On Uncoupling and Solving the Equations of Motion of Vibrating Linear Discrete Systems

Abstract

The equations of motion governing the vibrations of a linear, viscously damped, discrete system are generally mutually coupled. This article examines the problem, when the viscous damping is nonclassical, of how best to uncouple and solve by approximation the governing second-order differential equations of motion. It is shown that when the equations of motion are expressed in normal coordinates, the equations can then be transformed by an orthogonal coordinate transformation to a new generalized coordinate system in which a bound on the relative error introduced in the response by discarding all the coupling terms is a minimum. This approach extends the applicability of undamped modal analysis to certain types of nonclassically damped systems. The analytical results and the effectiveness of the proposed method are illustrated with four examples taken from other previously published approaches to the stated problem.