Summary
Two degree-of-freedom systems with weak quadratic nonlinearities are studied under weak external and parametric excitations respectively. All six possible cases, that arise in the presence of 1∶2 internal resonance, are investigated. The method of averaging is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order asymptotic approximation to the response. An analytical technique, based on Melnikov's method is used to predict the parameter range for which chaotic dynamics exists in the undamped averaged system. Numerical studies show that such chaotic responses are quite common in these quadratic systems, and they seem to persist even in the presence of damping.
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Bajaj, A., Chang, S., Johnson, J.: Amplitude modulated dynamics of a resonantly excited autoparametric two degree of freedom system. Nonlin. Dyn.5, 433–457 (1994).
Balachandran, B., Nayfeh, A.: Cyclic motion near a hopf bifurcation of a four dimensional system. Nonlin. Dyn.3, 19–39 (1992).
Banerjee, B., Bajaj, A., Davies, P.: Resonant dynamics of an autoparametric system. A study using higher order averaging. Int. J. Non-Linear Mech.31, 21–39 (1996).
Banerjee, B., Bajaj, A.: Chaotic responses in two degree-of-freedom systems with 1∶2 internal resonances. Fields Institute Comm.9, 1–21 (1996).
Feng, Z., Sethna, P.: Global bifurcation and chaos in parametrically forced systems with one-one resonance. Dyn. Stability Syst.5, 201–225 (1990).
Gu, X., Sethna, P.: Resonant surface waves and chaotic phenomena. J. Fluid Mech.183, 543–565 (1987).
Hatwal, H., Mallik, A., Ghosh, A.: Forced nonlinear oscillations of an autoparametric system—part 2: Chaotic responses. ASME J. Appl. Mech.50, 663–668 (1983).
Haxton, R., Barr, A.: The autoparametric vibration absorber. ASME J. Eng. Ind.94, 119–125 (1972).
Holmes, P.: Proof of non-integrability for the Henon-Heiles Hamiltonian near an exceptional integrable case. Physica D5, 335–347 (1982).
Holmes, P., Marsden, J.: Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom. Comm. Math. Phys.82, 523–544 (1982).
Markeyev, A.: Asymptotic trajectories and the stability of the periodic motions of an autonomous Hamiltonian system with two degrees of freedom. Prik. Mat. Mekh. (PMM U.S.S.R)52, 283–289 (1988).
Markeyev, A.: Resonances and asymptotic trajectories in Hamiltonian systems. Prik. Mat. Mekh. (PMM U.S.S.R)54, 169–173 (1990).
Miles, J.: Parametric excitation of an internally resonant double pendulum. J. Appl. Math. Phys. (ZAMP)36, 337–345 (1985).
Murdock, J.: Perturbations: theory and methods. New York: Wiley 1991.
Nayfeh, A.: Parametric excitation of two internally resonant oscillators. J. Sound Vibr.119, 95–109 (1989).
Nayfeh, A., Surface waves in closed basins under parametric and internal resonances. Phys. Fluids30, 2976–2983 (1987).
Nayfeh, A., Balachandran, B.: Modal interactions in dynamical and structural systems. Appl. Mech. Rev.42, S175-S201 (1989).
Nayfeh, A., Mook, D.: Nonlinear oscillations. New York: Wiley 1979.
Nayfeh, A., Zavodney, L.: The response of two degree-of-freedom systems with quadratic nonlinearities to a combination parametric resonance. J. Sound Vibr.107, 329–350 (1986).
Nayfeh, A., Mook, D., Marshall, L.: Nonlinear coupling of pitch and roll modes in ship motions. J. Hydronautics7, 145–152 (1973).
Sethna, P.: Vibrations of dynamical systems with quadratic nonlinearities. ASME J. Appl. Mech.32, 576–582 (1965).
Tien, W., Namachchivaya, N. S., Bajaj, A.: Nonlinear dynamics of a shallow arch under periodic excitation, part i-1∶2 internal resonance. Int. J. Non-Linear Mech.29, 349–366 (1994).
Wiggins, S.: Global bifurcations and chaos. New York: Springer 1988.
Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. New York: Springer 1990.
Yagasaki, K.: Chaotic dynamics of a quasi-periodically forced beam. ASME J. Appl. Mech.59, 161–167 (1992).
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Banerjee, B., Bajaj, A.K. Amplitude modulated chaos in two degree-of-freedom systems with quadratic nonlinearities. Acta Mechanica 124, 131–154 (1997). https://doi.org/10.1007/BF01213022
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DOI: https://doi.org/10.1007/BF01213022