Reverberation-ray matrix analysis of free vibration of piezoelectric laminates

Based on the theory of first-order ordinary differential equations, a dual relation between two solutions in the dual local coordinates for a single layer in the laminate is derived, which is further arranged in a manner that can avoid the numerical instability usually encountered in the state space method. Joint coupling relation can be established by the consideration of equilibrium and compatibility conditions at the interfaces/surfaces or nodes. The two relations correspond to the phase relation and scattering relation in the method of reverberation-ray matrix (MRRM), as demonstrated by considering the free vibration of orthotropic (or cross-ply) piezoelectric laminates in cylindrical bending. Another contribution of the paper is that the case of repeated eigenvalues of the coefficient matrix of the state equation is discussed, which has never been tackled before. The discussion completes the mathematical formulation of MRRM. The approach is first verified by comparing the results with those obtained from the state space method for laminates with simple supports. Calculations are then performed to show the dominating modes at some given frequencies and particular discussions are presented.