An Innovated Vibration Equation for Longitudinal Plate by Using the Symmetric and Asymmetric Spectral Decomposition

Thick wall structures involving longitudinal wave are typically utilized in aerospace engineering, nuclear power engineering, precision transmission device design, and pressure vessels design. Consequently, developing sophisticated dynamic models for thick plates is of paramount importance. However, the commonly used longitudinal vibration equation is of the second order, which is regarded as a plane stress problem. Its dispersion curve is a straight line, which cannot describe the actual dispersion in the plate. In this paper, the spectral analysis of Navier equation describing three-dimensional elasto-dynamics is carried out by using the symmetric and asymmetric spectral decomposition theory of differential operators and introducing the concept of virtual differential operators. The infinite product operator series describing longitudinal vibration are truncated into fourth order. The governing equation of longitudinal vibration consists of a fourth-order wave equation and a second-order wave equation. Owing to the fact that no a priori assumptions were introduced during the derivation of its dynamic equations, the proposed plate dynamic model boasts higher precision and is applicable across a broader frequency spectrum and for plates with greater thicknesses. This is a breakthrough in the longitudinal vibration equation of plates.