ISOSPECTRAL VIBRATING SYSTEMS

G. M. L. Gladwell

Department of Civil Engineering, University of Waterloo,
Waterloo, Ontario, Canada N2L 3G1
E-mail: ggladwell@uwaterloo. ca

Abstract

Two vibrating systems are said to be isospectral if they have the same natural frequencies. This paper reviews some recent results on isospectral conservative (i. e., undamped) discrete vibrating systems. The paper centres around two ways of creating isospectral systems: by QR factorisation with a shift, and by using the concept of isospectral flow. Both these procedures are illustrated by using FEM models.

Keywords: vibration, isospectral, QR factorisation, isospectral flow

1. Introduction

An undamped vibrating system has certain frequencies, called natural frequencies, at which it can vibrate freely, without the application of forces. An actual physical system has theoretically an infinity of such frequencies. A model of such a system may be either continuous or discrete, having respectively an infinity or a finite number of natural frequencies; in this paper we consider only discrete systems. Two systems with the same set of natural frequencies, spectrum, are said to be isospectral. In general, the spectrum of a system mirrors the system, but does not specify it completely: there can be many systems, an isospectral family, with the same spectrum. There are two broad classifications of problems relating to a system and its spectrum. In inverse problems, one attempts to construct a system with a given spectrum; in isospectral problems, one attempts to find another system, maybe a family of systems, having the same spectrum as a given system. In some ways, isospectral problems are easier than inverse problems: at least one is sure that there exists at least one system, the given system, with the specified spectrum; this is not always the case with inverse problems. For an in-depth discussion of inverse and isospectral problems, see Gladwell (2004).

Modelling of a physical system is usually done by means of some finite element method (FEM): the system is treated as a set of elements, connected in some way. Each element is specified by a set of generalised displacements, an element stiffness matrix Ke, and an element mass (inertia) matrix Me. In the process of assembling the elements, one constructs overall or global stiffness and mass matrices, K and M, and assembles the element displacements into a displacement vector u. The natural frequencies of the system appear as (the square roots, as = АД, of) the eigenvalues (A.;)" of the generalised eigenvalue problem

(K – AM)u = 0. (1)

A system is thus defined by a pair of matrices (K, M). We say that two systems (K, M) and (K’, M’) are isospectral if (K’ – AM’)u’ = 0 has the same spectrum of eigenvalues as (1).

In practice, the matrices K, M have specific forms. Let Mn denote the set of square matrices of order n, and Sn denote the subset of symmetric matrices. If the system is conservative, then

31

M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 31-38.

© 2006 Springer. Printed in the Netherlands.

Sx

•——— •—– •— … •——— •

1

Подпись: Fig. 1. The matrix J lies on the path Gt-

2 3 -1 n

K, M є Sn, K is positive semi-definite (PSD), i. e., uTKu > 0; M is positive definite (PD), i. e., uTMu > 0, for all u = 0.

The matrices K, M, and in particular their structure, i. e., the pattern of zero and non-zero entries, will depend on the choices of finite elements, and on how these elements are connected. It is con­venient to use concepts from graph theory: a (simple, undirected) graph G is a set of vertices Pi in a vertex set V, connected by edges (i, j)(= (j, i)) in an edge set E.

A matrix A є Sn is said to lie on G if aij = 0 whenever (i, j) Є E. For example, a symmetric tridiagonal matrix, sometimes called a Jacobi matrix J, and written

a b b a2 b2

Подпись: (2)J=

. bn—1 bn—1 an

lies on a graph G1 that is a path with V = {1, 2,… ,n} and E = {(1, 2), (2, 3),…, (n — 1, n)}, as shown in Figure 1.

Подпись: (3)

Подпись: a1 b2 b3 b4 b2 a2 0 0 b3 0 a3 0 b4 0 0 a4 Подпись: a1 b1 0 b4 b1 a2 b2 0 0 b2 a3 b3 b4 0 b3 a4

Figure 2 shows two simple graphs: a star, G2, and a ring, G3. The matrices A2 and A3 lie on G 2 , G 3 respectively.

The matrix A2 is an example of a bordered matrix, A3 is called a periodic Jacobi matrix. A Jacobi matrix is a particular case of a band matrix – it is a symmetric matrix with bandwidth 1. An important

Подпись: A4 ISOSPECTRAL VIBRATING SYSTEMS Подпись: (4)

subset of band matrices is the set of staircase matrices, an example of which is shown in (4); A4 lies of the graph $4.

The general isospectral problem is this: Given a system (K, M) with K, M є Sn, and K, M, lying on a graph G, find another (or all) isospectral system(s) (K’, M’) with K’, M’ lying on the same graph G.

This problem is very difficult, and is still open. To simplify it somewhat, we suppose that the systems have lumped mass, so that M, M’ are diagonal. In that case, if K is PD (PSD) then the Xi will be positive (non-negative) and K’ will be PD (PSD).

In practice, the problem is even more difficult because, instead of being just PD (PSD), K will have to satisfy other, usually positivity constraints, that state that the system is physically realisable. We need some more concepts from matrix theory.

Suppose A є Mn. Let a = {, i2,…, ik} be a sequence of k numbers taken from {1, 2,… ,n}. The submatrix of A with rows taken from a = {i1, i2,…, ik}, and columns taken from в = {j1, j2,…, jk} is denoted by A(afi). The determinnant

det(A(a|e)) = A(a; в)

is called a minor of A. A minor A(a; a) is called a principal minor of A. Suppose A є Mn:

— A is totally positive, TP, if all its minors are positive,

— A is totally non-negative, TN, if all its minors are non-negative; A is NTN if it is non-singular and TN,

— A is oscillatory, O, if A is TN and a power of A, AP, is TP.

It may be shown that A is O iff it is NTN, and its immediately off-diagonal entries ai, i+1 and ai+1,i, i = 1, 2,… ,n — 1 are positive, see Gladwell (1998). Note that TP is much stronger than PD: A є Sn is PD iff its principal minors are positive.

Note that the definition of TP, TN and O matrices applies to any matrix in Mn, not just to symmetric matrices, those in Sn. Such matrices have many important properties.

Define Z = diag(+1, —1,…, (—)n-1); the operation A ^ ZAZ = A changes the signs of the entries of A in a chequered pattern.

We list three properties:

— if A, B are O, so is AB,

— A—1 is O iff A is O; we say A is sign-oscillatory, SO,

— if A is O then it has n eigenvalues, and they are positive and distinct.

The last property is particularly important: recall that if A є Sn, all we can say is that it has n real eigenvalues; they may not be distinct. If we know only that A є Mn, then we do not know a priori, how many eigenvalues it has.