Abstract
The fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis. The system is in a ‘1−1’ internal resonance, i.e., it has two equal linearized eigenfrequencies, and it possesses ‘nonlinear normal modes.’ For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies. The analytical results are verified by a numerical integration of the equations of motion, and a discussion of the effects of the mode bifurcation on the dynamics of the system is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Nayfeh A. H. and Mook D. T., Nonlinear Oscillations, Wiley Interscience, New York, 1979.
Bogoliubov N. and Mitropolsky Y., Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach Science Publication, New York, 1961.
Sridhar S., Nayfeh A. H., and Mook D. T., ‘Nonlinear resonances in a class of multidegree-of-freedom systems’. Journal of the Acoustical Society of America 58(1), 1975, 113–123.
Nayfeh A. H., Mook D. T., and Sridhar S., ‘Nonlinear analysis of the forced response of structural elements’, Journal of the Acoustical Society of America 55(2), 1974, 281–291.
Sethna P.R., ‘Vibrations of dynamical systems with quadratic nonlinearities’, ASME Journal of Applied Mechanics 32, 1964, 576–582.
Yamamoto T. and Hayashi S., ‘Summed and differential harmonic oscillations in nonlinear vibratory systems’, Nagoya University Faculty of Engineering Memoires 18, 1966, 85–150.
Asfar K. R., Nayfeh A.H., and Barrash K. A., ‘A nonlinear oscillator Lanchester damper’, ASME Journal of Vibration, Acoustics, Stress and Reliability in Design 109, 1987, 343–347.
Nayfeh A. H., ‘The response of single-degree-of-freedom systems with quadratic and cubic nonlinearities to a subharmonic excitation’, Journal of Sound and Vibration 89(4), 1983, 457–470.
HaQuang N., Mook D. T., and Plaut R. H., ‘A nonlinear analysis of the interactions between parametric and external excitations’, Journal of Sound and Vibration 118(3), 1987, 425–439.
Nayfeh A. H., ‘Parametric excitation of two internally resonant oscillators’, Journal of Sound and Vibration 119(1), 1987, 95–109.
Ariaratnam S. T. and Namachchivaya N. S., ‘Periodically perturbed nonlinear gyroscopic systems’, Journal of Structural Mechanics 14(2), 1986, 153–175.
Rosenberg R., ‘On nonlinear vibrations of systems with many degrees-of-freedom’, Advances in Applied Mechanies 9, 1966, 155–242.
Month L. and Rand R., ‘The stability of bifurcating periodic solutions in a two degree-of-freedom nonlinear system’, ASME Journal of Applied Mechanics 44, 1977, 782–783.
Caughey T. K., Vakakis A.F., and Sivo J., ‘Analytical study of similar normal modes and their bifurcations in a class of strongly nonlinear systems’, International Journal of Non-Linear Mechanics 25(5), 1990, 521–533.
Caughey T. K. and Vakakis A.F., ‘A method for examining steady-state solutions of forced discrete systems with strong nonlinearities’, International Journal of Non-Linear Mechanics 26(1), 1991, 89–103.
Szemplinska-Stupnicka W., ‘The resonant vibration of homogeneous non-linear systems’, International Journal of Non-Linear Mechanics 15, 1980, 407–415.
Vakakis, A.F. and Rand, R., ‘Normal modes and global dynamies of a two-degree-of-freedom nonlinear system II: High energies’,, International Journal of Non-Linear Mechanics (to appear).
Press W.H., Flannery B. P., Teukolsky S.A., and Vetterling W. T., Numerical Recipes. Cambridge University Press, Cambridge, England, 1988.
Sethna P. R. and Bajaj A.K., ‘Bifurcations in dynamical systems with internal resonance’, Journal of Applied Mechanics 45, 1978, 895–902.
Bajaj, A. K. and Johnson, J. M., ‘Asymptotic techniques and complex dynamics in weakly non-linear forced mechanical systems’, International Journal of Non-Linear Mechanics 25, 211–226.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vakakis, A.F. Fundamental and subharmonic resonances in a system with a ‘1-1’ internal resonance. Nonlinear Dyn 3, 123–143 (1992). https://doi.org/10.1007/BF00118989
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00118989