求强非线性保守系统共振解的渐近法

强非线性保守系统经引入参数变换，并在一定的假设条件下可转化为弱非线性保守系统，再将其解展开为傅里叶级数，利用参数待定法可方便地求出强非线性保守系统的共振周期解．研究了Duffing方程的1／3亚谐共振和主共振周期解．这些例子表明近似解与数值解比较接近．用本文方法求强非线性保守系统共振周期解时，无须解微分方程和依靠消除永年项建立补充方程，求解过程简单，易于掌握，精度高．

A Asymptotic Method for Solving Resonance Solutions of Strongly Nonlinear Conservative Systems

By introducing a parameter transformation and based on a hypothesis, the strongly nonlinear conservative system was transformed into a weakly nonlinear conservative system, whose solution was expanded by Fourier series. Thus the approximate resonance cycle solution of the strongly nonlinear conservative system can be obtained by the undetermined parameter method. The 1/3 subharmonic and main harmonic resonance cycle solutions of Duffing equation were studied. The results show that the approximate solutions are close to the numerical solutions.Using this method to solve the resonance cycle solution of the strongly nonlinear conservative system, the solution process was changed into a series of algebraic operations, avoiding solving the differential equation and establishing additional equation to eliminate the secular terms.