Symmetric State-Space Method for a Class of Nonviscously Damped Systems

Abstract

Multiple-degree-of-freedom linear nonviscously damped systems are considered. It is assumed that the nonviscousdampingforcesdependonthepasthistoryofvelocitiesviaconvolutionintegralsoverexponentiallydecaying kernel functions. The traditional state-space approach, well known for viscously damped systems, is extended to such nonviscously damped systems using a set of internal variables. Suitable numerical examples are provided to illustrate the proposed approach. I. Introduction V ISCOUSdampingisthemostcommonmodelforthemodeling of vibration damping in linear systems. This model, e rst introduced by Rayleigh, 1 assumes that the instantaneous generalized velocities are the only relevant variables that determine damping. Viscous damping models are used widely for their simplicity and mathematical convenience, even though the behavior of real structural materials is, at best, poorly mimicked by simple viscous models.Forthisreason,itiswellrecognizedthat,ingeneral,aphysically realistic model of damping will not be viscous. Damping models in whichthedissipativeforcesdependonanyquantityotherthantheinstantaneous generalized velocities are nonviscous damping models. Mathematically, any causal model that makes the energy dissipation functional nonnegative is a possible candidate for a nonviscous damping model. Clearly, a wide range of choice is possible, either based on the physics of the problem or by selecting a model a priori and e tting its parameters from experiments. Here, we will use a particular type of damping model that is not viscous, and throughout the paper the terminology nonviscous damping will refer to this specie c model only. Possiblythemostgeneralwaytomodeldampingwithinthelinear range is to use nonviscous damping models that depend on the past history of motion via convolution integrals over kernel functions. 2 The equations of motion of an N-degree-of-freedom linear system with such damping can be expressed by