Stabilization of the Parametric Resonance of a Cantilever Beam by Bifurcation Control with a Piezoelectric Actuator

In this work, bifurcation control using a piezoelectric actuator isimplemented to stabilize the parametric resonance induced in acantilever beam. The piezoelectric actuator is attached to the surfaceof the beam to produce a bending moment in the beam. The dimensionlessequation of motion for the beam with the piezoelectric actuator on itssurface is derived and the modulation equations for the complexamplitude of an approximate solution are obtained using the method ofmultiple scales. We then acquire the bifurcation set that expresses theboundary of the stable and unstable regions. The bifurcation set ischaracterized by the modulation equations. Next, we determine the orderof feedback gains to modify these modulation equations. By actuating thepiezoelectric actuator under the appropriate feedback, bifurcationcontrol is carried out resulting in the shift of the bifurcation set andthe expansion of the stable region. The main characteristic of thestabilization method introduced above is that the work done by thepiezoelectric actuator is zero in the state where the parametricresonance is stabilized. Thus zero power control is realized in such astate. Experimental results show the validity of the proposedstabilization method for the parametric resonance induced in thecantilever beam.

[1]  Nobuharu Aoshima,et al.  Nonlinear Analysis of a Parametrically Excited Cantilever Beam : Effect of the Tip Mass on Stationary Response(Special Issue on Nonlinear Dynamics) , 1998 .

[2]  J. Thompson,et al.  Elastic Instability Phenomena , 1984 .

[3]  Nobuharu Aoshima,et al.  Stabilization of 1/3-Order Subharmonic Resonance Using an Autoparametric Vibration Absorber , 1999 .

[4]  Balakumar Balachandran,et al.  Experimental Verification of the Importance of The Nonlinear Curvature in the Response of a Cantilever Beam , 1994 .

[5]  Jon R. Pratt,et al.  A Nonlinear Vibration Absorber for Flexible Structures , 1998 .

[6]  A. Tondl,et al.  Non-linear Vibrations , 1986 .

[7]  Leonard Meirovitch,et al.  Dynamics And Control Of Structures , 1990 .

[8]  H. Yabuno.  Bifurcation Control of Parametrically Excited Duffing System by a Combined Linear-Plus-Nonlinear Feedback Control , 1997 .

[9]  Jerry H. Griffin,et al.  Transient response of joint dominated space structures - A new linearization technique , 1988 .

[10]  E. Abed,et al.  Local feedback stabilization and bifurcation control, I. Hopf bifurcation , 1986 .

[11]  M. R. Silva,et al.  Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. I. Equations of Motion , 1978 .

[12]  A. D. S. Barr,et al.  The Autoparametric Vibration Absorber , 1972 .

[13]  A. Ertas,et al.  Dynamics and bifurcations of a coupled column-pendulum oscillator , 1995 .

[14]  Amir Khajepour,et al.  Experimental Control of Flexible Structures Using Nonlinear Modal Coupling: Forced and Free Vibration , 1997 .

[15]  M. Farid Golnaraghi,et al.  Regulation of flexible structures via nonlinear coupling , 1991 .

[16]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[17]  E. Abed,et al.  Local feedback stabilization and bifurcation control, II. Stationary bifurcation , 1987 .

[18]  J. L. Fanson,et al.  Positive position feedback control for large space structures , 1990 .