Geometrical nonlinear vibration of beams and plates by the heirarchical finite element method

The geometrically nonlinear free and steady-state forced vibration of simply-supported and clamped-clamped isotropic beams, and of fully-clamped isotropic and laminated plates, is studied by the hierarchical finite element method (HFEM). Deflections' of the order of the structures' thicknesses, and therefore small rotations, are considered. Von Kármán's nonlinear strain-displacement relationships are employed and the middle plane in-plane displacements are included in the model. The equations of motion in the time domain are developed by applying the principle of virtual work. The high order polynomials that emerge in the HFEM are integrated by symbolic manipulation. The time vibration of the solution is expressed by means of a Fourier series and the harmonic balance method (HBM) is utilised to derive the equations of motion in the frequency domain. These equations are solved by different methods, but mainly by a continuation method, which is capable of calculating stable and unstable solutions, and of passing bifurcation and turning points. The stability of the steady-state solutions is studied by Floquet's theory.The convergence properties of the HFEM, the influences of the number of degrees of freedom and of the middle plane in-plane displacements are discussed. One element only is used to construct the models of the structures analysed. Results are compared with published ones and good agreement is found. It is demonstrated that the HFEM method requires far fewer degrees of freedom than the more common h-version of the FEM. This is a very important advantage in nonlinear analysis, as the time required to solve the nonlinear equations of motion increased significantly with the number of degrees of freedom.The convergence properties of the HBM are investigated. Internal resonances, i.e., modal coupling between modes with resonance frequencies related by a rational number, are discovered. They result in bifurcation points and in associated secondary branches, in loops or complex curvatures of the backbone curve (curve that relates the frequency with the amplitude of vibration) and frequency response curve. The large variation of the mode shape due to internal resonance is shown.