Mass, Stiffness, and Damping Matrices from Measured Modal Parameters

The theory of complex mode shapes for damped oscillatory mechanical systems is explained, using the matrix of transfer functions in the Laplace domain. These mode shapes are defined to be the solutions to the homogeneous system equation. It is shown that a complete transfer matrix can be constructed once one row or column of it has been measured, and hence that mass, stiffness, and damping matrices corresponding to a lumped equivalent model of the tested structure can also be obtained from the measured data. INTRODUCTION In recent years, there has been considerable activity in the study of elastic structure dynamics,. in an attempt to design structures that will function properly in a hostile vibration environment. Although much of the early work centered around fatigue and life testing, the latest efforts have been directed towards analytical modeling and simulation of mechanical structures. Distributed structures are generally modeled as networks of lumped mechanical elements, in an effort to predict failures more reliably and faster than is afforded by conventional life testing procedures. With the advent of the inexpensive mini-computer, and computing techniques such as the Fast Fourier Transform algorithm, it is now relatively easy to obtain fast, accurate, and complete measurements of the behavior of mechanical structures in various vibration environments. Modal responses of many modes can be measured simultaneously and complex mode shapes can be directly identified instead of relying upon and being constrained by the so called "normal mode" concept. Furthermore, the entire system response matrix, which comprises the mass, stiffness, and damping matrices of the lumped equivalent model, can be measured. The following material covers the theoretical background that is needed to understand these new measurement techniques. COMPLEX MODES AND THE TRANSFER MATRIX Let's assume that the motion of a linear physical system can be described by a set of n simultaneous second order linear differential equations in the time domain, given by Mx Cx Kx f && & + + = (1) where the dots denote differentiation with respect to time. f = f(t) is the applied force vector, and x = x(t) is the resulting displacement vector, while M, C, and K are the (n by n) mass, damping, and stiffness matrices respectively. In this discussion, our attention will be limited to symmetric matrices, and to real element values in M, C, and K. Taking the Laplace transform of the system equations gives B s X s F s ( ) ( ) ( ) = , where (2) B s Ms Cs K ( ) = + + 2 (3) Here, s is the Laplace variable, and now F(s) is the applied force vector and X(s) is the resulting displacement vector in the Laplace domain. B(s) is called the system matrix, and the transfer matrix H(s) is defined as H s B s 1 ( ) ( ) = − (4) which implies that H s F s X s ( ) ( ) ( ) = (5) Each element of the transfer matrix is a transfer function. The elements of B are quadratic functions of s, and since H B 1 = − , it follows that the elements of H are rational fractions in s, with det(B) as the denominator. Thus, H(s) can always be represented in partial fraction form. If it is assumed that the poles of H, i.e. the roots of det(B) = 0, are of unit multiplicity, then H can be expressed as