Asymptotic Analysis Of A Forced Oscillation Model In A Single Mechnical System

In this paper, we study a simple model of elastic vibration, which is presented by Arnold and Case. It can show many nonlinear phenomenons related to physical process. The object is analyzed and a nonlinear equation is presented as The initial value condition isIt brings great difficulty to obtain an exact solution of the above differential equation because of the nonlinear feature. In order to simplify the problem, a series of transformation is made and approximate formulation of the original problem is as followsIt shows that approximate equation is the type of Duffing Equation. Firstly, perturbation theory is applied to analyze the Duffing problem and a asymptotic expression can be determined. Then, what we will consider is to introduce the solution to the original problem and discuss the validity. The cases of ?2?(?)1-? and ?2=(?)1-? are analyzed when ?<1 in this paper.The system will not display the behaviour of resonances when the parameters satisfy ?2?(?)1-?. In this circumstance, the zero-order asymptotic solution of auxiliary problem is obtained using the multiple scales method. Next, we estimate the error between the asymptotic solution and the exact solution according to nonlinear Gronwall inequation. The conclusion is as follows:for arbitrary ?>0 and time variable ?, it exists constants ?0>0 and T>0, so that asymptotic solution can meet the requirement of accuracy for both Duffing problem and original problem when 0<???0 and 0<??<T. In order to improve the precision of solution, higher order approximate solution has been solved similarly and its validity has also been proved.When the parameters satisfy ?2=(?)1-?, the system displays the behaviour of resonances according to Forbes L K121. In this case, it becomes more complex. The Renormalization Group method has been applied to solve the Duffing problem and the approximate solution is proved to be bounded.