Electric potential response analysis of a piezoelectric shell under random micro-vibration excitations

The response characteristics of a spherically symmetric piezoelectric shell under random boundary micro-vibration excitations are analyzed and calculated. The equation for electric potential is integrated radially to obtain the electric potential as a function of displacement, so that the differential equations for the piezoelectric shell with electrical and mechanical coupling are converted into an equation only for the displacement. The displacement transformation is constructed to convert the random boundary conditions into homogeneous ones, and the transformed displacement is expanded in space to further convert the partial differential equation for the displacement into ordinary differential equations using the Galerkin method. The equations represent a multi-degree-of-freedom dynamic system with an asymmetric stiffness matrix under random micro-vibration excitations. The frequency-response function matrix, power spectral density matrix and correlation function matrix of the system response are derived from these equations based on the theory of random vibration. The expressions of mean-square displacement, stress and electric potential of the piezoelectric shell are finally obtained and illustrated by numerical results for random micro-vibration excitations. The random electrical and mechanical coupling properties, in particular the relations between boundary electric potential responses and micro-displacement excitations, are explored.