多尺度法的设解形式之探讨

Discussion of solutions supposed forms for method of multiple scales

当采用多尺度法研究非线性振动问题时,经常会遇到不同情形下系统响应的设解形式不同的问题.文中针对一类具有平方和立方非线性项的单自由度和多自由度非线性系统,得到不同设解形式下的一次近似解和二次近似解,并与数值解相比较,发现两种设解形式的近似解与数值仿真解的解曲线吻合很好,表明传统的各种设解形式在研究弱非线性系统时都有很好的近似性.

When the nonlinear vibration problems are investigated by the method of multiple scales, various forms of solution supposed in different circumstances are often observed. It has never been discussed in the related references that whether the results of solution supposed as different forms are the same or not and which form is better. The first approximation and second approximation are obtained for a single degree freedom and a multi-degree-freedom nonlinear system with quadratic and cubic nonlinearities. The results obtained by the traditional method of multiple scales with different supposed forms presented coincide very well with the results obtained by numerical integration for weakly nonlinear systems, which shows that the method of multiple scales method with the supposed forms is a quite efficient method to study the weakly nonlinear system.