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Research Papers

Application of Time-Delay Absorber to Suppress Vibration of a Dynamical System to Tuned Excitation

[+] Author and Article Information
W. A. A. El-Ganaini

Department of Physics
and Engineering Mathematics,
Faculty of Electronic Engineering,
Menoyfia University,
Menouf 32952, Egypt
e-mail: wa.elganaini@hotmail.com

H. A. El-Gohary

Department of Physics
and Engineering Mathematics,
Faculty of Electronic Engineering,
Menoyfia University,
Menouf 32952, Egypt
e-mail: elgohary_hany@yahoo.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 21, 2012; final manuscript received February 25, 2014; published online June 2, 2014. Assoc. Editor: Brian P. Mann.

J. Vib. Acoust 136(4), 041014 (Jun 02, 2014) (10 pages) Paper No: VIB-12-1205; doi: 10.1115/1.4027629 History: Received July 21, 2012; Revised February 25, 2014

In this work, we present a comprehensive investigation of the time delay absorber effects on the control of a dynamical system represented by a cantilever beam subjected to tuned excitation forces. Cantilever beam is one of the most widely used system in too many engineering applications, such as mechanical and civil engineering. The main aim of this work is to control the vibration of the beam at simultaneous internal and combined resonance condition, as it is the worst resonance case. Control is conducted via time delay absorber to suppress chaotic vibrations. Time delays often appear in many control systems in the state, in the control input, or in the measurements. Time delay commonly exists in various engineering, biological, and economical systems because of the finite speed of the information processing. It is a source of performance degradation and instability. Multiple time scale perturbation method is applied to obtain a first order approximation for the nonlinear differential equations describing the system behavior. The different resonance cases are reported and studied numerically. The stability of the steady-state solution at the selected worst resonance case is investigated applying Runge–Kutta fourth order method and frequency response equations via Matlab 7.0 and Maple11. Time delay absorber is effective, but within a specified range of time delay. It is the critical factor in selecting such absorber. Time delay absorber is better than the ordinary one as from the effectiveness point of view. The effects of the different absorber parameters on the system behavior and stability are studied numerically. A comparison with the available published work showed a close agreement with some previously published work.

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Figures

Grahic Jump Location
Fig. 1

Response of the main system without absorber at combined resonance case Ωk+Ωj≅ω1; λ1 = 0.00009,α = 0.0001,β = 0.2,c = 0.013,d = -0.0008,Ωj/ω1 = 1/2Ωk/ω1 = 1/2,F1 = 0.2,F2 = 0.3,F3 = 0.4

Grahic Jump Location
Fig. 2

Response of the main system and ordinary absorber at simultaneous internal and combined resonance cases (ω1≅ω2,Ωk+Ωj≅ω1); μ = 0.6,λ1 = 0.00009,λ2 = 0.01,α = 0.0001,β = 0.2,c = 0.013 d = -0.0008, Ωj/ω1 = 1/2, Ωk/ω1 = 1/2,F1 = 0.2,F2 = 0.3,F3 = 0.4

Grahic Jump Location
Fig. 3

Effects of time delay on the system and absorber

Grahic Jump Location
Fig. 4

Effects of the time delay tq on steady-state amplitude a1 against σ2

Grahic Jump Location
Fig. 5

Effects of the time delay tq on steady-state amplitude a2 against σ2

Grahic Jump Location
Fig. 6

Effects of the time delay tq on steady-state amplitude a1 against σ1

Grahic Jump Location
Fig. 7

Effects of the time delay tq on steady-state amplitude a2 against σ1

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