摘要
对非线性振动中的无阻尼Duffing方程自由振动频率的求解方法进行了探讨,应用Gauss-Chebyshev求积公式计算了Duffing方程的自由振动频率,得到了精确解析解表达式。为了便于实际应用,推导了几个简单的近似计算公式。对Duffing方程椭圆积分频率解进行了数值计算,以此结果为基准,通过绘制Duffing方程自由振动频率解的频率-振幅曲线,定性分析了各近似计算公式的精确度。为进行定量分析,提出了Duffing方程特征振幅的概念,分析了不同特征振幅下各近似计算公式的相对误差。结果表明,基于Gauss-Chebyshev求积公式的Duffing方程自由振动频率近似解具有形式简洁、精度高的优点,这为深入研究Duffing方程的特性提供了一种新的方法。
A new method for solving free vibration frequency of undamped Duffing equation is presented in this paper. First, the analytical solution of free vibration frequency is achieved by using the Gauss-Chebyshev quadrature formula. Approximate formulas are derived to facilitate its practical application. Then, the elliptic integral frequency solution of the Duffing equation is obtained by using numerical method. The aeeuracy of the approximate formulas are analyzed via the curve of frequency-amplitude by the method of qualitative investigation. The concept of the Duffing equation＇s characteristic amplitude is proposed for quantitative analysis. The relative errors of approximate formulas under different characteristic amplitude are analyzed. Results show that the frequency solution of the Duffing equation, based on the Gauss-Chebyshev quadrature formula, is characterized by its clarity in form and accuracy in calculation.
出处
《机械科学与技术》
CSCD
北大核心
2008年第12期1591-1594,共4页
Mechanical Science and Technology for Aerospace Engineering
