Nonlinear damping in nonlinear vibrations of rectangular plates: Derivation from viscoelasticity and experimental validation

Abstract

Even if still little known, the most significant nonlinear effect during nonlinear vibrations of continuous systems is the increase of damping with the vibration amplitude. The literature on nonlinear vibrations of beams, shells and plates is huge, but almost entirely dedicated to model the nonlinear stiffness and completely neglecting any damping nonlinearity. Experiments presented in this study show a damping increase of six times with the vibration amplitude. Based on this evidence, the nonlinear damanaping of rectangular plates is derived assuming the material to be viscoelastic, and the constitutive relationship to be governed by the standard linear solid model. The material model is then introduced into a geometrically nonlinear plate theory, carefully considering that the retardation time is a function of the vibration mode shape, exactly as its natural frequency. Then, the equations of motion describing the nonlinear vibrations of rectangular plates are derived by Lagrange equations. Numerical results, obtained by continuation and collocation method, are very successfully compared to experimental results on nonlinear vibrations of a rectangular stainless steel plate, validating the proposed approach. Geometric imperfections, in-plane inertia and multi-harmonic vibration response are included in the plate model.