Abstract
The purpose of this study is to understand the main differences between the deterministic and random response characteristics of an inextensible cantilever beam (with a tip mass) in the neighborhood of combination parametric resonance. The excitation is applied in the plane of largest rigidity such that the bending and torsion modes are cross-coupled through the excitation. In the absence of excitation, the two modes are also coupled due to inertia nonlinearities. For sinusoidal parametric excitation, the beam experiences instability in the neighborhood of the combination parametric resonance of the summed type, i.e., when the excitation frequency is in the neighborhood of the sum of the first bending and torsion natural frequencies. The dependence of the response amplitude on the excitation level reveals three distinct regions: nearly linear behavior, jump phenomena, and energy transfer. In the absence of nonlinear coupling, the stochastic stability boundaries are obtained in terms of sample Lyapunov exponent. The response statistics are estimated using Monte Carlo simulation, and measured experimentally. The excitation center frequency is selected to be close to the sum of the bending and torsion mode frequencies. The beam is found to experience a single response, two possible responses, or non-stationary responses, depending on excitation level. Experimentally, it is possible to obtain two different responses for the same excitation level by providing a small perturbation to the beam during the test.
Similar content being viewed by others
References
Cesari, L., Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Academic Press, New York, 1963.
Ibrahim, R. A. and Barr, A. D. S., ‘Parametric vibration, Part I: Mechanics of linear problems’, The Shock and Vibration Digest 10(1), 1978, 15‐29.
Roberts, J. W., ‘Simple models of complex vibrations’, International Journal of Mechanical Engineering Education 13(1), 1985, 55‐75.
Crespo da Silva, M. R. M. and Glynn, C. C., ‘Nonlinear flexural-flexural-torsion dynamics of inextensible beams, I: Equations of motion’, Journal of Structure Mechanics 6(4), 1978, 437‐448.
Crespo da Silva, M. R. M. and Glynn, C. C., ‘Nonlinear flexural-flexural-torsion dynamics of inextensible beams, II: Forced motions’, Journal of Structure Mechanics 6(4), 1978, 449‐461.
Cartmell, M., ‘The equations of motion of a parametrically excited cantilever beam’, Journal of Sound and Vibration 143(3), 1990, 395‐406.
Crespo da Silva, M. R. M. and Zaretsky, C. L., ‘Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics, I: Primary resonance’, Nonlinear Dynamics 5, 1994, 3‐23.
Zaretsky, C. L. and Crespo da Silva, M. R. M., ‘Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics, II: Combination resonance’, Nonlinear Dynamics 5, 1994, 161‐180.
Bolotin, V. V., The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, CA, 1964.
Dugundji, J. and Mukhopadhyay, V., ‘Lateral bending-torsion vibrations of a thin beam under parametric excitation’, ASME Journal of Applied Mechanics 40, 1973, 693‐698.
Cartmell, M. P. and Roberts, J. W., ‘Simulation of combination resonances in a parametrically excited cantilever beam’, Strain 23(3), 1987, 117‐126.
Ibrahim, R. A., Parametric Random Vibration, Wiley, New York, 1985.
Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, Dover, New York, 1944.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Khasminiskii, R. Z., Stochastic Stability of Differential Equations, Sijthoff & Noordhoof, Alphen aan den Rijn, The Netherlands, 1980.
Arnold, L., ‘A formula connecting sample and moment stability of linear stochastic systems’, SIAM Journal of Applied Mathematics 44(4), 1984, 793‐802.
Horsthemke, W. and Lefever, R., Noise-Induced Transition, Springer-Verlag, Berlin, 1984.
Horsthemke, W. and Lefever, R., ‘Noise-induced transition’, in Noise in Nonlinear Dynamical Systems, Vol. 2: Theory of Noise-Induced Processes in Special Applications, F. Moss and P. V. E. McClintock (eds.), Cambridge University Press, Cambridge, 1989, pp. 179‐208.
Ibrahim, R. A., ‘Stabilization and stochastic bifurcation with application to nonlinear ocean structures’, in Stochastically Excited Nonlinear Systems, M. Shlesinger and T. Sween (eds.), World Scientific Publishers, Singapore, 1998 (in press).
Kliemann, W. and Arnold, L., ‘Lyapunov exponent of linear stochastic systems’, Report No. 93, Forschungsschwerpunkt Dynamische Universität Bremen, June 1983.
Arnold, L. and Kliemann, W., ‘Qualitative theory of stochastic systems’, in Probabilistic Analysis and Related Topics, Vol. 3, A. T. Bharucha-Reid, (ed.), Academic Press, New York, 1983, pp. 1‐79.
Nemat-Nasser, S., ‘On stability under non-conservative loads’, Study No. 6, University of Waterloo, Ontario, 1972, Chapter 10, pp. 351‐384.
Ariaratnam, S. T. and Srikantaiah, T. K., ‘Parametric instability in elastic structures under stochastic loading’, Journal of Structural Mechanics 6(4), 1978, 349‐365.
Ariaratnam, S. T. and Xie, W. C., ‘Lyapunov exponents and stochastic stability of coupled linear systems under real noise excitation’, ASME Journal of Applied Mechanics 59, 1992, 664‐673.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ibrahim, R.A., Hijawi, M. Deterministic and Stochastic Response of Nonlinear Coupled Bending-Torsion Modes in a Cantilever Beam. Nonlinear Dynamics 16, 259–292 (1998). https://doi.org/10.1023/A:1008096810969
Issue Date:
DOI: https://doi.org/10.1023/A:1008096810969