An analytical method for Mathieu oscillator based on method of variation of parameter
An analytical method for Mathieu oscillator based on method of variation of parameter
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DOI：
10.1016/j.cnsns.2016.02.003
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发表时间：
2016-08
期刊：
影响因子：
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通讯作者：
Xianghong Li;Jingyu Hou;Jufeng Chen
中科院分区：
文献类型：
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作者：
Xianghong Li;Jingyu Hou;Jufeng Chen
基于参数变换法的思想，提出了一种简单而精确的受迫Mathieu振子分析方法。假设Mathieu振子的时变参数为常数，则很容易得到其精确的解析解。在得到的精确解析解中，用周期时变参数代替常参数，即可得到Mathieu振子的近似解析解。为了验证该分析方法的正确性和精度，还给出了一阶和九阶谐波平衡法的近似解。将所提出的方法与数值模拟和HBM的结果进行了比较，验证了所提出的解析方法与数值模拟的结果吻合得很好。此外，新的解析方法的精度不仅高于一阶HBM的近似解，而且在较大的系统参数范围内也优于九阶HBM的近似解。
A simple, but very accurate analytical method for forced Mathieu oscillator is proposed, the idea of which is based on the method of variation of parameter. Assuming that the time-varying parameter in Mathieu oscillator is constant, one could easily obtain its accurately analytical solution. Then the approximately analytical solution for Mathieu oscillator could be established after substituting periodical time-varying parameter for the constant one in the obtained accurate analytical solution. In order to certify the correctness and precision of the proposed analytical method, the first-order and ninth-order approximation solutions by harmonic balance method (HBM) are also presented. The comparisons between the results by the proposed method with those by the numerical simulation and HBM verify that the results by the proposed analytical method agree very well with those by the numerical simulation. Moreover, the precision of the proposed new analytical method is not only higher than the approximation solution by first-order HBM, but also better than the approximation solution by the ninth-order HBM in large ranges of system parameters.