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

Author-Highlights
• Viscoelastic Vibrating Systems can be approximated by certain viscous approach named Equivalent Viscous Model (EVM).
• The EVM shares oscillatory (complex) modes with the original viscoelastic systems and neglect non-viscous (real) modes contribution.
• The EVM eigenvectors are those of the undamped problems, and these lead to a proportional system.
• EVM of real systems is in agreement with VEM in terms of the FRF, especially for proportional or lightly non-proportional damping.

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.