The Generalized Harmonic Balance Method For Periodic Solutions Of Dynamical Systems With Strong Non-Linearity

Generally speaking,dynamic systems are nonlinear in practice.There is no general analytical method for nonlinear dynamics,especially for strongly nonlinear dynamic problems.Therefore,approximate analytical methods with high precision and high efficiency are of great important to the research of nonlinear dynamical systems.In this dissertation,the generalized harmonic balance method is used along with the Liapunov first approximation theory.By using c++ to program the nonlinear differential equations of several classes of nonlinear systems,a general solving procedure is obtained.By iterations and stability analysis,the corresponding approximate analytic solutions are obtained.The approximate periodic solutions based on the generalized harmonic balance method are compared with the numerical solutions obtained by MATLAB to verify the validity and generality of the method.The main contents are as follows:Firstly,the undamped free vibration of the strongly nonlinear Duffing system is studied.By applying the generalized harmonic balance method,the averaging equation and the Jacobian matrix are derived,and the stable periodic solutions are compared with the numerical solutions.Secondly,the forced vibration of the strongly nonlinear Duffing system with damping is studied.The approximate analytical solutions with period-m which change with excitation amplitude and frequency are presented in terms of the averaging equation and the Jacobian matrix.The approximate analytical solutions are in agreement with the numerical solutions,which show that the method is highly accurate and reliable.Finally,Mathieu-Duffing system with strong nonlinearity is studied.The approximate analytical solutions with period-m which change with excitation amplitude and frequency are presented in terms of the averaging equation and the Jacobian matrix.The comparison of the approximate analytical solutions and the numerical solutions shows that the method is validity and generality.Through this study,the two major factors that influence the generalized harmonic balance method are found to be harmonic frequency and number of harmonics,and feasible methods are given to choose them.From this,approximate analytical methods for dynamical systems with strong nonlinearity are enriched and progressed.