Periodic geometrically nonlinear free vibrations of circular plates

The geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi-degree-of-freedom model, which is based on thin plate theory and on Von Kármánʹs nonlinear strain–displacement relations. The middle plane in-plane displacements are included in the formulation and the common axisymmetry restriction is not imposed. The equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in-plane and transverse displacement fields are given by weighted series of spatial functions. These spatial functions are based on hierarchical sets of polynomials, which have been successfully used in p-version finite elements for beams and rectangular plates, and on trigonometric functions. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. Convergence with the number of shape functions and of harmonics is analysed. The numerical results obtained are presented and compared with available published results; it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom. Internal resonances are found and the ensuing multimodal oscillations are described.