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Research Papers

Vibration Suppression of a Four-Degrees-of-Freedom Nonlinear Spring Pendulum via Longitudinal and Transverse Absorbers

[+] Author and Article Information
M. Eissa

Department of Engineering Mathematics, Faculty of Electronic Engineering Menouf,  Menoufia University, Menouf 32952, Egyptmeissa46@yahoo.com

M. Kamel

Department of Engineering Mathematics, Faculty of Electronic Engineering Menouf,  Menoufia University, Menouf 32952, Egyptdr_magdi_kamel@yahoo.com

A. T. El-Sayed1

Department of Basic Sciences,  Modern Academy for Engineering and Technology, Mokatem 11585, Egyptashraftaha211@yahoo.com

1

Corresponding author.

J. Appl. Mech 79(1), 011007 (Dec 08, 2011) (11 pages) doi:10.1115/1.4004551 History: Received October 27, 2009; Revised April 24, 2011; Published December 08, 2011; Online December 08, 2011

An investigation into the passive vibration reduction of the nonlinear spring pendulum system, simulating the ship roll motion is presented. This leads to a four-degree-of-freedom (4-DOF) system subjected to multiparametric excitation forces. The two absorbers in the longitudinal and transverse directions are usually designed to control the vibration near the simultaneous subharmonic and internal resonance where system damage is probable. The theoretical results are obtained by applying the multiple scale perturbation technique (MSPT). The stability of the obtained nonlinear solution is studied and solved numerically. The obtained results from the frequency response curves confirmed the numerical results which were obtained using time history. For validity, the numerical solution is compared with the analytical solution. Effectiveness of the absorbers (Ea) are about 13 000 for the first mode (x) and 10 000 for the second mode (ϕ). A threshold value of linear damping coefficient can be used directly for vibration suppression of both vibration modes. Comparison with the available published work is reported.

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Figures

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Figure 1

Diagrammatic representation of the system

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Figure 2

Response of the system without absorbers at simultaneous subharmonic resonance case (Ω12≅2ω1,Ω21≅2ω2)

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Figure 3

Response of the system with absorbers at simultaneous subharmonic and internal resonance case (Ω12≅2ω1,Ω21≅2ω2,ω3≅2ω1,ω2≅2ω4)

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Figure 4

Response of the natural and excitation frequencies

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Figure 5

Frequency response curves at selected values (a1 against σ1) (c1=0.08,c3=0.0018,c5=11c3,α1=0.4,α2=0.6,β1=0.04,β2=0.045,ω1=5,ω2=4,ω3=2ω1,f12=10,r2=0.013,a2=0.00006,a3=0.009)

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Figure 8

Frequency response curves of the system without and with absorber

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Figure 7

(…) analytic solution (—) numerical solution with the same values of parameters at: (Ω12≅2ω1,Ω21≅2ω1, ω3≅2ω1,ω4≅2ω2). (a) For the first mode (x). (b) For the second mode (ϕ).

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Figure 6

Frequency response curves at selected values (a2 against σ2) (c2=0.03,c4=0.0013, α1=0.4, β1=0.04, β2=0.045, ω1=5, ω2=4, ω4=ω1/2,f21=10, r1=0.3,r2=0.013,a1=0.00003)

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