Abstract
The principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation is studied by using the method of multiple scales and numerical simulations. The first-order approximations of the solution, together with the modulation equations of both amplitude and phase, are derived. The effects of the frequency detuning, the deterministic amplitude, the intensity of the random excitation and the time delay on the dynamical behaviors, such as stability and bifurcation, are studied through the largest Lyapunov exponent. Moreover, the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time delay can broaden the stable region of the trivial steady-state solution and enhance the control performance. The theoretical results are well verified through numerical simulations.
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References
Hale, J.: Theory of Functional Differential Equations. Springer-Verlag, New York (1977)
Gopalsmy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer, Dordrecht (1992)
Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, Heidelberg (2002)
Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.-O.: Delay Equations, Functional, Complex, and Nonlinear Analysis. Springer-Verlag, New York (1995)
Hu, H.Y., Dowell, E.H., Virgin, L.N.: Resonances of a harmonically forced duffing oscillator with time delay state feedback. Nonlinear Dyn. 15, 311–327 (1998)
Maccari, A.: The response of a parametrically excited van der Pol oscillator to a time delay state feedback. Nonlinear Dyn. 26, 105–119 (2001)
Ji, J.C., Leung, A.Y.T.: Bifurcation control of a parametrically excited duffing system. Nonlinear Dyn. 27, 411–417 (2002)
Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)
Davies, H.G., Rajan, S.: Random superharmonic and subharmonic response: multiple time scaling of a duffing oscillator. J. Sound Vib. 126(2), 195–208 (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), 1–8 (1990)
Huang, Z.L., Zhu, W.Q., Ni, Y.Q., Ko, J.M.: Stochastic averaging of strongly non-linear oscillators under bounded noise excitation. J. Sound Vib. 254, 245–267 (2002)
Rong, H.W., Xu, W., Fang, T.: Principal response of duffing oscillator to combined deterministic and narrow-band random parametric excitation. J. Sound Vib. 210(4), 483–515 (1998)
Rong, H.W., Xu, W., Wang, X.D., Meng, G., Fang, T.: Maximal Lyapunov exponent and almost-sure sample stability for second order linear stochastic system. Int. J. Nonlinear Mech. 38, 609–614 (2003)
Grigoriu, M.: Control of time delay linear system with Gaussian white noise. J. Eng. Mech. 12, 89–96 (1997)
Paola, M.D., Pirrotta, A.: Time delay induced effects on control of linear systems under random excitation. J. Eng. Mech. 16, 43–51 (2001)
Wedig, W.V.: Invariant measures and Lyapunov exponents for generalized parameter fluctuations. Struct. Saf. 8, 13–25 (1990)
Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd edn. Springer-Verlag, New York (1997)
Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)
Oseledec, V.L.: A multiplicative Ergodic theorem – Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products. Academic Press, New York (1980)
Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics, Advanced Theory and Applications. McGraw-Hill, New York (1995)
Fang, T., Dowell, E.H.: Numerical simulations of jump phenomena in stable duffing systems. Int. J. Nonlinear Mech. 22, 267–274 (1987)
Shinozuka, M., Jan, C.-M.: Digital simulation of random processes and its applications. J. Sound Vib. 25, 111–128 (1972)
Zhu, W.Q.: Random Vibration. Science Press, Beijing (1992)
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Jin, Y., Hu, H. Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation. Nonlinear Dyn 50, 213–227 (2007). https://doi.org/10.1007/s11071-006-9152-7
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DOI: https://doi.org/10.1007/s11071-006-9152-7