-
摘要:
响应与稳定性分析一直是随机动力学研究的热点, 发展预测随机响应及判定系统响应性态的方法具有重要的科学意义与广阔的应用前景. 本文综述了有关多自由度非线性随机系统的响应与稳定性的研究. 首先简介用于随机系统响应预测的Fokker-Planck-Kolmogorov方程法、随机平均法、等效线性化法、等效非线性系统法和Monte Carlo模拟法, 评述其优缺点, 进而讨论了多自由度非线性随机系统响应的精确平稳解、近似瞬态解的研究现状. 然后介绍了随机系统稳定性分析的两类方法, 即Lyapunov函数法及Lyapunov指数法,并综述了多自由度非线性随机系统稳定性分析的研究现状. 最后给出几点发展建议.
-
关键词:
- 多自由度非线性系统 /
- 随机响应 /
- 随机稳定性 /
- 随机平均法 /
- Fokker-Planck-Kolmogorov方程
Abstract:The response prediction and stability analysis are always hot topics of research in stochastic dynamics. Developing the method for response prediction of nonlinear stochastic systems and determining the qualitative behavior of the system response are of important significance and extensive application potential. This paper reviews response and stability of multi-degree-of-freedom nonlinear stochastic systems. Firstly, main methods for the response prediction of stochastic systems are outlined, such as Fokker-Planck-Kolmogorov equation, stochastic averaging method, equivalent linearization, equivalent nonlinear system procedure and Monte Carlo simulation. The advantages and disadvantages of these methods are discussed, respectively. The state-of-the-art of the exact stationary solutions and approximately nonstationary solutions is also illustrated for the multi-degree-of-freedom nonlinear stochastic systems. Then, two effective procedures to evaluate the stochastic stability, i.e., the Lyapunov function and Lyapunov exponent, are briefly presented. Based on these two methods, the stochastic stability of multi-degree-of-freedom nonlinear stochastic systems is outlined. Finally, some suggestions are given for further research on the response and stability of nonlinear stochastic systems.
-
1 Dimentberg M F. Statistical Dynamics of Nonlinear and Time-Varying Systems. England: Reserch Sdudies Press LtD, 1988 2 Roberts J B, Spanos P D. Random Vibration and Statistical Linearization. New York: Wiley, 1990 3 朱位秋. 随机振动. 北京: 科学出版社, 1992 4 林家浩, 张亚辉. 随机振动的虚拟激励法. 北京: 科学出版 社, 2004 5 Soize C. The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solution. Singapore: World Scientific, 1994 6 Lin Y K, Cai G Q. Probabilistic Structural Dynamics: Advanced Theory and Applications. New York: McGrawHill, 1995 7 方同. 工程随机振动. 北京: 国防工业出版社, 1995 8 Arnold L. Random Dynamical Systems. Berlin: Springer,1998 9 朱位秋. 非线性随机动力学控制——Hamilton 理论体系框 架. 北京: 科学出版社, 2003 10 Fujimura K, Kiureghian A D. Tailequivalent linearization method for nonlinear random vibration. Probabilistic Engineering Mechanics, 2007, 22: 63-76 11 Proppe C, Pradlwarter H J, Schu¨eller G I. Equivalent linearization and Monte Carlo simulation in stochastic dynamics. Probabilistic Engineering Mechanics, 2003, 18:1-15 12 Socha L. Linearization in analysis of nonlinear stochastic systems: recent results-Part I: Theory. Applied Mechanics Reviews, 2005, 58: 178-205 13 Saha N, Roy D. The Girsanov linearization method for stochastically driven nonlinear oscillators. ASME Journal of Applied Mechanics, 2007, 74: 885-897 14 Caughey T K, Payne H J. On the response of a class of self excited oscillators to stochastic excitation. International Journal of Non-Linear Mechanics, 1967, 2: 125-151 15 Caughey T K, Ma F. The exact steady-state solution of a class of nonlinear stochastic systems. International Journal of Non-Linear Mechanics, 1982, 17: 137-142 16 Caughey T K, Ma F. The steady-state response of a class of dynamical-systems to stochastic excitation. ASME Journal of Applied Mechanics, 1982, 49: 629-632 17 Fuller A T. Analysis of nonlinear stochastic systems by means of the Fokker-Planck equations. International Journal of Control, 1969, 9: 603-655 18 Soize C. Steady state solution of Fokker-Planck equation in higher dimension. Probabilistic Engineering Mechanics, 1988, 3:196-206 19 Soize C. Exact stationary response of multidimensional nonlinear Hamiltonian dynamic-systems under parametric and external stochastic excitations. Journal of Sound and Vibration, 1991, 149: 1-24 20 Zhu W Q, Cai G Q, Lin Y K. On exact stationary solutions of stochastically perturbed Hamiltonian systems. Probabilistic Engineering Mechanics, 1990, 5: 84-87 21 ZhuWQ, Cai G Q, Lin Y K. Stochastically excited Hamiltonian systems. In: Bellomo N, Casciati F, eds. Proceeding of IUTAM Symposium in Nonlinear Stochastic Mechanics. Berlin: Springer-Verlag, 1992. 543-552 22 Zhu W Q, Yang Y Q. Exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems. ASME Journal of Applied Mechanics, 1996,63: 493-500 23 Huang Z L, Zhu W Q. Exact stationary solutions of stochastically and harmonically excited and dissipated integrable Hamiltonian systems. Journal of Sound and Vibration, 2000, 230: 709-720 24 Zhu W Q, Huang Z L. Exact stationary solutions of stochastically excited and dissipated partially integrable Hamiltonian systems. International Journal of NonLinear Mechanics, 2001, 36: 39-48 25 黄志龙. 几类非线性随机系统动力学与控制研究. [博士论 文]. 杭州: 浙江大学, 2005 26 Wang R, Yasuda K, Zhang Z. A generalized analysis technique of the stationary FPK equation in nonlinear systems under Gaussian white noise excitations. International Journal of Engineering Science, 2000, 38: 1315-1330 27 To CWS. Nonlinear Random Vibration: Analytical Techniques and Applications. Netherlands: Swets & Zeitlinger,2000 28 Huang Z L, Jin X L, Li J Y. Construction of the stationary probability density for a family of SDOF strongly nonlinear stochastic second-order dynamical systems. International Journal of Non-Linear Mechanics, 2008, 43(7):563-568 29 Huang Z L, Jin X L. Exact stationary solutions independent of energy for strongly nonlinear stochastic systems of multiple degrees of freedom. Science in China Series E: Technological Sciences, 2009, 52(8): 2424-2431 30 Caughey T K. Nonlinear theory of random vibration. Advances in Applied Mechanics, 1971, 11: 209-253 31 Ibrahim R A. Parametric Random Vibration. New York: Research studies Press Ltd, 1985 32 Garrido L, Masoliver J. On a Class of Exact Solutions to the Fokker-Plank Equations. Journal of Mathematical Physics, 1982, 23: 1155-1158 33 Sanmiguel M. Class of exactly solvable Fokker Planck equations. Zeitschrift Fur Physik B-Condensed Matter,1979, 33: 307-312 34 Atkinson J D. Eigenfunction expansions for randomly excited nonlinear-systems. Journal of Sound and Vibration,1973, 30: 153-172 35 Johnson J P, Scott R A. Extension of eigenfunctionexpansion solutions of a Fokker-Planck equation-II Second Order System. International Journal of Non-Linear Mechanics, 1980, 15: 41-56 36 Liu Q, Davies H G. The Nonstationary response probability density-functions of nonlinearly damped oscillators subjected to white-noise excitations. Journal of Sound and Vibration, 1990, 139: 425-435 37 Bonzani I, Riganti R, Zavattaro M G. Transient solution to the diffusion equation for nonlinear stochastically perturbed systems. Mathematical and Computer Modelling,1990, 13: 59-66 38 Zhang X F, Zhang Y M, Pandey M D, et al. Probability density function for stochastic response of non-linear oscillation system under random excitation. International Journal of Non-Linear Mechanics, 2010, 45: 800-808 39 Yu J S, Cai G Q, Lin Y K. A new path integration procedure based on Gauss-Legendre scheme. International Journal of Non-Linear Mechanics, 1997, 32: 759-768 40 Mamontov E, Naess A. An analytical-numerical method for fast evaluation of probability densities for transient solutions of nonlinear Ito's stochastic differential equations. International Journal of Engineering Science, 2009, 47:116-130 41 Mignolet M P, Fan G W W. Nonstationary response of some 1st-order nonlinear-systems associated with the seismic sliding of rigid structures. International Journal of Non-Linear Mechanics, 1993, 28: 393-408 42 Wojtkiewicz S F, Bergman L A, Spencer B F, et al. Numerical solution of the four-dimensional nonstationary Fokker-Planck equation. In: IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics. Madras, India, 1999 43 Liu Z H, Zhu W Q. Stochastic averaging of quasiintegrable Hamiltonian systems with delayed feedback control. Journal of Sound and Vibration, 2007, 299: 178-195 44 Wang Y, Ying Z G, Zhu W Q. Nonlinear stochastic optimal control of Preisach hysteretic systems, Probabilistic Engineering Mechanics, 2009, 24(3): 255-264 45 Huang Z L, Jin X L. Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative. Journal of Sound and Vibration, 2009, 319(3-5): 1121-1135 46 Huang Z L, Liu Z H, Zhu W Q. Stationary response of multi-degree-of-freedom vibro-impact systems under white noise excitations. Journal of Sound and Vibration,2004, 275: 223-240 47 Spanos P D, Sofi A, Di Paola M. Nonstationary response envelope probability densities of nonlinear oscillators. ASME Journal of Applied Mechanics, 2007, 74:315-324 48 Jin X L, Huang Z L. Nonstationary probability densities of nonlinear multi-degree-of-freedom systems under Gaussian white noise excitations. In: Zhu W Q, Lin Y K, Cai G Q, eds. Proceedings of the IUTAM Symposium on Nonlinear Stochastic Dynamics and Control. New York: Springer, 2010. 35-44 49 Kushner H J. Stochastic Stability and Control. New York: Academic Press, 1967 50 Kozin F. A survey of stability of stochastic systems. Automatica, 1969, 5: 95-112 51 Khasminskii R Z. Stochastic Stability of Differential Equations. Alphen aan den Rijn: Sijthoff & Noordhoff, 1980 52 Oseledec V I. A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Transactions of Moscow Mathematical Society, 1968, 19: 197-231 53 Khasminskii R Z. Necessary and sufficent conditions for asymptotic stability of linear stochastic systems. Theory of Probability and Its Applications, Ussr, 1967, 12: 144-147 54 Potapov V D. Stability of elastic systems under a stochastic parametric excitation. Archive of Applied Mechanics,2008, 78: 883-894 55 Ariaratnam S T, Xie W C. Lyapunov exponent and stochastic stability of coupled linear systems under real noise excitation, ASME Journal of Applied Mechanics,1992, 59: 664-673 56 Zhu WQ, Huang ZL, Suzuki Y. Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems. International Journal of Non-Linear Mechanics, 2002, 37: 419-437 57 Zhu W Q, Huang Z L. Lyapunov exponents and stochastic stability of quasi-integrableHamiltonian systems. ASME Journal of Applied Mechanics, 1999, 66: 211-217 58 Zhu W Q. Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems. International Journal of Non-Linear Mechanics, 2004, 39: 569-579 59 Ying Z G. Generalized Hamiltonian norm, Lyapunov exponent and stochastic stability for quasi-Hamiltonian systems. Physics Letters A, 2004, 333: 271-276 60 Liu Z H, Zhu W Q. Asymptotic Lyapunov stability with probability one of quasi-integrable Hamiltonian systems with delayed feedback control. Automatica, 2008, 44:1923-1928 61 Huang Z L, Zhu W Q. A new approach to almost-sure asymptotic stability of stochastic systems of higher dimension. International Journal of Non-Linear Mechanics, 2003, 38: 239-247 62 Kolmanovskii V B, Shaikhet L. General method of Lyapunov functionals construction for stability investigation of stochastic difference equations. In: Dynamical Systems and Applications. New Jersey: World Scientific Publishing,1995. 397-439 63 Bao H B, Cao J D. Exponential stability for stochastic BAM networks with discrete and distributed delays. Applied Mathematics and Computation, 2012, 218: 6188-6199 64 Huang Z L, Jin X L, Zhu W Q. Lyapunov functions for quasi Hamiltonian systems. Probabilistic Engineering Mechanics, 2009, 24(3): 374-381 65 Ling Q, Jin X L, Huang Z L. Stochastic stability of quasiintegrable Hamiltonian systems with time delay by using Lyapunov function method. Science China-Technological Sciences, 2010, 53(3): 703-712 66 Jin X L, Huang Z L, Chen G R, et al. Response of energy envelop in complex oscillator networks to external stochastic excitations. Journal of Physics A-Mathematical and Theoretical, 2010, 43(27): 275101 -
点击查看大图
计量
- 文章访问数: 2203
- HTML全文浏览量: 147
- PDF下载量: 2651
- 被引次数: 0
下载:
