Parametric instability in a taut string with a periodically moving boundary, which is governed by a one-dimensional wave equation with a periodically varying domain, is investigated. Parametric instability usually occurs when coefficients in governing differential equations of a system periodically vary, and the system is said to be parametrically excited. Since the governing partial differential equation of the string with a periodically moving boundary can be transformed to one with a fixed domain and periodically varying coefficients, the string is parametrically excited and instability caused by the periodically moving boundary is classified as parametric instability. The free linear vibration of a taut string with a constant tension, a fixed boundary, and a periodically moving boundary is studied first. The exact response of the linear model is obtained using the wave or d'Alembert solution. The parametric instability in the string features a bounded displacement and an unbounded vibratory energy, and parametric instability regions in the parameter plane are classified as period-$i$ ($i≥1$) parametric instability regions, where period-1 parametric instability regions are analytically obtained using the wave solution and the fixed point theory, and period-$i$ ($i>1$) parametric instability regions are numerically calculated using bifurcation diagrams. If the periodic boundary movement profile of the string satisfies certain condition, only period-1 parametric instability regions exist. However, parametric instability regions with higher period numbers can exist for a general periodic boundary movement profile. Three corresponding nonlinear models that consider coupled transverse and longitudinal vibrations of the string, only the transverse vibration, and coupled transverse and axial vibrations are introduced next. Responses and vibratory energies of the linear and nonlinear models are calculated for both stable and unstable cases using three numerical methods: Galerkin's method, the explicit finite difference method, and the implicit finite difference method; advantages and disadvantages of each method are discussed. Numerical results for the linear model can be verified using the exact wave solution, and those for the nonlinear models are compared with each other since there are no exact solutions for them. It is shown that for parameters in the parametric instability regions of the linear model, the responses and vibratory energies of the nonlinear models are close to those of the linear model, which indicates that the parametric instability in the linear model can also exist in the nonlinear models. The mechanism of the parametric instability is explained in the linear model and through axial strains in the third nonlinear model.

## References

1.
Xie
,
Wei-Chau
,
2006
,
Dynamic Stability of Structures
,
Cambridge University
,
New York
, pp.
1
3
.
2.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1995
,
Nonlinear Oscillations
,
Wiley-VCH
,
Hoboken, NJ
, pp.
258
259
.
3.
Zhu
,
W. D.
, and
Chen
,
Y.
,
2005
, “
Forced Response of Translating Media With Variable Length and Tension
,”
Proc. Inst. Mech. Eng., Part K: J. Multibody Dyn.
,
219
(1), pp.
35
53
.10.1243/146441905X10014
4.
Kuchment
,
P. A.
,
1993
,
Floquet Theory for Partial Differential Equations
,
Birkhauser-Verlag
,
Basel, Switzerland
, p.
260
.
5.
Zhu
,
W. D.
,
Song
,
X. K.
, and
Zheng
,
N. A.
,
2011
, “
Dynamic Stability of a Translating String With a Sinusoidally Varying Velocity
,”
ASME J. Appl. Mech.
,
78
(
6
), p.
061021
.10.1115/1.4003908
6.
Zhu
,
W. D.
, and
Wu
,
K.
,
2013
, “
Dynamic Stability of a Class of Second-Order Distributed Structural Systems With Sinusoidally Varying Velocities
,”
ASME J. Appl. Mech.
,
80
(
6
), p.
061008
.10.1115/1.4023638
7.
Dittrich
,
J.
,
Duclos
,
P.
, and
Seba
,
P.
,
1994
, “
Instability in a Classical Periodically Driven String
,”
Phys. Rev. E
,
49
(
4
), pp.
3535
3538
.10.1103/PhysRevE.49.3535
8.
Fermi
,
E.
,
1949
, “
On the Origin of the Cosmic Radiation
,”
Phys. Rev.
,
75
(
8
), pp.
1169
1174
.10.1103/PhysRev.75.1169
9.
Jose
,
J. V.
, and
Cordery
,
R.
,
1986
, “
Study of a Quantum Fermi–Acceleration Model
,”
Phys. Rev. Lett.
,
56
(
4
), pp.
290
293
.10.1103/PhysRevLett.56.290
10.
Dodonov
,
V. V.
,
2001
, “
Nonstationary Casimir Effect and Analytical Solutions for Quantum Fields in Cavities With Moving Boundaries
,”
Modern Nonlinear Optics, Part I
, Vol. 119, 2nd ed., M. W. Evans, ed., John Wiley & Sons, New York, pp.
309
394
.10.1002/0471231479.ch7
11.
Cooper
,
J.
,
1993
, “
Long-Time Behavior and Energy Growth for Electromagnetic Waves Reflected by a Moving Boundary
,”
IEEE Trans. Antennas Propag.
,
41
(
10
), pp.
1365
1370
.10.1109/8.247776
12.
Wegrzyn
,
P.
,
2007
, “
Exact Closed-Form Analytical Solutions for Vibrating Cavities
,”
J. Phys. B
,
40
(
13
), pp.
2621
2637
.10.1088/0953-4075/40/13/008
13.
Chen
,
L. Q.
,
2005
, “
Analysis and Control of Transverse Vibrations of Axially Moving Strings
,”
ASME Appl. Mech. Rev.
,
58
(2), pp.
91
116
.10.1115/1.1849169
14.
Chen
,
L. Q.
, and
Ding
,
H.
,
2010
, “
Steady-State Transverse Response in Coupled Planar Vibration of Axially Moving Viscoelastic Beams
,”
ASME J. Vib. Acoust.
,
132
(1), p.
011009
.10.1115/1.4000468
15.
Zhang
,
G. C.
,
Ding
,
H.
,
Chen
,
L. Q.
, and
Yang
,
S. P.
,
2011
, “
Galerkin Method for Steady-State Response of Nonlinear Forced Vibration of Axially Moving Beams at Supercritical Speeds
,”
J. Sound Vib.
,
331
(7), pp.
1612
1623
.10.1016/j.jsv.2011.12.004
16.
Zhu
,
W. D.
, and
Ni
,
J.
,
2000
, “
Energetics and Stability of Translating Media With an Arbitrary Varying Length
,”
ASME J. Vib. Acoust.
,
122
(3), pp.
295
304
.10.1115/1.1303003
17.
Zhu
,
W. D.
, and
Zheng
,
N. A.
,
2008
, “
Exact Response of a Translating String With Arbitrary Varying Length Under General Excitation
,”
ASME J. Appl. Mech.
,
75
(
3
), p.
031003
.10.1115/1.2839903
18.
Alligood
,
K. T.
,
Sauer
,
T. D.
, and
Yorke
,
J. A.
,
1996
,
Chaos: An Introduction to Dynamical Systems
,
Springer-Verlag
,
New York
,
p. 135
.
19.
Grove
,
E. A.
, and
,
G. E.
,
2004
,
Periodicities in Nonlinear Difference Equations
,
CRC Press
,
Boca Raton, FL
, pp.
52
53
.
20.
Strikwerda
,
J. C.
,
2007
,
Finite Difference Schemes and Partial Differential Equations
,
Society for Industrial and Applied Mathematics
,
, pp.
17
64
.
21.
Mote
, Jr.,
C. D.
,
1966
, “
On the Nonlinear Oscillation of an Axially Moving String
,”
ASME J. Appl. Mech.
,
33
(2), pp.
463
464
.10.1115/1.3625075
22.
Lax
,
P. D.
,
1954
, “
Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computations
,”
Commun. Pure Appl. Math.
,
7
(
1
), pp.
157
193
.10.1002/cpa.3160070112