Response’s Probabilistic Characteristics of a Duffing Oscillator under Harmonic and Random Excitations
Main Article Content
Abstract
Abstract: Response’s probabilistic characteristics of a Duffing oscillator subjected to combined harmonic and random excitations are investigated by a technique combining the stochastic averaging method and the equivalent linearization method. The harmonic excitation frequency is taken to be in the neighborhood of the system’s natural frequency. The original equation is averaged by the stochastic averaging method in Cartesian coordinates. Then the equivalent linearization method is applied to the nonlinear averaged equations so that the equations obtained can be solved exactly by the technique of auxiliary function. The theoretical analyses of Duffing oscillator are validated by numerical simulation.
Keywords: Duffing, averaging method, equivalent linearization, harmonic excitation, random excitation.References
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[5] R.Z. Khasminskii, A limit theorem for the solutions of differential equations with random right-hand sides. Theory Probab. Applic. 11 (1966) 390.
[6] J.B. Roberts, P.D. Spanos, Stochastic averaging: An approximate method of solving random vibration problems. Non-linear mechanics 21(2) (1986) 111.
[7] T.K. Caughey, Response of a nonlinear string to random loading. ASME J. Applied Mechanics 26 (1959) 341.
[8] F. Casciati, L. Faravelli, Equivalent linearization in nonliear random vibration problems. In Conference on Vibration problems in Eng, Xian, China (1986) 986.
[9] L. Socha, T.T. Soong, Linearization in analysis of nonlinear stochastic systems. Appl. Mech. Rev. 44(1991) 399.
[10] N.D. Anh, W. Schiehlen, An extension to the mean square criterion of Gaussian equivalent linearization, Vietnam J. Math. 25(2) (1997) 115.
[11] L. Socha, Probability density equivalent linearization technique for nonlinear oscillator with stochastic excitations, Z. Angew. Math. Mech. 78 (1998) 1087.
[12] J.B. Roberts, P.D. Spanos, Random Vibration and Statistical Linearization. Dover Publications, Inc., Mineola, New York (1999).
[13] R.C. Zhang, Work/energy-based stochastic equivalent linearization with optimized power. Sound and Vibration 230 (2000) 468.
[14] I. Elishakoff, L. Andrimasy, M. Dolley, Application and extension of the stochastic linearization by Anh and Di Paola. Acta Mech, 204 (2008) 89.
[15] N.D. Anh, N.N. Hieu, N.N. Linh, A dual criterion of equivalent linearization method for nonlinear systems subjected to random excitation. Acta Mechanica, 223 (3) (2012) 645.
[16] M.F. Dimentberg, Response of a non-linearly damped oscillator to combined periodic parametric and random external excitation. Int. J. Nonlinear Mechanics 11 (1976) 83.
[17] Y.A. Mitropolskii, N.V. Dao, N.D. Anh, Nonlinear oscillations in systems of arbitrary order. Kiev: Naukova- Dumka (in Russian) (1992).
[18] J.S. Yu, Y.K. Lin YK, Numerical path integration of a nonlinear oscillator subject to both sinusoidal and white noise excitations, Int. J. Non-Linear Mechanics 37 (2004) 1493.
[19] A.H. Nayfeh, S.J Serhan, Response statistics of nonlinear systems to combined deterministic and random excitations. Int. J. Nonlinear Mechanics 25 (5) (1990) 493.
[20] R. Haiwu, X. Wei, M. Guang, F. Tong, Response of a Duffing oscillator to combined determinstic harmonic and random exctaion J. Sound and Vibration 242(2) (2001) 362.
[21] C.S. Manohar, R.N. Iyengar, Entrainment in Van der Pol’s oscillator in the presence of noise. Int. J. Nonlinear Mechanics 26(5) (1991) 679.
[22] N.D. Anh, N.N. Hieu, The Duffing oscillator under combined periodic and random excitations. Probabilistic Engineering Mechanics 30 (2012) 27.
[23] N.D. Anh, Two methods of integration of the Kolmogorov-Fokker-Planck equations (English). Ukr. Math. J. 38 (1986) 331; translation from Ukr. Mat. Zh. 38(3) (1986) 381.
[24] L. Lutes, S. Sarkani, Stochastic Analysis of Structural Dynamics. Upper Saddle River, New Jersey: Prentice Hall (1997).