On the harmonic balance with linearization for asymmetric single degree of freedom non-linear oscillators
On the harmonic balance with linearization for asymmetric single degree of freedom non-linear oscillators
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DOI：
10.1007/s11071-012-0326-1
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发表时间：
2012-01
影响因子：
5.6
通讯作者：
S. B. Yamgoué
中科院分区：
文献类型：
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作者：
S. B. Yamgoué
本文采用线性化谐波平衡方法，研究保守非对称单自由度系统非线性振动的解析近似。这种技术包括在谐波平衡之前线性化控制方程，使我们能够避免求解复杂的非线性代数方程。但它只能应用于对称振荡，事实证明它非常简单且有效。此限制是由于该方法需要适当的初始近似解作为输入。对于非对称振荡，这种解决方案不容易被识别，与基波工作良好的对称情况相反。对于这些非对称振荡，我们在本文中建议考虑由小偏差加上基波组成的初始近似。通过将相应的谐波平衡方程分别展开到偏差的一阶和二阶，我们能够轻松地确定偏差，从而确定所需的初始近似解，该初始近似解在高阶上产生一致的解。我们使用三个例子来说明所提出的方法，并揭示其简单性和良好的收敛性。
This paper deals with analytical approximation of non-linear oscillations of conservative asymmetric single degree of freedom systems, using the method of harmonic balance with linearization. This technique which consists of linearizing the governing equations prior to harmonic balance permits us to avoid solving complicated non-linear algebraic equations. But it could be applied only to symmetric oscillations for which it proves to be very simple and effective. This restriction is due to the fact that the method requires an appropriate initial approximate solution as input. Such a solution could not be readily identified for nonsymmetric oscillations, contrary the symmetric case where the fundamental harmonic works well. For these nonsymmetric oscillations, we propose in this paper to consider an initial approximation which consists of a small bias plus the fundamental harmonic. By expanding the corresponding harmonic balance equations respectively to first and second order in the bias, we are able to easily determine the bias and thus the required initial approximate solution that yields consistent solution at higher order. We use three examples to illustrate the proposed approach and reveal its simplicity and its very good convergence.