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TECHNICAL PAPERS

Open-Loop Nonlinear Vibration Control of Shallow Arches via Perturbation Approach

[+] Author and Article Information
W. Lacarbonara

Dipartimento di Ingegneria Strutturale e Geotecnica, University of Rome La Sapienza, via Eudossiana, 18 Rome 00184, Italy

C.-M. Chin

VSAS Center General Motors Corporation, MC 480-305-200, 6440 E. 12 Mile Rd., Warren, MI 48090-9000

R. R. Soper

Westvaco Covington Research Laboratory, 752 N. Mill Road, Covington, VA 24426

J. Appl. Mech 69(3), 325-334 (May 03, 2002) (10 pages) doi:10.1115/1.1459069 History: Received October 18, 2000; Revised October 25, 2001; Online May 03, 2002
Copyright © 2002 by ASME
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References

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Chin, C. M., Nayfeh, A. H., and Lacarbonara, W., 1997, “Two-to-One Internal Resonances in Parametrically Excited Buckled Beams,” Proceedings of the 38th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials, Kissimmee, FL, Apr. 7–10, AIAA Paper No. 97-1081.
Fujino,  Y., Warnitchai,  P., and Pacheco,  B. M., 1993, “Active Stiffness Control of Cable Vibration,” ASME J. Appl. Mech. 60, pp. 948–953.
Gattulli,  V., Pasca,  M., and Vestroni,  F., 1997, “Nonlinear Oscillations of a Nonresonant Cable Under In-Plane Excitation With a Longitudinal Control,” Nonlinear Dyn. 14, pp. 139–156.
Oueini,  S. S., Nayfeh,  A. H., and Pratt,  J. R., 1998, “A Nonlinear Vibration Absorber for Flexible Structures,” Nonlinear Dyn. 15, pp. 259–282.
Yabuno, H., Kawazoe, J., and Aoshima, N., 1999, “Suppression of Parametric Resonance of a Cantiliver Beam by a Pendulum-Type Vibration Absorber,” Proceedings of the 17th Biennial ASME Conference on Mechanical Vibration and Noise, Las Vegas, NV, Sept. 12–15, ASME, New York, Paper No. DETC99/VIB-8072.
Maschke, B. M. J., and van der Schaft, A. J., 2000, “Port Controlled Hamiltonian Representation of Distributed Parameter Systems,” Proceedings of the IFAC Workshop Lagrangian and Hamiltonian Methods for Nonlinear Control, N. E. Leonard and R. Ortega, eds., Princeton University, Princeton, NJ, March 16–18, Elsevier Science, Oxford, UK.
Ortega, R., van der Schaft, A. J., and Maschke, B. M. J., 1999, “Stabilization of Port Controlled Hamiltonian Systems,” Stability and Stabilization of Nonlinear Systems, Vol. 246, D. Aeyels, F. Lamnabhi-Lagarrigue, and A. J. van der Schaft, eds., Springer-Verlag, New York, pp. 239–260.
Nayfeh, A. H., 1984, “Interaction of Fundamental Parametric Resonances with Subharmonic Resonances of Order One-Half,” Proceedings of the 25th Structures, Structural Dynamics and Materials Conference, Palm Springs, CA, May 14–16, AIAA, Washington, DC.
Soper, R. R., Lacarbonara, W., Chin, C. M., Nayfeh, A. H., and Mook, D. T., 2001, “Open-Loop Resonance-Cancellation Control for a Base-Excited Pendulum,” J. Vib. Control (in press).
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Figures

Grahic Jump Location
Shallow arch geometry with the disturbance and the control input
Grahic Jump Location
First and second-order shape functions: (a) Uc; (b) B; (c) χ1 and χ2; (d) χ3 and χ4; (e) χ5 and χ6; (f ) χ7; (g) χ8; (h) χ9; (i ) χ10; and ( j) χ11 when b=16.5,uB=1.0, and Uc=28.936
Grahic Jump Location
Regions of activation/nonactivation of the principal parametric resonance in the plane of the disturbance frequency detuning and gain when b=16.5 and μ=0.05
Grahic Jump Location
Frequency-response curve of the uncontrolled arch when b=16.5,μ=0.05, and uB=1. Solid (dashed) line indicates stable (unstable) solutions.
Grahic Jump Location
Regions (shaded) of nonactivation of the principal parametric resonance in the plane of the disturbance and control gains for different control phase angles when b=16.5 and μ=0.05
Grahic Jump Location
Uncontrolled (thin line) and optimally controlled (thick line) dynamic deflections at seven discrete times equally spaced within a period of oscillation when b=16.5,μ=0.05,uB=1,σ=−10, and Uc=28.936
Grahic Jump Location
Time histories of the uncontrolled and optimally controlled deflections (ψc=0 and Uc=28.936) at x=1/4 when uB=1,σ=−10,b=16.5,μ=0.05,p(0)=1.5, and q(0)=−0.95
Grahic Jump Location
Time histories of the controlled deflections (second-order solution) at x=1/4 in the detuned case when (a) Δσ=σ−σc=−9.55 and (b) Δσ=8.95 and ψc=0,Uc=28.936,σ=−10,b=16.5,μ=0.05,uB=1,p(0)=1.5, and q(0)=−0.95

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