The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part II: iterative and explicit analytical solution for non-linear coupled transverse and in-plane vibrations

The objective of this paper is to present a more realistic and complete study of the geometrically non-linear free vibrations of clamped immovable circular plates by taking into account the in-plane deformation, which has not been examined in Part I of this series of papers (J. Sound Vibration 265 (2003) 123). The problem is solved by a numerical iterative procedure in order to obtain more accurate results for vibration amplitudes up to twice the plate thickness. The numerical results thus obtained are presented and compared with the available published results, and with the ones calculated when neglecting the in-plane displacement for the first two non-linear axisymmetric mode shapes. An explicit analytical solution is then presented and its range of validity, for the fundamental non-linear mode, is determined via a detailed comparison with the solution based on the iterative procedure. The results obtained by the explicit method show the usefulness of the new approach in comparison with the single mode approach solution frequently used in non-linear vibration analysis of structures, and are expected to be easy to implement in fatigue models in order to make more realistic and secure predictions of the structural fatigue life.