Free-vibration analysis of compressed clamped circular plates

Small free vibrations of a thin elastic clamped circular plate under a compressive thrust applied at its edges is studied near the nonlinear static deformation state. Based on the solution of von Karman's plate equations for the static state of circular plates, the dynamic-eigenvalue or free-vibration problem is reduced to one of solving three algebraic equations leading to an analytic solution. This is achieved by introducing a coefficient of transformation related to the eigenfunctions and presenting the algebraic homogeneous equations in terms of the eigenvalues. The natural frequency versus the applied load is derived and plotted for various values of the load. The frequency of the small vibrations vanishes at the limit load. The advantage of the proposed analytic method lies in its ability to predict and describe the stability of the static equilibrium configuration and the postbuckling path.