Journal of Sound and Vibration (1990) 137(3), 433-442 DYNAMIC RESPONSE OF SOME DISSIPATIVE SYSTEMS BY M EA N S OF FUNCTIONS OF MATRICES L. Y. BAHAR AND H. G. KWATNY Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, Pennsylvania 19104, U.S.A. (Received 10 March 1989, and in revised form 20 June 1989) A certain class of linear, viscously damped dynamical systems, the matrix coefficients of which satisfy a certain commutativity condition, are known to exhibit the same normal modes as the ones that exist in the absence of damping. These systems are said to have classical normal modes. In the present study, such systems are shown to be particular cases of dynamical systems that possess a zero commutator, which form a subset of separable systems. The response of the latter is investigated by means of functions of matrices, and its connection with the classical eigenfunction expansion approach is established. The paper continues the authors’ previous work where functions of matrices were used to (a) develop an algorithm for the numerical integration of the equations of motion for such systems, and (b) to establish their conservation laws. 1. INTRODUCTION The dynamic response of linear non-conservative systems has been the subject of intensive research over many years. While it would appear that with such a rich literature, the subject could no longer lend itself to fertile avenues of research, this has not been the case. As examples of recent activity in this field one can cite the work by Newland [l] where new features of the modal analysis for non-conservative linear systems are pointed out, and that of ijzgiiven [2] in which a new method to determine the harmonic response of damped structures using undamped modal data is developed. For a more extensive listing of recent papers in this area, the reader should consult a review article by Nicholson [3] and the references contained therein. The necessity to control linear dynamical systems has brought new concerns into the study of this field, such as stabilizability, controllability and observability. When con- siderations of this nature are added to the older and better understood features, such as combinations between the damping and gyroscopic matrices as well as the stiffness and circulatory matrices on the one hand, and the integration methods based on modal superposition and direct integration on the other, it is not difficult to see that the analysis of linear systems will continue to be a fruitful area of research for some time to come. A text book by Inman [4] that has just appeared gives a lucid expository treatment of most of these subjects in a unifying fashion with an up-to-date list of references. The study to be described here represents a continued effort on the part of the authors to introduce the use of functions of matrices in the investigation of linear dynamical systems. Recent contributions directly related to the work to be pursued in this paper include that of Bahar and Kwatny [5] in which the conservation laws for systems possessing classical normal modes are derived, and Bahar and Law [6], where a numerical algorithm for the integration of the equations of motion governing such systems is developed. In the interest of brevity, the references quoted in references [5] and [6] will not be repeated 433 0022-460X/90/060433 + 10 $03.00/O 0 1990 Academic Press Limited