Response Bounds for Nonclassically Damped Mechanical Systems Under Transient Loads

Abstract

Time-decaying upper bounds are derived for the response of damped linear mechanical systems under impulsive loads and under step loads. The bounds are expressed in terms of the extreme eigenvalues of the symmetric, positive definite constituent system matrices. The system is assumed to exhibit nonclassical damping by which we mean that classical normal modes do not occur: i.e., the modes are coupled (complex). The governing system equation is first reduced to a particular version of “state form” suited for application of the one-sided Lipschitz constant. A formal bound for general transient loads is then presented. This is specialized to the case of impulsive loads. For step loading, an overshoot measure is introduced which generalizes the corresponding notion for single degree-of-freedom systems. A bound is derived for the overshoot and for the settling time of the system. A simple example is given.