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Research Papers

On the Transverse, Low Frequency Vibrations of a Traveling String With Boundary Damping

[+] Author and Article Information
Nick V. Gaiko

Department of Mathematical Physics,
Delft Institute of Applied Mathematics,
Delft University of Technology,
Delft 2628 CD, The Netherlands
e-mail: n.gaiko@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 August 27, 2014; final manuscript received January 23, 2015; published online March 12, 2015. Assoc. Editor: Philip Bayly.

J. Vib. Acoust 137(4), 041004 (Aug 01, 2015) (10 pages) Paper No: VIB-14-1316; doi: 10.1115/1.4029690 History: Received August 27, 2014; Revised January 23, 2015; Online March 12, 2015

In this paper, we study the free transverse vibrations of an axially moving (gyroscopic) material represented by a perfectly flexible string. The problem can be used as a simple model to describe the low frequency oscillations of elastic structures such as conveyor belts. In order to suppress these oscillations, a spring–mass–dashpot system is attached at the nonfixed end of the string. In this paper, it is assumed that the damping in the dashpot is small and that the axial velocity of the string is small compared to the wave speed of the string. This paper has two main objectives. The first aim is to give explicit approximations of the solution on long timescales by using a multiple-timescales perturbation method. The other goal is to construct accurate approximations of the lower eigenvalues of the problem, which describe the oscillation and the damping properties of the problem. The eigenvalues follow from a so-called characteristic equation obtained by the direct application of the Laplace transform method to the initial-boundary value problem. Both approaches give a complete and accurate picture of the damping and the low frequency oscillatory behavior of the traveling string.

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Figures

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Fig. 1

A schematic model of an axially moving string with a spring–mass–dashpot at the downstream boundary (pl)

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Fig. 2

Graphical analysis of eigenvalues for m = 1 and κ = 1

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Fig. 3

Transverse displacements w0 at (a) x = 0.5 and at (b) x = 1 against time t with the initial displacement φ(x) = 0.1sin(πx) and the initial velocity ψ(x) = 0.05sin(πx) for ɛ = 0.1,η = 0.5,m = 1,κ = 1, and N = 10

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Fig. 4

Transverse displacements w0 at (a) x = 0.5 and at (b) x = 1 against time t with the initial displacement φ(x) = 0.1sin(πx) and the initial velocity ψ(x) = 0.05sin(πx) for ɛ = 0.1,η = 2,m = 1,κ = 1, and N = 10

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Fig. 5

The approximate damping rates (snre(ɛ) ~ ɛs1,1,n) are plotted against the approximate frequencies (snim(ɛ) ~ s0,2,n + ɛs1,2,n) for ɛ = 0.1,m = 1,κ = 1, and η = 0.5(*),1(°),2(×),4(+)

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