Study On Several Nonlinear Properties Of Weakly Nonlinear Vibration Equations

In this paper,several nonlinear properties of weakly nonlinear oscillation equations are studied by using the multi-scale method.The first chapter introduces the research background,domestic and foreign research progress and research purpose of this article.In Chapter 2,we study the multi-wave initial value problem for the weakly nonlinear Klein-Gordon wave equation.By using the multi-scale method and introducing the formal asymptotic expansion of the solution,the quantitative relationship that the first approximate solution of the equation depend on the velocity of wave propagation is obtained.The results show that the propagation velocity(phase velocity)of wave is larger than that of wave propagation deduced by single initial wave.Finally,Mathematica is used for numerical simulation,and the simulation results show that the multi-scale method is efficient.On the basis of the research in the second chapter,the third chapter discusses the problem of single plane wave and double wave initial value with three-sided weak nonlinear terms and five weak nonlinear equations.In the first section,the problem of individual plane wave initial values with three-way weak nonlinear terms and five weak nonlinear equations is considered,and the frequency range characteristics of the first approximation solution and wave propagation of the equation are obtained by means of multiple-scale methods.In the second section,considering the problem of double-wave initial values with three-way weak nonlinear terms and five weak nonlinear terms,similarly,the first approximate solution of the equation is obtained by using multi-scale method,and the results show that the frequency characteristics of the solution become more complex due to nonlinear factors when multi-wave propagation.In Chapter 4,Duffing-van der Pol oscillation equation with cubic weak nonlinear term and external excitation term is studied.By using the multi-scale method and introducing the formal asymptotic expansion of the solution,the first approximate solution of the equation,the dependence of the amplitude and nonlinear effect of the system,and the dependence of the amplitude and frequency parameters of the system are obtained.