Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates

Nonlinear free axisymmetric vibration of simply supported isotropic circular plates is investigated by using the energy method and a multimode approach. In-plane deformation is included in the formulation. Lagrange's equations are used to derive the governing equation of motion. Using the harmonic balance method, the equation of motion is converted into a nonlinear algebraic form. The numerical iterative method of solution adopted here is the so-called linearized updated mode method, which permits the authors to obtain accurate results for vibration amplitudes up to three times the plate thickness. The percentage of participation of each out-of-plane basic function to the deflection shape and to the bending stress at the plate centre and of each in-plane basic function to the membrane stress at the centre are calculated in order to determine the minimum number of in- and out-of-plane basic functions to be used in order to achieve a good accuracy of the model. The nonlinear frequency, the nonlinear fundamental mode shape and their associated nonlinear bending and membrane stresses are determined at large amplitudes of vibration. The numerical results obtained here are presented and compared with available published results, based on various approaches and with the single-mode solution. The limit of validity of the single-mode approach is also investigated.