Cantilevered, Rectangular Plate Dynamics by Finite Difference Methods

TL;DR

This paper models the dynamics of a cantilevered rectangular plate using finite difference methods, incorporating damping and complex boundary conditions, with simulations and energy computations provided.

Contribution

It introduces a finite difference approach for simulating the complex dynamics of a rectangular plate with mixed boundary conditions and damping.

Findings

Successful numerical simulations of plate dynamics

Effective handling of high-order PDE with mixed boundary conditions

Energy analysis of the plate's motion

Abstract

In this technical note, we consider a dynamic linear, cantilevered rectangular plate. The evolutionary PDE model is given by the fourth order plate dynamics (via the spatial biharmonic operator) with clamped-free-free-free boundary conditions. We additionally consider damping/dissipation terms, as well as non-conservative lower order terms arising in various applications. Dynamical numerical simulations are achieved by way of a finite difference spatial approximation with a MATLAB time integrator. The rectangular geometry allows the use of standard 2D spatial finite differences, while the high spatial order of the problem and mixed clamped-free type boundary conditions present challenges. Dynamic energies are also computed. The relevant code is presented, with discussion of the model and context.