Vibration reduction of a three DOF non-linear spring pendulum

Eissa, M.; Sayed, M.

doi:10.1016/j.cnsns.2006.04.001

Cited by 52 publications

(28 citation statements)

References 14 publications

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“…(19) into Eqs. (9)- (12) and eliminating the secular terms, leads to the solvability conditions for the first order approximation noting that A 1 , A 2 , A 3 and A 4 are functions in T 1 we get…”

Section: Perturbation Analysismentioning

confidence: 98%

“…(13)-(15) into Eq. (10)- (12) and eliminating the secular terms to obtain the solutions are given by:…”

Section: Perturbation Analysismentioning

confidence: 99%

“…It can be applicable for all excitation frequencies. Asfar, Eissa, El-Bassiouny and Shitikova [5][6][7][8][9][10][11][12][13][14][15][16] showed how the passive vibration control is effective for reducing the vibrations at different resonance cases. Eissa, El-Bassiouny and Jaensch [17][18][19][20][21][22][23] showed how the active control is effective in vibration reduction at resonance at different modes of vibration.…”

Section: Introductionmentioning

confidence: 99%

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Vibration suppression in multi-tool ultrasonic machining to multi-external and parametric excitations

2009

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Ultrasonic machining (USM) is of particular interest for the machining of non-conductive, brittle materials such as engineering ceramics. In this paper, a multi-tool technique is used in USM to reduce the vibration in the tool holder and have reasonable amplitude for the tools. This can be done via dynamic absorbers. The coupling of four nonlinear oscillators of the tool holder and tools representing ultrasonic cutting process are investigated. This leads to a four-degree-of-freedom system subjected to multi-external and multi-parametric excitation forces. The aim of this work is to control the tool holder behavior at simultaneous primary, sub-harmonic and internal resonance condition. Multiple scale perturbation method is used to obtain the solution up to the second order approximations. The different resonance cases are reported and studied numerically. The stability of the system is investigated by using both phase-plane and frequency response techniques. The effects of the different parameters of the tools on the system behavior are studied numerically. Comparison with the available published work is reported. Keywords Control · Stability · Ultrasonic machining (USM)List of symbols c n (n = 1, 2, 3, 4) The damping coefficients of the tool holder and the tools k n (n = 1, 2, 3, 4) The stiffness of the tool holder and the tools and frequenciesThe masses of the tool holder and the tools ζ n = c n 2m 1 (n = 1, 2, 3, 4) The linear damping factors of the tool holder ζ 5 = c 5 m 1The quadratic damping factors of the tool holderThe damping factors of the tools η n = h n /m 1 (n = 1, 2, 3, 4)The coupling non-linear parameters of the tool holder η 5 = h 2 m 2 , η 6 = h 3 m 3 , η 7 = h 4 m 4The coupling non-linear parameters of the tools ω 2 n = k n m n (n = 1, 2, 3, 4) The natural frequencies of the tool holder and tools. γ

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Section: Perturbation Analysismentioning

confidence: 98%

“…(13)-(15) into Eq. (10)- (12) and eliminating the secular terms to obtain the solutions are given by:…”

Section: Perturbation Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Vibration suppression in multi-tool ultrasonic machining to multi-external and parametric excitations

2009

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“…A threshold value of linear damping has been obtained, where the system vibration can be reduced dramatically. Eissa and Sayed [17][18][19] and Sayed [20], studied the effects of different active controllers on simple and spring pendulum at the primary resonance via negative velocity feedback or its square or cubic. Hamed et al [21][22][23] studied USM model subject to multi-external or both multi-external and multi-parametric and both multi-external and tuned excitation forces.…”

Section: Introductionmentioning

confidence: 99%

Non-Linear Analysis of Vibrations of Non-Linear System Subjected to Multi-Excitation Forces via a Non-Linear Absorber

El-Ghareeb¹,

Hamed²,

Elkader³

2012

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The dynamic response of mechanical and civil structures subjected to high-amplitude vibration is often dangerous and undesirable. Sometimes controlled vibration is desirable as in the machinery used in the formation of rigid hard material as Ceramics and diamond. This process is done via the passive control methods. The main purpose of this paper is to how reduction of vibration of nonlinear system subjected to multi-excitation forces via a nonlinear absorber. The nonlinear differential equations describing the model, which describe ultrasonic cutting machine are solved by using perturbation method. The effects of different parameter on the response of the system are studied. The stability of the numerical solution is investigated by using frequency response equations and phase-plane method. The simulation results are achieved using Matlab and Maple programs. A comparison is made with the available published work.

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“…Eissa and Sayed [19][20][21] and Sayed [22], studied the effects of different active controllers on simple and spring pendulum at the primary resonance via negative velocity feedback or its square or cubic. Sayed and Hamed [23] studied the response of a two-degree-of-freedom system with quadratic coupling under parametric and harmonic excitations.…”

Section: Introductionmentioning

confidence: 99%

The Analytical and Numerical Solutions of Differential Equations Describing of an Inclined Cable Subjected to External and Parametric Excitation Forces

Elkader¹

2011

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The analytical and numerical solutions of the response of an inclined cable subjected to external and parametric excitation forces is studied. The method of perturbation technique are applied to obtained the periodic response equation near the simultaneous principal parametric resonance in the presence of 2:1 internal resonance of the system. All different resonance cases are extracted. The effects of different parameters and worst resonance case on the vibrating system are investigated. The stability of the system are studied by using frequency response equations and phase-plane method. Variation of the parameters α 2 , α 3 , β 2 , γ 2 , η 2 , γ 3 , η 3 , f 2 leads to multi-valued amplitudes and hence to jump phenomena. The simulation results are achieved using MATLAB 7.6 programs.

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Vibration reduction of a three DOF non-linear spring pendulum

Cited by 52 publications

References 14 publications

Vibration suppression in multi-tool ultrasonic machining to multi-external and parametric excitations

Vibration suppression in multi-tool ultrasonic machining to multi-external and parametric excitations

Non-Linear Analysis of Vibrations of Non-Linear System Subjected to Multi-Excitation Forces via a Non-Linear Absorber

The Analytical and Numerical Solutions of Differential Equations Describing of an Inclined Cable Subjected to External and Parametric Excitation Forces

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