Two-parameter dynamics of an impact oscillator with a pre-compressed spring

Lv Xiaohong,WANG Jipei,ZHANG Jintao

Journal of Vibration and Shock ›› 2024, Vol. 43 ›› Issue (4) : 12-19.

PDF(1966 KB)
PDF(1966 KB)
Journal of Vibration and Shock ›› 2024, Vol. 43 ›› Issue (4) : 12-19.

Two-parameter dynamics of an impact oscillator with a pre-compressed spring

  • Lv Xiaohong,WANG Jipei,ZHANG Jintao
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Abstract

A mechanical impact oscillator with a pre-compressed spring is considered. The Poincaré map composed of smooth flow map and discontinuous map is constructed, and a numerical calculation method of Floquet multipliers is given. The periodic attractor patterns and their parameter regions of the system in the two-parameter plane are obtained by numerical simulation. The bifurcation characteristics of period-1 attractors and the discontinuous bifurcation behaviors, such as discontinuous grazing bifurcation, bifurcations induced by grazing and period-doubling, subcritical period-doubling bifurcation and crisis, are studied by applying continuation shooting method and cell mapping method, and the formation mechanism of hysteresis and subharmonic inclusions regions is revealed. In the transition between adjacent 1–m and 1–(m+1) attractors, the hysteresis and subharmonic inclusions regions are created by discontinuous grazing bifurcations. However, they are respectively created by grazing induced saddle-node and period-doubling bifurcations as the pre-compression is equal to 0. The results can provide guidance for the parameter design and optimization of mechanical impact system with a pre-compressed spring.

Key words

Non-smooth system / numerical continuation / two-parameter dynamics / periodic attractor / bifurcation

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Lv Xiaohong,WANG Jipei,ZHANG Jintao. Two-parameter dynamics of an impact oscillator with a pre-compressed spring[J]. Journal of Vibration and Shock, 2024, 43(4): 12-19

References

[1] Gritli H., Belghith S. Diversity in the nonlinear dynamic behavior of a one-degree-of-freedom impact mechanical oscillator under OGY-based state-feedback control law: Order, chaos and exhibition of the border-collision bifurcation [J]. Mechanism and Machine Theory, 2018, 124: 1-41. [2] 张思进, 周利彪, 陆启韶. 线性碰振系统周期解擦边分岔的一类映射分析方法[J]. 力学学报,2007, 37(1): 132-136 Zhang Sijin, Zhou Libiao, Lu Qishao. A map method for grazing bifurcatin in linear vibro-impact system [J]. Chinese Journal of Theoretical and Applied Mechanics, 2007, 39(1): 132-136. [3] 金俐, 陆启韶, 王琪. 非光滑动力系统Floquet特征乘子的计算方法[J]. 应用力学学报, 2004, 3(21): 21–26. Jin Li, Lu Qishao, Wang Qi. Calculation Methods of Floquet multipliers for Non-Smooth Dynamic System [J]. Chinese Journal of Applied Mechanics, 2004, 3(21): 21–26. [4] 徐慧东, 谢建华. 一类单自由度分段线性系统的分岔和混沌控制[J]. 振动与冲击, 2008, 27(6): 20–24. Xu Huidong, Xie Jianhua. Bifurcation and chaos control of a single –degree-of-freedom system with piecewise-linearity [J]. Journal of Vibration and Shock, 2008, 27(6): 20–24. [5] Yin Shan, Shen Yongkang, Wen Guilin, et al. Analytical determination for degenerate grazing bifurcation points in the single-degree-of-freedom impact oscillator [J]. Nonlinear Dynamics, 2017, 90(1): 443–456. [6] Chávez JP, Brzeski P, Perlikowski P. Bifurcation analysis of non-linear oscillators interacting via soft impacts [J]. International Journal of Non-Linear Mechanics, 2017, 92: 76–83. [7] Zhang W , Li QH, Meng ZC. Complex bifurcation analysis of an impacting vibration system based on path-following method [J]. International Journal of Non-Linear Mechanics, 2021, 133: 103715. [8] Tan Z, Yin S, Wen G, et al. Near-grazing bifurcations and deep reinforcement learning control of an impact oscillator with elastic constraints [J]. Meccanica, 2022. [9] 苟向锋, 韩林勃, 朱凌云, 等. 单自由度齿轮传动系统安全盆侵蚀与分岔[J]. 振动与冲击, 2020, 39(2): 123-131. Gou Xiangfeng, Han Linbo, Zhu Lingyun, et al. Erosion and bifurcation of the safe basin for a single-degree-of-freedom spur gear system [J]. Journal of Vibration and Shock, 2020, 39(2): 123-131. [10] 刘莉, 徐伟, 岳晓乐, 韩群. 一类含非黏滞阻尼的Duffing单边碰撞系统的激变研究[J]. 物理学报, 2013, 62(20): 200501. Liu Li, Xu Wei, Yue XiaoLe, Han Qun. Global analysis of crises in a Duffing vibro-impact oscillator with non-viscously damping [J]. Acta Physica Sinica, 2013, 62(20): 200501. [11] 乐源,缪鹏程.一类碰撞振动系统的激变和拟周期-拟周期阵发性[J]. 振动与冲击, 2017, 36(7): 1–7. Yue Yuan,Miao Pengcheng. Crisis and quasiperiod- quasiperiod intermittency in a vibro-impact system [J]. Journal of Vibration and Shock, 2017, 36(7): 1–7. [12] Chong ASE, Yue Y, Pavlovskaia E, et al. Global dynamics of a harmonically excited oscillator with a play: Numerical studies [J]. International Journal of Non-Linear Mechanics, 2017, 94: 98–108. [13] 吕小红, 罗冠炜. 含间隙振动系统周期振动的多样性和转迁特征[J]. 振动工程学报, 2020, 33(04): 688–697. LÜ Xiaohong, LUO Guanwei. Diversity and transition characteristics of periodic vibration of a vibro-impact system with a clearance [J]. Journal of Vibration Engineering, 2020, 33(04): 688–697. [14] 侍玉青, 杜三山, 吕小红, 等. 含间隙振动系统低频周期冲击振动的模式类型及分岔特征[J].振动与冲击, 2019, 38(06): 218-225. Shi Yuqing, Du Sanshan, Lü Xiaohong, et al. Pattern types and bifurcation characteristics of the low frequency periodic impact vibration of a periodically forced system with a clearance [J]. Journal of Vibration and Shock, 2019, 38(06): 218-225.
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