A novel mathematical method to analyze the free vibration of eccentric annular plates

Abstract

Vibration analysis of nominally axisymmetric plate structures with either imperfections or geometric asymmetries due to practical motivations is of interest in designing and developing some mechanical structures. Semi-analytical methods to model such structures suffer from either choosing inappropriate admissible functions or both plausible convergence issues and additional computations owing to employing the addition theorem of Bessel functions. Therefore, the present study aims at developing a new mathematical method to analyze the vibrational behavior of circular plates with geometric asymmetries. The suggested approach makes use of the separation of variables to determine general solutions of the partial differential equation of the plate transverse displacement while defining multiple polar coordinate systems each of which offers a formulation of the plate deformation. Moreover, closed-form geometric equations and the chain rule for determining derivatives are implemented to move from one coordinate system to the other to satisfy boundary conditions without any need for the cumbersome transformation involved in using the addition theorem. A finite element model is also constructed to evaluate the validity of the proposed method before studying the effects of the cutout location and size on natural frequencies and mode shapes of eccentric annular plates.