Nonlinear saturation control of an oscillatory cantilever beam excited transversely at its free end

Document Type : Original Article

Authors

1 Electronics and Electrical Communications Engineering, Faculty of Electronic Engineering, Menoufia University,

2 1Department of Applied and Computational Mathematics, Institute of Basic and Applied Sciences, Egypt-Japan University of Science and Technology (E-JUST), New Borg El-Arab City, 21934, Alexandria, Egypt

3 Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menouf, 32952, Menoufia University, Egypt

Abstract

The oscillations of a cantilever beam with transversely energized free under external and parametric excitation forces are studied. When this kind of excitation occurs, external and parametric forces manifest, leading to the intensification of undesired nonlinear oscillations of the beam, especially in resonant states. The nonlinear differential equation that describes the cantilever beam’s vibrations can be approximately solved analytically and numerically. To reduce these vibrations, the Nonlinear Saturation Controller (NSC) algorithm is applied using a piezoelectric (PZT) actuator. The PZT actuator is placed throughout the beam’s surface to overcome the primary resonance case. Furthermore, the multiple time scales perturbation method is applied to visualize the overall properties of the beam so that the effectiveness of the controller can be assessed. Moreover, the Routh-Hurwitz criterion and Lyapunov's first (indirect) technique are used to verify the stability of the steady-state oscillation amplitude and phase. The cantilever beam bifurcation analysis is presented before and after control, allowing for a comparison of the beam's behavior. Time responses and phase portraits have been used to perform numerical verifications that validate the implemented control method.

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