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Research Papers

On Parametric Stability of a Nonconstant Axially Moving String Near Resonances

[+] Author and Article Information
Rajab A. Malookani

Department of Mathematical Physics,
Delft Institute of Applied Mathematics,
Delft University of Technology,
Delft 2628 CD, The Netherlands
e-mail: R.Ali@tudelft.nl

Wim T. van Horssen

Department of Mathematical Physics,
Delft Institute of Applied Mathematics,
Delft University of Technology,
Delft 2628 CD, The Netherlands
e-mail: w.t.vanhorssen@tudelft.nl

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 2, 2015; final manuscript received August 15, 2016; published online October 25, 2016. Assoc. Editor: Walter Lacarbonara.

J. Vib. Acoust 139(1), 011005 (Oct 25, 2016) (12 pages) Paper No: VIB-15-1505; doi: 10.1115/1.4034628 History: Received December 02, 2015; Revised August 15, 2016

The stability of an axially moving string system subjected to parametric excitation resulting from speed fluctuations has been examined in this paper. The time-dependent velocity is assumed to be a harmonically varying function around a (low) constant mean speed. The method of characteristic coordinates in combination with the two timescales perturbation method is used to compute the first-order approximation of the solutions of the equations of motion that governs the transverse vibrations of an axially moving string. It turns out that the system can give rise to resonances when the velocity fluctuation frequency is equal (or close) to an odd multiple of the natural frequency of the system. The stability conditions are investigated analytically in terms of the displacement-response and the energy of the system near the resonances. The effects of the detuning parameter on the amplitudes of vibrations and on the energy of the system are also presented through numerical simulations.

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Figures

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Fig. 1

A schematic model of a moving belt system between two fixed supports

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Fig. 2

Integration in the characteristic ξ (or σ) direction

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Fig. 3

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, δ = 0.01, and x = 0.05. (a) The unstable first-order asymptotic approximation (v0) and (b) the unstable numerical solution (u).

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Fig. 4

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, and δ = 0.01. (a) Approximated energy and (b) energy of the system.

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Fig. 5

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, δ = 2α, and x = 0.05. (a) The unstable first-order asymptotic approximation (v0) and (b) the unstable numerical solution (u).

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Fig. 6

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, and δ = 2α. (a) Approximated energy and (b) energy of the system.

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Fig. 7

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, δ = − 2α, and x = 0.05. (a) The unstable first-order asymptotic approximation (v0) and (b) the unstable numerical solution (u).

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Fig. 8

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, and δ = −2α. (a) Approximated energy and (b) energy of the system.

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Fig. 9

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, δ = 15, and x = 0.05. (a) The stable first-order asymptotic approximation (v0) and (b) the stable numerical solution (u).

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Fig. 10

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, and δ = 15. (a) Approximated energy and (b) energy of the system.

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Fig. 11

Approximate instability region in the (α, Ω)-plane for ε = 0.1: the amplitude α of the velocity fluctuation of the belt versus the frequency Ω of the velocity fluctuation of the belt. The boundaries of the instability region are given by Ω=(2m−1)π+εδ with |δ|=2α and m = 1, 2,….

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