Critical relationships in nonviscous systems with proportional damping

Abstract

[EN] Materials with time-dependent dissipative behavior currently play an important role in the design of new mechanisms for vibration control in civil, automotive, aeronautical and mechanical engineering. Damping forces are assumed to depend on the past history of the velocity response via convolution integrals over multiple exponential hereditary kernels. Hence, the computational complexity increases both in time and frequency domain with respect to the widely used viscous models. The derivations of this article are carried out under the hypothesis of nonviscous proportional damping, that is, the time-dependent damping matrix becomes diagonal in the modal space of the undamped system. In this context, the damping parameters, which control the behavior of dissipative forces, will be considered as variable. Critical manifolds can be defined as hypersurfaces in the domain of the damping parameters, which represent boundaries between oscillatory and non-oscillatory motion. In particular, critical manifolds of two parameters are the so-called critical curves. In this paper a new method to obtain critical curves in proportionally damped nonviscous multiple degree-of-freedom systems is presented. It is proved that the conditions of critical damping lead to relationships that enables an analytical determination of such critical curves, in parametric form. In addtion, it is demonstrated that modal critical regions arise as the intersection of the critical curves. The proposed method is validated through two numerical examples involving discrete and continuous system with generalized proportional damping. (C) 2020 Elsevier Ltd. All rights reserved.