Two Harmonics Method for Strongly Nonlinear Duffing Equation

Two harmonics method is presented for strongly nonlinear dynamic-system.In a periodic oscillation,the periodic solutions can be expressed in the form of basic harmonics and bifurcate harmonics.Thus,an oscillation system which is described as a second order ordinary differential equation,can be expressed as a set of non-linear algebraic equations with a frequency and amplitudes as the independent variables using Ritz-Galerkin's method.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.Two examples are given by two harmonics method.In example one,the results are compared with analytic method.In example two,the results are compared with the numerical integration method.The examples are given at end of this paper.In example one,the phase trajectories of Duffing equation are computed for a hardening spring,a softening spring,and Ueda genre.The result agrees very well with the numerical integration method.