Vibration of viscously damped linear dynamic systems

Abstract

A general theory of vibration of damped linear dynamic systems is given. The limitations on the use of the usual normal mode theory in determining the response of damped systems were first studied systematically by Caughey when he derived necessary and sufficient conditions for the uncoupling of systems in N-space. Systems which cannot be uncoupled in N-space may still be solvable by modal methods on transforming them to 2N-space and using the results of Foss. However there exist systems which cannot be solved by the usual modal techniques in either N-space or 2N-space. Such systems which include some passive physically realizable systems require the general theory for a complete determination of their motion. For weakly coupled systems the simple perturbation analysis presented gives surprisingly accurate approximations to the actual response of the systems. In any design problem questions of stability arise, particularly when dealing with nor, symmetric systems, and therefore a discussion on the stability of these systems is given. The second part of the thesis is concerned with linear continuous systems. Exactly solvable continuous systems are rare and in general recourse must be had to numerical methods. The interchangeability of the differential and integral formulation of continuous systems is noted. As in the discrete systems constructive necessary and sufficient conditions are derived for a damped system to possess the same set of complete eigenfunctions as the undamped system. In the discretization of continuous systems the main problem of practical interest is the error bounds on the solution of these discrete approximations when compared to the exact solution. Unfortunately the literature is very poor in this area but what is known is applied to the systems under discussion.