Vibration Analysis of Kirchhoff Micro Plate with C2 Continuity

In this article, the analysis of free and forced vibrations of a microplate based on the Kirchhoff gradient strain theory with clamped boundary conditions is presented using the finite element formulation with  continuity. Due to the presence of higher-order derivatives,  continuous interpolation functions are employed to provide a standard formulation. Subsequently, the stiffness and mass matrices of the microplate element are extracted using the Hamiltonian principle, and the formulation is applied to a four-noded quadrilateral element with 36 degrees of freedom. The natural frequencies of the desired microplate are obtained using the finite element method and compared with analytical results. Additionally, by refining the element dimensions (mesh refinement), the accuracy of this method is compared by examining the obtained frequencies with the finite element method and the desired analytical results. Furthermore, the natural frequencies for microplates with different boundary conditions are provided, and the influence of the  ratio on the natural frequencies of microplates with various boundary conditions is investigated.Subsequently, vibration diagrams for the microplate with specified initial conditions and applied forces are presented. The effects of boundary conditions and the  ratio on various microplate parameters are examined and discussed.