The Transient Response of Flat Plates Using the Finite Element Method

Abstract

In the finite element method a discrete system is derived to approximate a continuum. A discretizing method is justified for this investigation by comparing the dynamic analysis of a discrete system with that of a plate continuum using the variational calculus. The transient response of the former may be determined by solving a matrix eigenvalue problem followed by modal analysis. In contrast the equation of motion of a plate is derived as a fourth order partial differential equation. The finite element method is described with particular reference to plate bending by considering two rectangular elements and one triangular element. The equation of motion for free vibration of the approximate system is in the form of a matrix eigenvalue problem indicating that the plate has been discretized. The digital computer is ideally suited for the manipulation of large matrices and the necessary programming is explained. Results are presented of the natural frequencies determined for several plate systems and in every case the value given for fundamental frequency is satisfactory. For higher mode frequencies discretizing with the rectangular elements gives the most accurate results and accuracy improves as the number of degrees of freedom of the approximate system increases. Satisfactory results are also obtained for frequencies of a clamped circular plate discretized using the triangular element. Computer graphical output showing the response of a rectangular cantilever plate is compared with the results of practical tests. Close agreement is obtained during the first cycle of the lowest frequency component. The computer predicted response of a clamped circular plate is less accurate but the maximum amplitude is less than 10% high. The necessity for dynamic analysis is demonstrated by a numerical investigation of the effect of reduction in the rise time of the force. Higher mode frequencies are excited and cause the peak amplitudes.