Abstract
This paper is concerned with the nonzero mean stationary probability density function (PDF) solution for nonlinear oscillators under external Gaussian white noise. The PDF solution is governed by the well-known Fokker–Planck–Kolmogorov (FPK) equation and this equation is numerically solved by the exponential-polynomial closure (EPC) method. Different types of oscillators are further investigated in the case of nonzero mean response. Either weak or strong nonlinearity is considered to show the effectiveness of the EPC method. When the polynomial order equals 2, the results of the EPC method are identical with those given by equivalent linearization (EQL) method. These results obtained with the EQL method differ significantly from exact solution or simulated results. When the polynomial order is 4 or 6, the PDFs obtained with the EPC method present a good agreement with the exact solution or simulated results, especially in the tail regions. The numerical analysis also shows that the nonzero mean PDF of the response is nonsymmetrically distributed about its mean unlike the case of the zero mean PDF reported in the references.
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References
Baber, T.T.: Nonzero mean random vibration of hysteretic systems. ASCE J. Eng. Mech. 110, 1036–1049 (1984)
Roberts, J.B., Spanos, P.D.: Random Vibration and Statistical Linearization. Dover, Mineola (2003)
Baratta, A., Zuccaro, G.: Analysis of nonlinear oscillators under stochastic excitation by the Fokker-Planck-Kolmogorov equation. Nonlinear Dyn. 5, 255–271 (1994)
Caughey, T.K., Ma, F.: The exact steady-state solution of a class of non-linear stochastic systems. Int. J. Non-Linear Mech. 17, 137–142 (1982)
Dimentberg, M.F.: An exact solution to a certain non-linear random vibration problem. Int. J. Non-Linear Mech. 17, 231–236 (1982)
Huang, Z.L., Jin, X.L., Li, J.Y.: Construction of the stationary probability density for a family of SDOF strongly non-linear stochastic second-order dynamical systems. Int. J. Non-Linear Mech. 43, 563–568 (2008)
Sobczyk, K., Trębicki, J.: Maximum entropy principle and nonlinear stochastic oscillators. Physica A 193, 448–468 (1993)
Cai, G.Q., Lin, Y.K.: A new approximate solution technique for randomly excited non-linear oscillators. Int. J. Non-Linear Mech. 23, 409–420 (1988)
Lin, Y.K., Cai, G.Q.: Exact stationary response solution for second order nonlinear systems under parametric and external white-noise excitations: Part II. ASME J. Appl. Mech. 55, 702–705 (1988)
Stratonovich, R.L.: Topics in the Theory of Random Noise, vol. 1. Gordon & Breach, New York (1963)
Cai, G.Q., Lin, Y.K.: Random vibration of strongly nonlinear systems. Nonlinear Dyn. 24, 3–15 (2001)
Crandall, S.H.: Perturbation techniques for random vibration of nonlinear systems. J. Acoust. Soc. Am. 35, 1700–1705 (1963)
Dunne, J.F., Ghanbari, M.: Extreme-value prediction for non-linear stochastic oscillators via numerical solutions of the stationary FPK equation. J. Sound Vib. 206, 697–724 (1997)
Shinozuka, M.: Monte Carlo solution of structural dynamics. Comput. Struct. 2, 855–874 (1972)
Caughey, T.K.: Response of a nonlinear string to random loading. ASME J. Appl. Mech. 26, 341–344 (1959)
Iyengar, R.N., Dash, P.K.: Study of the random vibration of nonlinear systems by the Gaussian closure technique. ASME J. Appl. Mech. 45, 393–399 (1978)
Er, G.K.: An improved closure method for analysis of nonlinear stochastic systems. Nonlinear Dyn. 17, 285–297 (1998)
Er, G.K., Zhu, H.T., Iu, V.P., Kou, K.P.: PDF solution of nonlinear oscillators subject to multiplicative Poisson pulse excitation on displacement. Nonlinear Dyn. 55, 337–348 (2009)
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Er, G.K., Zhu, H.T., Iu, V.P. et al. Nonzero mean PDF solution of nonlinear oscillators under external Gaussian white noise. Nonlinear Dyn 62, 743–750 (2010). https://doi.org/10.1007/s11071-010-9758-7
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DOI: https://doi.org/10.1007/s11071-010-9758-7