Dynamic Research Of Strongly Nonlinear Duffing Oscillators By The Homotopy Method (Ham)

The strongly nonlinear Duffing type system model is a typical nonlinear vibration system model. The nonlinear Duffing equation which is obtained from the typical model describes the resonance and chaotic phenomenon. As a simple mathematical model it is often used to study harmonic vibration, quasi periodic vibration and strange attractor. In science and engineering many nonlinear vibration problems can be transformed into the Duffing equation to research. From a certain point, the research about strongly nonlinear Duffing type system is the foundation to study many other complex dynamics. Its important significance has attracted a large number of scholars. As an example, we solve the generalized Duffing equation in form of u+u+α3u3+α5u5+α7u7+…+αnun=0(n is odd) by using the homotopy analysis method (HAM) in this study. The values of odd n in the generalized Duffing equation mean different mathematical models in numerous disciplines. For any arbitrary power of odd n, both the frequencies ω and periodic solutions u(t) of the generalized nonlinear Duffing equation can be explicitly and analytically formulated by the HAM. Furthermore the optimized homotopy analysis method is used to speed up the convergence of the series solution.This paper will discuss the nonlinear Duffing equations for n=5and n=7respectively and obtain the frequencies co and periodic solutions u(t) for the nonlinear Duffing equations by the HAM. Based on some numerical simulation, we obtain the numerical results and draw some tables and figures. The numerical results by the HAM are compared with exact integration and published analytical solutions to verify the accuracy and correctness of this approach. By solving the practical problems in engineering it shows the strong advantages of applying the HAM to strongly nonlinear Duffing systems. Besides, the optimal HAM approach are introduced to accelerate the convergence of solutions. Thus the homotopy analysis method will be a simple and effective tool to solve the strongly nonlinear Duffing equations.