An analysis of the conditions under which a damped linear system possesses classical normal modes is presented. It is shown that a necessary and sufficient condition for the existence of classical normal modes is that the damping matrix be diagonalized by the same transformation that uncouples the undamped systems. Sufficient though not necessary conditions on the damping matrix are developed, and it is shown that Rayleigh's solution is a special case of the present theory.IT IS well known that undamped linear dynamic systems possess normal modes, and that in each normal mode the various parts of the system vibrate in the same phase. In damped systems, however, this property is generally violated and classical normal modes do not exist; in such cases the more general treatment of K. A. Foss 1 is required. Rayleigh' showed that if the damping matrix is a linear combination of the stiffness and inertia matrices, the damped system will have classical normal modes. The purpose of this paper is to determine the general conditions under which a damped dynamic system possesses classical normal modes. It will be shown that a necessary and sufficient condition that a damped dynamic system possess classical normal modes is that the damping matrix be diagonalized by the same transformation which uncouples the undamped s3'stem.
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