Abstract
The primary objective of this paper is to examine the random response characteristics of coupled nonlinear oscillators in the presence of single and simultaneous internal resonances. A model of two coupled beams with nonlinear inertia interaction is considered. The primary beam is directly excited by a random support motion, while the coupled beam is indirectly excited through autoparametric coupling and parametric excitation. For a single one-to-two internal resonance, we used Gaussian and non-Gaussian closures, Monte Carlo simulation, and experimental testing to predict and measure response statistics and stochastic bifurcation in the mean square. The mean square stability boundaries of the coupled beam equilibrium position are obtained by a Gaussian closure scheme. The stochastic bifurcation of the coupled beam is predicted theoretically and experimentally. The stochastic bifurcation predicted by non-Gaussian closure is found to take place at a lower excitation level than the one predicted by Gaussian closure and Monte Carlo simulation. It is also found that above a certain excitation level, the solution obtained by non-Gaussian closure reveals numerical instability at much lower excitation levels than those obtained by Gaussian and Monte Carlo approaches. The experimental observations reveal that the coupled beam does not reach a stationary state, as reflected by the time evolution of the mean square response. For the case of simultaneous internal resonances, both Gaussian and non-Gaussian closures fail to predict useful results, and attention is focused on Monte Carlo simulation and experimental testing. The effects of nonlinear coupling parameters, internal detuning ratios, and excitation spectral density level are considered in both investigations. It is found that both studies reveal common nonlinear features such as bifurcations in the mean square responses of the coupled beam and modal interaction in the neighborhood of internal resonances. Furthermore, there is an upper limit for the excitation level above which the system experiences unbounded response in the neighborhood of simultaneous internal resonances.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ibrahim R. A., ‘Multiple internal resonance in a structure-liquid system’, ASME Journal of Engineering for Industry 98, 1976, 1092–1098.
Bux S. L. and Roberts J. W., ‘Nonlinear vibrators interactions in systems of coupled beam’, Journal of Sound and Vibration 104(3), 1986, 497–520.
Cartmell M. P. and Roberts J. W., ‘Simultaneous combination resonances in an autoparametrically resonant system’, Journal of Sound and Vibration 123, 1988, 81–101.
Balachandran B. and Nayfeh A. H., ‘Nonlinear oscillations of a harmonically excited composite structure’, Composite Structures 16, 1990, 323–339.
Balachandran B. and Nayfeh A. H., ‘Observations of modal interactions in resonantly forced beam-mass structure’, Nonlinear Dynamics 2, 1991, 77–117.
Nayfeh T. A., Asrar W., and Nayfeh A. H., ‘Three-mode interaction in harmonically excited system with quadratic nonlinearities’, Nonlinear Dynamics 3, 1992, 385–410.
Tadjbakhsh I. G. and Wang Y., ‘Wind-driven nonlinear oscillations of cables’, Nonlinear Dynamics 1, 1990, 265–291.
Lee C. L. and Perkins N. C., ‘Three-dimensional oscillations of suspended cables involving simultaneous internal resonances’, ASME WAM Symposium on Nonlinear Vibrations 50, 1992, 59–67.
Lee, C. L. and Perkins, N. C., ‘Experimental investigation of isolated and simultaneous internal resonances in suspended cables’, ASME Journal of Vibration and Acoustics, in print.
Fujino Y., Warnitchai P., and Pacheco B. M., ‘An experimental and analytical study of autoparametric resonance in a 3DOF model of cable-stayed-beam’, Nonlinear Dynamics 4(2), 1993, 111–138.
Benedettini F., Rega G., and Alaggio R., ‘Nonlinear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions’, Journal of Sound and Vibration 182(5), 1995, 775–798.
Kunitsyn A. L., ‘On the stability of pure imaginary roots in the critical case in the presence of intrinsic resonance’, Journal of Applied Mathematics and Mechanics (PMM) 35, 1971, 164–167.
Khazina G. G., ‘Certain stability questions in the presence of resonances’, Journal of Applied Mathematics and Mechanics (PMM) 38(1), 1974, 43–51.
Goltser Ia. M. and Kunitsyn A. L., ‘On the stability of autonomous systems with intrinsic resonance’, Journal of Applied Mathematics and Mechanics (PMM) 39(6), 1975, 974–984.
Kunitsyn A. L. and Matveyev M. V., ‘The stability of a class of reversible systems’, Journal of Applied Mathematics and Mechanics 55(6), 1991, 780–788.
Van der Aa E. and Sanders J. A., ‘The 1:2:1-resonance, its periodic orbits and integrals,’ in Lecture Notes in Mathematics, Vol. 711, Asymptotic Analysis from Theory to Application, F. Verhulst (ed.), Springer-Verlag, Berlin, 1979, pp. 187–208.
Kunitsyn A. L. and Markeev A. P., ‘Stability in resonant cases,’ in Achievements of Science and Technology, General Mechanics, Vol. 4, Moscow, Viniti, 1979.
Kunitsyn A. L. and Perezhogin A. A., ‘The stability of neutral systems in the case of a multiple fourth-order resonance,” Journal of Applied Mathematics and Mechanics (PMM) 49(1), 1985, 53–57.
Kunitsyn A. L. and Tuyakbayev A. A., ‘The stability of hamiltonian systems in the case of a multiple fourth-order resonance,’ Journal of Applied Mathematics and Mechanics 56(4), 1992, 572–576.
Kunitsyn A. L. and Muratov A. S., ‘The stability of a class of quasi-autonomous periodic systems with internal resonance’, Journal of Applied Mathematics and Mechanics (PMM) 57(2), 1993, 247–255.
Zhuravlev V. F., ‘Oscillations shape control in resonant systems’, Journal of Applied Mathematics and Mechanics (PMM) 56(5), 1992, 725–735.
Ibrahim R. A. and Roberts J. W., ‘Broad band random excitation of a two-degree-of-freedom system with autoparametric coupling’, Journal of Sound and Vibration 44(3), 1976, 335–348.
Roberts J.W., ‘Random excitation of a vibratory system with autoparametric interaction’, Journal of Sound and Vibration 69(1), 1980, 101–116.
Ibrahim R. A. and Sullivan D. G., ‘Experimental investigation of structural autoparametric interaction under random excitation’, AIAA Journal 28(2), 1990, 338–344.
Ibrahim R. A., Yoon Y. J., and Evans M. G., ‘Random excitation of nonlinear coupled oscillators’, Nonlinear Dynamics 1, 1990, 91–116.
Ibrahim R. A., Parametric Random Vibration, Wiley, New York, 1985.
Clough R. W. and Penzien J., Dynamics of Structures, McGraw-Hill, New York, 1975.
Scott D. W., Tapia R. W., and Thompson J. R., ‘Nonparametric probability density estimation by discrete penalized-likelihood criteria’, The Annals of Statistics 8, 1980, 820–832.
Bendat, J. S. and Piersol, A. G., Random Data: Analysis and Measurement Procedures, Wiley, 1986.
Yoon Y. J. and Ibrahim R. A., ‘Parametric random excitation of nonlinear coupled oscillators’, Nonlinear Dynamics 8(3), 1995, 385–413.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Afaneh, A.A., Ibrahim, R.A. Random excitation of coupled oscillators with single and simultaneous internal resonances. Nonlinear Dyn 11, 347–400 (1996). https://doi.org/10.1007/BF00045332
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00045332