Abstract
This paper presents a study on the performance of a positive position feedback (PPF) controller to suppress the vibration of a horizontal beam under vertical excitation. Time delays in the control loop are taken into consideration to study their effects on the controller performance and the stable region. The integral iterative method is conducted to obtain a second-order approximate solution and the corresponding amplitude equations for the considered system. The stability of the steady-state solutions is ascertained using a combination of Floquet theory and Hill’s determinant. The maximum limits of time delays at which the system remains stable have been determined for different values of control parameters. And the effects of various control parameters on the existence of multiple-solution region are investigated. The analysis illustrates that the appearance of time delay and the elimination of controller damping coefficient are the two main factors to enhance the nonlinear characteristics of the controlled system. The points at which the steady-state amplitude of the main system reaches its minimum are studied analytically. The analyses show that the analytical results are in excellent agreement with the numerical simulations.
Highlights
Active vibration control has been used to suppress the undesired vibrations in different systems for many years
Jun [2] employed an active linear absorber based on positive position feedback (PPF) control to reduce the high-amplitude vibration of the single-mode of the flexible beam when subjected to primary resonance excitation
U€ + 2ζω2u_ + ω22u αv t − τ2, where v denotes the response of the main system, u denotes the response of the PPF controller, ω1 is the natural frequency of the main system, μ is the damping ratio of the main system, β is the curvature nonlinearity coefficient, δ denotes the inertia nonlinearity coefficient, ζ is the damping ratio of the controller, ω2 is the natural frequency of the controller, y0 and Ω represent the amplitude and frequency of the support motion, c denotes the control signal gain, α denotes the feedback signal gain, and τ1, τ2 are time delays
Summary
Active vibration control has been used to suppress the undesired vibrations in different systems for many years. Jun [2] employed an active linear absorber based on PPF control to reduce the high-amplitude vibration of the single-mode of the flexible beam when subjected to primary resonance excitation. It showed that the control scheme possessed a wide suppression bandwidth if the absorber’s frequency was properly tuned. Kandil and El-Ganaini [17] utilized a time-delayed PPF controller to reduce the nonlinear oscillations of the compressor blade system subjected to a primary excitation at 1 : 1 internal resonance. A positive position feedback controller is utilized to suppress the vibration of a nonlinear horizontal beam under vertical excitation. U€ + 2ζω2u_ + ω22u αv t − τ2, where v denotes the response of the main system (the horizontal beam), u denotes the response of the PPF controller, ω1 is the natural frequency of the main system, μ is the damping ratio of the main system, β is the curvature nonlinearity coefficient, δ denotes the inertia nonlinearity coefficient, ζ is the damping ratio of the controller, ω2 is the natural frequency of the controller, y0 and Ω represent the amplitude and frequency of the support motion, c denotes the control signal gain, α denotes the feedback signal gain, and τ1, τ2 are time delays
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