Frequency equation for the in-plane vibration of a clamped circular plate

The in-plane vibration of the circular plate structure is important in the transmission of high-frequency vibration, but none have produced an exact frequency equation of the in-plane vibration of a clamped circular plate. Therefore, this paper focuses on deriving the frequency equation for the in-plane vibration of the clamped circular plate of uniform thickness with an isotropic material in the elastic range. To derive the frequency equation for the clamped circular plate in the cylindrical coordinate, kinetic and potential energy for in-plane behavior were first obtained by using the stress–strain–displacement expressions and applying Hamilton's principle, which led to two sets of highly coupled differential equations for the equations of motion. Substitution of Helmholtz decomposition for the coupled differential equations produced uncoupled equations of motion. The assumption of a harmonic solution for the uncoupled equations led to wave equations. Using the separation of the variable, the general solutions for the wave equations were obtained. The solutions generated the in-plane displacements in the r and θ directions. Finally, the application of boundary conditions yielded the frequency equation for the clamped circular plate. The derived frequency equation was validated by finite element analysis and by comparison of previously reported results.