Letter to the Editor

A novel formulation of the receptance matrix of non-proportionally damped dynamic systems

The calculation of the inverse of the summation of matrices is an operation that one comes across frequently in different scientific disciplines. One of the most known examples is the receptance matrix which plays a very important role in the mechanical vibration area.

As is known the receptance matrix (also called the frequency response matrix) is an important matrix which interrelates the input and output of a damped linear discrete mechanical system which is subject to harmonical forcing as input. There are many publications in the literature on this subject. Some of the recent publications are Refs. [1], [2], [3]. Yang presented in Ref. [1] an exact method for evaluating the receptances of non-proportionally damped dynamic systems. Based on a decomposition of the damping matrix, an iteration procedure is developed which does not require matrix inversion. In Ref. [2], Lin and Lim developed a new and effective method to derive structural design sensitivities which include both frequency response function sensitivities and eigenvalue and eigenvector sensitivities from limited vibration test data. The study of Mottershead [3] was concerned with the zeros of structural frequency response functions and their sensitivities.

The recent study in Ref. [4] is concerned with a viscously damped linear mechanical system, the co-ordinates of which are assumed to be subject to several constraint equations. The frequency response matrix of the constrained system described above is established in terms of the frequency response matrix of the unconstrained system and the coefficient vectors of the constraint equations.

This study has been motivated by Yang's article [1]. In his paper, an iterative method was developed for the calculation of the receptance matrix when the damping matrix was decomposed into the sum of dyadic products. It was specifically pointed out that there was no need to applying an inverse operation of matrices during the iteration procedure. In Yang's method, first the iteration process starts with the receptance matrix of the undamped system. Then the application of iteration, using as many iterations as the number of dyadic products in the damping matrix, gives the receptance matrix of the damped system. In getting to the final form of the iteration equation, a formula which is known as the Sherman–Morrison formula [5] in the matrix literature, used for obtaining the inverse of the summation of a regular matrix, and a dyadic product was also utilized.

The aim of this study is to show that it is possible to obtain the receptance matrix directly without using the iterations which were used in Ref. [1]. In this frame, a more general procedure of obtaining the inverse of the sum of a regular matrix and any symmetric, positive semi-definite matrix, is taken into account. Subsequently, a new formula is given to obtain the inverse of the general problem mentioned before. The present formulation does not require any iterations but it needs one more inverse operation in addition to Yang's procedure. Actually, both Ref. [1] and the present procedure require the inverse operation for obtaining the receptance matrix of the undamped system. The new formulation is based on the fact that a symmetric and positive semi-definite matrix can be expressed as the sum of dyadic products [6].

As it is known, the Sherman–Morrison formula is useful in obtaining the inverse of a regular matrix and only one dyadic product (i.e., rank 1). On the other hand, the Woodbury formula gives the inverse of the sum of a regular matrix and a matrix product whose rank can be grater than 1 (rank r⩾1) [5].

The new methodology formulated in this study makes it possible to put any number of dyadic products by expressing it in terms of a matrix product (r⩾1) into a form where the Woodbury formula can be used.

In the following section, after a brief introduction, the procedure followed in Ref. [1] will be summarized, both, from the point of completeness and that of clarity for the readers’ understanding. In Section 3, the derivation of the new formula which was mentioned above will be given. In the last section, the calculation of receptance matrix by using the new formula without iteration will be given.