Abstract
The paper presents analytical and numerical studies of the primary resonance and the 1/3 subharmonic resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay. By using the method of multiple scales, the first order approximations of the resonances are derived and the effect of time delay on the resonances is analyzed. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. In order to numerically solve the problem of history dependence prior to the start of excitation, the concepts of the Poincaré section and fixed points are generalized. Then, a modified shooting scheme associated with the path following technique is proposed to locate the periodic motion of the delayed system. The numerical results show the efficacy of the first order approximations of the resonances.
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Hale, J., Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer, Dordrecht, 1992.
Diekmann, O., van Gils, S. A., Verduyn Lunel, S. M., and Walther, H.-O., Delay Equations, Functional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
Belair, J. and Campbell, S. A., 'Stability and bifurcation of equilibria in a multi-delayed differential equation', SIAM Journal on Applied Mathematics 54(5), 1994, 1402-1424.
Palkovics, L. and Venhovens, P. J. Th., 'Investigation on stability and possible chaotic motions in the controlled wheel suspension system', Vehicle System Dynamics 21(5), 1992, 269-296.
Stepan, G. and Haller, G., 'Quasiperiodic oscillations in robot dynamics', Nonlinear Dynamics 8(4), 1995, 513-528.
Moiola, J. L., Chiacchiarini, H. G., and Desages, A. C., 'Bifurcations and Hopf degeneracies in nonlinear feedback systems with time delay', International Journal of Bifurcation and Chaos 6(4), 1996, 661-672.
Plaut, R. H. and Hsieh, J. C., 'Non-linear structural vibrations involving a time delay in damping', Journal of Sound and Vibration 117(3), 1987, 497-510.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
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Hu, H., Dowell, E.H. & Virgin, L.N. Resonances of a Harmonically Forced Duffing Oscillator with Time Delay State Feedback. Nonlinear Dynamics 15, 311–327 (1998). https://doi.org/10.1023/A:1008278526811
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DOI: https://doi.org/10.1023/A:1008278526811