Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

Abstract In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass–spring–damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.

[1]  J. Hunt,et al.  The broadband dynamic vibration absorber , 1982 .

[2]  M. F. Golnaraghi,et al.  Experimental implementation of the internal resonance control strategy , 1996 .

[3]  J. Shaw,et al.  On the response of the non-linear vibration absorber , 1989 .

[4]  A. Maccari,et al.  Vibration amplitude control for a van der Pol–Duffing oscillator with time delay , 2008 .

[5]  Henry J. Rice,et al.  Combinational ínstability of the non-linear vibration absorber , 1986 .

[6]  Ricardo A. Burdisso,et al.  Control of internal resonances in vibration isolators using passive and hybrid dynamic vibration absorbers , 2005 .

[7]  Attilio Maccari.  Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback , 2003 .

[8]  Guanrong Chen,et al.  Resonance Control for a Forced Single-Degree-of-Freedom Nonlinear System , 2004, Int. J. Bifurc. Chaos.

[9]  A. Maccari.  Vibration Control for the Primary Resonance of a Cantilever Beam by a Time Delay State Feedback , 2003 .

[10]  Kefu Liu,et al.  The damped dynamic vibration absorbers: revisited and new result , 2005 .

[11]  B. Schmalhorst,et al.  Optimization of a non-linear dynamic vibration absorber , 1985 .

[12]  H. Nijmeijer,et al.  Application of a Dynamic Vibration Absorber to a Piecewise Linear Beam System , 2004 .

[13]  Jinchen Ji,et al.  Local Bifurcation Control of a Forced Single-Degree-of-Freedom Nonlinear System: Saddle-Node Bifurcation , 2001 .

[14]  S. Chatterjee.  Vibration control by recursive time-delayed acceleration feedback , 2008 .

[15]  Mark J. Schulz,et al.  Non-linear vibration absorbers using higher order internal resonances , 2000 .

[16]  Ali H. Nayfeh,et al.  A Theoretical and Experimental Implementation of a Control Method Based on Saturation , 1997 .

[17]  S. M. Holzer,et al.  Book Reviews : SYSTEM DYNAMICS Katsuhiko Ogata Prentice-Hall, Inc., Englewood Cliffs, NJ, 1978 , 1980 .

[18]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[19]  Ivan Jordanov,et al.  Optimal design of linear and non-linear dynamic vibration absorbers , 1988 .

[20]  D. J. Wilkins,et al.  Optimal design of a non-linear dynamic absorber , 1990 .

[21]  Guanrong Chen,et al.  Bifurcation Control: Theories, Methods, and Applications , 2000, Int. J. Bifurc. Chaos.

[22]  S. Natsiavas,et al.  Steady state oscillations and stability of non-linear dynamic vibration absorbers , 1992 .

[23]  D. J. Mead.  Passive Vibration Control , 1999 .

[24]  Jiashi Tang,et al.  A time delay control for a nonlinear dynamic beam under moving load , 2008 .

[25]  Waion Wong,et al.  H∞ and H2 optimizations of a dynamic vibration absorber for suppressing vibrations in plates , 2009 .

[26]  R. Bishop.  Mechanical Vibration , 1958, Nature.

[27]  J. Hunt.  Dynamic vibration absorbers , 1979 .

[28]  Henry J. Rice,et al.  On practical implementations of the non-linear vibration absorber , 1986 .