Non-linear free vibration of isotropic plates with internal resonance

The geometrically non-linear free vibration of thin isotropic plates is investigated using the hierarchical finite element method (HFEM). Von Karman's non-linear strain–displacement relationships are employed and the middle plane in-plane displacements are included in the model. The equations of motion are developed by applying the principle of virtual work and the harmonic balance method (HBM), and the solutions are determined using a continuation method. The convergence properties of the HFEM and of the HBM are analyzed. Internal resonances are discovered. The variation of the plate's mode shape with the amplitude of vibration and during the period of vibration is demonstrated.

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