Abstract
The nonlinear vibration control of a nonlinear dynamical system modeled as the well-known Duffing oscillators is investigated within this article. The conventional Positive Position Feedback (PPF) controller is proposed to mitigate the considered system nonlinear vibrations. The whole system mathematical model is analyzed by applying the multiple time scales perturbation method. The slow-flow modulation equations that govern the oscillation amplitudes of both the main system and controller are derived. The stability analysis is investigated based on Lyapunov’s first method. The effects of the different control parameters on both the main system and controller are explored. The obtained analytical and numerical results illustrated that the PPF controller can eliminate the main system nonlinear vibrations once the controller natural frequency is tuned to be the same value as the external excitation frequency, otherwise, the controller adds excessive vibrational energy to the main system rather than suppressing it. In addition, the PPF controller can destabilize the main system motion when excited by strong excitation force. Therefore, a modified version of the PPF controller named the Adaptive Positive Position Feedback (APPF) controller is proposed to overcome the main drawbacks of the conventional PPF controller. The idea is to track the external excitation frequency using an adaptive frequency measurement technique to update continuously the PPF controller natural frequency to become the same value of the excitation frequency. Based on this strategy, the system mathematical model is analyzed again by making the controller’s natural frequency equal to the external excitation frequency. The obtained analytical and numerical simulations showed that the adaptive positive position feedback controller can suppress the main system nonlinear vibration close to zero regardless of the excitation force amplitude and excitation frequency.
Highlights
This article is an open access articleThe occurrence of nonlinear vibration in different engineering systems such as aircraft wings, bridges, satellites, tall structures, robotic arms, rotating machinery, micro-electromechanical systems . . . etc. is an undesirable phenomenon
The positive position feedback (PPF) controller is one of the feasible control techniques that has been applied extensively to suppress the nonlinear vibration of a wide range of dynamical systems
The Adaptive Positive Position Feedback (APPF) controller is a conventional PPF controller that uses a real-time frequency tracking method to update its natural frequency to be the same value as the excitation frequency of the main system
Summary
The occurrence of nonlinear vibration in different engineering systems such as aircraft wings, bridges, satellites, tall structures, robotic arms, rotating machinery, micro-electromechanical systems . . . etc. is an undesirable phenomenon. The positive position feedback (PPF) controller is one of the feasible control techniques that has been applied extensively to suppress the nonlinear vibration of a wide range of dynamical systems. Applied the adaptive positive position feedback controller to mitigate the nonlinear vibrations of a frequency varying the structure. The author confirmed that the applied controller has high efficiency in suppressing the system vibrations once the controller’s natural frequency is properly tuned to the main system’s natural frequency. The numerical and experimental results confirmed that the fractional-order PPF controller is the best control algorithm for suppressing the modal vibrations without spillover occurrence. The APPF controller is a conventional PPF controller that uses a real-time frequency tracking method to update its natural frequency to be the same value as the excitation frequency of the main system. The schematic diagram describes the connection of the main system and the PPF controller
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