Two Harmonics Method for Strongly Nonlinear Dynamic Systems

Strongly nonlinear problems are difficult to solve by classical procedures such as perturbation methods,and two harmonics method is presented for them.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.In example one,The periodic solutions of a quintic nonlinear system are studied.The results are compared with the numerical integration method,and the agreements are very good,too.