Instability inspection of parametric vibrating rectangular Mindlin plates lying on Winkler foundations under periodic loading of moving masses

Parametric resonance is one of the most important issues in the study of dynamical behavior of structures. In this paper, dynamic instability of a moderately thick rectangular plate on an elastic foundation is investigated in the case of parametric and external resonances due to periodic passage of moving masses. The governing coupled partial differential equations (PDEs) of the system, with consideration of the first-order shear deformation theory (FSDT) or Mindlin plate theory, are presented and they are reduced to a set of ordinary differential equations (ODEs) with time-dependent coefficients using the Galerkin procedure. All inertial components of the moving masses are adopted in the dynamical formulation. Instability survey is carried out for three different loading trajectories considerably interested in many practical applications of the issue, i.e. rectilinear, diagonal and orbiting trajectories. In order to analyze the resonance conditions, the incremental harmonic balance (IHB) method is introduced to calculate instability boundaries, as well as external resonance curves in parameters plane. A comprehensive study is done to assess effects of thickness ratio and foundation stiffness on the resonance conditions. It is found that an increase in the plate’s thickness ratio leads to a reduction in values of critical parameters. Moreover, it is observed that increasing the foundation stiffness moves the instability regions and resonance curves to higher frequencies of the moving masses and also leads to further stability of the parametrically excited system at lower frequencies. Time response simulations done via Runge–Kutta method confirmed the results predicted by IHB method.