Abstract
The approximate nonstationary probability density of a nonlinear single-degree-of-freedom (SDOF) oscillator with time delay subject to Gaussian white noises is studied. First, the time-delayed terms are approximated by those without time delay and the original system can be rewritten as a nonlinear stochastic system without time delay. Then, the stochastic averaging method based on generalized harmonic functions is used to obtain the averaged Itô equation for amplitude of the system response and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the nonstationary probability density of amplitude is deduced. Finally, the approximate solution of the nonstationary probability density of amplitude is obtained by applying the Galerkin method. The approximate solution is expressed as a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. The proposed method is applied to predict the responses of a Van der Pol oscillator and a Duffing oscillator with time delay subject to Gaussian white noise. It is shown that the results obtained by the proposed procedure agree well with those obtained from Monte Carlo simulation of the original systems.
Similar content being viewed by others
References
Caughey, T.K.: Nonlinear theory of random vibration. Adv. Appl. Mech. 11, 209–253 (1971)
Caughey, T.K., Dienes, J.K.: Analysis of a nonlinear first-order system with a white noise input. J. Appl. Phys. 32, 2476–2479 (1961)
Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin (1983)
Andronov, A.A., Pontryagin, L.S., Witt, A.A.: On the statistical investigation of dynamical systems. J. Exp. Theor. Phys. 3, 165–180 (1933) (in Russian)
Kramers, H.A.: Brownian motion in a field of force and diffusion model of chemical reactions. Physica 7, 284–304 (1940)
Caughey, T.K., Ma, F.: The steady-state response of a class of dynamical systems to stochastic excitation. ASME J. Appl. Mech. 49, 629–632 (1982)
Dimentberg, M.F.: An exact solution to a certain non-linear random vibration problem. Int. J. Non-Linear Mech. 17, 231–236 (1982)
Yong, Y., Lin, Y.K.: Exact stationary-response solution for second order nonlinear systems under parametric and external white-noise excitations. ASME J. Appl. Mech. 54, 414–418 (1987)
Lin, Y.K., Cai, G.Q.: Exact stationary-response solution for second order nonlinear systems under parametric and external excitations. Part II. ASME J. Appl. Mech. 55, 702–705 (1988)
Zhu, W.Q., Yang, Y.Q.: Exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems. ASME J. Appl. Mech. 63, 493–500 (1996)
Roberts, J.B.: Energy method for nonlinear systems with non-white excitation. In: Hennig, K. (ed.) Proceedings of the IUTAM Symposium on Random Vibrations and Reliability, pp. 285–294. Akademie, Berlin (1983)
Zhu, W.Q.: Nonlinear stochastic dynamics and control in Hamiltonian formulation. Appl. Mech. Rev. 59, 230–248 (2006)
Zhu, W.Q., Huang, Z.L., Suzuki, Y.: Response and stability of strongly non-linear oscillators under wide-band random excitation. Int. J. Non-Linear Mech. 36, 1235–1250 (2001)
Spanos, P.D.: Non-stationary random vibration of a linear structure. Int. J. Solids Struct. 14, 861–867 (1978)
Iwan, W.D., Spanos, P.D.: Response envelope statistics for nonlinear oscillators with random excitation. ASME J. Appl. Mech. 45, 170–174 (1978)
Spanos, P.D.: Stochastic analysis of oscillators with non-linear damping. Int. J. Non-Linear Mech. 13, 249–259 (1978)
Spanos, P.D.: A method for analysis of non-linear vibrations caused by modulated random excitation. Int. J. Non-Linear Mech. 16, 1–11 (1981)
Spanos, P.D., Sofi, A., Di Paola, M.: Nonstationary response envelope probability densities of nonlinear oscillators. ASME J. Appl. Mech. 74, 315–324 (2007)
Niculescu, S.I.: Delay Effects on Stability: A Roust Control Approach. Lecture Notes in Control and Information Sciences, vol. 269. Springer, Berlin (2001)
MacDonald, N.: Biological Delay Systems. Cambridge University Press, Cambridge (1989)
Malek-Zavarej, M., Jamshidi, M.: Time-Delay Systems, Optimization and Applications. North-Holland, New York (1987)
Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, Berlin (2002)
Agrawal, A.K., Yang, J.N.: Effect of fixed time delay on stability and performance of actively controlled civil engineering structures. Earthquake Eng. Struct. Dyn. 26, 1169–1185 (1997)
Pu, J.P.: Time delay compensation in active control of structure. ASCE J. Eng. Mech. 124, 1018–1028 (1998)
Swain, A.K., Mendes, E.M.A.M., Nguang, S.K.: Analysis of the effects of time delay in nonlinear systems using generalised frequency response functions. J. Sound Vib. 294, 341–354 (2006)
Grigoriu, M.: Control of time delay linear systems with Gaussian white noise. Probab. Eng. Mech. 12, 89–96 (1997)
Di Paola, M., Pirrotta, A.: Time delay induced effects on control of linear systems under random excitation. Probab. Eng. Mech. 16, 43–51 (2001)
Bilello, C., Di Paola, M., Pirrotta, A.: Time delay induced effects on control of non-linear systems under random excitation. Meccanica 37, 207–220 (2002)
Elbeyli, O., Sun, J.Q., Ünal, G.: A semi-discretization method for delayed stochastic systems. Commun. Nonlinear Sci. Numer. Simul. 10, 85–94 (2005)
Pirrotta, A., Zingales, M.: Stochastic analysis of dynamical systems with delayed control forces. Commun. Nonlinear Sci. Numer. Simul. 11, 483–498 (2006)
Guillouzic, S., L’Heureux, I., Longtin, A.: Small delay approximation of stochastic delay differential equations. Phys. Rev. E 59, 3970–3982 (1999)
Frank, T.D., Beek, P.J.: Stationary solutions of linear stochastic delay differential equations: applications to biological systems. Phys. Rev. E 64, 021917 (2001)
Liu, Z.H., Zhu, W.Q.: Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control. J. Sound Vib. 299, 178–195 (2007)
Xu, Z., Chung, Y.K.: Averaging method using generalized harmonic functions for strongly non-linear oscillators. J. Sound Vib. 174, 563–576 (1994)
Khasminskii, R.Z.: On the averaging principle for Itô stochastic differential equations. Kibernetika 4, 260–279 (1968) (in Russian)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jin, X.L., Huang, Z.L. Nonstationary probability densities of strongly nonlinear single-degree-of-freedom oscillators with time delay. Nonlinear Dyn 59, 195–206 (2010). https://doi.org/10.1007/s11071-009-9532-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-009-9532-x