Inverse Operator Method And Its Application In Nonlinear Dynamic Analysis Of Mechanical System

ABSTRACTThe inverse operator method and its application in nonlinear dynamic analysis of mechanical systems are studied in this paper.Firstly, by adopting the thought of Adomian's decomposition method, the generic dynamic model in mechanical systems are transformed into a standard first-orderdifferential-equations, and then the inverse operator method (TOM) for the approximate analytic solution of nonlinear mechanical system is developed based on the exact solution in form. In accordance with the characteristics of the nonlinear mechanical models, the inverse operator method which can be used directly for the higher-order-differential-equations is presented hereby. The convergence of IOM is proved, which established the theoretical foundation of the method.Secondly, the symbolic-numeric (S-N) method on the bases of the IOM is proposed for the first time in this paper. The one-step-inverse-operator-method (IOM- 1) and the improved IOM- 1 which are simple and practicable are therefore derived. In such a way that the prices time integration algorithm (P11) becomes a special case in the S-N method.Thirdly, the three typical nonlinear mechanical systems are investigated by using TOM-i method. The investigation of the geared rotor-bearing system with clearance non-linearities shows the period-doubling route to chaos. The investigation regarding the dynamic responses of a nonlinear cam-follower systems to several input functions shows that the non-linearities of spring has little influence to the kinematics factors of the dynamic responses. The S-N method for the solution of flexible multibody dynamic equations is demonstrated, and the numerical results on two experimental models shows that TOM- 1 method is of high accuracy and high efficiency for solving nonlinear stiff equations.Finally, the periodic solutions of a self-excited vibration systems with nonlinear damping, the van der Pol equation and the undamped Duffing equation are discussed. At the preliminary stage for analyzing the periodic responses, the saw-tooth time transformation and triangular time transformation are employed. And thereafter, by using IOM, the analytical approximate solution of the transformed systems are developed.