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Introduction

Let us consider small transverse oscillations of a linear elastic continuum (e.g., a string or a rod), spread along the x-axis ranging from $ x=a$ to $ x=b$. The natural harmonic oscillation of the continuum is given by:

$\displaystyle y(x,t)= \varphi(x) \sin(pt+\alpha).$    

The function $ y(x,t)$ denotes the deflection at point $ x$ at time $ t$, $ \varphi(x)$ stands for the amplitude at point $ x$, $ \alpha$ for the phase and $ p$ for the frequency of the oscillation. A so-called segmental continuum has the following main oscillation properties (from [83, Introduction]):
  1. All the frequencies $ p$ are simple. (This means that the amplitude function of a given frequency is uniquely determined up to a constant factor.)
  2. At frequency $ p_i$ the oscillation has exactly $ i$ nodes. (Suppose $ x$ to be a node, this means that $ \varphi(x)=0$.)
  3. The nodes of two successive overtones alternate.
  4. When superposing natural oscillations with frequencies $ p_k <
p_l< \ldots < p_m$, the number of sign changes of the deflection fluctuates with time within the limits from $ k$ to $ m$.

Now we will construct a so-called oscillation matrix, and in [83] it is shown that all the properties mentioned above are strictly connected to the properties of the oscillation matrix. Suppose our oscillation system consists of $ n$ masses $ m_1,m_2,\ldots,
m_n$, located at points $ s_1,s_2,\ldots,s_n$. Note that the masses are not fixed but movable in the direction of the $ y$-axis. With $ K(x,s)$ we denote the so-called influence function. It denotes the deflection at point $ x$ under the influence of a unit force at point $ s$. Denote $ a_{ik}=K(x_i,s_k)$. Assume $ y_1,y_2,\ldots,y_n$ to be the deflection of these masses, under the influence of a force:

$\displaystyle -m_k \frac{d^2 y_k}{dt^2} \quad (k=1,2,\ldots,n).$    

We have that the deflection at point $ x$ at time $ t$ can be written as

$\displaystyle y(x,t)=- \sum_{k=1}^{n} K(x,s_k)m_k \frac{d^2 y_k}{dt^2}.$    

For $ x$ equal to $ s_1,s_2,\ldots,s_n$ we have

$\displaystyle y_i=- \sum_{k=1}^{n} a_{ik} m_k \frac{d^2 y_k}{dt^2}.$    

When denoting the amplitude of the deflection as $ u_i=\varphi(s_i)$ in the harmonic oscillation equation we get

$\displaystyle y_i = u_i \sin(pt + \alpha)$    

which leads after differentiation of $ y_i$ to the following system of equations:

$\displaystyle u_i = p^2 \sum_{k=1}^{n} a_{ik} m_k u_k \quad (k=1,2,\ldots,n)$    

which can be rewritten, to obtain the following system of equations:

% latex2html id marker 32210
$\displaystyle \left\{ \begin{array}{rcl} (1-p^2 a_...
...^2 a_{n2} m_2 u_2 - \cdots +(1-p^2 a_{nn} m_n) u_n & = & 0. \end{array} \right.$    

Which has a solution if:

% latex2html id marker 32212
$\displaystyle \det \left( \begin{array}{rrcr} 1-p^...
...n1} m_1 & -p^2 a_{n2} m_2 & \cdots & 1-p^2 a_{nn} m_n \\  \end{array} \right)=0$    

revealing thereby the possible frequencies of the oscillation.

As already mentioned, in [124] Krein observed that the oscillation properties, given above, are closely related to the influence coefficients $ a_{ik}$. Even more, the oscillation properties are related to the fact that all minors (a minor is the determinant of a submatrix of the given matrix) of the matrix $ (a_{ik})$ (of all orders) have to be nonnegative. The theory connected to these matrices is called the theory of oscillation matrices.

Definition 34   An oscillation matrix $ A$ is a matrix such that all minors of all orders of this matrix (principal and nonprincipal) are nonnegative. A minor of $ A$ is the determinant of a square submatrix of $ A$.

Important properties of these oscillation matrices, and their connections with the oscillation properties where investigated in the papers [80,81,82]. Also other authors show their interest in this field of matrices [64,75,76], especially the article [64] contains several references related to positive and nonnegative matrices. The choice of the term oscillation matrix by the authors of the book [83] comes from the following circumstance: (citation from [83], Introduction, page 3)

As soon as for a finite system of points, the matrix of coefficients of influence of a given linear elastic continuum is an oscillation matrix (as it always is in the case for a string or a rod supported at the endpoints in the usual manner), this automatically implies the oscillation properties of the vibration of the continuum, for any distribution of masses at these points.

The theory of oscillation matrices (sometimes also other matrix theories), has an analogue in the theory of integral equations. For the oscillation matrices this corresponds to the theory of the following integral equation:

$\displaystyle \varphi(x) = \lambda \int_a^b K(x,s) \varphi(s) d\sigma(s),$    

with an oscillation kernel $ K(x,s)$.

Definition 35   A kernel $ K(x,s)$ is called an oscillation kernel if for every choice of $ x_1 < x_2 < \ldots < x_n$ in the interval $ [a,b]$, the matrix $ K(x_i,x_k)$, with $ (i,k \in \{1,2,\ldots,n\})$ is an oscillation matrix.

It follows from the definition that an oscillation kernel is characterized by the following inequalities:

This type of integral equation for $ d\sigma=ds$ was investigated in [118,119]. The more general case for a nonsymmetric kernel and an increasing function $ \sigma(s)$ is studied for example in [79]. In Section 3.3 of this chapter we will show that discretization of this integral equation also results in solving a system of equations with a semiseparable matrix as coefficient matrix.

For the purpose of our thesis this introduction into the theory of oscillation matrices is enough. In the remainder of this section we will take a closer look at two types of oscillation matrices: Jacobi matrices and one-pair matrices. Jacobi matrices are a special type of tridiagonal matrices, namely irreducible ones, and one-pair matrices are a special type of semiseparable matrices, namely symmetric generator representable semiseparable matrices of semiseparability rank $ 1$. Also attention will be paid to the most important properties of these matrices, which are of interest in the theory of oscillation matrices.


next up previous contents index
Next: Definition and examples Up: Oscillation matrices Previous: Oscillation matrices   Contents   Index
Raf Vandebril 2004-05-03