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27 July 1998 Control of a single-degree-of-freedom system under principal parametric excitation
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We consider the problem of suppressing the vibrations of a structure that is subjected to a principal parametric excitation. The vibration amplitudes resulting from such resonance cannot be fully controlled by conventional techniques, such as the addition of linear damping through velocity feedback or by the implementation of conventional mass absorbers. However, it has been shown that the growth of the response is limited by nonlinearities. In this work, we capitalize on this fact and devise a simple nonlinear feedback law to suppress the vibrations of a cantilever beam when subjected to a parametric resonance. We model the dynamics of the first mode of the beam with a second-order nonlinear ordinary-different equation. The model accounts for viscous damping, air drag, and inertia and geometric nonlinearities. We propose a control law based on cubic position and velocity feedback. We use the method of multiple scales to derive two first-order ordinary- differential equations that govern the time variation of the amplitude and phase of the response. We conduct a stability study and analyze the effect of the control gains on the response of the system. The results show that cubic velocity feedback leads to effective vibration suppression and bifurcation control.
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Shafic S. Oueini and Ali H. Nayfeh "Control of a single-degree-of-freedom system under principal parametric excitation", Proc. SPIE 3329, Smart Structures and Materials 1998: Smart Structures and Integrated Systems, (27 July 1998);

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