Abstract
An efficient numerical procedure is proposed to obtain mean-square stability regions for both single-degree-of-freedom and two-degree-of-freedom linear systems under parametric bounded noise excitation. This procedure reduces the stability problem to a matrix eigenvalue problem. Using this approach, ranges of applicability to the well-known stochastic averaging method are discussed. Numerical results show that the small parameter size in the stochastic averaging method can have a significant effect on the stability regions. The influence of noise on the shape of simple and combination parametric resonances is studied.
Keywords:
random vibration, stochastic averaging, mean square stability, bounded noiseReferences
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[2] M.I. Freidlin, A.D. Wentzell. Random Perturbations of Dynamical Systems, Springer, New York, 1998.
[3] Y.K. Lin, G.Q. Cai. Probabilistic Structural Dynamics. McGraw-Hill, New York, 2004.
[4] A.V. Skorokhod. Asymptotic Methods in the Theory of Stochastic Differential Equations. AMS, Providence, 1989.
[5] M.F.Dimentberg. Statistical Dynamics of Non-linear and Time-varying Systems. Research Studies Press, Taunton, 1988.
[6] R.A. Ibrahim. Parametric Random Vibration. Research Studies Press, Letchworth, 1985.
[7] K. Sobczyk. Stochastic Differential Equations, Kluwer, New York, 1991.
[8] J.B. Roberts, P.D. Spanos. Stochastic averaging: An approximate method of solving random vibration problems. Int. J. Non-Linear Mech., 21: 111–134, 1986.
[9] W.Q. Zhu. Stochastic averaging methods in random vibration. ASME Appl. Mech. Rev., 41: 189–199, 1988.
[10] W.Q. Zhu. Recent developments and applications of the stochastic averaging method in random vibration, ASME Appl. Mech. Rev., 49: s72–s80, 1996.
[11] H.J. Kushner. A cautionary note on the use of singular perturbation method for “small noise” models. Stochastics, 6: 117–120, 1981.
[12] K.Y.R. Billah. Stochastic averaging versus physical consistency. J. Sound Vib., 189: 289–297, 1996.
[13] G. Blankenship, G.C. Papanicolaou. Stability and control of systems with wide-band noise disturbances. SIAM J.Appl. Math., 34: 423–476, 1978.
[14] R.V. Bobryk, A. Chrzeszczyk. Transitions in Duffing oscillator excited by random noise. Nonlinear Dyn., 51: 541–550, 2008.
[15] M.F. Dimentberg, D.V. Iourtchenko. Stochastic and/or chaotic response of a vibration system to imperfectly periodic sinusoidal excitation. Int. J. Bif. Chaos, 15: 2057–2061, 2005.
[16] Z.H. Feng, X.J. Lan, X.D. Zhu. Principal parametric resonances of a slender cantilever beam subject to axial narrow-band random excitation of its base. Int. J. Non-Linear Mech. 42: 1170–1185, 2007.
[17] Z.I. Huang, W.Q. Zhu. Stochastic averaging of quasi-integrable Hamiltonian systems under bounded noise excitations. Probab. Eng. Mech., 19: 219–228, 2004.
[18] W.C. Xie. Moment Lyapunov exponents of a two-dimensional system under bounded noise parametric excitation. J. Sound Vib., 263: 593–616, 2003.
[19] R.V. Bobryk. Closure method and asymptotic expansions for linear stochastic systems. J. Math. Anal. Appl., 329: 703–711, 2007.
[20] Z. Kotulski, K. Sobczyk. On the moment stability of vibratory systems with random impulsive parametric excitation. Arch. Mech., 40: 465–475, 1988.
[21] R.H. Cameron, W.T. Martin. Transformations of Wiener integrals under translations. Ann. Math., 45: 386–396, 1944.
[22] S.T. Ariaratnam, T.K. Srikantaiah. Parametric instabilities in elastic structures under stochastic loading. J. Struct. Mech., 6: 349–365, 1978.
[23] V.V. Bolotin. Random Vibration of Elastic Systems. Kluwer, Dordrecht, 1984.
[24] V.A. Yakubovich, V.M. Starzhinskii. Linear Differential Equations with Periodic Coefficients, vols. 1 and 2. Wiley, New York, 1975.
