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Research Papers

# Dynamic Stability of a Translating String With a Sinusoidally Varying Velocity

[+] Author and Article Information
W. D. Zhu1

Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250wzhu@umbc.edu

X. K. Song

Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250

N. A. Zheng

Principal Product Engineer A.O. Smith Electrical Products Company, 531 North 4th Street, Tipp City, OH 45371

1

Corresponding author.

J. Appl. Mech 78(6), 061021 (Sep 15, 2011) (11 pages) doi:10.1115/1.4003908 History: Received May 07, 2010; Revised March 10, 2011; Published September 15, 2011; Online September 15, 2011

## Abstract

A new parametric instability phenomenon characterized by infinitely compressed, shocklike waves with a bounded displacement and an unbounded vibratory energy is discovered in a translating string with a constant length and tension and a sinusoidally varying velocity. A novel method based on the wave solutions and the fixed point theory is developed to analyze the instability phenomenon. The phase functions of the wave solutions corresponding to the phases of the sinusoidal part of the translation velocity, when an infinitesimal wave arrives at the left boundary, are established. The period number of a fixed point of a phase function is defined as the number of times that the corresponding infinitesimal wave propagates between the two boundaries before the phase repeats itself. The instability conditions are determined by identifying the regions in a parameter plane where attracting fixed points of the phase functions exist. The period-1 instability regions are analytically obtained, and the period-i $(i>1)$ instability regions are numerically calculated using bifurcation diagrams. The wave patterns corresponding to different instability regions are determined, and the strength of instability corresponding to different period numbers is analyzed.

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## Figures

Figure 1

Schematic of a translating string with a constant length and a sinusoidally varying velocity

Figure 2

Compression of an infinitesimal wave after reflection from the boundary x=0

Figure 3

Plot of φ(t¯n)-2kπ/ω¯ (k=1) versus t¯n, with a neutral fixed point t¯f; ν¯0=0.5, ω¯=2.7097, and ν¯1=0.25

Figure 4

Plot of φ(t¯n)-2kπ/ω¯ (k=1) versus t¯n, with a repelling fixed point t¯f and an attracting fixed point t¯f*; ν¯0=0.5, ω¯=2.5, and ν¯1=0.25

Figure 5

Period-1 (k=1,2,3,5,6), period-2, and period-3 instability regions of a translating string with ν¯0=0.5; there are six period-2 and 12 period-3 instability regions in the parameter plane shown

Figure 6

Plots of φ2(t¯n)-t¯n-2kπ/ω¯ for k=1 and various (ω¯,ν¯1) in the period-2 case: (a) ω¯=1.0893,ν¯1=0.2368; (b) ω¯=1.1107,ν¯1=0.2368; (c) ω¯=1.1322,ν¯1=0.2368; (d) ω¯=1.1536,ν¯1=0.2368; (e) ω¯=1.1666,ν¯1=0.0526; (f) ω¯=1.1704,ν¯1=0.0526; (g) ω¯=1.1741,ν¯1=0.0526; and (h) ω¯=1.1779,ν¯1=0.0526

Figure 7

Bifurcation diagrams near: (a) the period-1 instability region with k=1, (b) the period-2 instability region with k=1, and (c) the period-3 instability region with k=1; ν¯1=0.2368

Figure 8

Wave shapes at t¯=15 for ν¯1=0.2368 and: (a) ω¯=2.4678 which is inside the period-1 instability region with k=1; (b) ω¯=4.8703 which is inside the period-1 instability region with k=2; (c) ω¯=7.1739 which is inside the period-1 instability region with k=3; and (d) ω¯=11.8926 which is inside the period-1 instability region with k=5. The initial conditions of the string are u¯(x¯,0)=sin(πx¯) and u¯t¯(x¯,0)=0.

Figure 9

Wave shapes for ω¯=1.1322 and ν¯1=0.2368, which is in the period-2 instability region with k=1, at (a) t¯=15, and (b) t¯=150. The wave shape for ω¯=3.4943 and ν¯1=0.2368, which is in the period-2 instability region with k=3, at t¯=150 is shown in (c). The initial conditions are u¯(x¯,0)=sin(πx¯) and u¯t¯(x¯,0)=0.

Figure 10

Wave shapes for ω¯=0.7559 and ν¯1=0.2368, which is in the period-3 instability region with k=1, at (a) t¯=15, and (b) t¯=500. The initial conditions are u¯(x¯,0)=sin(πx¯) and u¯t¯(x¯,0)=0.

Figure 11

The time histories of the displacement of the point at x¯=0.5 with ω¯=2.4678 and ν¯=0.2368, which is in the period-1 instability region, with (a) k=1, and (b) k=2. The initial conditions are u¯(x¯,0)=sin(πx¯) and u¯t¯(x¯,0)=0.

Figure 12

The vibratory energy of the string for: (a) t¯∈[0,15], with ω¯=2.4678 and ν¯1=0.2368, which is in the period-1 instability region with k=1; (b) t¯∈[0,130], with ω¯=1.1322 and ν¯1=0.2368, which is in the period-2 instability region with k=1; and (c) t¯∈[0,150], with ω¯=0.7559 and ν¯1=0.2368, which is in the period-3 instability region with k=1. The initial conditions are u¯(x¯,0)=sin(πx¯) and u¯t¯(x¯,0)=0.

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