Nonlinear vibrations of Duffing system under the combination of constant excitation and harmonic excitation

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Journal of Vibration and Shock ›› 2020, Vol. 39 ›› Issue (4) : 49-54.

HOU Lei1,2，LUO Gang1，SU Xiaochao1，LI Hongliang1，CHEN Yushu1

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Abstract

This paper focuses on the basic dynamical characteristics of a Duffing system under the combination of constant excitation and harmonic excitation.The Harmonic Balance method was employed to solve the motion equation of the system.The amplitude-frequency relationship was obtained, and the backbone curve and the stability of the obtained periodic solution were analyzed as well.The amplitude-frequency curves and the backbone curves were used to show the main dynamical characteristics of the system.These dynamical characteristics affected by the constant excitation and the amplitude of the harmonic excitation were discussed significantly.In the vibration response of the system, it was shown that when the excitation frequency increases the constant term changes synchronous with the amplitude of the harmonic component, but towards an opposite direction.Nevertheless, the backbone curves for them both bend slightly to the left at the first stage and bend rightward after that.As a result, in some parameter regions, the system may have at most five periodic solutions, three of them are stable and the other two are unstable.By increasing the constant excitation, an effect of “stiffness enhancement” is presenting in the system, but it is accompanied by a more significant “stiffness softening” characteristic.Consequently, for a certain harmonic excitation, the increasing of the constant excitation may change the amplitude-frequency curve from a pure soft spring characteristic to a coexistence of soft and hard spring characteristics, and even to a pure hard spring characteristic finally.In a larger scale, however, the influence of the constant excitation to the backbone curve is mainly reflected at the down part.In other word, with the increase of the harmonic excitation, the effect of the constant excitation on the excitation frequency becomes weak, the backbone curves for different values of constant excitation tend to be similar.

Key words

constant excitation / Duffing system / backbone curve / soft and hard spring characteristics coexistence / vibration jump phenomenon

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HOU Lei1,2，LUO Gang1，SU Xiaochao1，LI Hongliang1，CHEN Yushu1. Nonlinear vibrations of Duffing system under the combination of constant excitation and harmonic excitation[J]. Journal of Vibration and Shock, 2020, 39(4): 49-54

References

[1] Wu Zhi-qiang, Chen Yu-shu. Effects of constant excitation on local bifurcation [J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(2): 161–166.
[2] 侯磊, 陈予恕, 李忠刚. 一类两自由度参激系统在常数激励下的响应研究[J]. 物理学报, 2014, 63(13): 134501.
Hou Lei, Chen Yu-shu, Li Zhong-gang. Constant-excitation caused response in a class of parametrically excited systems with two degrees of freedom [J]. Acta Physica Sinica, 2014, 63(13): 134501.
[3] Yang Yong-feng, Chen Hu, Jiang Ting-dong. Nonlinear response prediction of cracked rotor based on EMD [J]. Journal of the Franklin Institute, 2015, 352(8): 3378–3393.
[4] Lu Zhen-yong, Hou Lei, Chen Yu-shu, et al. Nonlinear response analysis for a dual-rotor system with a breathing transverse crack in the hollow shaft [J]. Nonlinear Dynamics, 2015, 83(1): 169–185.
[5] Bai Chang-qing, Zhang Hong-yan, Xu Qing-yu. Effects of axial preload of ball bearing on the nonlinear dynamic characteristics of a rotor-bearing system [J]. Nonlinear Dynamics, 2007, 53(3): 173–190.
[6] Zhang Zhi-yong, Chen Yu-shu, Cao Qing-jie. Bifurcations and hysteresis of varying compliance vibrations in the primary parametric resonance for a ball bearing [J]. Journal of Sound and Vibration, 2015, 350: 171–184.
[7] Jin Yu-lin, Yang Rui, Hou L, et al. Experiments and numerical results for varying compliance vibrations in a rigid-rotor ball bearing system [J]. ASME Journal of Tribology, 2017, 139(4): 041103.
[8] Duchemin M, Berlioz A, Ferraris G. Dynamic behavior and stability of a rotor under base excitation [J]. ASME Journal of Vibration and Acoustics, 2006, 128(5): 576–585.
[9] Han Qin-kai, Chu Fu-lei. Parametric instability of flexible rotor-bearing system under time-periodic base angular motions [J]. Applied Mathematical Modelling, 2015, 39(15): 4511–4522.
[10] Hou Lei, Chen Yu-shu. Dynamical simulation and load control of a Jeffcott rotor system in Herbst maneuvering flight [J]. Journal of Vibration and Control, 2016, 22(2): 412–425.
[11] Hou Lei, Chen Yu-shu, Cao Qin-gjie. Nonlinear vibration phenomenon of an aircraft rub-impact rotor system due to hovering flight [J]. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(1): 286–297.
[12] Hou Lei, Chen Yu-shu, Li Zhong-gang. Super-harmonic responses analysis for a cracked rotor system considering inertial excitation [J]. Science China Technological Sciences, 2015, 58(11): 1924–1934.
[13] Hou Lei, Chen Yu-shu. Analysis of 1/2 sub-harmonic resonance in a maneuvering rotor system [J]. Science China Technological Sciences, 2014, 57(1): 203–209.
[14] 朱顺泉, 谢发根. Duffing方程的分叉结构及其标度特性[J]. 物理学报, 1992, 41(10): 1638–1646.
Zhu Shun-quan, Xie Fa-gen. Bifurcation structure and scaling property of Duffing equation [J]. Acta Physica Sinica, 1992, 41(10): 1638–1646.
[15] Cao Qing-jie, Zhang Tian-de, Li Jiu-ping. A study of the static and global bifurcations for Duffing equation [J]. Applied Mathematics and Mechanics (English Edition), 1999, 20(12): 1413–1420.
[16] 毕勤胜, 陈予恕. Duffing系统解的转迁集的解析表达式[J]. 力学学报, 1997, 29(5): 573–581.
Bi Qin-sheng, Chen Yu-shu. Analytical expression of transition boundaries of the solution of Duffing systems [J]. Acta Mechanica Sinica, 1997, 29(5): 573–581.
[17] 王坤, 关新平, 丁喜峰, 等. Duffing振子系统周期解的唯一性与精确周期信号的获取方法[J]. 物理学报, 2010, 59(10): 6859–6863.
Wang Qun, Guan Xi-ping, Ding Xi-feng, et al. Acquisition method of precise periodic signal and uniqueness of periodic solutions of Duffing oscillator system [J]. Acta Physica Sinica, 2010, 59(10): 6859–6863.
[18] 刘晓君, 洪灵, 江俊. 非自治分数阶Duffing系统的激变现象[J]. 物理学报, 2016, 65(18): 226–233.
Liu Xiao-jun, Hong Ling, Jiang Jun. Crises in a non-autonomous fractional-order Duffing system [J]. Acta Physica Sinica, 2016, 65(18): 226–233.
[19] Hou Lei, Chen Yu-shu, Fu Yi-qiang, et al. Nonlinear response and bifurcation analysis of a Duffing type rotor model under sine maneuver load [J]. International Journal of Non-Linear Mechanics, 2016, 78(1): 133–141.
[20] Saeed NA, El-Gohary HA. On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force [J]. Nonlinear Dynamics, 2017, 88(1): 293–314.
[21] Kovacic I, Brennan MJ, Lineton B. On the resonance response of an asymmetric Duffing oscillator [J]. International Journal of Non-Linear Mechanics, 2008, 43(9): 858–867.
[22] Friedmann P, Hammond CE. Efficient numerical treatment of periodic systems with application to stability problems [J]. International Journal for Numerical Methods in Engineering, 1977, 11(7): 1117–1136.
[23] Hou Lei, Chen Yu-shu, Fu Yi-qiang, et al. Application of the HB-AFT method to the primary resonance analysis of a dual-rotor system [J]. Nonlinear Dynamics, 2017, 88(4): 2531–2551.