Free Vibration Analysis of a Circular Plate with Multiple Circular Holes by Using the Multipole Trefftz Method

Abstract

This paper presents the multipole Trefftz method to derive an analytical model describing the free vibration of a circular plate with multiple circular holes. Based on the addition theorem, the solution of multipoles centered at each circle can be expressed in terms of multipoles centered at one circle, where boundary conditions are specified. In this way, a coupled infinite system of simultaneous linear algebraic equations is derived for the circular plate with multiple holes. The direct searching approach is employed in the truncated finite system to determine the natural frequencies by using the singular value decomposition (SVD). After determining the unknown coefficients of the multipole representation for the displacement field, the corresponding natural modes are determined. Some numerical eigensolutions are presented and further utilized to explain some physical phenomenon such as the dynamic stress concentration. No spurious eigensolutions are found in the proposed formulation. Excellent accuracy, fast rate of convergence and high computational efficiency are the main features of the present method thanks to the analytical procedure.