The Universal Program To Solve The Strongly Nonlinear Vibration System And Application

As theoretical basis of undetermined fundamental frequency method, normal form theory is one of the important tools for simplification and dimensionality reduction of high dimensional dynamical systems. The emergence of undetermined fundamental frequency method extended the application of normal form effectively, and it has also obtained great achievements in static and dynamic bifurcation analysis in strongly nonlinear vibration systems, and static and dynamic bifurcation analysis in low-dimensional systems was carried out as future work. However, the undetermined fundamental frequency method has many unsolved problems in solving steady-state asymptotic solution in systems with multi-degree of freedom and in non-classical mechanical systems with engineering background, mainly includes, the research object mainly focused on classical dynamical systems with only cubic nonlinear terms rather than universal systems with extensive nonlinear terms, which restricted application of theory in practical engineering background. And the calculation of coefficients of near-identity nonlinear transformations in solving process of normal form was repetitive and lack of universal procedure. Therefore, in order to deal with those two above-mentioned problems, the research contents and achievements in this paper are as follows:⑴The universal solving procedure to calculate steady-state asymptotic solution of strongly nonlinear vibration systems with multi-degree of freedom was presented, by using the undetermined fundamental frequency method and computer algebra Mathematica. The procedure was divided into many computing units through introduction of modular thinking, and procedure interfaces were set up between every two units, which obtained data transmission between units of selecting parameters, calculating normal form, getting steady-state solution and so on. The procedure enhanced researchers'operation speed in analyzing complex problem efficiently, and laid a good foundation for further study of stability behaviors and bifurcation characteristics in high-dimensional complex systems. Steady-state asymptotic solution and phase diagrams of some strongly nonlinear systems with single degree of freedom, two and three degrees of freedom were achieved by using this procedure, which had better agreement with the solution of numerical integration more than the traditional normal form approach without undecided fundamental frequencies.⑵The procedure also be used to analyze strongly nonlinear vibration of a class of coupled double-walled carbon nanotube with two degree of freedom, and then the influence of exciting force and viscous damping coefficient on vibration amplitude were discussed. Finally the results of quantitative analysis obtained by procedure, numerical results and initial normal form results were compared to verify the accuracy of the programming methodology. And it also reflected the effect of universal design thoughts in improving the efficiency of undetermined fundamental frequency method to analyze complex problems, which was helpful to promote the use of dynamic theory in problems with practical background.