Exact solutions for the in-plane vibrations of rectangular Mindlin plates using Helmholtz decomposition

Deriving frequency equations for in-plane vibration of a rectangular plate with different boundary conditions and uniform thickness in the elastic range is the goal of this research. To derive frequency equations, the kinetic and potential energy for in-plane behavior initially are obtained by using the stress-strain-displacement expressions according to the theory of Mindlin plates in Cartesian coordinates by applying the Hamilton's principle, which leads to five sets of highly coupled differential equations for the equations of motion. Replacement of Helmholtz decomposition for the coupled differential equations creates uncoupled equations of motion. The hypothesis of a harmonic solution for the uncoupled equations lead to wave equations. The general solutions for the wave equations are obtained by using the separation of variables. Finally, the application of boundary conditions yields the frequency equations for the rectangular plate. The natural frequencies are compared and validated by finite element analysis and previously reported results.