An Explicit Time-Domain Solution for Nonlinear Harmonic Vibration Analysis of Mindlin Viscoelastic Plates

Abstract

This study examines the geometrically nonlinear vibration of viscoelastic Mindlin plates under harmonic transverse loading. A novel time-domain semi-analytical formulation is developed to simultaneously capture both transient and steady-state responses of nonlinear viscoelastic systems. The model incorporates von Karman strain–displacement relations and employs the Boltzmann superposition principle with a two-term Prony series to characterize linear viscoelastic behavior under the assumption of a constant bulk modulus. Through the Laplace–Carson transformation and a new inverse approximation, a simplified time-domain representation is derived. Nonlinear frequencies are estimated using a pseudo-frequency-based approach, while nonlinear viscous damping is determined iteratively via eigenvalue analysis. The steady-state response is obtained through the homotopy perturbation method in conjunction with harmonic balance. Moreover, a novel closed-form time-domain expression is derived for the total response, explicitly capturing resonance, higher-order harmonic effects, and viscoelastic decay in a unified formulation. To the best of the author’s knowledge, such a comprehensive time-domain function has not been previously reported in the literature for geometrically nonlinear vibration of viscoelastic plates.