The moment Lyapunov exponents of a single degree-of-freedom viscoelastic system under the excitation of a wideband noise are studied in this paper. A realistic example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The method of averaging, both first order and second order, is applied. The averaged Itô differential equation governing the pth norm is established and the pth moment Lyapunov exponent is then obtained. White noise and real noise are considered as models of wideband noises. The variations of the moment Lyapunov exponents with the change of different parameters are discussed.

Issue Section:
Research Papers
Keywords:
differential equations, Lyapunov methods, noise, stability, vibrations, viscoelasticity
Topics:
Excitation, Noise (Sound), Stability, Differential equations, Stress, White noise
1.
Christensen
,
R. M.
, 1982,
Theory of Viscoelasticity: An Introduction
,
Academic
,
New York
.
2.
Roscoe
,
R.
, 1950, “
Mechanical Models for the Representation of Visco-Elastic Properties
,”
Br. J. Appl. Phys.
0508-3443,
1
, pp.
171
–
173
.
Crossref
Search ADS
 
3.
Ariaratnam
,
S. T.
, 1993, “
Stochastic Stability of Linear Viscoelastic Systems
,”
Probab. Eng. Mech.
0266-8920,
8
, pp.
153
–
155
.
Crossref
Search ADS
 
4.
Larionov
,
G. S.
, 1969, “
Investigation of the Vibrations of Relaxing Systems by the Averaging Method
,”
Mech. Compos. Mater.
0191-5665,
5
(
5
), pp.
714
–
720
.
5.
Potapov
,
V. D.
, 1994, “
On Almost Sure Stability of a Viscoelastic Column Under Random Loading
,”
J. Sound Vib.
0022-460X,
173
(
3
), pp.
301
–
308
.
Crossref
Search ADS
 
6.
Potapov
,
V. D.
, 1997, “
Numerical Method for Investigation of Stability of Stochastic Integro-Differential Equations
,”
Appl. Numer. Math.
0168-9274,
24
, pp.
191
–
201
.
Crossref
Search ADS
 
7.
Stratonovich
,
R. L.
, 1963,
Topics in the Theory of Random Noise
, Vol.
1
,
Gordon and Breach
,
New York
.
8.
Stratonovich
,
R. L.
, 1967,
Topics in the Theory of Random Noise
, Vol.
2
,
Gordon and Breach
,
New York
.
9.
Khasminskii
,
R. Z.
, 1966, “
On Stochastic Processes Defined by Differential Equations With a Small Parameter
,”
Theor. Probab. Appl.
0040-585X,
11
(
2
), pp.
211
–
228
.
Crossref
Search ADS
 
10.
Khasminskii
,
R. Z.
, 1966, “
A Limit Theorem for the Solutios of Differential Equations With Random Right-Hand Sides
,”
Theor. Probab. Appl.
0040-585X,
11
(
3
), pp.
390
–
406
.
Crossref
Search ADS
 
11.
Hijawi
,
M.
,
Ibrahim
,
R. A.
, and
Moshchuk
,
N.
, 1997, “
Nonlinear Random Response of Ocean Structures Using First- and Second-Order Stochastic Averaging
,”
Nonlinear Dyn.
0924-090X,
12
(
2
), pp.
155
–
197
.
Crossref
Search ADS
 
12.
Lin
,
Y. K.
, and
Cai
,
G. Q.
, 2000, “
Some Thoughts on Averaging Techniques in Stochastic Dynamics
,”
Probab. Eng. Mech.
0266-8920
15
(
1
), pp.
7
–
14
.
Crossref
Search ADS
 
13.
Xie
,
W. C.
, and
Huang
,
Q.
, 2006, “
Simulation of Moment Lyapunov Exponents for Linear Homogeneous Stochastic Systems
,”
ASME J. Appl. Mech.
0021-8936, submitted.
14.
Kloeden
,
P. E.
, and
Platen
,
E.
, 1992,
Numerical Solution of Stochastic Differential Equations
,
Springer-Verlag
,
Berlin
.
Copyright © 2008
 by American Society of Mechanical Engineers
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