中文摘要：

提出了求解无阻尼达芬方程自由振动的新方法—时间尺度函数法。基于高斯-切比雪夫求积公式得到了达芬方程自由振动频率的精确解,消除了后续求解过程中振动频率未知的问题。将自由振动解表述为关于时间尺度函数的简谐函数,代入相轨迹方程导出了时间尺度微分方程,改变了达芬方程的求解途径。应用分离变量法和幂级数展开进行分析,发现达芬方程的时间尺度函数应当包含时间的线性项和周期函数项,进而提出了恰当的时间尺度函数表达式。通过引入求解条件确定了待定系数,得到了无阻尼达芬方程自由振动新解,并证明了该解在平衡点和极限位移点上没有误差。由于没有引入任何与弱非线性相关的近似假设,本方法在强非线性条件下具有高精度,当α=1、ε=100、A=10000时,自由振动位移的最大相对误差仅为0.011%左右。

英文摘要：

A new method for solution of free vibration of undamped Duffing equation was proposed, which is called timescale function method. The exact solution of free vibration frequency of undamped Duffing e- quation was obtained by using Gauss-Chebyshev quadrature formula, which eliminates the problem that caused by unknown frequency in the further solving process. The solving path of Duffing equation was changed by expressing the free vibration solution in simple harmonic function of timescale function and de- riving the differential equation of timescale function from equation of phase trajectory. This differential e- quation was analyzed with variable separation method and power series expansion. It was found that the timescale function of Dufffing equation should contain linear and periodical terms of time, a fit expression of timescale function was selected consequently, in which undetermined coefficients were fixed by solving conditions. A new free vibration solution of undamped Duffing equation was gained. It was proved that this solution is error-free in the equilibrium and limit displacement points. Since no approximate hypothe- sis relating to weak nonlinearity is adopted in this method, this solution has high accuracy in the condition of strong nonlinearity, when a = 1, = 100, A = 10000, its maximum relative error of free vibration dis- placement is about 0. 011%.