宽带噪声激励下碰撞摩擦系统的随机响应和稳定性的研究

马兰; 田丽丽; 刘莉

doi:10.21656/1000-0887.440313

宽带噪声激励下碰撞摩擦系统的随机响应和稳定性的研究

doi: 10.21656/1000-0887.440313

马兰^1,,
田丽丽¹,
刘莉^{1, 2, ,}

1.
宁夏大学数学统计学院，银川 750021
2.
西北工业大学数学与统计学院，西安 710072

基金项目:

宁夏自然科学基金 2020AAC03064

国家自然科学基金 12262032

详细信息

作者简介:
马兰(2000—)，女(E-mail: ml2676649359@163.com)

通讯作者:
刘莉(1979—)，女，副教授，博士(通讯作者. E-mail: liuli0951@126.com)

中图分类号: O211.63
计量
- 文章访问数: 78
- HTML全文浏览量: 36
- PDF下载量: 33
- 被引次数: 0
出版历程
- 收稿日期: 2023-10-18
- 修回日期: 2024-05-12
- 刊出日期: 2024-09-01

Stochastic Responses and Stability Analysis of Vibro-Impact Systems With Friction Under Wideband Noise Excitation

MA Lan^1
,,
TIAN Lili¹,
LIU Li^{1, 2
, ,}

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, P.R.China
2.
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, P.R.China

摘要

摘要: 研究了宽带噪声激励下碰撞摩擦系统的随机响应和概率为1渐近稳定性. 基于非光滑变换和随机平均法得到了碰撞摩擦系统响应的稳态概率密度，并通过与Monte Carlo数值模拟结果对比，验证了上述解析方法的有效性. 讨论了摩擦力和碰撞恢复系数对系统稳态概率密度的影响. 基于平均Itô微分方程，得到其线性化方程的最大Lyapunov指数的表达式，通过Lyapunov指数确定系统平凡解的稳定性. 结果表明，改变碰撞恢复系数和摩擦系数能调整系统的随机稳定性.
- 碰撞摩擦系统 /
- 随机平均法 /
- 稳态响应 /
- 随机稳定性
Abstract: The stochastic responses and the asymptotic stability with probability 1 of vibro-impact systems with friction under wideband noise excitation were investigated. The Zhuravlev non-smooth transformation and the stochastic averaging method were extended to obtain the steady-state probability density functions of the system. The accuracy of the method was verified through comparison of the theoretical results with those from the Monte Carlo simulations. The effects of the friction force and the vibro-impact restitution coefficient on the system responses were studied. Furthermore, The Lyapunov exponent of the linearized averaged Itô equation was derived and the stability of the trivial solution was determined with the Lyapunov exponent. The results show that, changing the frictional coefficient and the vibro-impact restitution coefficient could adjust the system stochastic stability.
- vibro-impact system with friction /
- stochastic averaging method /
- steady state response /
- stochastic stability

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图 1 不同摩擦系数下系统稳态响应的概率密度函数

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 1. PDFs of system responses under different values of μ

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图 2 不同碰撞系数下系统稳态响应的概率密度函数

Figure 2. PDFs of system responses under different values of r

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图 3 不同非线性阻尼系数β和γ下系统稳态响应的概率密度函数

Figure 3. PDFs of system responses under different values of β and γ

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图 4 Lyapunov指数随碰撞恢复系数r和随摩擦系数μ的变化

Figure 4. The variations of TLE with r and μ

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图 5 系统在(r, μ)平面内的概率为1渐近稳定区域

Figure 5. The asymptotic stability region with probabilityone on the plane of (r, μ)

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参考文献(19)

[1]	KUNZE M. Non-Smooth Dynamical Systems[M]. New York: Springer, 2000.
[2]	张伟, 胡海岩. 非线性动力学理论与应用的新进展[M]. 北京: 科学出版社, 2009. ZHANG Wei, HU Haiyan. New Advances in Nonlinear Dynamics Theory and Applications[M]. Beijing: Science Press, 2009. (in Chinese)
[3]	FENG Q, PFEIFFER F. Stochastic model on a rattling system[J]. Journal of Sound and Vibration, 1998, 215 (3): 439-453. doi: 10.1006/jsvi.1998.1646
[4]	FENG Q, HE H. Modeling of the mean Poincaré map on a class of random impact oscillators[J]. European Journal of Mechanics A: Solids, 2003, 22 (2): 267-281. doi: 10.1016/S0997-7538(03)00015-9
[5]	叶正伟, 邓生文, 梁相玲. Gauss白噪声激励下的永磁同步电动机模型的分岔分析[J]. 应用数学和力学, 2023, 44 (7): 884-894. YE Zhengwei, DENG Shengwen, LIANG Xiangling. Bifurcation analysis of the permanent magnet synchronous motor model under white Gaussian noises[J]. Applied Mathematics and Mechanics, 2023, 44 (7): 884-894. (in Chinese)
[6]	JING H S, YOUNG M. Random response of a single-degree-of-freedom vibro-impact system with clearance[J]. Earthquake Engineering & Structural Dynamics, 1990, 19 (6): 789-798.
[7]	JING H S, SHEU K C. Exact stationary solutions of the random response of a single-degree-of-freedom vibro-impact system[J]. Journal of Sound and Vibration, 1990, 141 (3): 363-373. doi: 10.1016/0022-460X(90)90632-A
[8]	HUANG Z L, LIU Z H, ZHU W Q. Stationary response of multi-degree-of-freedom vibro-impact systems under white noise excitations[J]. Journal of Sound and Vibration, 2004, 275 (1): 223-240.
[9]	RONG H W, WANG X D, XU W, et al. Resonant response of a non-linear vibro-impact system to combined deterministic harmonic and random excitations[J]. International Journal of Non-Linear Mechanics, 2010, 45 (5): 474-481. doi: 10.1016/j.ijnonlinmec.2010.01.005
[10]	ZHU H T. Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises[J]. Journal of Sound and Vibration, 2014, 333 (3): 954-961. doi: 10.1016/j.jsv.2013.10.002
[11]	ZHU H T. Stochastic response of a vibro-impact Duffing system under external Poisson impulses[J]. Nonlinear Dynamics, 2015, 82 (1/2): 1001-1013.
[12]	孙娇娇, 徐伟, 林子飞, 等. 高斯色噪声激励下含黏弹力摩擦系统的随机响应分析[J]. 应用数学和力学, 2001, 22 (8): 852-861. SUN Jiaojiao, XU Wei, LIN Zifei, et al. Random response analysis of friction systems with viscoelastic forces under Gaussian colored noise excitation[J]. Applied Mathematics and Mechanics, 2001, 22 (8): 852-861. (in Chinese)
[13]	TIAN Y P, WANG Y, JIN X L, et al. Optimal load resistance of a randomly excited nonlinear electromagnetic energy harvester with Coulomb friction[J]. Smart Materials and Structures, 2014, 23 (9): 095001.
[14]	VIRGIN L N, BEGLEY C J. Grazing bifurcations and basins of attraction in an impact-friction oscillator[J]. Physica D: Nonlinear Phenomena, 1999, 130 (1/2): 43-57.
[15]	ANDREAUS U, CASINI P. Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle[J]. International Journal of Non-Linear Mechanics, 2002, 37 (1): 117-133
[16]	SU M, XU W, YANG G D. Stochastic response and stability of system with friction and a rigid barrier[J]. Mechanical Systems and Signal Processing, 2019, 132 : 748-761.
[17]	SU M, XU W, YANG G D. Response analysis of Van der Pol vibro-impact system with Coulomb friction under Gaussian white noise[J]. International Journal of Bifurcation and Chaos, 2018, 28 (13): 1830043
[18]	ZHURAVLEV V. A method for analyzing vibration-impact systems by means of special functions[J]. Mechanics of Solids, 1976, 11 : 23-27.
[19]	OSELEDEC V I. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems[J]. Transactions of the Moscow Mathematical Society, 1968, 19 (2): 197-231.