Bifurcate solutions of free vibration to a circular plate for various boundary conditions

The nonlinear dynamic equation of free vibration to a circular plate for various boundary conditions, with the geometric nonlinear effects taken into account. By using Galerkin's method, the governing partial differential equation was reduced to three standard types of Duffing's equation. Two harmonics method is presented for strongly nonlinear dynamic-system. An oscillation system which is described as a second order ordinary differential equation, is expressed as a set of non-linear algebraic equations with a frequency and amplitudes as the independent variables. Considering binding equation of initial conditions, they constitutes a complete set of non-linear algebraic equations with a frequency and amplitudes as the independent variables. Using Maple program, the algebraic equations can be solved conveniently. The results show that two harmonics method can solve both a symmetry problem of free vibration and a non-symmetry problem of free vibration. The backbone curve and bifurcation diagram about amplitude-frequency of Duffing-Holmes equation were gained. Via contrast it can be clearly seen that the results are believed to be rather general and complete.