中文摘要：

应用同伦分析法研究单自由度三次强非线性系统受迫振动问题,不同于其他解析近似方法,该方法从根本上克服摄动理论对小参数的过分依赖,其有效性与所研究的非线性问题是否含有小参数无关,适用范围广。同伦分析法提供选取基函数的自由,可以选取较好的基函数,更有效地逼近问题的解,通过引入辅助参数和辅助函数来调节和控制级数解的收敛区域和收敛速度,同伦分析法为非线性问题的解析近似求解开辟一个全新的途径,特别适用于强非线性问题。通过具体算例,将同伦分析法与四阶龙格库塔方法数值解做了比较,结果表明,该方法不仅能求解稳态解而且也能计算非稳态解并且具有较好的计算精度。

英文摘要：

A strongly cubic nonlinear forced vibration system with single degree of freedom is investigated by means of homotopy analysis method（HAM）. Different from other approximate computational method, the HAM is totally independent of small physical parameters, and thus is suitable for most nonlinear problems. The HAM provides us with a great freedom to choose base functions of solution series, so that a nonlinear problem may be approximated more effectively. The HAM adjusts and controls progression solution convergence region and the convergence rate through the introduction of auxiliary parameters and auxiliary functions. Therefore it opens up a new approach to the solution of analytical approximation of non-linear problems, and is especially suitable for strong non-linear problems. The computation results indicate that this method can not only solve thesteady state solution but also calculate the unsteady state solution, and has good computational accuracy.