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

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Journal of Vibration and Shock ›› 2023, Vol. 42 ›› Issue (10) : 289-296.

SUI Peng1, SHEN Yongjun1,2, WANG Xiaona3

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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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Complexification-averaging method / Multi-scale method / Harmonic balance method / Nonlinear system / Quasiperiodic responses

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SUI Peng1, SHEN Yongjun1,2, WANG Xiaona3. Similarities and differences between the complexification-averaging method and other approximation methods[J]. Journal of Vibration and Shock, 2023, 42(10): 289-296

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