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TECHNICAL PAPERS

# An Experimental Investigation of the Motion of Flexible Strings: Whirling

[+] Author and Article Information
Bisen Lin

Center for Mechanics of Solids, Structures and Materials, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712-1085

K. Ravi-Chandar1

Center for Mechanics of Solids, Structures and Materials, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712-1085kravi@mail.utexas.edu

1

To whom correspondence should be addressed

J. Appl. Mech 73(5), 842-851 (Dec 13, 2005) (10 pages) doi:10.1115/1.2172270 History: Received September 02, 2005; Revised December 13, 2005

## Abstract

Whirling of strings has been studied both theoretically and experimentally for many decades. According to linear theory, a heavy string can exhibit steady-state whirl only at its natural frequencies which form a discrete spectrum. The nonlinear theory, however, suggests that a string can undergo steady whirl at any frequency larger than the fundamental frequency and further that for each frequency between the $nth$ and the $(n+1)th$ eigenvalue, there exist $n$ distinct whirling modes. Quantitative experimental observations on such whirling have never been reported, although anecdotal evidence suggests the possibility of such whirl. In this paper, we examine the whirling of a string with negligible bending stiffness through experiments utilizing a stereo-vision imaging system. It is shown that steady motion exists only when the string whirls at its natural frequencies and that whirling motions for other frequencies exhibit rich dynamics that needs further exploration.

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## Figures

Figure 1

Motion configuration of the string

Figure 2

Variation of the tip amplitude with frequency of whirling for the different modes. The first mode has a horizontal asymptote at A∕l∼1 and hence is not shown completely.

Figure 3

Mode shapes determined from the nonlinear analysis for modes two to seven; mode shapes are plotted corresponding to a frequency of 60rad∕s. The first mode appears nearly horizontal and hence is not shown in this figure.

Figure 4

Variation in the tension (normalized by the weight) in the string along its length for different modes

Figure 5

Variation in the amplitude with the frequency for forced motion of the string near the second and third natural frequencies. The stable branches may be identified as in the simple case of the Duffing equation.

Figure 6

Time-averaged images of the rubber string corresponding to four steady-state modes are compared with the predictions of the linearized theory (dashed lines) and the nonlinear theory (solid lines)

Figure 7

Time-averaged images of the rubber string corresponding to frequencies in between the different steady modes

Figure 8

Locus of motion of the tip point of the string. The dark points in the interior of the annulus are obtained from other points along the length of the string that could not be eliminated in the optical arrangement. (ω: rad/s).

Figure 9

Patterns formed by the tip point at different rotational frequencies. Solid line in the first image is drawn with ωm∕ω=8 (see Eq. 26).

Figure 10

Variation of the amplitude of the tip point as a function of the driving frequency

Figure 11

Time variation of the position of the tip at a driving frequency of 24.91rad∕s

Figure 12

Time variation of the position of the tip at a driving frequency of 26.83rad∕s

Figure 13

Time variation of the X3 position of the tip at different driving frequencies

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