x- and y- directions, respectively. Each substructure is made up of three nodes and two orthogonal segments of x- and ybeams. The displacement vector {S)(,,,of substructure (j, k) is made up of three nodal displacement vectors, namely,

where the nodal displacement vector can be expressed as

and w , 0, and 0, denote, respectively, the transverse dwplacement and two

Structures with Bi-periodicity in Two Directions

angular rotations, i.e. 8,

=& l a

and 8,

179

= &I 9 . As the serial number of the

first node of substructure (j, k) is also (j, k), it can be stipulated that w , ( ~ =, ~w)( ~ , ~ ) ,

'' x ( j , k ) and e y l ( j , k ) '' y ( j , k ) . By using the conventional assembly process, the stiffness matrix [ K ] , , of a typical substructure can be obtained from the two stiffness matrices of the x- and ybeam elements 'xl(j,k)

where the sub-matrices [ k u ] (i,j=l, 2,3) can be expressed as

[

:]

-12 6a 0 [k,,] = K , - 6a 2a2 ,

[k131

-12 = K2[-:b

: 0

6b 2:2]

(4.3.4c)

and

K

/ a 3,

K 2 = Ely / b 3

(4.3.5)

When the substructure includes a segment of the stiffening girder, the stiffness parameter K, in Eq. (4.3.4) must be replaced by (1+ y ) K , . The stiffness

180

Exact Analysis of Bi-periodic Structures

parameter (1+ y ) K , can be divided into K , and yK, corresponding to the regular and additional stiffnesses, respectively. In order to make the structure cyclic monoperiodic, the opposite of the internal forces caused by the additional stiffness can be regarded as additional external loads. Note that the substructure (j,k) ( k = 1,1+ p, . ,1+ (2n - l ) p ) includes a segment of the stiffening girder. The additional load vector for substructure (j,k) can be expressed as

where

and yK,[KIe denotes the additional stiffness associated with the stiffening girder. The matrix [ K ] " is given as

Structures with Bi-periodicity in Two Directions

181

The additional loads associated with the stiffening girder depend on the nodal displacements w and 8, of the nodes at the stiffening girders. The total potential energy of the equivalent system with 2M x 2 N subsystems can be expressed as

n

where TI j,k denotes the potential energy of substructure (j, k). In general, TI

,,

can

be defined as

where the superior bar denotes complex conjugation and { F ) ( j , k denotes , the load vector for substructure (j, k). The load vector { F ) ( j , k ,should include both the external loads and the additional loads associated with the stiffening girder, i.e.

in which {F);,,,, denotes the external load vector for substructure (j,k). Without

loss of generality, the load vector {F);,,,, only includes the load acting at the first node of substructure (j, k), i.e.

where F(j,k)denotes the transverse external load at node (j, k). The external loads for the equivalent system must satisfy the anti-symmetric condition, namely

182

Exact Analysis of Bi-periodic Structures

The spatial relationship between adjacent substructures requires the following conditions to be satisfied 2'(

)(j,k)

=1'(

)(j+I,k)

3

3'(

)(j,k) = I'{

j = 1,2-.-,2M; k = 1,2...,2N

)(j,k+l)

(4.3.10)

where {'I)(.?hf+l,k) {'l)(l,k) and {Sl)(j,2N+I) {Sl)(j,l) due to periodicity. For the present case, the double U-transformation and its inverse can be given as

and

By applying the double U-transformation (4.3.11) to Eqs. (4.3.8) and (4.3.10),

Structures with Bi-periodicity in Two Directions

183

we have

and

which indicate that {q, ) ( r , s ) and { q , ) ( r , s ) depend on {q, I ( , , ) and { q l ) ( r , s ) are independent variables where r = 1.2, --. ,2M and s = 1.2,...,2N . The continuity condition (4.3.15) can be rewritten as 9

where the matrix

is made up of unit matrices I,,, of order three as

Substituting Eq. (4.3.16a) into Eq. (4.3.13), we have

where

184

Exact Analysis of Bi-periodic Structures

The necessary condition to make the total potential energy a minimum can be expressed as

which is equivalent to the nodal equilibrium equation. Introducing Eqs. (4.3.3), (4.3.4) and (4.3.16b) into Eq. (4.3.18), we have

24Kl(1-cosry/l)+24Kz(1-cossy/z) 12K1aisinry/, 12Kzbisinsy, 4KIa2(2+ cosry,) 0 - 12KIaisinry/, 0 4 ~ , b ' ( 2+ cossyl,) - 12Kzbisin s y/,

1

Obviously, [ K ] ; , , ~is, a Hermitian matrix. Substituting Eqs. (4.3.14), (4.3.9), (4.3.6) and (4.3.16b) into Eq. (4.3.19), we have

where

Structures with Bi-periodicity in Two Directions

185

Introducing the anti-symmetric condition (4.3.9) in Eq. (4.3.22e), we have

that leads to

Because of the anti-symmetry of the nodal displacements, it can be shown that

In Eq. (4.3.20), {f);,,3, is dependent on the unknown nodal displacements associated with the stiffening girder besides external loads. The deformation compatibility condition of the stiffening girder needs to be satisfied.

4.3.2 Static Solution At first, the deformation compatibility condition in terms of the nodal displacements associated with the stiffening girder must be established. Then the compatibility equation can be uncoupled by applying the double U-transformation again, which leads to the explicit solution. Inserting Eqs. (4.3.22a), (4.3.22d), (4.3.23b) and (4.3.23~)into Eq. (4.3.20), we have

Substituting Eqs. (4.3.21) and (4.3.22) into Eq. (4.3.20), the solution for {q,) (,,., can be expressed as

186

Exact Analysis of Bi-perionic Structures

where

and

r=1,2;-.,2M; s=1,2,---,2Nand ( r ,s ) + ( 2 M , 2 N )

(4.3.25b)

e l ( , , ,= -(IKl - cos rty1)'(2 + cossly2) D(r,s)

(4.3.25e)

Structures with Bi-periodicity in Two Directions

187

Introducing Eq. (4.3.25a) into the double U-transformation (4.3.11a), we have

in which

and

), and Bi(j,k, denote the basic solution for the grid without where w ; ~ , ~Bl(j,k)

stiffening girders, while w:,,~,, 6,$k) and B;(,,,, account for the effects of the stiffening girders. The latter depends on the nodal displacements of the nodes at the stiffening girders. The deformation compatibility condition of the stiffening girders can be obtained from Eqs. (4.3.27) and (4.3.25). Substituting Eqs. (4.3.27b) and (4.3.25b) into Eq. (4.3.27a) and replacing the subscript (j,k) by (j,1+ (k - 1)p) , we have

188

Exact Analysis ofBi-periodic Structures

where

In the compatibility equation (4.3.28), the unknown variables W ( j , k ) and @ x ( j , k ) (j=1,2;-.,2M ) denote the nodal displacements of the k-th stiffening girder. The matrix

[ P ( j,k ,(,, v , ]

possesses cyclic periodicity, i.e.

and

The different matrices are [ P ( j , k ) ( l , l ) ] (j=1,2;..,2M; k=1,2;..,2n ). In view of Eq. (4.3.30), Eq. (4.3.28) can be uncoupled by using the double U-transformation. Let

Structures with Bi-periodicity in Two Directions

189

with p = n / n = p y 2 and y, = n / M . Premultiplying both sides of Eq. (4.3.28) by and applying the cyclic periodicity of

1

y,fe-i(j-l)ryl j=, k=l

.I%,/%

e-i(k-~)~p

[P(j,k)(u,v)] ,we have

with

It can be shown that

and

represent the generalized displacements of The vectors [Q,Q,]:,~,and [bl,b2]LXs) the stiffening girders with y + 0 and y = 0 , respectively.

Exact Analysis of Bi-periodic Structures

190

Noting that 1

2M

-CCe 4MN

2n

i(j'-I)(r'-r)v,

ei(V-l)(s'-s)p

V=l

-

r' = rand s'=s, s+2n,..., s+(p-1)2n r' z r or S' z S, s+2n, -,s+(p-l)2n

with r=1,2,. ..,2M; s=1,2,. ..,2n and substituting Eq. (4.3.29b) into Eq. (4.3.32a), we have

Now the compatibility equation becomes a set of two-degree-of-ffeedom equations by using the double U-transformation. Introducing Eq. (4.3.29~)into Eq. (4.3.34), the solution for Q(r,s, and Q x ( r , s ) of Eq. (4.3.34) can be expressed as

where

Whenthe structural parameters K,, K 2 , y , a , b , M , N (or n ) , p and

Structures with Bi-periodicity in Two Directions

the nodal loads are given, the generalized displacements Q(r,s,and

191

can be

calculated from the relevant formulas derived above. given in Eq. (4.3.29a), and Noting the definitions of W(,,,, and OXQsk) comparing Eqs. (4.3.25b) with (4.3.3 lb), we have

When Q(r,s, and

( r = I 7 2 ; . . , 2 M ; s = 1 , 2 , . - . a )are found, {q,):r,s,

( r=1,2,. ..,2M , s=1,2,...,2N ) can be easily obtained from Eqs. (4.3.37) and

(4.3.25). Therefore, substituting Eqs. (4.3.27b), (4.3.27c), (4.3.26) and (4.3.37) into Eq. (4.3.27a), the exact solution for w(j,k,, BXo:,)and BYQsk, can be found. 4.3.3 Example

In order to explain the computational procedure and to verify the exactness of the derived formulas, a specific example of rectangular grid is chosen for detailed discussion. Consider a uniform square grid with a 6 x 6 mesh and two internal stiffening girders as shown in Fig. 4.3.4. The grid is simply supported at the four sides and subjected to a concentrated load with magnitude P acting at the center node. The structural parameters are given as

and

The external nodal loads for the original grid can be expressed as

192

Exact Analysis of Bi-periodic Structures

Figure 4.3.4 A grid with a 6 x 6 square mesh and two interior stiffening girders subjected to a concentrated load acting on the center node Introducing Eqs. (4.3.38) and (4.3.39) into Eq. (4.3.23a), we have

P . r7t . srr , f(r7s) =-- 3 sin 2 sin 2

r , s = 1,2;..,12

(4.3.40)

Substituting Eqs. (4.3.38), (4.3.40), (4.3.26) and (4.3.2511) into Eq. (4.3.27c), all nodal displacements for the equivalent system with y = 0 , namely w ; ~ ,, ~ ) and , can be found in explicit form. The results for the original grid are summarized in Table 4.3.1. The other nodal displacements for the extended grid can

Structures with Bi-periodicizy in Two Directions

193

be obtained from anti-symmetry, namely

Substituting Eq. (4.3.38) and the results shown in Table 4.3.1 and Eq. (4.3.41) into Eq. (4.3.32b),we have

bl(r,s)= 0 ,

r is even or s # 1,5

(4.3.24~)

b,(,,,

r i s evenor s + 1,5

(4.3.420

=0,

Inserting Eqs. (4.3.42), (4.3.36), (4.3.25e), (4.3.250 and p=2 into Eq. (4.3.39, results in

194

Exact Analysis of Bi-periodic Sfrucfures

(TO

be continued)

Structures with Bi-periodicity in Two Directions

195

(continued)

Q,,,s,= 0 ,

=0 ,

r is even or s # 1,5

(4.3.43d)

r is even or s # 1,5

(4.3.4311)

196

Exact Analysis of Bi-periodic Structures

Finally, introducing Eqs. (4.3.37), (4.3.43), (4.3.25) and (4.3.38) into Eq. (4.3.27b), the effects of the stiffening girders, namely wPjSk),e:(i,k, and B;(~,,, , can be found. The nodal displacements for the original grid are symmetric, i.e.,

The results for nodes (j, k) (j, k=1,2,3,4) of a quarter of the grid are summarized in Table 4.3.2. The solution for the nodal displacements of the original grid can therefore be obtained from Eq. (4.3.27a) using the results given in Tables 4.3.1 and 4.3.2 together with Eq. (4.3.44). It can be proved that the results satisfy all of the nodal equilibrium equations and simply supported boundary condition without any error. Therefore, the solution is shown to be exact. Consider now the effect of y on the nodal displacements.When y approaches zero, the effects of the stiffening girders w&), O(j,k) and 8;(j,,, as shown in Table 4.3.2 approach zero. Obviously the solution with y = 0 is applicable to the grid without stiffening girders. When y approaches infinity, w:~,~),8:j,k) and t9i(j,k, approach finite values as their limits respectively. These limiting values are given in Table 4.3.3. Comparing Table 4.3.1 and Table 4.3.3, it is obvious that

which indicates that the stiffening girders have no flexural deformations. The solution with y approaching infinity is therefore also applicable to a continuous grid with equidistant line-supports. Finally, let us consider the deflections of the nodes on the axis of symmetry orthogonal to the stiffening girders. These nodal deflections can be expressed as

Structures with Bi-periodicity in Two Directions

197

where

The numerical results for w(,,,, (k given in Table 4.3.4.

= 1,2,..-,7)with

increasing values of y are

198

Exact Analysis of Bi-periodic Structures

- 529y(640 + 48 l y ) 320B(y)

- 529y(640 + 479y) 160B(y)

Notes: " B ( y ) = 25600+ 25600y + 4 8 1 3 ;~

+

- 529y(280 21 l y )

80B(y

C ( y ) = 47(94 + 53y)

Structures with Bi-periodicity in Two Directions

Notes: " B ( y ) a 25600+ 25600y + 4813y2 ; b C ( y ) = 47(94 + 5 3 y )

199

200

Exact Analysis of Bi-periodic Structures

Structures with Bi-periodicity in Two Directions

201

202

Exact Analysis of Bi-periodic Structures

Table 4.3.4

The effect of y on w(,,,, / ( P / K )

Chapter 5 NEARLY PERIODIC SYSTEMS WITH NONLINEAR DISORDERS The mode localization phenomenon in infinite periodic mass-spring systems with one disorder was investigated by Cai et a1 [20] using the U-transformation technique. The localized mode in mono-coupled periodic mass-spring systems having two nonlinear disorders was studied [2 11 by means of the U-transformation and L-P method. Recently the same method was applied to analyze the localized modes in a twodegree-coupling periodic system with a nonlinear disordered subsystem [22] and the forced vibration for the damped periodic system having one nonlinear disorder [23]. 5.1

Periodic System with Nonlinear Disorders - Mono-coupled System 1211

Consider the system shown in Fig. 5.1.1. It consists of infinite number of subsystems connected to each other by means of the linear springs having stiffness &kc. Each subsystem consists of a mass M connected to a rigid foundation by a spring with linear stiffness (for ordered ones) or nonlinear stiffness (for disordered ones). In Fig. 5.1.1, jl and j 2 denote the ordinal number of the disordered subsystem.

Figure 5.1.1 Periodic system with nonlinear disorders

204

Exact Analysis of Bi-periodic Structures

It is assumed that the system energy is large enough in order to realize the localized motion mode and both the coupling stiffness Ek, and the coefficient ~y of the cubic term of the nonlinear stiffness in the disordered subsystems are weak, i.e., E is a positive nondimensional small parameter. This section is aimed at analyzing the mode localization phenomena. The localized modes of the system with infinite number of subsystems are hardly affected by the conditions at infinity. Consequently the system under consideration may be regarded as a cyclic periodic one. 5.1.1

Governing Equation

At the outset a cyclic periodic system with N number of subsystems is considered. Then by adopting a limiting process with N approaching infinity, the governing equation will be applicable for a cyclic periodic system with infinite number of subsystems. Applying Newton's second law to every mass M, we can write the differential equations of motion as follows: Mi,

+ Kx,

- dc, (x,+, + x,-, - 2xk)= 0

k

Mi, +Kx, -dc,(x,+, +x,-, - 2 x , ) + ~ & = 0

~

(5.1.1a) ~ 2

k = j l , j2

(5.1.1b)

#~

1

where x, denotes the longitudinal displacement of the k-th mass and x ~ --=+xl,~x0 = xN due to cyclic periodicity. The superior double dot denotes the second derivative with respect to the time variable t. K and M denote the stiffness and mass for the ordered subsystems. The nonlinear terms play the role of disorder. Introducing the time substitution

into Eq. (5.1.1) results in m2x; + 0;x,

2

n

2

k M

+

= & L ( X , + ~Xk-l - 2Xk)

kc M

Y

O X ~ + W ~ X ~ = E - ( X ~ + ~ + X ~ - ~ - ~ X ~ ) - E - k X= ~

M

j l , j Z (5.1.3b)

where the prime sign designates differentiation with respect to z and

Nearly Periodic Systems with Nonlinear Disorders

205

The w in Eqs. (5.1.2) and (5.1.3) is the fundamental frequency to be determined. The periodicity condition may be expressed as x, (Z+ 2n) = xk (z)

k = 1,2;-., N

(5.1.5)

According to the L-P method, the solution of Eq. (5.1.3) is assumed to have the form

and w is given as

Substituting Eqs. (5.1.6) and (5.1.7) into Eq. (5.1.3), the coefficients of equal powers of & on both sides of Eq. (5.1.3) must be equal, i.e.,

Without loss of generality, we can assume that the initial velocity for each subsystem is zero, i.e.,

Then the solution of Eq. (5.1.8a) may be expressed as

206

Exact Analysis of Bi-periodic Structures

x,,, = A,, cos r

k

(5.1.10)

= 1,2,...,N

Inserting Eq. (5.1.10) into Eqs. (5.1.8b) and (5.1.8~)results in k 2womlA,, + - & 4 , + , , ,+ A,-,, - 2Rk,,)] sos r k*j19j2

(5.1.11a)

and

1 In the second equation the identical relation cos3 r = -(3 cos r + cos 3.r) has been 4 used. In order to eliminate the secular term from xk, ( k = 1,2,... ,N ) we must set the coefficients of cos T on the right hand sides of Eqs. (5.1.11a) and (5.1.11b) equal to zero, i.e.,

kc 2wowlAko+-(A,+~,, M

+ A,,,

- 2Ak,,)--

3~ 4M

A:, = 0

k = j , , j,

(5.1.12b)

One can now apply the U-transformation technique to Eq. (5.1.12). The U and inverse U transformations may be expressed as

Nearly Periodic Systems with Nonlinear Disorders

207

2~ with y~ = - , i = f i and a,-, = a, in which the superior bar denotes N complex conjugation. By using the U-transformation, Eq. (5.1.12) becomes

Consequently a,, takes the form as

Substituting Eq. (5.1.15) into Eq. (5.1.13a) yields

Noting that the imaginary part of the summation of the series in the square brackets is equal to zero, Eq. (5.1.16a) may be rewritten as

By letting N approach infinity, the limit of the series summation in the square brackets on the right hand side of Eq. (5.1.16b) becomes the definite integral [5], i.e.,

208

Exact Analysis of Bi-periodic Structures

the definite integral shown in Eq. (5.1.17) is in existence. The condition (5.1.18) is equivalent to

where w2 = + 2wOw,.5+ o ( E ' ) and w, , w, denote the upper and lower limits of the pass band for the ordered system ( y = 0 ), i.e.,

Eq. (5.1.19) indicates that w lies in the stop band of the ordered periodic system. Because the frequency corresponding to localized mode must lie in the stop band, we have no interest for the case that the integral shown in Eq. (5.1.17) is not in existence. Consider now the system having infinite number of subsystems. Consequently Eq. (5.1.16b) becomes

where yield

P(,-,, has been defined by Eq. (5.1.17). Setting

k = j, and j, respectively,

Nearly Periodic Systems with Nonlinear Disorders

209

and

The unknown amplitudes A,, can be obtained by substituting the nontrivial solution of Eq. (5.1.22) into Eq. (5.1.21). Before we discuss the solutions of Eqs. (5.1.22a) and (5.1.22b), let us calculate the definite integral Pishown in Eq. (5.1.17).

where

It can be verified that

where

210

Exact Analysis of Bi-periodic Structures

It is obvious that

4

and a are of opposite sign and

(51< 1 .

5.1.2 Localized Modes in the System with One Nonlinear Disorder The periodic system with infinite number of subsystems in which only one subsystem having nonlinear stiffness is considered. The governing equation (5.1.22) can be reduced to

There is a nontrivial solution of the above equation if and only if

Inserting Eqs. (5.1.23) and (5.1.27) into Eq. (5.1.31) results in

where

Noting that w 2 = wi + 2&wOw,+ o(E'), it can be proved that if y > 0 (hardening

K K 4% and y < 0 (softening spring) w < spring) w 2 > -+. Consequently M M M we can come to the conclusion that if y > 0 there is one localized mode with w greater than w, and if y < 0 there is one localized mode with w less than w, . For the localized mode the amplitudes of all subsystems may be obtained from Eqs. (5.1.21) and (5.1.30) as

Nearly Periodic Systems with Nonlinear Disorders

21 1

It indicates that the amplitudes decay exponentially on either side of the nonlinear disorder. The attenuation rate 6 of localized mode may be found by substituting Eqs. (5.1.27) and (5.1.32) into Eq. (5.1.29) as

5 5

is the odd hnction of nondimensional parameter 77. When 17 approaches zero, also approaches zero. It indicates that if kc decreases or y increases or increases, the mode will approach strongly localized state. The localized level of the modes is dependent on not only the structural parameters kc and y but also the amplitude Ajl, related to the initial condition or the total energy of system. Ail ,o can be determined by applying the energy conservation to the considered system. Ignoring the O(E) term it can be proved that the following equation

is a necessary condition for the energy conservation. p2 is an energy-like quantity 1 which is related to the total energy E of the system by -~ p +' O(E) = E . 2 Inserting Eqs. (5.1.34), (5.1.35) and (5.1.33) into Eq. (5.1.36) results in

21 2

Exact Analysis of Bi-periodic Structures

which indicates that when kc/y is given, in order that a localized mode would occur, however small the amplitude, the energy constant p2 must be greater than 18kC/3yl.If the system energy is below the required level it is impossible to realize any localized mode. This property is different from that for the linear periodic system. 5.1.3 Localized Modes in the System with Two Nonlinear Disorders

For the system under consideration,the governing equation (5.1.22) becomes

1

1

in which n = j, - j, and

6 has been defined by Eqs. (5.1.29) and (5.1.27).

Eq. (5.1.38a) minus and plus Eq. (5.1.38b) yield

The above equations are equivalent to

Nearly Periodic Systems with Nonlinear Disorders

21 3

respectively. The simultaneous equations (5.1.40) may be divided into three sets of equations, i.e,

(ii)

(iii)

'jlo

-fl,(l-<")(~i, 3~ 8kc

= 'j20

(5.1.42a)

+ AjloAjzo+A:,)='

(5.1.43a)

Let us discuss in detail the case of j, = j, + 1 , i.e., n = 1 . (i) The first set of equations, i.e., Eq. (5.1.41), describes the symmetric localized modes. Inserting Eq. (5.1.4 1a) and n = 1 into Eq. (5.1.41b) yields

Recalling Eq. (5.1.33), substituting Eqs. (5.1.23), (5.1.27) and (5.1.29) into Eq. (5.1.44) results in

where 7 is the nondimensional parameter as shown in Eq. (5.1.33). It is obvious that if 7 > O (i.e., y > O ) , wow,>-2kc M

(i.e., w>w,),

there is a symmetric

2 14

Exact Analysis of Bi-periodic Structures

localized mode with higher frequency (i.e., w > w,) and if -1 < TI < 0 , w < w, there is a symmetric localized mode with lower frequency (i.e., w < w, ). Recalling that j2 = j, + 1 and Ail, = -Ajto , the symmetric localized mode can be obtained from Eq. (5.1.2 1) as

which indicates that the amplitudes on either side decay exponentially away from the nonlinear disorders. The attenuation rate c of the amplitudes of symmetric mode may be found by substituting Eqs. (5.1.27) and (5.1.45) into Eq. (5.1.29), as

The sum of the squares of the amplitudes should be equal to constant pZ, i.e.,

Inserting Eqs. (5.1.47) and (5.1.33) into the above equation results in

4k p 2 >-- 3% (for y < 0 ) , p2>> (for y > 0 ) (5.1.49) 3Y 3Y In order that the symmetric localized mode may occur, p2 must be greater than max

[

2]

-2 -

3:;

Nearly Periodic Systems with Nonlinear Disorders

21 5

The stability analysis of the symmetric localized mode shown in Eqs. (5.1.46) and (5.1.47) is investigated by superposing the small perturbations on the mode solution and applying the Floquet theory [24]. It is concluded that when 77 lies in the region [O, 21, the symmetric localized mode is unstable and when 77 > 2 or -1 < 77 < 0 the symmetric localized mode is stable. If q l - 1 , the symmetric localized mode does not exist. (ii) The second set of equations shown in Eq. (5.1.42) describes the antisymmetric localized modes. Inserting Eq. (5.1.42a) and n = 1 into Eq. (5.1.42b) yields

Substituting Eqs. (5.1.23), (5.1.27) and (5.1.29) into Eq. (5.1.50) results in

8k -2 . . Recalling 77 = L A j l o ,it is concluded that if 7 < 0 ( y < 0 ), wowl < 0 ( w < w, ) 3Y there is an antisymmetric localized mode with lower frequency ( w < w,); and if 2kc ( w > w, ) there is an antisymmetric localized mode with 0 < 7 < 1, wowl > M higher frequency ( w > w, ). The amplitudes for the antisymmetric localized mode can be found by inserting j, = j, + 1 and Ail, = AjXOinto Eq. (5.1.21) as

Substituting Eqs. (5.1.27) and (5.1.51) into Eq. (5.1.29) results in

For the present case Eq. (5.1.48) is also applicable in which

< should be defined as

2 16

Exact Analysis of Bi-periodic Structures

in Eq. (5.1.53) instead of Eq. (5.1.47). Substituting Eqs. (5.1.53) and (5.1.33) into Eq. (5.1.48) results in

32kc

p2 >-

(for y > O ) ,

p 2 >-- 4kc

3Y

(for y
(5.1.54)

3Y

In order that the antisymmetric localized mode may occur p2 must be greater than

By applying the Floquet theory to the stability analysis of the antisymmetric localized mode shown in Eqs. (5.1.52) and (5.1.53), it is concluded that when 71 lies in the region [-2, 01, the antisymmetric localized mode is unstable, and when 7 < -2 or 0 < 77 < 1 the antisymmetric localized mode is stable. If 77 2 1, it is impossible for the antisymmetric localized mode to occur. (iii) The third set of equations shown in Eq. (5.1.43) describes the nonsymmetric localized modes. By inserting Eqs. (5.1.23), (5.1.29) and n = 1 into Eq. (5.1.43), it can be proved that Eq. (5.1.43) is equivalent to

where a is defined as Eq. (5.1.27). Without loss of generality, we assume A

1 2 1 A 1 ( 14 2 2 ). Eliminating

Aj2, from Eq. (5.1.55b) and substituting Eq. (5.1.27) into it results in

Nearly Periodic Systems with Nonlinear Disorders

217

where 17 is given in Eq. (5.1.33). It is concluded that if y > 0 ( 77 > 0 ) then

a > a, ( a o w l >

%),

therefore the non-symmetric localized mode with higher

frequency ( w > w , ) exists and if y < 0 ( 7 < 0 ) then w < w, ( w, < 0 ), therefore the non-symmetric localized mode with lower frequency ( w < w , ) exists. The amplitudes for the non-symmetric mode may also be obtained from Eq. (5.1.43). For the case of j , = j, + 1 ,Eq. (5.1.2 1) becomes

and

Recalling Eqs. (5.1.28) and (5.1.38) with n = 1 , Eq. (5.1.57) may be rewritten as

Aj,-k,O= S ~ A ~ , , , k = 0,1,2,...,m and

where A,,, and AjZOmust satisfy Eq. (5.1.55a). Substituting Eqs. (5.1.56) and (5.1.27) into Eq. (5.1.29) results in

Noting Eq. (5.1.58), the sum of the amplitude squares may be expressed as

Inserting Eqs. (5.1.59) and (5.1.55a) in the above equation results in

(5.1.58a)

/

Figure 5.1.2 Localized modes (A,:, A,,,)

versus q

( 2 1 q = -A&

mode 1 - symmetric mode mode 2 - antisymmetric mode mode 3 - non-symmetric mode - stable mode; ----- unstable mode

8k Recalling 11 = 2 . 4 ; ; ,

. it is in~possibleto find the closed fornl solution for

A:, of

3Y 32k, Eq. (5.1.61). But it can be proved that if and only if p 2 2 -, the solution of 9Y Eq. (5.1.61) exists. i.e.. there are two non-symmetric localized modes ( A , ~ ~ / A ~ , ~ and

A~,O/A,:O

=-vI~).

It can be concluded that when

1 ~ 51 2

in which

t;l=

8kC -A:

there are two

3Y

non-symmetric stable localized modes where one mode is the symmetric image of the other. The condition for each type of localized mode to occur is shown in Fig. 5.1.2. 5.2

Periodic System with One Nonlinear Disorder

-Two-degree-coupling System 1221

Consider the two-degree-coupling periodic system with infinite number of subsystems as shown in Fig. 5.2.1, where K and M denote the linear stiffness and mass for the ordered subsystem respectively; &, and 2&, denote the stiffness for two kinds of coupling springs in longitudinal and inclined directions respectively; s denotes the ordinal number of the disordered subsystem with nonlinear stiffness.

Figure 5.2.1 Two-coupling periodic system wit11 a = lr / 4

220

Exact Analysis of Bi-periodic Structures

It is assumed that the system energy is large enough to realize the localized motion mode and that the coupling stiffness and the coefficient &yo of the cubic term of the nonlinear stiffness in the disordered subsystem are weak, i.e., E is a positive nondirnensional small parameter. This section is aimed at analyzing the localized modes. The localized modes in the system with infinte number of subsystems are hardly affected by the conditions at infinity. Consequently the system under consideration may be regarded as a cyclic periodic one. At the outset a cyclic periodic system with N number of subsystems is considered. Then by adopting a limiting process with N approaching infinity, the limiting solution will be applicable to the system having infinite number of subsystems.

5.2.1 Governing Equation Applying Newton's second law to every mass M, the differential equations of motion can be expressed as

j = 1,2;.., N and j

#s

(5.2.la)

j = 1,2,..., N and j

#s

(5.2.lb)

where xIj and x Z j denote the longitudinal displacements of two masses in the j-th (k=1,2) due to cyclic periodicity. The subsystem and x,,,+, = x,, , x,, = ,x superior double dot denotes the second derivative with respect to the time variable t. The nonlinear terms play the role of "load in the above equation. In order to apply the L-P method, it is necessary to transfer Eq. (5.2.1) into the standard form with the uncoupled linear terms. One can now apply the U-transformation to Eq. (5.2.1). The U- and inverse Utransformations may be expressed as

Nearly Periodic Systems with Nonlinear Disorders

221

and

with

y/

27r

= - and i = N

fi.

Applying the U-transformation to Eq. (5.2.1), i.e., premultiplying both sides of N

Eq. (5.2.1) by the operator

e-i(j-l)mv

fi

, we obtain

,=I

where w, denotes the natural frequency of the single subsystem, i.e.,

Conveniently the frequency equation for the linear system corresponding to the differential equation (5.2.3) with yo vanishing can be obtained as

222

Exact Analysis o/Bi-periodic Structures

where w denotes the natural frequency of the ordered system. The solutions for w2 of Eq. (5.2.5) are

The lower and upper bound (w, and w, ) of the pass band can be obtained from Eq. (5.2.6), as

Introducing the time substitution 5=wt into Eq. (5.2.3) results in w2q1:, + 002qlrn= -E

-x:s

M

~~~l~+ wiq2m= -E

{2 -[(2

-cosmy)qIm - ( c o ~ m v ) q+~ ~ ]

(q]],q12,".,q]N)

{2

-[-(COS

m y)q,,

= 1>2>".,

(5.2.9a)

+ (2 - cos m y/)q2,] +

in which w denotes the fimdamental frequency to be determined and the prime

Nearly Periodic Systems with Nonlinear Disorders

223

symbol designates differentiation with respect to the new time variable z . According to the L-P method, the solution of Eq. (5.2.9) is assumed to have the form

and w is given as

Substituting Eqs. (5.2.10) and (5.2.11) into Eq. (5.2.9), the coefficients of equal powers of & on both sides of Eq. (5.2.9) must be equal, i.e.,

....... ....... where x,,,, and x,,,, denote zero-order approximation for xIsand xZs which can

224

Exact Analysis of Bi-periodic Structures

be expressed in terms of the generalized displacements with zero-order approximation as

Without loss of generality, we can assume that the initial velocity for each mass is equal to zero, i.e.,

5.2.2

Perturbation Solution

The solution for q,,,, and q2,,, of Eq. (5.2.12) with initial conditions shown in Eq. (5.2.15a) can be expressed as

Inserting Eq. (5.2.16) in Eq. (5.2.14) yields

where

Nearly Periodic Systems with Nonlinear Disorders

225

Substituting Eq. (5.2.17) into the right hand sides of Eq. (5.2.13) results in

---YO 4~

1

-i(s-,)my

A:,,

fie

cos 32

In order to eliminate the secular terms from q,,,, and q,,,, , the coefficients of

cos z on the right sides of Eq. (5.2.19) must be equal to zero, i.e., 2k 2kc [ 2w0w, - L ( 2 - cos m y/)lalrn,o+ -(cosm M M

2kc -(COS rn ty)a,,,, M

+ [2wowl--(22k

M

rn

The unknowns a,,,, and a,,,,

3 ~ 01 v)a,,, = --e 4M f i

-i(s-l)mv

- cos rn ~ y ) ] a , ~=,-~YO 1 -i(s-l)mv

= 1,2;..,

4~

N

can be expressed formally as

fie

A;,,

3 A2,,0

(5.2.21)

Exact Analysis of Bi-periodicStructures

226

where A, = (-)4kc

M

Here the nondimensional parameter

2

R(R + cos m y )

(5.2.23a)

0 is defined as

Substituting Eqs. (5.2.22) and (5.2.23) into Eq. (5.2.1 8) results in ' I j , ~= a j s ' L , o

+Pjs'is.0

= Pjs'Z.0

+a j s ' i , o

'2j,o

in which

pp, -

3 ~ 0l 1 $[e<j-s,rnw 32kc R N

cosm y, I R+cosm y

Consider now the system with infrnite number of subsystems. By letting N approach infinity, the limits of the series summation on the right sides of Eq. (5.2.26) become the definite integral, i.e.,

Nearly Periodic Systems with Nonlinear Disorders

c

l N lim N+m

N

227

ZR + cos 8

ei( j - s ' m y

R+cosB

m=l

where

The definite integral B, can be expressed in terms of the elementary functions as

(R(> 1 is necessary and sufficient condition for that the definite integral B, is in existence. Let us consider the physical meaning of the condition 1521 > 1. Recalling the definition of

shown in Eq. (5.2.24),

IRI

> 1 is equivalent to

228

Exact Analysis of Bi-periodic Structures

w;

+ 2&w0w1< w ; (02)

which indicate w ( w , + ~ w , lies ) in the stop band of the ordered periodic system. It is well known that the frequency corresponding to localized mode must lie in the stop band. It can be proved that B, has the property [211

with

151 is always less than one. Inserting Eq. (5.2.27) in Eq. (5.2.26) with N approaching infinity yields

YO 1 P Is. =-(--ajs 32k, SZ

+Bj-,)

Setting j = s in Eq. (5.2.25) gives 4s-0

= a, A:,,, + P,, A:,,,

j = 1,2;-0

(5.2.34b)

Nearly Periodic Systems with Nonlinear Disorders

where a, and

229

psscan be obtained from Eq. (5.2.34) as

The localized modes in the system under consideration must satisfy Eqs. (5.2.35) and (5.2.36), where R and Bo are as shown in Eqs. (5.2.24) and (5.2.30a) respectively. 5.2.3

Localized Modes

The periodic system with infinite number of subsystems and having one nonlinear disorder is considered. The governing equation for the localized modes has been given as shown in Eqs. (5.2.35) and (5.2.36). Eq. (5.2.35a) minus or plus Eq. (5.2.35b) yields

A,,, + AzS,o = (a,

+ ps)(AA,o

+

A;~,O)

(5.2.37b)

The above equations are equivalent to

and

respectively. The simultaneous equations (5.2.38a,b) may be divided into three sets of equations; that is

230

Exact Analysis of Bi-periodic Structures

(111) A:,,,

1

2

+ A,s,oA2s,o + Azs,o = ass

- P,

There are three types of localized modes corresponding to the above three sets of equations, respectively. Let us discuss each type of localized modes in detail. (I) The first set of Eqs. (5.2.39a,b) describes the symmetric localized mode about the longitudinal centre line. Inserting Eq. (5.2.39a) in Eq. (5.2.39b) gives 1

A:,, = ass

with

AlS,O

= AZs,O

+ Pss

.

Substituting Eq. (5.2.36) into the right side of Eq. (5.2.42a) yields

Inserting Eq. (5.2.30a) into the above equation gives

Nearly Periodic Systems with Nonlinear Disorders

23 1

is a nondimensional parameter. Substituting Eq. (5.2.24) into Eq. (5.2.43) results in

Noting w2 = m i +2~w,w,+ 0 ( c 2 ) , it can be proved that if yo >O (hardening spring), w2 > 0:

+%

M

and if yo < 0 (softening spring), w2 < wi .

It can be concluded that if yo > 0 , there is one symmetric localized mode with w greater than w, and if yo < 0, there is one symmetric localized mode with w less than w, . For the symmetric localized mode, the amplitudes of all masses can be found by substituting Eqs. (5.2.34) and (5.2.39a) into Eq. (5.2.25) as

Inserting Eq. (5.2.32b) into the right sides of the above equations yields

which indicates that the amplitudes decay exponentially on either side of the nonlinear disorder. The attenuation constant of localized mode can be found by inserting Eqs. (5.2.24) and (5.2.45) into Eq. (5.2.33) as

c

232

Exact Analysis of Bi-periodic Structures

5

is the odd function of 77 as shown in Fig.5.2.2. When 77 approaches zero, 6 also approaches zero. Recalling Eq. (5.2.44), it is concluded that if kc decreases or yo increases or A,,,, increases, the mode will approach a strongly localized state and if yo is greaterlless than zero, then 6 is lesslgreater than zero; i.e., the displacements of two adjacent subsystems are of oppositelsame signs in the corresponding localized motion mode. The localized level of the modes is dependent on not only the structural parameters kc and yo but also the amplitude A,,,, related to the initial condition or the total energy of the system. AIS,Ocan be determined by applying the energy conservation to the considered system. Ignoring the O ( E )term, the conservation of energy can be expressed as

where E denotes the total energy of system. Inserting Eq. (5.2.47) into Eq. (5.2.49) yields

Substituting Eq. (5.2.48) into the above equation results in

Recalling Eq. (5.2.44), the solution for A : , of Eq. (5.2.51) can be found as

which indicates that when the structural parameters K, kc and yo are given, in

Nearly Periodic Systems with Nonlinear Disorders

233

order that a symmetric localized mode would occur, however small amplitude, the 16k, K energy constant E must be greater than -. If the system energy is below the 31.1 required level, it is impossible to realize any localized mode with symmetry. The stability analysis of the symmetric localized mode shown in Eq. (5.2.47) is investigated by superposing the small perturbation on the mode solution and applying the Floquet theory [24]. It is concluded that when < 2&, the

1

ltl

symmetric localized mode is unstable and when Iql t 2 f i , the mode is stable. (11) The second set of equations shown in Eqs. (5.2.40a7b) describes the antisymmetric localized mode about the longitudinal centre line. Inserting Eqs. (5.2.40a) and (5.2.36) into Eq. (5.2.40b) yields

Recalling the definition of R shown in Eq. (5.2.24), Eq. (5.2.53) becomes

It is concluded that if -1 < 77 < 0 (yo < 0), wowl < 0 (w < w,) , there is an antisymmetric localized mode with lower frequency (w < w,) and if 4kc (w > w,) , there is an anti-symmetric localized mode 0 < q < 1(yo > 0), wowl > M with higher frequency (w > w,) . The amplitudes for the anti-symmetric localized mode can be obtained by substituting Eqs. (5.2.34) and (5.2.40a) into Eq. (5.2.25) as

which indicates that all of the subsystems are motionless except for the disordered

234

Exact Analysis of Bi-periodic Structures

one. This is a strongly localized mode. The amplitudes for the disordered subsystem can also be determined by using the conservation law of energy shown in Eq. (5.2.49), namely

E A:,, = -

K

which indicates that any amount of energy E, however small, is sufficient to cause one anti-symmetric localized mode. By applying the Floquet theory to stability analysis for the anti-symmetric localized mode shown in Eq. (5.2.55), one comes to the conclusion that when < 1 , the anti-symmetric localized mode is stable. If 2 1 , the anti-symmetric localized mode does not exist. (111) The third set of equations, i.e., Eqs. (5.2.41a,b), describes the nonsyrnmetric localized mode. Eq. (5.2.41a) minus and plus Eq. (5.2.41b) yields

Ivl

14

and

A:,,, + 4 ,=

ass

a:

-Pi

respectively. Substituting Eq. (5.2.36) into Eqs. (5.2.57a;b) results in

Recalling Eq. (5.2.30a) and multiplying the left (right) side of Eq. (5.2.58b) by the left (right) of Eq. (5.2.58a), we have

Nearly Periodic Systems with Nonlinear Disorders

235

Introducing the nondimensional mode parameter

and eliminating A,,,, from Eqs. (5.2.59) and (5.2.60) results in

Recalling Eq. (5.2.44), the above equation can be rewritten as

which indicates that the amplitude ratio R must be greater than zero, i.e., A,,,, and

A,,,, are of the same signs. Considering Eqs. (5.2.30a) and (5.2.44), dividing both sides of Eq. (5.2.58a) by A:,, gives

The solution for R of Eq. (5.2.62) can be expressed as

which leads to

Noting that the sum of two terms in the round brackets on the right side of Eq. (5.2.64) is always greater than two, it is concluded that if y , < 0,

236

Exact Analysis of Bi-periodic Structures

4k wowl < 0 (w < w,) and if yo > 0, wowl > L ( w > w,) , i.e., w lies in the stop M band. For arbitrary positive value of R, the nonsymmetric localized mode is in existence. Replacing R in Eq. (5.2.64) by R-' , wowl is invariable. Therefore when all of the structural parameters are given, there are two nonsymmetric localized modes with the same frequency and different amplitude ratio. All of the amplitudes for the nonsyrnmetric localized mode can be found by substituting Eqs. (5.2.34) and (5.2.32b) into Eq. (5.2.25) as

and

where

The attenuation constant 5 has been defined as shown in Eq. (5.2.33), but R i n . Eq. (5.2.33) must correspond to nonsymmetric localized mode. Substituting Eq. (5.2.63) into Eq. (5.2.33) results in

It is obvious that

is always less than one.

Nearly Periodic Systems with Nonlinear Disorders

237

Eqs. (5.2.65a-d) indicate that the nonsymmetric localized mode may be divided into symmetric and anti-symmetric components, the amplitudes of the symmetric component decay exponentially on either side of the nonlinear subsystem but the anti-symmetric mode component is confined to the disordered one with nonzero amplitude. According to the parameter equation shown in Eqs. (5.2.67) and (5.2.61b), the 6 - 7 curve can be obtained as shown in Fig. 5.2.2. Substituting Eqs. (5.2.65a-d) into the equation of energy conservation, i.e., Eq. (5.2.49), results in

mode 3

Figure 5.2.2

5

versus 17 curves

238

Exact Analysis of Bi-periodic Structures

Recalling Eqs. (5.2.66), (5.2.60), (5.2.61a) and (5.2.67), the above equation can be expressed in terms of R as

The nonsymrnetric localized mode for arbitrary value of R is stable. The conditions for each type of localized mode to occur are shown in Fig. 5.2.3 where the longitudinal coordinate T,I is redefined as

A

................. .. ...unstable

b

-3

-2

-1

1

(I

2

3

mode 2

Figure 5.2.3

A2,,0 versus q

AlS.O

curves

Nearly Periodic Systems with Nonlinear Disoders

with A,

= max(~~,,,~I, IA,,~I) , instead

239

of Eq. (5.2.44). A pair of nonsymmetric

localized modes with A,,, l A,,, being equal to R and R-I , are corresponding to the same value of 77 defined in Eq. (5.2.70). According to the new definition of 77 shown in Eq. (5.2.70), R should be less than or equal to one in Eq. (5.2.61b) and

l ~ ( ,=,2~f i . When 14 > 2 f i , the

nonsymmetric localized mode does not exist. 5.3

Damped Periodic Systems with One Nonlinear Disorder [23]

Consider the system shown in Fig. 5.3.l(a) which consists of n number of subsystems connected to each other by means of a linear spring having stiffness &kc. Each subsystem is made up of a mass M connected to both a dashpot with a nondimensional damping coefficient €6, and a spring with linear stiffness K (for x ~ one), where ordered subsystems) or nonlinear stifmess K + ~ ~ (for, disordered E is a positive small parameter. In Fig. 5.3.l(a), s denotes the ordinal number of the disordered subsystem and x j denotes the longitudinal displacement of the j-th mass. In order to apply the U-transformation to uncouple the linear terms of the governing equation, an equivalent system with cyclic periodicity must be created. It is necessary to extend the original system by its symmetrical image and apply the antisymmetric loading on the corresponding extended part as shown in Fig. 5.3.l(b) in which the first and last (2n-th) masses are imaginarily jointed by a spring with stiffness &kc. This imaginary spring is not subjected to any load for antisymmetric vibration. If and only if the dynamic response of the extended system is antisymmetric, two extreme end conditions of the original system are satisfied in the extended one, i.e., the extended system is equivalent to the original one. The response of the first half (i.e., substructures 1 n ) of the equivalent system is the same as that of the original system.

-

5.3.1

Forced Vibration Equation

Applying Newton's second law to every mass in the equivalent system, one can write the differential equations of motion as follows

(a) Original system

Centre line

(b) Equivalent system

Figure 5.3.1 Damped periodic system with a nonlinear disorder

Nearly Periodic Systems with Nonlinear Disorders

241

and

where the superior dot denotes the derivative with respect to the time variable t, w, denotes the natural frequency for the single ordered subsystem and x ~ , , +=~xI,xO= xZn due to cyclic periodicity, EFjo denotes the amplitude of the harmonic force acting on the j-th mass and R denotes the driving frequency, &yo is the coefficient of the cubic term of the nonlinear stiffness in the disordered subsystem. The external excitation for the equivalent system must satisfy the antisymmetry condition, i.e.,

-

where F,,, F,,, indicate the real excitation acting on the original system. If the initial conditions are antisymmetric, then the dynamic displacements are also antisymmetric, i.e.,

One can now apply the U-transformation to the governing equation (5.3.1). The U- and inverse U- transformations may be expressed as

and

242

Exact Analysis of Bi-periodic Structures

with y = - and i = f i , where 2n denotes the total number of subsystems for n the equivalent system. Noting that the displacements are always real variables, it can be proved that the generalized displacements q, (m = 1,2;-.,2n) have the following property iT

and qn , qzn must be real variables, in which the superior bar denotes complex conjugation. By using the U-transformation, i.e., premultiplying both sides of Eq. (5.3.1) by 1 2n the operator - e'"-""* , Eq. (5.3.1) becomes

4%

j=I

where

w m ( m = 0,1,2,. n - 1) is the (m+l)-th natural frequency for the undamped periodic system without any disorder. The lower and upper bounds of the pass band can be expressed respectively as a * ,

Nearly Periodic @stems with Nonlinear Disorders

243

Introducing the time substitution

Rt=r+v into Eq. (5.3.7) results in

and V, =

2 wm -

+

n2 a2

E

2k,(1- cosm y) M R ~

where the prime symbol designates differentiation with respect to the new time variable z and y, is an unhown phase angle. Consider now the case of primary resonance, i.e., R = wo . By letting

Eq. (5.3.13) can be written as

where

77,=770+

2k, (1 - cos m v / ) MR2

Inserting Eq. (5.3.15) in Eq. (5.3.12), gives

4: + q m= G m

m = 1,2,...,2n

244

Exact Analysis of Bi-periodic Structures

in which 1

I

cos(r + a,) -cos(r --)myyox:(q1 ,q2,...,q2n) 2

Gm=

According to the perturbation method, we seek a solution of Eq. (5.3.17) in the form of power series in & not only for q m ( r ),but also for a,. Hence, let q m ( r )= q m 0 ( r + ) E q m l ( ~+)& 7 , z ( r ) +...

(5.3.19)

p = p o + & P I+ E2 p2 +.-.

(5.3.20)

Eq. (5.3.19) is equivalent to x j ( r )=

x ~ ~ ( ~ ) + E x ~ ~ ( ~ ) + E ~ x ~ ~ ( ~ ) + (5.3.21) . . .

with

Cei(i-~mrqmp

1 x ( r )= -

J2n

2n

r

= 0,1,2,--.

(5.3.22)

,=I

Substituting Eqs. (5.3.19) and (5.3.20) into Eqs. (5.3.17) and (5.3.18), the coefficients of equal powers of & on both sides of Eq. (5.3.17) must be equal, i.e.,

Nearly Periodic Systems with Nonlinear Disorders

245

5.3.2 Perturbation Solution

The solution of Eq. (5.3.23a) may be expressed as qmo= a,, cos 7 + b,, sin 7

m

-

-

= 1,2;..,2n

(5.3.24)

-

with a2,-,,, = urn,,and b2,-,,, =b,,, due to q2,-,,,,o= q,,,, where a,, and b,, ( m = 1,2,. ..,2n ) are complex constants to be determined. The physical displacements corresponding to q,, shown in Eq. (5.3.24) can be obtained from Eq. (5.3.22) with ~0 as

xj,

= Ajo cos r

+ Bjosin 7

j

= 1,2,-..,2n

(5.3.25a)

where

Ajo and Bjo are real numbers and A,,-,,, = A,, , B,,-,,, = B j o , which lead to -

.

X 2 n - j , ~- n j ~

Without loss of generality, we can assume that the initial velocity for the

246

Exact Analysis of Bi-periodic Structures

disordered subsystem is zero besides the antisymmetry for both initial displacement and velocity, which leads to

Bso = 0

(5.3.26)

xS0 = Aso cos r

(5.3.27)

and

In order to prevent secular terms, the coefficients of cos T and sin T on the right side of Eq. (5.3.23b) must be zero. Introducing Eqs. (5.3.24) and (5.3.27) into Eq. (5.3.23b), letting the coefficients of cos r and sin r be equal to zero, give 2eiimv

I

[(g

MD2G

1 Fjo cos(j - -)m y cos 9, - cos(s - -)m y 2A; 2 " 4 2

I

Consider now a specific loading condition as that there is no excitation acting on each subsystem except the disordered one, i.e.,

Fjo = 0

j +s

and

j=1,2;-.,n

(5.3.29a)

Inserting Eq. (5.3.29) into Eq. (5.3.28), the solution for am, and bmo of simultaneous equations (5.3.28a7b)can be expressed as

Nearly Periodic Systems with Nonlinear Disorders

1 2 a,,,, = --e 2kc

I

i-mv

fi

b m"

- 1 2kc

2 -e2

J2n

i'mv

1 cos(s - $rn

( D + l - c ~ s m y ) I ,+CI,

247

(5.3.30a)

( ~ + 1 - c o s m y )+~c 2

1 CI, -(D+l-cosmy)I, COS(S---)my (5.3.30b) 2 (D+I-cos~~)~+c~

in which

C and D are two nondimensional parameters. They are dependent on the nondirnensional frequency, stifiess and damping constant, i.e.,

fi 3 and w, ' K

EC, . In Eqs. (5.3.31a) and (5.3.31b), A,, and p, are unknown variables. Substituting Eq. (5.3.30) into Eqs. (5.3.25b) and (5.3.25~)results in

where

248

Exact Analysis of Bi-periodic Structures

Consider now the s-th set of simultaneous equations in Eq. (5.3.32). Inserting j=s and Eq. (5.3.26) in Eq. (5.3.32), yields

Noting the definitions of I, and I , shown in Eqs. (5.3.31a,b), Eqs. (5.3.34a,b) may be rewritten as

F,, sin po = 2kcPsAs, a,'+ P,' From the above equation, we can find the phase angle with zero-order approximation as 2kc Ps

po = tan-'

3

%as + q ~ o ~ : o(a: +

a:

and the frequency response curve as

in which a, and

P,

are dependent on 0 . They can be expressed as

Nearly Periodic Systems with Nonlinear Disorders

249

where C and D are dependent on R l w o besides the structural parameters as shown in Eqs. (5.3.31~)and (5.3.31d). If the parameters of the system and loading are given, the response As, for the loaded subsystem can be calculated from Eq. (5.3.37), and the other Ajo and Bjo can be obtained by substituting Eq. (5.3.34) into Eq. (5.3.32) as

R The characteristic of the frequency response ( (Aso--) curve is similar to

I

W0

that for the single nonlinear subsystem, i.e., the jump phenomenon may occur. For = 30 the specific case of &yo = 0.2 , E<, = 0.2 , Ek, = 0.25, K = 2.5 and with n approaching inf~nity,the frequency response curve is as shown in Fig. 5.3.2. Introducing Eqs. (5.3.27) and (5.3.24) into Eq. (5.3.23b), noting the coefficients of cosz and sin z on the right side of Eq. (5.3.23b) vanishing, yields ilm,

4:1+4,1

= - 2e

Ma2

J2n

1 Yo A; cos 3r cos(s - -)my 2 4

(5.3.40)

The solution for q,, of Eq. (5.3.40) can be expressed as

qml= amlcos z

+ b,,

sin z +

1

1

eiimv cos(s - -)m y cos3r

6 &~

~

~2

(5.3.41)

I

Figure 5.3.2 The frequency response ( ] A , , versus

R ) curve Wo

Substituting Eq. (5.3.41) into Eq. (5.3.22) with r = 1 results in I

Y,A:, I =ei(j-T)mv x,, = A , , c o s r + BJ.I sinr+-1 6 ~ 2n ~ ' j

where

= 1,2;..,2n

1 cos(s - -)m 2

y/

cos 31

Nearly Periodic Systems with Nonlinear Disorders

251

Noting the initial velocity vanishing for the nonlinear subsystem and 2n 1 cos2(s - -)m y I n ,inserting j = s into Eq. (5.3.42) gives 2 m=l

1

yoA" cos 3r xS1= As, cos z + 32MQ2

and

Substituting Eqs. (5.3.41), (5.3.44) and (5.3.27) into Eq. (5.3.23~)and letting the coefficients of cos z and sin z on the right side of Eq. (5.3.23~)be equal to zero gives I 1 2elimVcos(s - -)m a,, = 2kc& 2

v

.I +my 1 1 b,, = 2e COS(S - -)m 2kCG 2

v

(D+I-COS~~)I,*+CI; ( D + ~ - c o s ~ Y+ )c ~2 CI,* - ( ~ + l - c o s m ~ ) I f

( ~ + l - c o s m y )+~c 2

in which

If

= PI cos Po

F,O

Substituting Eq. (5.3.46) into Eq. (5.3.43) results in

(5.3.46a)

(5.3.46b)

252

Exact Analysis of Bi-periodic Structures

in which the definitions of ajand Pj are as shown in Eq. (5.3.33). Recalling Eq. (5.3.45), inserting j = s into Eq. (5.3.48) yields

Introducing Eq. (5.3.47) into Eq. (5.3.49) gives

Inserting Eq. (5.3.49) into Eq. (5.3.48) yields

The forced response with first-order-approximation can be obtained by substituting Eqs. (5.3.25a), (5.3.39), (5.3.42) and (5.3.51) into Eq. (5.3.21), as

Nearly Periodic Systems with Nonlinear Disorders

X . I

=

ajas +Pips (A,,

a:

+ G q S l ) ~+0 Pjas ~ ~ - a Ps j

+P E2

af + P,'

(A,, + &,I

253

sin

with z=SZt-(q,+&p,). Consider now the sum of series in the square brackets as follows

Introducing the above result into Eq. (5.3.52), noting a,,-,+, = a j and

P2n-j+l= Bj , yields

j

= 1,2;-.,n

and

j

;t s

(5.3.54a)

It is obvious that the forced vibration shown in Eqs. (5.3.54a,b) satisfies the antisymmetric condition shown in Eq. (5.3.4), i.e., the solution x j ( j = 1,2;..,n) is applicable to the original system. In Eq. (5.3.54a), a j and Pj are dependent on the total number of subsystems besides the parameters C and D. Consider now the periodic system with infinite number of subsystems. By letting n approach mfinity, the limit of the series

254

Exact Analysis of Bi-periodic Structures

summation on the right sides of Eqs. (5.3.33a) and (5.3.33b) become the definite integral [5] respectively, i.e.,

and

When a finite periodic system is considered, i.e., n is a finite number, a j and

pj can be expressed exactly as the series forms shown in Eq. (5.3.33). The series form can be regarded as the rectangular integration formula for the definite integral shown in Eqs. (5.3.55a7b),where the integration interval [O,27r] is divided into 2n subintervals, i.e., each subinterval is y . If the integral form is adopted instead of the series form, the error is in agreement with that for rectangular integral formula, i.e., 0 ( n - I ) . Generally, there are infinite subsystems between the disordered subsystem and the extreme one at infinity, i.e., ( j+ s - 1) is an infinite number. Introducing the Riemann lemma into Eqs. (5.3.55a7b)yields D+1-cose 1 cos k~ dB, k = 0,+1,_+2,-.. (5.3.56a) 2~ o (D+I-COSB)~ +c2

a,+,= -

C 1 cos k0 d0, k 2a ( D + ~ - c o s ~+ )c ~2

1 p,+k = -

2R

o

= 0,+1,+_2,

. - 0

(5.3.56b)

The above definite integrals can be expressed in terms of elementary hnctions, such as

Nearly Periodic Systems with Nonlinear Disorders

255

a,-, = a,,, = [(D + 1)2+ c2]Eo- (D + 1)E, - 1

(5.3.57b)

Ps-l = Ps+I = CEI

(5.3.57~)

where

cos I3 d 8 (~+l-cosB)~+~~

All of a,,, and Eo and E l .

P,,,

(k = 1,2,...) can be expressed as the linear combination of

5.3.3 Localized Property of the Forced Vibration Mode

The periodic response with zero-order approximation shown in Eq. (5.3.25a) can be written as x, = A,,

COST+ B,, sinz = X, cos(z-8,)

j = 1,2,...,oo

(5.3.59)

256

Exact Analysis of Bi-periodic Structures

where

It is clear that X , = A,,

and

Os = 0

(5.3.61)

Because of a,- = a,, and P,-j = P , + j ,we have Xs-j=Xs+j

and

8,-j=8,+j

(5.3.62)

which indicate the symmetry of the forced vibration about the nonlinear subsystem. The localized level of the mode is dependent on the attenuation rate of the amplitudes. Let

and

118, indicates the phase difference between the corresponding displacements in (s + k) -th and (s + k - 1) -th subsystems. 5, and AOk are only dependent on three nondimensional parameters, i.e., Q/w, , &,/K and EC,, and independent from the nonlinear parameter q,.The numerical results are given as shown in Tables 5.3.1 and 5.3.2. The accurate numerical results show that

Nearly Periodic Systems with Nonlinear Disorders

257

which are in agreement with those obtained from the linear periodic system. By using the results shown in Eqs. (5.3.64a) and (5.3.64b), Eq. (5.3.59) can be written as

(61 is always less than one except that E<, = 0 and R lies in the pass band as shown in Table 5.3.1. The case of 161= 1 represents that the corresponding mode is not localized. Moreover let us consider the forced vibration with first-order approximation. By using the above results, Eqs. (5.3.54a,b) can be expressed as

where r = R t - (p, + E V ))~ and S,, denotes the Kronecker symbol. For the zero-order approximation shown in Eq. (5.3.65), the amplitudes decay exponentially on either side of the nonlinear disorder and for the first order approximation shown in Eq. (5.3.66), the same conclusion can be obtained except the nonlinear subsystem. From the above property, the forced mode for the considered system looks like the linear one. On the other hand, the frequency response curve for every subsystem has the nonlinear property as shown in Fig. 5.3.2. Finally, let us discuss the results shown in Tables 5.3.1 and 5.3.2. Table 5.3.1 leads to the conclusion: (1) when R lies at far from the pass band, 5 is much less than one and the effect of damping on 6 is very weak; (2) when R lies in the pass band and E<, is very small, 4 is approximately equal to one but 5 decreases rapidly with increasing E<, . From Table 5.3.2, one comes to the

2dc conclusion that (3) when n2= W: (1 + 2) , AQ, is identically equal to 90°, K which can be proved mathematically as follows: Inserting Q' = o : ( l + s ) into K Eqs. (5.3.31d) and (5.3.58b) yield D = -1 and El = 0 , leading to a, = 0 and

Ps+,= 0 , then introducing these results into Eq. (5.3.60b) gives

A0, E 0,+,= 90°,

258

Exact Analysis of Bi-periodic Structures

which is independent of the damping; (4) when fi2 < wi(1+ -2Ac ) , A8, increases K 2Ac with increasing E<,,; and when R2 >@;(I+-), AO, decreases with K increasing EL,,. Table 5.3.1 (a) Attenuation constant

(,

(4, ) for -- 0.1

" The numerical results in the round brackets denote 6, ; bCaseof R = w , ; Case of R = w, .

K

Nearly Periodic Systems with Nonlinear Disorders

Table 5.3.1 (b) Attenuation constant

6, (6,) for

&k = 0.1 K

" The numerical results in the round brackets denote 5, ; b C a ~ e o fR = w , ; ' Case of SZ = w, .

259

260

Exact Analysis of Bi-periodic Structures

Table 5.3.2 (a) Difference of phase angles AB, (AB, ) (degree) for

" The numerical results in the round brackets denote d B , ; Case of R = w, ; Case of R = a,.

&k = 0.1 K

Nearly Periodic Systems with Nonlinear Disorders

Table 5.3.2

(b) Difference of phase angles AO, (AO,) (degree) for

" The numerical results in the round brackets denote A@, ; case of R = w , ; Case of R = w, .

&k K

261

3 = 0.1

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REFERENCES H.C. Chan, C.W. Cai and Y.K. Cheung, Exact analysis of structures with periodicity using U-transformation. World Scientific, Singapore, 1998. C.W.Cai, Y.K.Cheung and H.C.Chan, Uncoupling of dynamic equations for periodic structures, Journal of Sound and Vibration, 139(2), 253-263, 1990. C.W.Cai and F.G.Wu, On the vibration of rotationally periodic structures, Acta ScientiarumNaturalium Universitatis Sunyatseni, 22(3), 1-9, 1983. C.W.Cai, On analytical solution of rotationally periodic structures, Acta Scientiarum Naturalium Universitatis Sunyatseni, 25(2), 64-67, 1986. C.W.Cai, Y.K.Cheung and H.C.Chan, Dynamic response of infinite continuous beams subjected to a moving force -- an exact method, Journal of Sound and Vibration, 123(3), 461-472, 1988. Y.K.Cheung, H.C.Chan and C.W.Cai, Exact method for static analysis of periodic structures, Journal of Engineering Mechanics, ASCE, 115(2), 4 15434,1989. C.W.Cai, Y.K.Cheung and H.C. Chan, Transverse vibration analysis of plane trusses by analytical method, Journal of Sound and Vibration, 133(1), 139-150, 1989. H.C.Chan, C.W.Cai and Y.K.Cheung, A static solution of stiffened plates, Thin-Walled Structures, 11,291-303, 1991. H.C. Chan and C.W. Cai, Dynamics of nearly periodic structures, Journal of Sound and Vibration, 213(1), 89-106,1998. C.W. Cai, H.C. Chan and Y.K. Cheung, Exact method for static and natural vibration analyses of bi-periodic structures, Journal of Engineering Mechanics, ASCE, 124(8), 836-841,1998. C.W. Cai, H.C. Chan and J.K. Liu, Analytical solution for plane trusses with equidistant supports, Journal of Engineering Mechanics, ASCE, 126(4), 333339,2000. C.W. Cai, J.K. Liu, F.T.K. Au and L.G. Tham, Dynamic analysis of continuous plane trusses with equidistant supports, Journal of Sound and Vibration, 246(1), 157-174,2001. H.C.Chan, C.W.Cai and Y.K.Cheung, Moments and deflections of simply supported rectangular grids -- an exact method, International Journal of Space Structures, 4(3), 163-173, 1989. H.C.Chan, C.W.Cai and Y.K.Cheung, An analflcal method for static analysis of double layer grids, International Journal of Space Structures, 4(2), 107-116, 1989. Y.K.Cheung, H.C.Chan and C.W.Cai, Natural vibration analysis of rectangular

264

Exact Analysis of Bi-periodic Structures

networks, International Journal of Space Structures, 3(3), 139-152, 1988. 16. Y.K.Cheung, H.C.Chan and C.W.Cai, Dynamic response of orthogonal cable networks subjected to a moving force, Journal of Sound and Vibration, 156(2), 337-347, 1992. 17. H.C.Chan, C.W.Cai and Y.K.Cheung, Convergence studies of dynamic analysis by using the finite element method with lumped mass matrix, Journal of Sound and Vibration, 165(2), 193-207, 1993. 18. H.C. Chan, C.W. Cai and J.K. Liu, Exact static solution of grillwork with periodic supports, Journal of Engineering Mechanics, ASCE, 126(5), 480-487, 2000. 19. H.C. Chan, Y.K. Cheung and C.W. Cai, Exact solution for vibration analysis of rectangular cable networks with periodically distributed supports, Journal of Sound and Vibration, 218(1), 29-44, 1998. 20. C.W.Cai, Y.K.Cheung and H.C.Chan, Mode localization phenomena in nearly periodic systems, Journal of Applied Mechanics, ASME, 62(1), 141-149, 1995. 21. C.W. Cai, H.C. Chan and Y.K. Cheung, Localized modes in periodic systems with nonlinear disorders, Journal of Applied Mechanics, ASME, 64(4), 940945,1997. 22. C.W. Cai, H.C. Chan and Y.K. Cheung, Localized modes in a two-degreecoupling periodic system with a nonlinear disordered subsystem, Chaos, Solitons and Fractals, 11(10), 1481-1492,2000. 23. H.C. Chan, C.W. Cai and Y.K. Cheung, Forced vibration analysis for damped periodic systems with one nonlinear disorder, Journal of Applied Mechanics, ASME, 67(1), 2000. 24. A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations, John Wiley and Sons, New York, 1979.

NOMENCLATURE loading vector unit matrix stiffness matrix mass matrix U-matrix [see Eqs. (1.1.8a, b)] displacement vector elementary cyclic matrix [see Eq. (1.1.12)] flexural rigidity of beam loading vectors for subsystems j and ( j , k) ,respectively generalized loads for systems with cyclic periodicity in one and two directions, respectively imaginary unit generalized displacements for systems with cyclic periodicity in one and two directions, respectively displacement vectors for subsystems j and ( j , k ) , respectively periods of cyclic bi-periodic system attenuation constant for localized modes vibration frequency lower and upper bounds of pass band, respectively potential energy of whole system potential energy of subsystems j and ( j , k) , respectively transposed matrix of [ ] inverse matrix of [ ] second derivative of ( ) with respect to time variable t complex conjugate of ( )

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INDEX

attenuation constant, 23 1,236 attenuation rate, 21 1,214,256

bi-periodic structure (s), 15,27, 81, 125 bi-periodic system (s), 30,44, 59 bi-periodicity, 27,45,69

damped periodic system, 203 diagonal matrix, 5 Dirac delta function, 117 disordered subsystem, 203, 204, 219, 220,234,239,24 1,246,254 double U-transformation, 13, 14,21, 23, 24, 125, 128, 132, 137, 140, 141, 153, 161, 162, 165-167, 172, 182,185,187, 188,190

C cable network (s), 125 complex conjugation, 3, 84, 159, 181, 207,242 concentrated load, 94, 121, 169, 170, 191,192 continuity condition, 86,87,162,183 continuous beam, 115 continuous truss, 8 1,99, 103, 104, 106,107 cyclic bi-periodic equation, 15, 18, 20,26 cyclic bi-periodic system, 34, 45, 69, 80,82,126, 159,175 cyclic bi-periodicity, 27, 45, 60, 71, 80,83, 115, 125, 158, 177,204 cyclic matrix, 4 cyclic periodic equation, 2, 16, 128 cyclic periodicity, 2-5, 11-15, 18,21, 24,27,30, 35,40,44,48,49, 84, 86,91,92, 101, 117, 120, 125, 127, 128, 132, 152, 157, 158, 161, 167,182, 188, 189,220,239,241

eigenvalue, 9 eigenvectors, 9 energy conservation, 21 1,232,237 equilibrium equation, 14,28, 29,30, 34,88,96,99, 116, 128, 129, 139, 159,163,173, 184,196 equivalent system, 27, 44, 45, 60, 64, 67, 69, 72, 91, 108, 109, 115-117, 119, 121, 125, 128, 141, 149, 181, 192,239,241,242

forced vibration, 39, 107, 203,253, 256,257 frequency equation, 36,37,51-53,55, 63, 75-77, 80, 100, 102, 103, 106, 142-144,221 frequency response, 58, 69, 113, 155, 248-250,257

268

Exact Analysis of Bi-periodic Structures

generalized displacement, 7,9, 29, 38, 39, 53, 64, 67, 77, 87, 100, 108, 119,129, 143,189,191,224,242 governing equation, 11, 12, 15,27, 31,45,90, 92,204,210,212, 229, 239,241 grid, 175, 176, 187, 191, 192, 196 grillwork, 156-159, 161, 164, 169, 172,173

harmonic load (s), 40, 107, 112, 149, 151,153 Hermiltian matrix, 6

influence coefficient (s), 30, 35 inverse U-transformation, 7, 12, 18, 28,86

jump phenomenon, 249

linear periodic structure, 27 linear periodic system, 44,212,257 localized mode (s), 203, 204, 208, 210-217, 219, 220, 228-231, 233, 234,236-239 L-P method, 203,205,220,223

mass-spring system (s), 27, 45, 58, 203 mode subspace, 10,27

natural frequency, 9-11,34,39,44, 58,67,76,79, 100, 115, 142, 143, 145,146,221,222,241,242 natural mode, 9, 10,37-39,44,53-55, 64, 65, 77,79, 80, 102, 106, 142, 145 natural vibration, 9, 34,44, 99, 125, 139 nonlinear disorder, 203, 211, 214, 229,23 1,240,257

orthogonal basis, 3

pass band, 11, 104-106,208,222, 242,257 periodic structure, 1,2, 3, 7,9, 11,27, 81,84,125 periodicity, 4, 15,20,27,86, 125, 128,205 phase difference, 3, 10, 11,256 potential energy, 84, 87, 159, 163, 181,184

Index

Q quasi-diagonal matrix, 4

R resonance, 58, 113, 115, 155,243 rotating mode, 3, 10, 11

stability analysis, 215, 216,233,234 stable mode, 2 18 static problem, 115 stiffening girder (s), 175, 179-181, 185, 187-189, 191, 192, 196 stiffness matrix, 84, 85, 159, 160, 179 stop band, 208,228,236

269

U U-matrix, 2 , 3 unit matrix, 3,4,20, 87, 163 unitary space, 3 unstable, 215,216,218,233 U-transformation, 6-8, 11-14, 16-18, 20, 21, 23, 24, 26-28, 30-32, 34, 35,40,41,44,45,48-51, 61, 62, 69, 73, 74, 81, 84, 86, 91, 92, 96, 100, 101, 109, 110, 115, 117, 119, 120, 141, 167, 175,203,206,207, 220,221,239,241,242

Warren truss, 8 1, 94, 102, 112

By using the U-transformation method, it i s possible t o uncouple linear simultaneous equations, either algebraic or differentiat with cyclic periodicity. This book presents a procedure for applying the U-transformation technique twice t o uncouple the two sets of unknown variabies i n a doubly periodic structure t o achieve an analytical exact solution. Expticit exact solutions for the static and dynamic analyses for cerbin engineering structures with doubly periodic properties - such as a 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 - can be found in the book. The avai1abiiit.y of these exact solutions not only helps with the checking of the convergence and accuracy of numerical solutions, b u t also provides a basis for optimization design for these types of

structures. The study of the force vibration and mode shape of periodic systems with nonLinear disorder i s yet another research area which has attained considerable success by the U-transformation method. This book illustrates the analytical approach and procedure for the problems of locahzation of the mode shape of nearly periodfc systems together with the results. l

World Scientific WWW.w~rldscientifi~~ corn 4940 hc

Exact Analysis of

Bi-Periodic