A Numerov–Galerkin Framework for the Transient Dynamics of Anisotropic Plates on Vlasov Foundations

In this study, a high-order Galerkin–Numerov approach is presented to solve the transient vibration problem of anisotropic Kirchhoff plates supported by a uniform Vlasov foundation. A discretization of the governing fourth-order plate equation is derived based on a mixed boundary value problem and a hybrid Hermite–sine Galerkin formulation, which maintains the C1-continuity properties of classical plate theory. The resulting reduced-order modal system is integrated in time with the Numerov scheme, which is fourth-order accurate, and has a small numerical dispersion and good phase-preserving properties for oscillatory dynamics. The proposed methodology is evaluated using stability and convergence tests and parametric investigations. The fourth-order temporal convergence and rapid spectral-like spatial convergence of the numerical results are validated, and the long-time accuracy and robustness of the formulation is confirmed by the negligible phase error and bounded energy drift. The results from the parametric study indicate that the thickness of the plates and the stiffness of the Winkler foundation are the two most important mechanisms for vibration suppression, while the orthotropic coupling and the Vlasov shear interaction have substantial effects on the modal redistribution and transient deformation properties. The proposed method is compared with the conventional lower-order integration schemes, and it is observed that the method gives better phase fidelity and computational efficiency, and it is possible to predict the vibration amplitude and vibration timing accurately. In addition to the numerical benefits, the framework also provided physical insights on the coupled effect of anisotropy, foundation interaction and boundary restraint. The suggested model is directly applicable for composite floor systems, aerospace panels, foundation supported slabs, biomechanical plate analogs, etc., and smart vibration control platforms. This work thus lays the groundwork for future studies of nonlinear behavior, adaptive foundations and digital twin simulation of structural systems and presents a strong and scalable computational tool for the study of plate–foundation dynamics.