Abstract
Random colored noise excitations, more close to real environmental excitations, are considered. In this paper, probabilistic solutions of nonlinear SDOF systems excited by colored noise are analyzed. With the aid of filtering method, the initial equation of motion is transferred to four coupled first-order SDEs. As a result, a Markov process of enlarged system responses comes into being, and traditional FPK equation method can be employed. However, this way makes the corresponding FPK equation become four-dimensional. In order to solve this problem, the developed and improved exponential polynomial closure (EPC) method is employed. In the solution procedure, an approximate exponential function is extended to include four state variables. Then additional unknown coefficients associated with extended state variables are produced, and the solution procedure becomes more complicated. On the other hand, since there is no exact solution, Monte Carlo simulation (MCS) is performed to verify the efficiency of the developed EPC method. Four examples associated with Gaussian and non-Gaussian colored noise excitations are considered. Numerical results show that the developed EPC method provides good agreement with the MCS method. Moreover, the effect of bandwidth for colored noise on system responses is taken into account.
Similar content being viewed by others
References
Kanai, K.: Semi-empirical formula for the seismic characteristics of the ground motion. Bull. Earthq. Res. Inst. Univ. Tokyo 35, 309–325 (1957)
Tajimi, H.: A statistical method of determining the maximum response of a building structure during an earthquake. In: Proceedings of the second World Conference on Earthquake Engineering Japan, pp. 781–798 (1960)
Lin, Y.K., Yong, Y.: Evolutionary Kanai-Tajimi earthquake models. ASCE J. Eng. Mech. 113, 1119–1137 (1987)
Daqaq, M.F.: Transduction of a bistable inductive generator driven by white and exponentially correlated Gaussian noise. J. Sound Vib. 330, 2554–2564 (2011)
Rahman, M.: Stationary solution for the color-driven Duffing oscillator. Phy. Rev. E 53, 6514–6550 (1996)
Dykman, M.I., Mannella, R., McClintock, P.V.E., Stein, N.D., Stocks, N.G.: Probability distributions and escape rates for systems driven by quasimonochromatic noise. Phy. Rev. E 47, 3996–4009 (1993)
Cetto, A.M., de la Peña, Velasco, R.M.: Approximate Fokker–Planck equation with colored Gaussian noise. Phy. Rev. A 39, 2747–2748 (1989)
Fox, R.F.: Uniform convergence to an effective Fokker-Planck equation for weakly colored noise. Phy. Rev. A 34, 4525–4527 (1986)
Kumar, P., Narayanan, S., Gupta, S.: Finite element solution of Fokker-Planck equation of nonlinear oscillators subjected to colored non-Gaussian noise. Prob. Eng. Mech. 38, 143–155 (2014)
Yang, C.Y.: Random Vibration of Structures. Wiley, New York (1986)
Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics. McGraw-Hill, Singapore (1995)
Marmarelis, V.Z.: Nonlinear Dynamic Modeling of Physiological Systems. Wiley, Hoboken (2012)
Xu, Y., Xu, W., Mahmoud, G.M., Lei, Y.M.: Beam-beam interaction models under narrow-band excitation. Phy. A 346, 372–383 (2005)
Xu, Y., Zhang, H.Q., Xu, W.: On stochastic complex beam-beam interaction models with Gaussian colored noise. Phy. A 384, 259–272 (2007)
Xu, Y., Guo, R., Jia, W.T., Li, J.J.: Stochastic averaging for a class of single degree of freedom systems with combined Gaussian noise. Acta Mech. 225, 2611–2620 (2014)
Cai, G.Q., Lin, Y.K.: Response spectral densities of strongly nonlinear systems under random excitations. Prob. Eng. Mech. 12, 41–47 (1997)
Cross, E.J., Worden, K.: Approximation of the Duffing oscillator frequency response function using the FPK equation. J. Sound Vib. 330, 743–756 (2011)
Cai, G.Q., Lin, Y.K.: Generation of non-Gaussian stationary stochastic processes. Phy. Rev. E 54, 299–303 (1996)
Klosek-Dygasm, M.M., Matkowsky, B.J., Schuss, Z.: Colored noise in dynamical systems. SIAM J. Appl. Math. 48, 425–441 (1988)
Booton, R.C.: Nonlinear control systems with random inputs. IRE Trans. Circuit Theory 1(1), 9–18 (1954)
Caughey, T.K.: Response of a nonlinear string to random loading. ASME J. Appl. Mech. 26, 341–344 (1959)
Anh, N.D., Hung, L.X., Viet, L.D., Thang, N.C.: Global-local mean square error criterion for equivalent linearization of nonlinear systems under random excitation. Acta. Mech. 226, 3011–3029 (2015)
Bover, D.C.C.: Moment equation methods for nonlinear stochastic systems. J. Math. Anal. Appl. 65, 306–320 (1978)
Ibrahim, R.A.: Stationary response of a randomly parametric excited nonlinear system. ASME J. Appl. Mech. 45, 910–916 (1978)
Wu, W.F., Lin, Y.K.: Cumulant-neglect closure for non-linear oscillator under random parametric and external excitations. Int. J. Non-Linear Mech. 19, 349–362 (1984)
Floris, C.: Mean square stability of a second-order parametric linear system excited by a colored Gaussian noise. J. Sound Vib. 336, 82–95 (2015)
Er, G.K.: An improved closure method for the analysis of nonlinear stochastic systems. Nonlinear Dyn. 17, 285–297 (1998)
Guo, S.S., Er, G.K.: The probabilistic solution of stochastic oscillators with even nonlinearity under Poisson excitations. Cent. Eur. J. Phys. 10, 702–707 (2012)
Guo, S.S., Lam, C.C., Er, G.K.: Probabilistic solutions of nonlinear oscillators excited by correlated external and velocity-parametric Gaussian white noises. Nonlinear Dyn. 77, 597–604 (2014)
Er, G.K.: Methodology for the solutions of some reduced Fokker–Planck equations in high dimensions. Ann. Phys. 523, 247–258 (2011)
Zhu, H.T.: Probabilistic solution of a multi-degree-of-freedom Duffing system under nonzero mean Poisson impulses. Acta Mech. 226, 3133–3149 (2015)
Shinozuka, M.: Monte Carlo solution of structural dynamics. Int. J. Numer. Methods Eng. 14, 855–874 (1972)
Pirrotta, A.: Non-linear systems under parametric white noise input: digital simulation and response. Int. J. Non-Linear Mech. 40, 1088–1101 (2005)
Er, G.K.: A consistent method for the solution to reduced FPK equation in statistical mechanics. Phys. A 262, 118–128 (1999)
Richard, K., Anand, G.V.: Non-linear resonance in strings under narrow band random excitation, part I: planar response and stability. J. Sound Vib. 86, 85–98 (1983)
Davies, H.G., Nandlall, D.: Phase plane for narrow band random excitation of a Duffing oscillator. J. Sound Vib. 104, 277–283 (1986)
Iyengar, R.N.: Stochastic response and stability of the Duffing oscillator under narrowband excitation. J. Sound Vib. 126, 255–263 (1988)
Davies, H.G., Liu, Q.: The response envelope probability density function of a Duffing oscillator with random narrow-band excitation. J. Sound Vib. 139, 1–8 (1990)
Robert, J.B.: Multiple solutions generated by statistical linearization and their physical significance. Int. J. Non-Linear Mech. 26, 945–959 (1991)
Zhu, W.Q., Lu, M.Q., Wu, Q.T.: Stochastic jump and bifurcation of a Duffing oscillator under narrow-band excitation. J. Sound Vib. 165, 285–304 (1993)
Di Paola, M., Floris, C.: Iterative closure method for non-linear systems driven by polynomials of Gaussian filtered processes. Comput. Struct. 86, 1285–1296 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guo, SS., Shi, Q. Probabilistic solutions of nonlinear oscillators to subject random colored noise excitations. Acta Mech 228, 255–267 (2017). https://doi.org/10.1007/s00707-016-1715-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-016-1715-1