Abstract
Stability and bifurcation behaviors for a model of a flexible beam undergoing a large linear motion with a combination parametric resonance are studied by means of a combination of analytical and numerical methods. Three types of critical points for the bifurcation equations near the combination resonance in the presence of internal resonance are considered, which are characterized by a double zero and two negative eigenvalues, a double zero and a pair of purely imaginary eigenvalues, and two pairs of purely imaginary eigenvalues in nonresonant case, respectively. The stability regions of the initial equilibrium solution and the critical bifurcation curves are obtained in terms of the system parameters. Especially, for the third case, the explicit expressions of the critical bifurcation curves leading to incipient and secondary bifurcations are obtained with the aid of normal form theory. Bifurcations leading to Hopf bifurcations and 2-D tori and their stability conditions are also investigated. Some new dynamical behaviors are presented for this system. A time integration scheme is used to find the numerical solutions for these bifurcation cases, and numerical results agree with the analytic ones.
Similar content being viewed by others
References
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)
Mosekilde, E.: Topics in Nonlinear Dynamics. World Scientific, Singapore (1996)
Thomson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos. Wiley, New York (1986)
Knudsen, C., Slivsgaard, E., Rose, M., True, H., Feldberg, R.: Dynamics of a model of a railroad wheelset. Nonlinear Dyn. 6, 215–236 (1994)
Sorensen, C.B., Mosekilde, E., Granazy, P.: Nonlinear dynamics of a thrust vectored aircraft. Phys. Scr. 67, 176–183 (1996)
Thomson, J.M.T.: Complex dynamics of compliant off-shore structures. Proc. R. Soc. Lond. A 387, 407–427 (1983)
Shy, A.: Prediction of jump phenomena in roll-coupled maneuvers of airplanes. J. Aircr. 14, 375–382 (1977)
Malhotra, N., Namachivaya, N.S.: Chaotic motion of shallow arch structures under 1:1 internal resonance. J. Eng. Mech. 123, 620–627 (1997)
Namachchivaya, N.S., Van Roessel, H.J.: Unfolding of degenerate Hopf bifurcation for supersonic flow past a pitching wedge. J. Guid. Control Dyn. 9, 413–418 (1986)
Namachchivaya, N.S., Van Roessel, H.J.: Unfolding of double-zero eigenvalue bifurcations for supersonic flow past a pitching wedge. J. Guid. Control Dyn. 13, 343–347 (1990)
Kane, T.R., Ryan, R.R., Banerjee, A.K.: Dynamics of a cantilever beam attached to a moving base. J. Guid. Control Dyn. 10(2), 139–151 (1987)
Hyun, S.H., Yoo, H.H.: Dynamic modeling and stability analysis of axially oscillating cantilever beams. J. Sound Vib. 228(3), 543–558 (1999)
Chin, C.M., Nayfeh, A.H.: Three-to-one internal resonances in hinged-clamped beams. Nonlinear Dyn. 12(2), 129–154 (1997)
Chin, C.M., Nayfeh, A.H.: Three-to-one internal resonances in parametrically excited hinged-clamped beams. Nonlinear Dyn. 20(2), 131–158 (1999)
Feng, Z.H., Hu, H.Y.: Dynamic stability of a slender beam with internal resonance under a large linear motion. Acta Mech. Sin. 34(3), 389–399 (2002) (in Chinese)
Feng, Z.H., Hu, H.Y.: Principal parametric and three-to-one internal resonances of flexible beams undergoing a large linear motion. Acta Mech. Sin. 19(4), 355–364 (2003)
Feng, Z.H., Hu, H.Y.: Largest Lyapunov exponent and almost certain stability analysis of slender beams under a large linear motion of basement subject to narrowband parametric excitation. J. Sound Vib. 257(4), 733–752 (2002)
Yu, P.: Computation of norm forms via a perturbation technique. J. Sound Vib. 211, 19–38 (1998)
Yu, P.: Symbolic computation of normal forms for resonant double Hopf bifurcations using a perturbation technique. J. Sound Vib. 247, 615–632 (2001)
Yu, P.: Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales. Nonlinear Dyn. 27, 19–53 (2002)
Yu, P., Bi, Q.: Analysis of non-linear dynamics and bifurcations of a double pendulum. J. Sound Vib. 217, 691–736 (1998)
Yu, P., Huseyin, K.: Static and dynamic bifurcations associated with a double-zero eigenvalue. Dyn. Stab. Syst. 1, 1–42 (1986)
Yu, P., Zhang, W., Bi, Q.: Vibration analysis on a thin plate with the aid of computation of normal forms. Int. J. Non-linear Mech. 36, 597–627 (2001)
Zhou, L.Q., Chen, F.Q.: Stability and bifurcation analysis for a model of a nonlinear coupled pitch-roll ship. Math. Comput. Simul. (in press)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, X., Chen, F. & Zhou, L. Stability and bifurcation for a flexible beam under a large linear motion with a combination parametric resonance. Nonlinear Dyn 56, 101–119 (2009). https://doi.org/10.1007/s11071-008-9382-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-008-9382-y