Fundamental and Subharmonic Resonances of Harmonically Oscillation with Time Delay State Feedback

Time delays occur in many physical systems. In particular, when automatic control is used with structural or mechanical systems, there exists a delay between measurement of the system state and corrective action. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. We investigate the fundamental resonance and subharmonic resonance of order one-half of a harmonically oscillation under state feedback control with a time delay. By using the multiple scale perturbation technique, the first order approximation of the resonances are derived and the effect of time delay on the resonances is investigated. The fixed points correspond to a periodic motion for the starting system and we show the external excitation-response and frequency-response curves. We analyze the effect of time delay and the other different parameters on these oscillations.

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The Vibration reduction of a cantilever beam subjected to parametric excitation using time delay feedback

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v13i2.5971 ◽

2017 ◽

Vol 13 (2) ◽

pp. 7186-7193

Author(s):

Y A Amer

Keyword(s):

Time Delay ◽

Cantilever Beam ◽

Parametric Excitation ◽

Order Approximation ◽

Perturbation Technique ◽

Dynamical Behavior ◽

Multiple Scale ◽

Steady State Solution ◽

State Feedback Control ◽

Resonance Case

In this paper, dynamical behavior of a cantilever beam subject to parametric excitation under state feedback control with time delay is analyzed. The method of multiple scale perturbation technique is applied to obtain the solution up to the first order approximation. We obtain equations for the amplitude and phase. We studied all resonance cases numerically. Stability of the steady state solution for the selected resonance case is studied applying Rung-Kutta fourth method and frequency response equation via Matlab 7.0 and maple 16. From the results, it can be seen that the frequency and amplitude responses for the selected resonance case can be affected by the time delayed control. Effects of different parameters of the system are studied.

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Vibration Control of a Cantilever Beam with Time Delay State Feedback

Zeitschrift für Naturforschung A ◽

10.1515/zna-2006-1204 ◽

2006 ◽

Vol 61 (12) ◽

pp. 629-640 ◽

Cited By ~ 4

Author(s):

Atef F. El-Bassiouny

Keyword(s):

Time Delay ◽

Vibration Control ◽

Cantilever Beam ◽

State Feedback ◽

Multiple Scales ◽

State Feedback Control ◽

Response Curves ◽

Flow Equations ◽

Equivalent Damping ◽

The Stability

The primary and subharmonic resonance of order one-third of a cantilever beam under state feedback control with a time delay are investigated. Using the method of multiple scales, we obtain two slow flow equations for the amplitude and phase. The first-order approximate solution is derived and the effect of time delay on the resonance is investigated. The concept of an equivalent damping, related to the delay feedback, is proposed and an appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. The fixed points corresponding to the periodic motion of the starting system are determined, and the frequency-response and external excitation-response curves are shown. Bifurcation analysis is conducted in order to examine the stability of the system.

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Vibration Suppression of Subharmonic Resonance Response Using a Nonlinear Vibration Absorber

Journal of Vibration and Acoustics ◽

10.1115/1.4029268 ◽

2015 ◽

Vol 137 (2) ◽

Cited By ~ 2

Author(s):

A. T. EL-Sayed ◽

H. S. Bauomy

Keyword(s):

Nonlinear System ◽

Vibration Suppression ◽

Order Approximation ◽

Perturbation Technique ◽

Multiple Scale ◽

Internal Resonances ◽

State Response ◽

Averaged Equations ◽

Resonance Response ◽

Scale Perturbation

This paper is concerned with the vibration of a two degree-of-freedom (2DOF) nonlinear system subjected to multiparametric excitation forces. The vibrating motion of the system is described by the coupled differential equations having both quadratic and cubic terms. The aim of this work is to use a nonlinear absorber to control the vibration of the nonlinear system near the simultaneous subharmonic and internal resonances, where the vibrations are severe. Multiple scale perturbation technique (MSPT) is applied to obtain the averaged equations up to the second-order approximation. The steady-state response and their stability are studied numerically for the nonlinear system at the simultaneous subharmonic and internal resonances. Some recommendations regarding to the different system parameters are given following studying the effects of various parameters. Comparison with the available published work is made.

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Vibration Control in MEMS Resonator Using Positive Position Feedback (PPF) Controller

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v12i11.1114 ◽

2016 ◽

Vol 12 (11) ◽

pp. 6821-6834

Author(s):

Y A Amer ◽

A.T EL Sayed ◽

A.M. Salem

Keyword(s):

Order Approximation ◽

Perturbation Technique ◽

Multiple Scale ◽

Steady State Solution ◽

Positive Position Feedback ◽

Mems Resonator ◽

Position Feedback ◽

Resonator System ◽

The Stability ◽

First Order Approximation

In this paper, the vibration of a micro-electromechanical resonator with positive position feedback controller is studied. The analytical results are obtained to the first order approximation by using the multiple scale perturbation technique. The stability of the steady-state solution is presented and studied applying frequency response equations near the simultaneous primary and internal resonance cases. The effects of the controller and some system parameters on the vibrating system are studied numerically. The main result of this paper indicates that it is possible to reduce the vibration for the resonator system.

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