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| Similarities and differences between the complexification-averaging method and other approximation methods |
| SUI Peng1, SHEN Yongjun1,2, WANG Xiaona3 |
1.Department of Mechanical Engineering，Shijiazhuang Tiedao University，Shijiazhuang 050043，China;
2.State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures，Shijiazhuang Tiedao University，Shijiazhuang 050043，China;
3.Hebei Vocational College of Rail Transportation，Shijiazhuang 050021，China |
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Abstract The complexification-averaging method has received lots of attention from researchers because of its generality and practicality, but it will produce some errors in solving the system response. This paper aims 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 are applied to obtain the analytical solutions of single degree-of-freedom autonomous and non-autonomous systems. The Duffing oscillator is 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 are derived. The amplitude and root mean square are used as evaluation indicators to describe the precision of the system response. Some important conclusions are obtained. 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 taking the root mean square as an indicator. When the amplitude is used as an indicator, the harmonic balance method presents higher degrees of approximation.
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Received: 22 March 2022
Published: 28 May 2023
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