复变量平均法与其他近似方法的异同

Similarities and differences between the complexification-averaging method and other approximation methods

复变量平均法因其通用性和实用性受到学界的大量关注,但在求解系统响应时会产生一定误差。该研究旨在通过比较不同近似方法间的区别揭示各方法的精度差异和适用条件。应用复变量平均法、多尺度法和谐波平衡法获得单自由度自治和非自治系统的近似解析解,并以Duffing振子为算例进行数值验证。随后针对二自由度非线性能量阱系统,推导出系统稳态响应的半解析解,以振幅和均方根值为评价指标描述系统的响应情况。结果表明:对于单自由度系统,复变量平均法和多尺度法得到的衰减振动瞬态解相同,不同于谐波平衡法;三种方法获得的受迫振动稳态解相同。三者对于弱非线性自治系统和非自治系统响应的近似准确率较高。复变量平均法和谐波平衡法均能良好地描述二自由度耦合系统的稳态周期运动且精度较高。当出现拟周期运动时,以均方根值为指标,复变量平均法的解析效果更好;以振幅为指标,谐波平衡法的近似程度更高。

The complexification-averaging method has received lots of attention because of its generality and practicability,but there are some errors in solving the system response.This paper aimed to reveal the differences in accuracy and the applicability conditions of each method by comparing the differences between the different approximation methods.The complexification-averaging method,multi-scale method,and harmonic balance method were applied to obtain the analytical solutions of single degree-of-freedom autonomous and non-autonomous systems.The Duffing oscillator was used as an example for numerical verification.Semi-analytical solutions of the steady-state response of a two-degree-of-freedom nonlinear energy sink system were derived.The amplitude and root mean square were used as evaluation indicators to describe the precision of the system response.The results show that for single-degree-of-freedom systems,the decay vibration transient solutions derived by the complexification-averaging method and multi-scale method are identical,differing from that by the harmonic balance method.The forced steady-state solutions obtained by the three methods are the same.The three methods have high accuracy in approximating the response of both weakly nonlinear autonomous systems and non-autonomous systems.The steady-state periodic motion of the coupled two-degree-of-freedom system is well described with high accuracy using the complexification-averaging method and harmonic balance method.When quasi-periodic motion occurs,the complexification-averaging method has better analytical results while taking the root mean square as an indicator.The harmonic balance method presents higher performance while the amplitude is used as an indicator.