A viscous approach based on oscillatory eigensolutions for viscoelastically damped vibrating systems

摘要

Dissipative mechanisms in linear multiple-degree-of-freedom viscoelastic systems are characterized by the dependence on the history of the velocity degrees-of-freedom via convolution integrals over kernel functions. As a result, damping matrices are frequency-dependent and, in general, state-space approach based methods are widely used for their resolution. This paper proposes a new approach suitable for modelling such systems via a viscous model with proportional damping, which approximates the response of the original viscoelastic system using the undamped eigenvectors together with the complex eigenvalues with oscillatory nature, neglecting those modes with non-viscous nature (negative real eigenvalues). It is rigorously demonstrated that the transfer function of any viscoelastic system can be expanded as the sum of the transfer function of certain viscous model and a residual term. The obtained viscous model, named Equivalent Viscous Model, is characteristic of each viscoelastic system. In order to study the bound of the residual term two indexes are introduced to describe, on one hand, the non-proportionality, or the modal decoupling capability of the damping matrix and, on the other hand, the damping level of the system. The proposed viscous model is validated through numerical examples simulating different conditions of proportionality and damping.