【摘要】 将同伦理论和参数变换技术相结合提出了一种可适用于求解强非线性动力系统响应的新方法,即PE-HAM方法(基于参数展开的同伦分析技术).其主要思想是通过构造合适的同伦映射,将一非线性动力系统的求解问题,转化为一线性微分方程组的求解问题,然后借助于参数展开技术消除长期项,进而得到系统的解析近似解.为了检验所提方法的有效性,研究了具有精确周期的保守Duffing系统的响应,求出了其解析的近似解表达式.在与精确周期的比较中,可以得出:在非线性强度α很大,甚至在α→∞时,近似解的周期与原系统精确周期的误差也只有2.17%.数值模拟结果说明了新方法的有效性.更多还原

【Abstract】 A new homotopy technique based on the parameter expansion (PE-HAM) was proposed to strongly nonlinear oscillation. By means of the technique of parameter expansion and the theory of homotopy, we transformed the original non-linear dynamical system into a set of linear differential equations which can be solved easily. This method is a more general one in which the magnitude of the non-linear need not be a small parameter. A typical cubic system in the form of oscillator was employed to show its feature. Not only the zero-th and first-th approximation of the conservative Duffing oscillator but also the approximate period were obtained by the method. The results verify that when α is not a small parameter, even when α→∞, the relative error between the exact period and the approximate period exceeds no more than 3% . The analytical results obtained by the method agreed well with the numerical result obtained by the forth order Runge-Kutta method.更多还原