This paper presents an approach to obtaining higher order approximations to limit cycles of an autonomous multi-degree-of-freedom system with a single cubic nonlinearity based on a first approximation involving first and third harmonics obtained with the harmonic balance method. This first approximation, which is similar to one which has previously been reported in the literature, is an analytical solution, except that the frequency has to be obtained numerically from a polynomial equation of degree 16. An improved solution is then obtained in a perturbation procedure based on the refinement of the harmonic balance solution. The stability of the limit cycles obtained is then investigated using Floquet analysis.

The capability of this approach to refine the results obtained by the harmonic balance first approximation is demonstrated, by direct comparison with time domain simulation and frequency components obtained using the Discrete Fourier Transform. The particular case considered was based on an aeroelastic analysis of an all-moving control surface with a nonlinearity in the torsional degree-of-freedom of the root support, and parameters corresponding to air speed, together with linear stiffness and viscous damping of the root support were varied. It is also shown, for the cases considered, how the method can reveal further bifurcational behaviour of the system beyond the initial Hopf bifurcations which first lead to the onset of limit cycle oscillations.

本文提出了一种基于谐波近似方法获得的包含一次和三次谐波的第一近似值，从而获得具有单立方非线性的自治多自由度系统的极限环的高阶近似方法。这个第一近似值类似于先前在文献中报道的近似值，是一种解析解，只是必须从一个次数为16的多项式方程中数值获得频率，然后在一个扰动过程中获得一个改进的解。基于谐波平衡解决方案的细化。然后使用Floquet分析研究获得的极限环的稳定性。

通过与时域仿真和使用离散傅立叶变换获得的频率分量进行直接比较，证明了该方法改进通过谐波平衡第一近似获得的结果的能力。所考虑的特殊情况是基于对全运动控制表面的气动弹性分析，该表面在根部支撑件的扭转自由度方面具有非线性，并且具有对应于空气速度的参数以及根部的线性刚度和粘性阻尼支持多种多样。对于所考虑的情况，还示出了该方法如何揭示除了最初的霍普夫分叉之外的系统的进一步的分叉行为，该霍普夫分叉首先导致极限循环振荡的开始。