Parametrically Excited Instability of Periodic Visco-Elastomer Sandwich Plate with Supported Masses under Quadrilateral Longitudinal Harmonic Excitations

The parametrically excited instability of the spatially periodic visco-elastomer sandwich plate with supported masses under quadrilateral longitudinal temporal harmonic excitations is studied. The improvability of the parametrically excited instability by periodic distribution parameters is explored. The direct eigenvalue analysis approach for solving the parametrically excited stability problem of the periodic sandwich plate system under longitudinal harmonic excitations is proposed. The spatial periodic distribution of facial layer thickness and core layer modulus of the sandwich plate is considered. The non-linear partial differential equations of longitudinal and transverse coupling motions of the periodic visco-elastomer sandwich plate with supported masses under biaxial longitudinal boundary excitations are derived. The longitudinal displacements of the sandwich plate are separated into two parts and the longitudinal boundary excitations relevant to symmetric part are incorporated into the sandwich plate system. Then the partial differential equations with boundary excitations are converted into parametrically excited system equations and further converted into ordinary differential equations with time-varying parameters, which describe the parametrically excited vibration with multi-mode coupling of the periodic sandwich plate system. The fundamental perturbation solution to the equations is expressed as the product of periodic and exponential parts based on the Floquet theorem. The ordinary differential equations with harmonic parameters are converted into a set of algebraic equations using the harmonic balance method. Then the parametrically excited instability of the periodic sandwich plate system is determined directly by matrix eigenvalues. The overall instability characteristics of parametrically excited vibration with multi-mode coupling of the system under longitudinal harmonic excitations are illustrated by numerical results on unstable regions. The parametrically excited instability can be improved by the spatially periodic distribution of geometrical and physical parameters. The proposed approach is applicable to general sandwich structures with spatial distribution parameters in multi-mode-coupling parametrically excited vibrations for overall instability analysis on continuous frequency band.