Abstract
Response of two-degrees-of-freedom nonlinearsystem to narrow-band random parametric excitation isinvestigated. The method of multiple scales is used todetermine the equations of modulation of amplitude andphase. The effect of detunings and amplitude areanalyzed. Theoretical analyses and numerical simulationsshow that the nontrivial steady-state solution may changeform a limit cycle to a diffused limit cycle as theintensity of the random excitation increase. Under someconditions, the system may have two steady-statesolutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lyon, R. H., Heckl, M., and Hazelgrove, C. B., 'Response of hard-spring oscillator to narrow-band excitation', Journal of the Acoustical Society of America 33, 1961, 1404–1411.
Dimentberg, M. F., 'Oscillations of a system with a nonlinear cubic characteristic under narrow-band excitation', Mekhanica Tveidogo Tela 6, 1970, 150–154.
Iyengar, R. N., 'Random vibration of a second order nonlinear elastic system', Journal of Sound and Vibration 40, 1975, 155–165.
Richard, K. and Anand, G. V, 'Nonlinear resonance in strings under narrow band random excitation, Part I: Planar response and stability', Journal of Sound and Vibration 86, 1983, 85–98.
Davies, H. G. and Nandall, D., 'Phase plane for narrow band random excitation of a buffing oscillator', Journal of Sound and Vibration 104, 1986, 277–283.
Iyengar, R. N., 'Stochastic response and stability of the Duffing oscillator under narrow-band excitation', Journal of Sound and Vibration 126, 1988, 255–263.
Iyengar, R. N., 'Response of nonlinear systems to narrow-band excitation', Structural Safety 6, 1989, 177–185.
Lennox, W. C. and Kuak, Y C., 'Narrow band excitation of a nonlinear oscillator', Journal of Applied Mechanics 43, 1976, 340–344.
Grigoriu, M., 'Probabilistic analysis of response of Duffing oscillator to narrow band stationary Gaussian excitation', in Proceedings of First Pan American Congress of Applied Mechanics, Rio de Janeiro, Brazil, C. R. Steele and L. Berilacqua (eds.), American Academy of Mechanics, 1989, pp. 616–619.
Davis, H. G. and Liu, Q., 'The response probability density function of a Duffing oscillator with random narrow band excitation', Journal of Sound and Vibration 139, 1990, 1–8.
Kapitaniak, T., 'Stochastic response with bifurcations to non-linear Duffing's oscillator', Journal of Sound and Vibration 102, 1985, 440–441.
Fang, T. and Dowell, E. H., 'Numerical simulations of jump phenomena in stable Duffing systems', International Journal of Non-Linear Mechanics 22, 1987, 267–274.
Zhu, W. Q., Lu, M. Q., and Wu, Q. T., 'Stochastic jump and bifurcation of a Duffing oscillator under narrowband excitation', Journal of Sound and Vibration 165, 1993, 285–304.
Wedig, W. V., 'Invariant measures and Lyapunov exponents for generalized parameter fluctuations', Structural Safety 8, 1990, 13–25.
Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981.
Rajan, S. and Davies, H. G., 'Multiple time scaling of the response of a Duffing oscillator to narrow-band excitations', Journal of Sound and Vibration 123, 1988, 497–506.
Nayfeh, A. H. and Serhan, S. J., 'Response statistics of nonlinear systems to combined deterministic and random excitations', International Journal of Non-Linear Mechanics 25 1990, 493–509.
Rong, H. W., Xu, W., and Fang, T., 'Principal response of Duffing oscillator to combined deterministic and narrow-band random parametric excitation', Journal of Sound and Vibration 210, 1998, 483–515.
Zhu, W. Q., Random Vibration, Science Press, Beijing, 1992.
Shinozuka, M., 'Simulation of multivariate and multidimensional random processes', Journal of Sound and Vibration 19, 1971, 357–367.
Shinozuka, M., 'Digital simulation of random processes and its applications', Journal of Sound and Vibration 25, 1972, 111–128.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Hagedorn, P., Non-Linear Oscillations, Wiley, New York, 1981.
Nayfeh, A. H. and Balachandran, B., Applied Nonlinear Dynamics, Wiley, New York, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wei, X., Qun, H., Haiwu, R. et al. One-to-Two Internal Resonance in Two-Degrees-of-Freedom Nonlinear System with Narrow-Band Excitations. Nonlinear Dynamics 27, 385–395 (2002). https://doi.org/10.1023/A:1015254902478
Issue Date:
DOI: https://doi.org/10.1023/A:1015254902478