一类随机边界刚性约束悬臂梁系统周期运动的稳定性分析
Stability Analysis of Periodic Motion for a Class of Cantilever Beam System with Rigid and Random Constraints
DOI: 10.12677/AAM.2022.114171, PDF,    国家自然科学基金支持
作者: 王泽华, 张建文：太原理工大学数学学院，山西 太原；徐慧东：太原理工大学机械与运载工程学院，山西 太原
关键词: 悬臂梁系统；随机过程；碰撞；稳定性；Cantilever Beam System； Stochastic Process； Collision； Stability
摘要: 本文研究了一类二阶可微随机约束下的碰撞悬臂梁系统周期解的稳定性。通过推导含参数的随机零时间不连续映射给出相应的跳跃矩阵，结合跳跃矩阵和光滑流映射的基解矩阵得到了随机线性化矩阵。基于随机线性化矩阵探讨了随机约束对系统稳定性的影响，进一步调查了周期解失稳之后的倍化分岔现象，数值仿真验证了理论的有效性。
Abstract: This paper studies the stability of periodic solution for a class of second-order differentiable cantilever beam system with rigid and random constraints. The saltatory matrix is presented by deducing the random zero-time discontinuous mapping with parameters. The random linearization matrix is obtained by combining the saltatory matrix and the fundamental solution matrix of smooth flow mapping. Based on the random linearization matrix, the influence of stochastic constraints on the system stability is discussed, and the doubling bifurcation phenomenon after the instability of the periodic solution is further investigated. The validity of the theory is verified by numerical simulations.
文章引用：王泽华, 徐慧东, 张建文. 一类随机边界刚性约束悬臂梁系统周期运动的稳定性分析[J]. 应用数学进展, 2022, 11(4): 1567-1577. https://doi.org/10.12677/AAM.2022.114171

参考文献

[1] Qian, J.M. and Chen, L.C. (2021) Stochastic P-Bifurcation Analysis of a Novel Type of Unilateral Vibro-Impact Vibration System. Chaos, Solitons & Fractals, 149, Article ID: 111112. [Google Scholar] [CrossRef] 
[2] Qi, W.C. and Qiu, Z.P. (2012) A Collocation Interval Analysis Method for Interval Structural Parameters and Stochastic Excitation. Science China Physics, Mechanics and Astronomy, 55, 66-77. [Google Scholar] [CrossRef] 
[3] Feng, C.S. and Zhu, W.Q. (2008) Stochastic Optimal Control of Strongly Nonlinear Systems under Wild-Band Random Excitation with Actuator Saturation. Acta Mechanica Solida Sinica, 21, 116-126. [Google Scholar] [CrossRef] 
[4] Feng, J.Q., Xu, W. and Wang, R. (2008) Stochastic Responses of Vibro-Impact Duffing Oscillator Excited by Additive Gaussian Noise. Journal of Sound and Vibration, 309, 730-738. [Google Scholar] [CrossRef] 
[5] Feng, J.Q. and Liu, J. (2015) Chaotic Dynamics of the Vibro-Impact System under Bounded Noise Perturbation. Chaos, Solitons & Fractals, 73, 10-16. [Google Scholar] [CrossRef] 
[6] Li, C., Xu, W., Feng, J.Q. and Wang, L. (2013) Response Probability Density Functions of Duffing-Vander Pol Vibro-Impact System under Correlated Gaussian White Noise Excitations. Physica A: Statistical Mechanics and Its Applications, 392, 1269-1279. [Google Scholar] [CrossRef] 
[7] Rong, H.W., Wang, X.D., Luo, Q.Z., Xu, W. and Fang, T. (2011) Subharmonic Response of Single-Degree-of-Freedom Linear Vibro-Impact System to Narrow-Band Random Excitation. Applied Mathematics and Mechanics, 32, Article No. 1159. [Google Scholar] [CrossRef] 
[8] Fang, T., Leng, X.L. and Song, C.Q. (2003) Chebyshev Polynomial Approximation for Dynamical Response Problem of Random System. Journal of Sound and Vibration, 266, 198-206. [Google Scholar] [CrossRef] 
[9] Simpson, D.J.W., Hogan, S.J. and Kuske, R. (2018) Stochastic Regular Grazing Bifurcations. SIAM Journal on Applied Dynamical Systems, 12, 533-559. [Google Scholar] [CrossRef] 
[10] Simpson, D.J. and Kuske, R. (2015) Stochastic Perturbations of Periodic Orbits with Sliding. Journal of Nonlinear Science, 25, 967-1014. [Google Scholar] [CrossRef] 
[11] Simpson, D.J. and Kuske, R. (2016) The Influence of Localized Randomness on Regular Grazing Bifurcations with Applications to Impacting Dynamics. Journal of Vibration and Control, 24, 407-426. [Google Scholar] [CrossRef] 
[12] Staunton, E.J. and Piiroinen, P.T. (2020) Estimating the Dynamics of Systems with Noisy Boundaries. Nonlinear Analysis: Hybrid Systems, 36, Article ID: 100863. [Google Scholar] [CrossRef] 
[13] Staunton, E.J. and Piiroinen, P.T. (2020) Discontinuity Mappings for Stochastic Non-Smooth Systems. Physica D: Nonlinear Phenomena, 406, Article ID: 132405. [Google Scholar] [CrossRef] 
[14] Zhang, X.F. and Yuan, R. (2021) A Stochastic Chemostat Model with Mean-Reverting Ornstein-Uhlenbeck Process and Monod-Haldane Response Function. Applied Mathematics and Computation, 394, Article ID: 125833. [Google Scholar] [CrossRef] 
[15] Riccardo, B., Ioannis, K. and Gianluca, F. (2021) Moment-Matching Approximations for Stochastic Sums in Non-Gaussian Ornstein-Uhlenbeck Models. Insurance: Mathematics and Economics, 96, 232-247. [Google Scholar] [CrossRef] 
[16] Barbarra, B.O., Krzysztof, C. and Tomasz, K. (2010) The Effect of Discretization on the Numerical Simulation of the Vibrations of the Impacting Cantilever Beam. Communications in Nonlinear Science and Numerical Simulation, 15, 3073-3090. [Google Scholar] [CrossRef] 
[17] Oksendal, B. (2013) Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin.
[18] 徐慧东. 非光滑动力系统周期解的分岔研究[D]: [博士学位论文]. 成都: 西南交通大学, 2008.

为你推荐