摘要
应用谐波平衡法计算了一个恢复力与因变量成反比的非线性振子的近似频率和近似周期解。与Mickens的方法不同，直接求解了非线性奇异二阶微分方程。一阶和二阶谐波解所对应的非线性恢复力的傅立叶级数展开式的系数容易由相应的积分得到。由二阶谐波平衡法得到的非线性代数方程组很容易用符号运算软件求出。得到的一近似频率与精确频率的百分比误差是12.8%，而二阶近似频率与精确频率的百分比误差小于1.28%。与数值方法给出的“精确”周期解比较，二阶近似解析解要比一阶近似解析解精确得多。高阶谐波平衡法一般需要求解复杂的非线性代数方程组，但是借助于Matlab和Mathematica等符号运算软件，这一困难可以得到一定程度的克服.
Abstract
The method of harmonic balance is used to calculate analytical approximations to the periodic solutions of the nonlinear oscillator in which the restoring force is inversely proportional to the dependent variable. Unlike Mickens method, the second order nonlinear singular differential equation is directly solved. The Fourier series expansion coefficients of the nonlinear restoring forces corresponding to the first and second order harmonic balance solutions are easily calculated using corresponding integrals. The nonlinear algebraic equations for the second order approximate solution are solved by using symbolic computation software. The percentage error of the first approximate frequency in relation to the exact one is 12.8%, and the percentage error for the second approximate frequency is lower than 1.28%. A comparison of the first and second analytical approximate periodic solutions with the numerically exact solutions shows that the second analytical approximate periodic solution is much more accurate than the first analytical approximate periodic solution. It is usually rather difficult to use the harmonic balance method to produce higher order analytical approximations because it requires solutions of sets of complicated nonlinear algebraic equations. This difficulty may be overcome to some extent by using symbolic computation software such as Matlab or Mathematica. The derivation in this paper can be considered as a typical example.
关键词
角频率 /
非线性奇异振子 /
符号运算软件 /
谐波平衡法
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Key words
angular frequency /
nonlinear singular oscillator /
symbolic computation software /
the method of harmonic balance
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胡辉;郭源君;郑敏毅.
一个非线性奇异振子的谐波平衡解 [J]. 振动与冲击, 2009, 28(2): 121-123
HU Hui;GUO Yuan-jun;ZHENG Min-yi.
HARMONIC BALANCE APPROACHES TO A NONLINEAR SINGULAR OSCILLATOR[J]. Journal of Vibration and Shock, 2009, 28(2): 121-123
中图分类号：
O322
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