摘要
高斯白噪声下作用下的非线性系统响应的瞬态概率密度一直是随机动力学研究的热点与难点,本文针对这一问题的分为两个不同的非线性系统进行了有益的探讨。
第一部分研究高斯白噪声激励的有强非线性刚度带分数阶导数的单自由度振子的近似瞬态响应。应用基于广义谐和函数的随机平均法,并将幅值的瞬态概率密度的近似解表示为拉盖尔正交基函数的级数和,其中系数是随时间变化的,用Galerkin法可得到幅值的近似瞬态概率密度,从而得到状态变量响应的近似瞬态概率密度,通过与使用Monte Carlo数值模拟得到的结果比较表明该方法的有效性。
第二部分研究受高斯白噪声作用下的非线性阻尼耦合的两个杜芬-范德波振子响应的近似瞬态概率密度。使用本文介绍的方法将幅值的瞬态概率密度的近似解表示为系数随时间变化的拉盖尔正交基函数的级数和的形式,通过取权函数积分可得到幅值的近似瞬态概率密度和状态变量响应的近似瞬态概率密度,数值模拟结果表明该方法有很好的适用性及精度。
最后总结前面的分析,说明本文使用的方法具有较好的精度与可行性并可推广应用于更一般的非线性随机系统。
The nonstationary probability densities for responses of Nonlinear Stochastic Systems subject to Gaussian white noise excitations are interesting and difficult topics in the field of stochastic dynamical systems. The approximate nonstationary responses of a single-degree-of-freedom(SDOF) nonlinear stochasic system with fractional derivative and the coupled two-degree-of-freedom(TDOF) nonlinear stochastic system are investigated in the present thesis.
The first part is mainly about the nonstationary probability densities of SDOF stochastic system with fractional derivative which is of great significance in engineering field. Firstly, the stochastic averaging method based on generalized harmonic functions is applied. Then, the nonstationary probability density for amplitude response is approximately expressed as a series expansion in terms of a set of Laguerre orthogonal basis functions with time-dependent coefficients. Finally, the approximate solution of the nonstationary probability density for amplitude response is derived with the application of the Galerkin method, and the results from analytical solution are compared to with numerical simulation of the original system.
In the second part of the thesis, the approximate nonstationary probability densities for responses of two Duffing-Van der Pol oscillators with coupled nonlinear damping subject to Gaussian white noise excitations are investigated. Firstly, the stochastic averaging method based on generalized harmonic functions is applied. Then, the nonstationary probability density for amplitude response is approximately expressed as a series expansion in terms of a set of Laguerre orthogonal basis functions with time-dependent coefficients. Finally, the approximate solution of the nonstationary probability density for amplitude response is derived with the application of the Galerkin method. The results obtained from proposed procedures are compared with those obtained by Monte Carlo simulation of the original systems, and it is shown that the proposed method is of high precision and applicability.
It is concluded that the method in the thesis can be applied to obtain nonstationary probability densities of nonlinear Stochastic Systems subject to Gaussian white noise excitations.
引文
[1]李佳颖.单自由度非线性随机系统的响应[硕士学位论文].2007,3-4
[2]Wojtkiewicz SF, Bergman LA, Spencer BF, Johnson EA. Numerical solution of the four-dimensional nonstationary Fokker-Planck equation. In:IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics. Madras, India 1999
[3]Spanos PD. Stochastic Analysis of Oscillators with Nonlinear Damping. International Journal of Non-Linear Mechanics.1978,13:249-260
[4]Micaletti RC, Cakmak AS, Nielsen SRK, Koyluglu HU. A solution method for linear and geometrically nonlinear MDOF systems with random properties subject to random excitation. Probabilistic Engineering Mechanics. 1998,13:85-95
[5]Zhang L, Zu JW, Zheng Z. The stochastic Newmark algorithm for random analysis of multi-degree-of-freedom nonlinear systems. Computers & Structures.1999,70:557-568
[6]Ohtori Y, Spencer BF. Semi-implicit integration algorithm for stochastic analysis of multi-degree-of-freedom structures. Journal of Engineering Mechanics-Asce.2002,128:635-643
[7]Smyth AW, Masri SF. Nonstationary response of nonlinear systems using equivalent linearization with a compact analytical form of the excitation process. Probabilistic Engineering Mechanics.2002,17:97-108
[8]Saha N, Roy D. The Girsanov linearization method for stochastically driven nonlinear oscillators. Journal of Applied Mechanics-Transactions of the Asme. 2007,74:885-897
[9]金肖玲.多自由度强非线性随机系统的响应与稳定性研究多自由度强非线性随机系统的响应与稳定性研究[博士学位论文].浙江大学,2009,10-12
[10]Caughey TK. Nonlinear Theory of Random Vibration. Advances in Applied Mechanics.1971,11:209-253
[11]Nafeh A H. Perturbation methods. New York:John& Wiley,1976.
[12]陈予恕.非线性振动.北京：高等教育出版社,2002.
[13]朱位秋.随机振动.北京：科学出版社,1992,1998.
[14]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.
[15]Rong H W, Meng G, Wang X D, et al., Response of a strongly non-linear oscillator to narrowband random excitations. Journal of Sound and Vibration, 2003,266(4):875-887.
[16]Yang X L, Xu W, Sun Z K, et al., Responses of strongly non-linear oscillator parametrically excited by random narrow-band noise. Applied Mathematics and Computation,2005,171(2):887-901.
[17]Rong H W, Meng G, Wang X D, et al.. Response statistic of strongly non-linear oscillator to combined determinnistic and random excitation. International Journal of Non-Linear Mechanics,2004,39(6):871-878.
[18]Atkinson J D. Eigenfunction Expansions for Randomly Excited Nonlinear-Systems. Journal of Sound and Vibration,1973,30:153-172.
[19]Iwan W D, Spanos P T. Response Envelope Statistics for Non-Linear Oscillators with Random-Excitation. Journal of Applied Mechanics,1978, 45:170-174.
[20]Spanos P D. A Method for Analysis of Non-Linear Vibrations Caused by Modulated Random-Excitation. International Journal of Non-Linear Mechanics,1981,16:1-11.
[21]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.
[22]Spanos P D, Sofi A, Di Paola M. Nonstationary response envelope probability densities of nonlinear oscillators. Journal of Applied Mechanics, 2007,74:315-324.
[23]Hau S.A, Iwan W.D. Oscillator of a self-excited nonlinear system. Journal of Applied Mechanics,ASME,1984,51 (12),892-898.
[24]Storti D.W, Rand R.H. Dynamics of two strongly coupled Van der Pol oscillators, International Journal of Non-linear Mechanics,1982,17(3), 143-152
[25]Zhu W Q, Huang Z L, Suzuki Y. Response and stability of strongly non-linear oscillators under wide-band random excitation. International Journal of Non-Linear Mechanics,2001,36:1235-1250.
[26]Zhu W Q. Nonlinear Stochastic dynamics and control in Hamiltonian formulation. Applied Mechanics Reviews,2006,59:230-248.
[27]Xu Z, Cheung Y K. Averaging Method Using Generalized Harmonic Functions For Strongly Non-Linear Oscillators. Journal of Sound and Vibration,1994, 174:563-576.
[28]Caughey TK, Dienes JK. Analysis of a Nonlinear First-Order System with a White Noise Input. Journal of Applied Physics.1961,32:2476-2479
[29]Gardiner CW. Handbook of stochastic methods for physics, chemistry and the natural sciences. Berlin:Springer-Velarg,1983
[30]Johnson JP, Scott RA. Extension of Eigenfunction-Expansion Solutions of a Fokker-Planck Equation.2.2nd Order System. International Journal of Non-Linear Mechanics.1980,15:41-56
[31]Wen YK. Approximate Method for Nonlinear Random Vibration. Journal of the Engineering Mechanics Division-Asce.1975,101:389-401
[32]_Bonzani I, Riganti R, Zavattaro MG. Transient Solution to the Diffusion Equation for Nonlinear Stochastically Perturbed Systems. Mathematical and Computer Modelling.1990,13:59-66
[33]Yu JS, Cai GQ, Lin YK. A new path integration procedure based on Gauss-Legendre scheme. International Journal of Non-Linear Mechanics. 1997,32:759-768
[34]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
[35]Mignolet MP, Fan GWW. 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
[36]徐明瑜,谭文长.中间过程、临界现象——分数阶算子理论、方法、进展及其在现代力学中的应用.中国科学G辑.2006,36：225-238
[37]Oldham KB, Spanier J. The Fractional Calculus:Theory and Applications of Differentiation and Integration to Arbitrary Order. New York:Academic Press, 1974
[38]Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. New York:John Wiley & Sons Inc,1993
[39]Ross B. Fractional calculus and its applications. Berlin:Springer-Verlag,1975
[40]Mandelbrot BB. The fractal geometry of Nature. New York:W H Freeman, 1982
[41]Chen W, Holm S. Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. Journal of the Acoustical Society of America.2004,115:1424-1430
[42]Bounaim A, Chen W. Computations for a Breast Ultrasonic Imaging Technique and Finite Element Approach for a Fractional Derivative Modeling the Breast Tissue Acoustic Attenuation. International Journal of Tomography and Statistics.2008,10:31-43
[43]Chen W. A speculative study of 2/3-order fractional Laplacian modeling of turbulence:Some thoughts and conjectures. Chaos.2006,16:1-6
[44]Laskin N. Fractional market dynamics. In:229th WE Heraeus Seminar on Economics Dynamics from the Physics Point of View. Bad Honnef, Germany 2000
[45]Jumarie G. Fractionalization of the complex-valued Brownian motion of order n using Riemann-Liouville derivative. Applications to mathematical finance and stochastic mechanics. Chaos Solitons & Fractals.2006,28:1285-1305
[46]Yuan LX, Agrawal OP. A numerical scheme for dynamic systems containing fractional derivatives. Journal of Vibration and Acoustics-Transactions of the Asme.2002,124:321-324
[47]Shokooh A, Suarez L. A comparison of numerical methods applied to a fractional model of damping materials. Journal of Vibration and Control. 1999,5:331-354
[48]Zhu ZY, Li GG, Cheng CJ. A numerical method for fractional integral with applications. Applied Mathematics and Mechanics-English Edition. 2003,24:373-384
[49]Suarez LE, Shokooh A. An eigenvector expansion method for the solution.of motion containing fractional derivatives. Journal of Applied Mechanics-Transactions of the Asme.1997,64:629-635
[50]Li GG, Zhu ZY, Cheng CJ. Dynamical stability of viscoelastic column with fractional derivative constitutive relation. Applied Mathematics and Mechanics-English Edition.2001,22:294-303
[51]Wahi P, Chatterjee A. Averaging for oscillations with light fractional order damping. In:ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Chicago 2003
[52]Wahi P, Chatterjee A. Averaging oscillations with small fractional damping and delayed terms. Nonlinear Dynamics.2004,38:3-22
[53]Jumarie G. Path integral for the probability of the trajectories generated by fractional dynamics subject to Gaussian white noise. Applied Mathematics Letters.2007,20:846-852
[54]Metzler R, Klafter J. The random walk's guide to anomalous diffusion:a fractional dynamics approach. Physics Reports-Review Section of Physics Letters.2000,339:1-77
[55]Spanos PD, Zeldin BA. Random vibration of systems with frequency-dependent parameters or fractional derivatives. Journal of Engineering Mechanics-Asce.1997,123:290-292
[56]Rudinger F. Tuned mass damper with fractional derivative damping. Engineering Structures.2006,28:1774-1779
[57]Agrawal OP. An analytical scheme for stochastic dynamic systems containing fractional derivatives. In:Proceedings of the 1999 ASME Design Engineering Technical Conferences. Las Vegas 1999
[58]Agrawal OP. Stochastic analysis of a 1-D system with fractional damping of order 1/2. Journal of Vibration and Acoustics-Transactions of the Asme. 2002,124:454-460
[59]Agrawal OP. Stochastic analysis of dynamic systems containing fractional derivatives. Journal of Sound and Vibration.2001,247:927-938
[60]Agrawal OP. Analytical solution for stochastic response of a fractionally damped beam. Journal of Vibration and Acoustics-Transactions of the Asme. 2004,126:561-566
[61]Drozdov AD. Fractional oscillator driven by a Gaussian noise. Physica a-Statistical Mechanics and Its Applications.2007,376:237-245
[62]Ye K, Li L, Tang JX. Stochastic seismic response of structures with added viscoelastic dampers modeled by fractional derivative. Earthquake engineering and engineering vibration.2003,2:133-139
[63]Z.L.Huang, X.L.Jin Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative. Journal of Sound and Vibration 2009,319:1121-1135
[64]Bagley RL, Torvik PJ. A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity. Journal of Rheology.1983,27:201-210
[65]Bagley RL, Torvik PJ. Fractional Calculus-a Different Approach to the Analysis of Viscoelastically Damped Structures. Aiaa Journal. 1983,21:741-748
[66]Koeller RC. Applications of Fractional Calculus to the Theory of Viscoelasticity. Journal of Applied Mechanics-Transactions of the Asme. 1984,51:299-307
[67]Bagley RL, Torvik PJ. Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures. Aiaa Journal.1985,23:918-925
[68]Depollier C, Fellah ZE, Fellah M. Propagation of transient acoustic waves in layered porous media:Fractional equations for the scattering operators. Nonlinear Dynamics.2004,38:181-190
[69]Agrawal OP. Application of fractional derivatives in thermal analysis of disk brakes. Nonlinear Dynamics.2004,38:191-206
[70]Adolfsson K. Nonlinear fractional order viscoelasticity at large strains. Nonlinear Dynamics.2004,38:233-246
[71]Deng R, Davies P, Bajaj AK. A case study on the use of fractional derivatives: The low-frequency viscoelastic uni-directional behavior of polyurethane foam. Nonlinear Dynamics.2004,38:247-265
[72]Chen YQ, Vinagre BM, Podlubny I. Fractional order disturbance observer for robust vibration suppression. Nonlinear Dynamics.2004,38:355-367
[73]Agrawal OP. Response of a diffusion-wave system subjected to deterministic and stochastic fields. Zeitschrift Fur Angewandte Mathematik Und Mechanik. 2003,83:265-274