Abstract
A uniform cantilever beam under the effect of a time-periodic axial force is investigated. The beam structure is discretized by a finite-element approach. The linearised equations of motion describing the planar bending vibrations of the beam structure lead to a system with time-periodic stiffness coefficients. The stability of the system is investigated by a numerical method based on Floquet’s theorem and an analytical approach resulting from a first-order perturbation. It is demonstrated that the parametrically excited beam structure exhibits enhanced damping properties, when excited near a specific parametric combination resonance frequency. A certain level of the forcing amplitude has to be exceeded to achieve the damping effect. Upon exceeding this value, the additional artificial damping provided to the beam is significant and works best for suppression of vibrations of the first vibrational mode of the cantilever beam.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abadi, .: Nonlinear dynamics of self-excitation in autoparametric systems, PhD thesis. Utrecht University, Netherlands (2003)
Bilharz H. (1944). Bemerkung zu einem Satze von Hurwitz. Zeitschrift für angewandte Mathematik und Mechanik (ZAMM) 24(2): 77–82
Bolotin, V.V.: Dynamic stability of elastic systems, Holden Day, San Francisco (1964)
Cartmell M. (1990). Introduction to Linear, Parametric and Nonlinear Vibrations. Chapman and Hall, London
Chen C.-C. and Yeh M.-K. (1995). Parametric instability of a cantilevered column under periodic loads in the direction of the tangency coefficient. JSV 183(2): 253–267
Dohnal, F.: Damping of mechanical vibrations by parametric excitation. PhD thesis, Vienna University of Technology (2005)
Dohnal, F.: General parametric stiffness excitation–anti-resonance frequency and symmetry. Acta Mechanica (published online) (2007)
Dohnal, F.: Damping by parametric stiffness excitation—parametric resonance and anti-resonance. J Vibrat Control (accepted) (2007)
Dohnal, F., Ecker, H., Tondl, A.: Vibration control of self-excited oscillations by parametric stiffness excitation. In: Proceedings of the 11th International Congress of Sound and Vibration (ICSV11), St. Petersburg, pp. 339–346 (2004)
Ecker, H., Dohnal, F., Springer, H.: Enhanced Damping of a Beam Structure by Parametric Excitation. In Proceedings of the 5th EUROMECH Nonlinear Dynamics Conference (ENOC-2005), Eindhoven, August, pp. 22-271 (2005)
Ewins, D.J.: Modal Testing–Theory, Practice and Applications, 2nd edn. Research Studies Press Ltd, (2000)
Fatimah S. and Verhulst F. (2003). Suppressing flow-induced vibrations by parametric excitation. Nonlinear Dynamics 23: 275–297
Gèradin M. and Rixen D. (1994). Mechanical Vibrations—Theory and Application to Structural Dynamics. Wiley, New York
Habib M.S. and Radcliffe C.J. (1991). Active parametric damping of distributed parameter beam transverse vibration. J Dyn Syst Meas Control 113: 295–299
Iwatsubo T., Sugiyama Y. and Ogino S. (1974). Simple and combination resonances of columns under periodic axial loads. JSV 33: 211–221
Nayfeh A.H. and Pai P.F. (2004). Linear and Nonlinear Structural Mechanics. Wiley, New York
Parkus H. (1966). Mechanik fester Körper, 2nd edn. Springer, Wein
Petyt M. (1990). Introduction to finite element vibration analysis. Cambridge University Press, London
Rahn C.D. and Mote C.D. (1996). Axial Force Stabilization of Transverse Vibration in Pinned and Clamped Beams. J Dyn Syst Meas Control 118: 379–380
Seyranian, A., Mailybaev, A.: Multiparameter stability theory with mechanical applications, Series on Stability, Vibration and Control of Systems 13, World Scientific (2003)
Springer, H., Kovyrshin, S.: Active parametric vibration control of a smart beam. In: Proceedings of 8th International Conference on Vibrations in Rotating Machinery, University of Wales, Swansea, Sept. 7–9. IMECHE, C623/017, pp. 703–712 (2004)
Tondl A. (1965). Some problems of rotor dynamics. Chapman and Hall, London
Tondl, A.: On the interaction between self-excited and parametric vibrations, in Monographs and Memoranda 25, National Research Institute for Machine Design, Prague (1978)
Tondl, A.: Quenching of self-excited vibrations, Academy of Sciences Czech Republic (1991)
Tondl A. (1998). To the problem of quenching self-excited vibrations. Acta Technica CSAV 43: 109–116
Tondl, A., Ecker, H.: Cancelling Of Self-Excited Vibrations By Means Of Parametric Excitation. Proceedings of 1999 ASME Design Engineering Technical Conferences (DETC), Sept. 12–15, 1999, Las Vegas (1999)
Yakubovich V.A. and Starzhinskii V.M. (1975). Linear differential equations with periodic coefficients, vols.1 and 2. Wiley, London
Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Springer, Heidelberg (2000)
Verhulst, F.: Methods and Applications of Singular Perturbations, Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dohnal, F., Ecker, H. & Springer, H. Enhanced damping of a cantilever beam by axial parametric excitation. Arch Appl Mech 78, 935–947 (2008). https://doi.org/10.1007/s00419-008-0202-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-008-0202-0