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.

## References

1.
Chen
,
L. Q.
, 2005,
“Analysis and Control of Transverse Vibrations of Axially Moving Strings,”
Appl. Mech. Rev.
,
58
, pp.
91
116
.
2.
Miranker
,
W. L.
, 1960,
“The Wave Motion in a Medium in Motion,”
IBM J. Res. Dev.
,
4
, pp.
36
42
.
3.
Swope
,
R. D.
, and
Ames
,
W. F.
, 1963,
J. Franklin Inst.
,
275
, pp.
36
55
.
4.
Ram
,
Y. M.
, and
Caldwell
,
J.
, 1996,
“Free Vibration of a String With Moving Boundary Conditions by the Methods of Distorted Images,”
J. Sound Vib.
,
194
(
1
), pp.
35
47
.
5.
Zhu
,
W. D.
, and
Guo
,
B. Z.
, 1998,
“Free and Forced Vibration of an Axially Moving String With an Arbitrary Velocity Profile,”
ASME J. Appl. Mech.
,
65
, pp.
901
907
.
6.
Pakdemirli
,
M.
,
Ulsoy
,
A. G.
, and
Ceranoglu
,
A.
, 1994,
“Transverse Vibration of an Axially Accelerating String,”
J. Sound Vib.
,
169
, pp.
179
196
.
7.
Pakdemirli
,
M.
, and
Ulsoy
,
A. G.
, 1997,
“Stability of an Axially Accelerating String,”
J. Sound Vib.
,
203
, pp.
815
832
.
8.
Parker
,
R. K.
, and
Lin
,
Y.
, 2001,
“Parametric Instability of Axially Moving Media Subjected to Multi-frequency Tension and Speed Fluctuations,”
ASME J. Appl. Mech.
,
68
, pp.
49
57
.
9.
Xu
,
G. Y.
, and
Zhu
,
W. D.
, 2010,
“Nonlinear and Time-Varying Dynamics of High-Dimensional Models of a Translating Beam With a Stationary Load Subsystem,”
ASME J. Vibr. Acoust.
,
132
(
6
),
061012
.
10.
Cooper
,
J.
, 1993,
“Asymptotic Behavior for the Vibration String With a Moving Boundary,”
J. Math. Anal. Appl.
,
174
, pp.
67
87
.
11.
Zhu
,
W. D.
, and
Ni
,
J.
, 2000,
“Energetics and Stability of Translating Media With an Arbitrarily Varying Length,”
ASME J. Vibr. Acoust.
,
122
, pp.
295
304
.
12.
Zhu
,
W. D.
, and
Zheng
,
N. A.
, 2008,
“Exact Response of a Translating String With Arbitrarily Varying Length Under General Excitation,”
ASME J. Appl. Mech.
,
75
(
3
),
031003
.
13.
Alligood
,
K. T.
,
Sauer
,
T. D.
, and
Yorke
,
J. A.
, 1996,
Chaos: An Introduction to Dynamical Systems
,
Springer-Verlag
,
New York
, p.
135
.
14.
Hirsch
,
M. W.
,
Smale
,
S.
, and
Devaney
,
R. L.
, 2004,
Dynamical Systems and an Introduction to Chaos
, 2nd ed.,
Elsevier
,
New York
, pp.
328
330
.