Pages from jam 12-1185 author-proof-2

PROOF COPY [JAM-12-1185]
Vladimir Stojanovic´1
Department of Mechanical Engineering,
University of Nisˇ,
Medvedeva 14,
18000 Nisˇ, Serbia
e-mail: stojanovic.s.vladimir@gmail.com
Marko Petkovic´
Department of Science and Mathematics,
University of Nisˇ,
Visˇegradska 33,
18000 Nisˇ, Serbia
Moment Lyapunov Exponents
1 and Stochastic Stability of a
Three-Dimensional System
2 on Elastic Foundation Using
3 a Perturbation Approach4
5 In this paper, the stochastic stability of the three elastically connected Euler beams on
elastic foundation is studied. The model is given as three coupled oscillators. Stochastic
stability conditions are expressed by the Lyapunov exponent and moment Lyapunov expo-
nents. It is determined that the new set of transformation for getting Itô differential equa-
tions can be applied for any system of three coupled oscillators. The method of regular
perturbation is used to determine the asymptotic expressions for these exponents in the
presence of small intensity noises. Analytical results are presented for the almost sure
and moment stability of a stochastic dynamical system. The results are applied to study
the moment stability of the complex structure with influence of the white noise excitation
due to the axial compressive stochastic load. [DOI: 10.1115/1.4023519]
Keywords: Winkler elastic layer, stochastic stability, perturbation, moment Lyapunov
6 exponent
7 1 Introduction
8 Frequent uncontrolled influences of the earthquakes, winds, and
9 waves, theoretical knowledge of the behavior of complex struc-
10 tures become more important.AQ1 Vibration and especially problems
11 of stochastic instability of beams or beam-columns on elastic
12 foundations occupy an important place in many fields of structural
13 and foundation engineering. This problem is very often encountered
14 in aeronautical, mechanical, and civil engineering applications. The
15 present work contains two parts in introduction to describe the fol-
16 lowing model. In the first is given past work of researchers who did
17 investigation of elastically connected beams, and the second part
18 focuses on the research in the field of stochastic stability.
19 Fundamental early work was conducted by Hyer et al. [1,2],
20 who made the theoretical and experimental investigation in non-
21 linear vibrations of the three-beam system with viscoelastic cores.
22 Researchers also have studied the vibrations of elastically con-
23 nected triple beams with effects of rotary inertia and shear [3].
24 Kelly and Srinivas [4] investigated the problem of the free vibra-
25 tions of a set of n axially loaded stretched Bernoulli–Euler beams
26 connected by elastic layers and connected to a Winkler-type foun-
27 dation. A normal-mode solution is applied to the governing partial
28 differential equations to derive a set of coupled ordinary differen-
29 tial equations, which are used to determine the natural frequencies
30 and mode shapes. It is shown that the set of differential equations
31 can be written in self-adjoint form with an appropriate inner prod-
32 uct. An exact solution for the general case is obtained, but numeri-
33 cal procedures must be used to determine the natural frequencies
34 and mode shapes. The numerical procedure is difficult to apply,
35 especially in determining higher frequencies. For the special case
36 of identical beams, an exact expression for the natural frequencies
37 is obtained in terms of the natural frequencies of a corresponding
38set of unstretched beams and the eigenvalues of the coupling
39matrix. Basic theoretical fundaments were used for a triple system
40of sandwich beams [5]. Stojanovic´ et al. [6] presented a general
41procedure for the determination of the natural frequencies and
42static stability for a set of beam systems under compressive axial
43loading using Timoshenko and high-order shear deformation
44theory. Matsunaga [7] studied buckling instabilities of a simply
45supported thick elastic beam subjected to axial stresses. Taking
46into account the effects of shear deformations and thickness
47changes, buckling loads and buckling displacement modes of
48thick beams are obtained. Based on the power series expansion of
49displacement components, a set of fundamental equations of a
50one-dimensional higher order beam theory was derived through
51the principle of virtual displacement. Several sets of truncated
52approximate theories are applied to solve the eigenvalue problems
53of a thick beam. Faruk [8] analyzed dynamic behavior of Timo-
54shenko beams on a Pasternak-type viscoelastic foundation sub-
55jected to time-dependent loads using the Laplace transformation
56and the complementary functions method to calculate exactly the
57dynamic stiffness matrix of the problem. Ma et al. [9] studied
58the static response of an infinite beam supported on a unilateral (ten-
59sionless) two-parameter Pasternak foundation. On the basis of the
60Bernoulli–Euler beam theory, the properties of free transverse vibra-
61tion and buckling of a double-beam system under compressive axial
62loading are investigated in the paper of Zhang et al. [10]. Explicit
63expressions are derived for the natural frequencies and the associated
64amplitude ratios of the two beams, and the analytical solutions of the
65critical buckling load is obtained. The influences of the compressive
66axial loading on the responses of the double-beam system are dis-
67cussed. It is shown that the critical buckling load of the system is
68related to the axial compression ratio of the two beams and the Win-
69kler elastic layer, and the properties of free transverse vibration of
70the system greatly depend on the axial compressions.
71Dynamic stability and instability of continuous systems under
72time-dependent deterministic or stochastic loading has been stud-
73ied by many authors for the last 40 years. The theory of random
1
Corresponding author.
Manuscript received May 6, 2012; final manuscript received December 29, 2012;
accepted manuscript posted January 29, 2013; published online xx xx, xxxx. Assoc.
Editor: Wei-Chau Xie.
J_ID: JAM DOI: 10.1115/1.4023519 Date: 9-February-13 Stage: Page: 1 Total Pages: 11
ID: sambasivamt Time: 12:10 I Path: S:/3b2/JAM#/Vol00000/130036/APPFile/AS-JAM#130036
Journal of Applied Mechanics MONTH 2013, Vol. 00 / 000000-1Copyright VC 2013 by ASME

74 dynamic systems and a comprehensive list of references can be
75 found in Arnold et al. [11]. Khasminskii and Moshchuk [12]
76 obtained an asymptotic expansion of the moment Lyapunov expo-
77 nents of a two-dimensional system under white noise parametric
78 excitation in terms of the small fluctuation parameter e, from
79 which the stability index was obtained. Kozic´ et al. [13] investi-
80 gated the Lyapunov exponent and moment Lyapunov exponent of
81 a double-beam system without connected damping coefficient
82 fluctuated by white noise considered as a separate viscosity sys-
83 tem. The method of regular perturbation was used to obtain
84 explicit expressions for these exponents in the presence of small
85 intensity noises. Xie [14] obtained weak noise expansions of the
86 moment Lyapunov exponents of a two-dimensional system
87 under real noise excitation, an Ornstein–Uhlenbeck process. Sri
88 Namachchivaya and Van Roessel [15] used a perturbation
89 approach to calculate the asymptotic growth rate of a stochasti-
90 cally coupled two-degree-of-freedom system. The noise was
91 assumed to be white and of small intensity in order to calculate
92 the explicit asymptotic formulas for the maximum Lyapunov
93 exponent. Sri Namachchivaya et al. [16] used a perturbation
94 approach to obtain an approximation for the moment Lyapunov
95 exponents of two coupled oscillators with commensurable fre-
96 quencies driven by small intensity real noise with dissipation. The
97 generator for the eigenvalue problem associated with the moment
98 Lyapunov exponents was derived without any restriction on the
99 size of the pth moment.
100 In the present study, instability of the complex system of the
101 beams and weak noise expansion for the moment Lyapunov expo-
102 nents are investigated for the six-dimensional stochastic system. It
103 is determined that the new set of transformation for getting Itô
104 differential equations for a system of three DOFAQ2 as a form of six
105 Stratonovich differential equations. The noise is assumed to be
106 white noise of small intensity such that one can obtain an asymp-
107 totic growth rate. The Lyapunov exponent is then obtained using
108 the relationship between the moment Lyapunov exponents and the
109 Lyapunov exponent. These results are applied to study the pth
110 moment stability and almost sure stability of a system on the elas-
111 tic foundation.
112 2 Application to Beams Under Stochastic Loads
113 Physical problems of real engineering can apply the further
114 investigation of the transverse vibration instability of a complex
115 system on elastic foundation subjected to stochastic compressive
116 axial loading.AQ3 It is assumed that the three beams of the system are
117 under the stochastic excitation. The rotary inertia and shear defor-
118 mation should be negligible in motion of the beams is governed
119 by the partial differential Eqs. (1a), (1b), and (1c).AQ4 This theory is
120 based on the assumption that plane cross-sections of a beam
121 remain plane during flexure and that the radius of curvature of a
122 bent beam is larger than the beam’s depth. It is valid only if the
123ratio of the depth to the length of the beam is small. We can obtain
124the general equations for transverse vibrations of elastically con-
125nected beams shown in Fig. 1 if we set m ¼ 3, ignore rotary inertia
126and shear effects, and include viscous damping in Eqs. (8a), (8b),
127and (8c) given in Stojanovic´ et al. [6].
EI1
@4
w1
@z4
þ qA1
@2
w1
@t2
þ ec0 2
@w1
@t
À
@w2
@t

þ F1ðtÞ
@2
w1
@z2
þ eKð2w1 À w2Þ ¼ 0 (1a)
EI2
@4
w2
@z4
þ qA2
@2
w2
@t2
þ ec0 2
@w2
@t
À
@w1
@t
À
@w3
@t

þ F2ðtÞ
@2
w2
@z2
þ eKð2w2 À w1 À w3Þ ¼ 0 (1b)
EI3
@4
w3
@z4
þ qA3
@2
w3
@t2
þ ec0
@w3
@t
À
@w2
@t

þ F3ðtÞ
@2
w3
@z2
þ eKðw3 À w2Þ ¼ 0 (1c)
128where w1, w2, and w3 are transverse beam deflections, which are
129positive if downward; I1; I2, and I3 are the second moments of
130inertia of the beams; A1; A2, and A3 are the cross-sectional area
131of the beams; and E and q are Young’s modulus and the mass den-
132sity. Thus, c0 represents the damping coefficients per unit axial
133length, respectively, and K is the stiffness modulus of a Winkler
134elastic layer. F1ðtÞ; F2ðtÞ, and F3ðtÞ are stochastically varying
135static loads. Simply supported ends for the same length l of the
136beams are satisfied with boundary conditions wið0; tÞ ¼ wiðl; tÞ
¼ @2
wið0; tÞ=@z2
¼ @2
wiðl; tÞ=@z2
¼ 0; i ¼ 1; 2; 3: Using the
137Galerkin method only for fundamental modes are considered; AQ5
138boundary conditions are satisfied by taking and substituting
wiðz; tÞ ¼ TiðtÞ sinðpz=lÞ; ði ¼ 1; 2; 3Þ into equations of motion
139Eqs. (1a), (1b), and (1c). Unknown time functions can be
140expressed as
€T1 þ
c0e
qA1
ð2 _T1 À _T2Þ þ
EI1
qA1
p4
l4
þ
2Ke
qA1
À
p2
qA1l2
F1ðtÞ

T1
À
2Ke
qA1
T2 ¼ 0 (2)
€T2 þ
c0e
qA2
ð2 _T2 À _T3 À _T1Þ þ
EI2
qA2
p4
l4
þ
2Ke
qA2
À
p2
qA2l2
F2ðtÞ

T2
À
Ke
qA2
T3 À
Ke
qA2
T1 ¼ 0 (3)
€T3 þ
c0e
qA3
_T3 À
c0e
qA3
_T2 þ
EI3
qA3
p4
l4
þ
Ke
qA3
À
p2
qA3l2
F3ðtÞ

T3
À
Ke
qA3
T2 ¼ 0 (4)
Fig. 1 Geometry of complex three-beam system on elastic foundation
000000-2 / Vol. 00, MONTH 2013 Transactions of the ASME

141 Using the further substitutions,
x2
1 ¼
EI1
qA1
p4
l4
; x2
2 ¼
EI2
qA2
p4
l4
; x2
3 ¼
EI3
qA3
p4
l4
; b1 ¼
C0
qA1
;
b2 ¼
C0
qA2
; b3 ¼
C0
qA3
K1 ¼
p2
qA1l2
; K2 ¼
p2
qA2l2
; K3 ¼
p2
qA3l2
; H1 ¼
K
qA1
;
H2 ¼
K
qA2
; H3 ¼
K
qA3
142 and assume that the compressive axial forces are stochastic white-
143 noise processes with small intensity F1ðtÞ ¼ F2ðtÞ ¼ F3ðtÞ
¼
ffiffi
e
p
cðtÞ, we have an oscillatory system in the form
d2
T1
dt2
þ x2
1T1 þ eb1 2
dT1
dt
À
dT2
dt

þ eH1ð2T1 À T2Þ
À
ffiffi
e
p
K1cðtÞT1 ¼ 0
d2
T2
dt2
þ x2
2T2 þ eb2 2
dT2
dt
À
dT3
dt
À
dT1
dt

þ eH2ð2T2 À T3 À T1Þ
À
ffiffi
e
p
K2cðtÞT2 ¼ 0
d2
T3
dt2
þ x2
3T3 þ eb3
dT3
dt
À
dT2
dt

þ eH3ðT3 À T2Þ
À
ffiffi
e
p
K3cðtÞT3 ¼ 0 (5)
144 The system consists of unknown generalized coordinates in func-
145 tions of time Ti; natural frequencies xi, and viscous damping
146coefficients ebi; ði ¼ 1; 2; 3Þ. The stochastic term
ffiffi
e
p
cðtÞ presents
147a white-noise process with small intensity. Dynamic stability of
148a oscillatory system can be known to determine the maximal
149Lyapunov exponent and the pth moment Lyapunov exponent,
150which is described as
kT ¼ lim
t!1
1
t
log Tðt; T0Þk k
151where Tðt; T0Þ is the solution process of a linear dynamic system.
152The almost sure stability depends upon the sign of the maximal
153Lyapunov exponent, which is an exponential growth rate of the
154solution of the randomly perturbed dynamic system. A negative
155sign of the maximal Lyapunov exponent implies the almost sure
156stability, whereas a nonnegative value indicates instability. The
157exponential growth rate E Tðt; T0; _T0Þ

pÂ Ã
is provided by the
158moment Lyapunov exponent, defined as
KTðpÞ ¼ lim
t!1
1
t
log E Tðt; T0Þk kp
½ Š
159where E½ Š denotes the expectation. If KTðpÞ 0; then, by defini-
160tion, E Tðt; T0; _T0Þ

pÂ Ã
! 0 as t ! 0, and this is referred to
161as pth moment stability. Although the moment Lyapunov expo-
162nents are important in the study of the dynamic stability of the
163stochastic systems, the actual evaluations of the moment Lyapu-
164nov exponents are very difficult, and the almost sure and moment
165stability of the equilibrium state T ¼ _T ¼ 0 of Eq. (32). Using the
166transformation T1 ¼ x1; _T1 ¼ x1x2; T2 ¼ x3; _T2 ¼ x2x4; T3 ¼ x5;
_T3 ¼ x3x6, the Eq. (1) can be represented in the first-order form
167by Stratonovich differential equations,
d
x1
x2
x3
x4
x5
x6
8

:
9
=
;
¼
0 x1 0 0 0 0
À2h1e À x1 À2eb1 eh1 eb1x2=x1 0 0
0 0 0 x2 0 0
h2e eb2x1=x2 À2h2e À x2 À2eb2 h2e eb2x3=x2
0 0 0 0 0 x3
0 0 eh3 eb3x2=x3 Àeh3 À x3 Àeb3
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
x1
x2
x3
x4
x5
x6
8

:
9
=
;
dt
þ
ffiffi
e
p
0 0 0
k1 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 k2
0 0 0
0 0 0
0 0 0
0 0 0
0 k3 0
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
x1
x2
x3
x4
x5
x6
8

:
9
=
;
dc tð Þ
(6)
168 where h1 ¼ H1=x1; h2 ¼ H2=x2; h3 ¼ H3=x3; k1 ¼ K1=x1;
k2 ¼ K2=x2; k3 ¼ K3=x3 Á c tð Þ is the white-noise process with
169 zero mean and autocorrelation function,
Rcc t1; t2ð Þ ¼ E c t1ð Þc t1ð Þ½ Š ¼ r2
d t2 À t1ð Þ (7)
170 where r is the intensity of the random process c tð Þ; dðÞ is the Dirac
171 d function, and E½ Š denotes expectation. Using corresponding
172 transformation,
x1 ¼ a cos u1 cos u2 cos h1; x2 ¼ Àa cos u1 cos u2 sin h1;
x3 ¼ a cos u1 sin u2 cos h2 x4 ¼ Àa cos u1 sin u2 sin h2;
x5 ¼ a sin u1 cos h3; x6 ¼ Àa sin u1 sin h3
P ¼ ap
¼ ðx2
1 þ x2
2 þ x2
3 þ x2
4 þ x2
5 þ x2
6Þ
p
2
À1 p 1; 0 hi 2p; 0 uj
p
2
; i ¼ 1; 2; 3; j ¼ 1; 2
Itô’s rule gets the set of equations for the pth power of the
173norm of the response and phase variables. Trigonometric trans-
174formation represents a as a norm of the response, where h1; h2,
175and h3 are the angles of the three oscillators and u1 and u2
176describe the coupling or exchange of energy between the
177oscillators.
dh1 ¼ m1 h1; h2; h3; u1; u2ð Þdt þ r11 h1; h2; h3; u1; u2ð ÞdW tð Þ
du1 ¼ m4 h1; h2; h3; u1; u2ð Þdt þ r41 h1; h2; h3; u1; u2ð ÞdW tð Þ
du2 ¼ m5 h1; h2; h3; u1; u2ð Þdt þ r51 h1; h2; h3; u1; u2ð ÞdW tð Þ
dP ¼ Pm6 h1; h2; h3; u1; u2ð Þdt þ Pr61 h1; h2; h3; u1; u2ð ÞdW tð Þ
(8)
Journal of Applied Mechanics MONTH 2013, Vol. 00 / 000000-3

178 where W tð Þ is the standard Weiner process and
m1 h1; h2; h3; u1; u2ð Þ
¼ x1 À eb1 sin 2h1 À
x2
x1
cos h1 sin h2 tan u2

þ eh1 1 þ cos2h1 À cos h1 cos h2 tan u2ð Þ
À e
k2
1r2
8
cos3h1 sin h1 þ 2 sin 2h1 þ cosh1 sin 3h1ð Þ (9)
r11 h1; h2; h3; u1; u2ð Þ ¼ À
ffiffi
e
p
k1r cos2
h1 (10)
m2 h1;h2;h3;u1;u2ð Þ
¼ x2 Àeb2

sin2h2 À
x1
x2
cosh2 cotu2 sinh1
À
x3
x2
cosh2 sinh3
tanu1
sinu2

þeh2

1þcos2h2 Àcosh1 cosh2 cotu2
À
tanu1
sinu2
cosh2 sinh3

Àe
k2
2r2
8

cos3h2 sinh2 þ2sin2h2
þcosh2 sin3h2

(11)
r21 h1; h2; h3; u1; u2ð Þ ¼ À
ffiffi
e
p
k2r cos2
h2 (12)
m3 h1; h2; h3; u1; u2ð Þ
¼
x3
16
13 þ cos2h3 þ
5 þ 4cos2h3 À cos4h3
2 1 þ cos2h3ð Þ

À
eb3
8
3 sin 2h3 þ tan h3 1 þ cos 2h3ð Þ
À 4
x2
x3
sin u2 cot u1 sin h2 cos h3 þ
1
cos h3
À tan h3
!
þ
eh3
2

1 þ cos 2h3 À cos h2 cos h3 cot u1 sin u2
À
cos h2
cos h3
cot u1 sin u2 þ cos h2 cot u1 sin h3 sin u2 tan h3

À e
k2
3r2
8
cos 3h3 sin h3 þ 2 sin 2h3 þ cos h3 sin 3h3ð Þ (13)
r31 h1; h2; h3; u1; u2ð Þ ¼ À
ffiffi
e
p
k3r cos2
h3 (14)
m4 h1; h2; h3; u1; u2ð Þ ¼ e
(
h1
4
Â
cos h2 sin h1 sin 2u1 sin 2u2 À sin 2h1
À
cos 2u2 sin 2u1 þ sin 2u1
ÁÃ
þ
b1
4

sin 2u1 1 À cos 2h1ð Þ 1 þ cos 2u2ð Þ À
x2
x1
sin h1 sin h1 sin 2u2
!'
þ
h2
4
sin 2u1 cos 2u2 sin 2h2 À sin 2h2 þ sin 2u2 cos h1 sin h2ð Þ þ 2 sin h2 cos h3 sin u2 1 À cos 2u1ð Þ½ Š
þ
b2
4

sin 2u1 1 À cos 2h2ð Þ 1 À cos 2u2ð Þ À
2x3
x2
1 À cos 2u1ð Þ sin h2 sin h3 sin u2
À
x1
x2
sin h1 sin h2 sin 2u1 sin 2u2
!'
þ
h3
4
sin 2h3 sin 2u1 À 2 cos h2 sin h3 sin u2 1 þ cos 2u1ð Þ½ Š
þ
b3
4
sin 2u1 cos 2h3 À 1ð Þ þ
2x2
x3
sin h2 sin h3 sin 2u2 1 þ cos 2u1ð Þ
!
þ
r2
64

2k2
1 cos2
u2
Â
sin 4u1 sin2
2h1 cos2
u2 À 8 cos2
h1 sin 2u1
À
cos2
h1 À cos 2u2 sin2
h1
ÁÃ
þ 2k2
2 sin2
u2 sin 4u1 sin2
2h2 sin2
u2 À 8cos2
h2 sin 2u1 cos2
h2 þ cos 2u2 sin2
h2
À ÁÂ Ã
þ 16k2
3 sin 2u1cos2
h3 cos 2h3 þ cos 2u1sin2
h3
À Á
þ k1k2 sin2
2u2 sin 2h1 sin 2h2 4 sin 2u1 þ sin 4u1ð Þ
À 4k1k3 cos2
u2 sin 4u1 sin 2h1 sin 2h3 À 4k2k3 sin2
u2 sin 4u1 sin 2h2 sin 2h3
')
(15)
r41 h1; h2; h3; u1; u2ð Þ ¼
ffiffi
e
p
r
8
sin 2u1 k1 sin 2h1 1 þ cos 2u2ð Þ þ k2 sin 2h2 1 À cos 2u2ð Þ À 2k3 sin 2h3 sin2u1Š½ (16)
m5 h1; h2; h3; u1; u2ð Þ ¼ e
(
h1
Â
sin h1 sin u2
À
cos h2 sin u2 À 2 cos h1 cos u2
ÁÃ
þ b1
Â
sin h1 sin u2
À
2 cos u2 sin h1 À
x2
x1
sin h2 sin u2
ÁÃ
À h2
Â
sin h2 cos u2
À
cos h1 cos u2 À 2 cos h2 sin u2 þ cos h3 tan u1
ÁÃ
þ b2 sin h2

x1
x2
cos2
u2 sin h1 À sin h2 sin 2u2 þ
x3
x2
cos u2 sin h3 tan u1
!
À
r2
16
f½4k2
1 cos2
h1 sin 2u2ðcos 2h1 À cos 2u2 sin2
h1ÞŠ À ½4k2
2 cos2
h2 sin 2u2ðcos 2h2 þ cos 2u2 sin2
h2ÞŠ
þ ½k1k2 sin 2h1 sin 2h2 sin 4u2Š
É
)
(17)
r51 h1; h2; h3; u1; u2ð Þ ¼
ffiffi
e
p
r
4
sin 2u2 k1 sin 2h1 À k2 sin2h2Š½ (18)

m6 h1; h2; h3; u1; u2ð Þ ¼ ep

h1
Â
cos2
u1 cos u2 sin h1
À
2 cos h1 cos u2 À cos h2 sin u2
ÁÃ
þ b1 cos2
u1 cos u2 sin h1
Â

x2
x1
sin h2 sin u2 À 2 cos u2 sin h1
!
þ
h2
2
Â
sin u2 sin h2
À
À cos h3 sin 2u1 À 2 cos2
u1
À
cos h1 cos u2
À 2 cos h2 sin u2
ÁÁÃ
þ
b2
2
sin u2 sin h2

x3
x2
sin h2 sin 2u1 þ 2 cos2
u1

x1
x2
cos u2 sin h1 À 2 sin h2 sin u2
!
þ
r2
k2
1
128
cos2
u1
À
2 cos2
h1 cos 2u2
À
10 þ 3p þ 4ð2 þ pÞ À ðp À 2Þ cos 2h1ð3 þ 4 cos 2u2ÞÞ þ ðp À 2Þ sin2
2h1
Â ð8 cos 2u1 cos4
u2 þ cos 4u2Þ þ
r2
k2
2
128
cos2
u1ð2 cos2
h2 cos 2u2ð10 þ 3p À 4
À
2 þ p
Á
þ ðp À 2Þ cos 2h2Þ
Â ð4 cos 2u2 À 3ÞÞ þ ðp À 2Þ sin2
2h2
À
cos 4u2 þ 8 cos 4u1 sin4
u2Þ
Á
þ
r2
k2
3
64
ð2 cos2
h3ð4 cos 2u1 À 3Þ cos 2h3
Â p À 2ð Þ þ 3p þ 2 À 4p cos 2u1Þ þ ðp À 2Þ cos 4u1 sin2
2h3 þ
r2
k1k2
16
ðp À 2Þ sin 2h1 sin 2h2 cos4
u1 sin2
2u2
þ
r2
k1k3
16
À
p À 2
Á
sin 2h1 sin 2h3 cos2
u2 sin2
2u1 þ
r2
k1k3
16
À
p À 2
Á
sin 2h2 sin 2h3 sin2
u2 sin2
2u1
'
(19)
r61 h1; h2; h3; u1; u2ð Þ ¼
Àp
ffiffi
e
p
r
2
À
k1 sin 2h1 cos2
u1 cos2
u2
þ k2sin2h2 cos2
u1 sin2
u2
þ k3 sin 2h3 sin2
u1
Á
(20)
179 Applying a linear stochastic transformation, given by Wedig [17],
S ¼ T h1; h2; h3; u1; u2ð ÞP; P ¼
S
T h1; h2; h3; u1; u2ð Þ
(21)
180 introducing the new norm process S by means of the scalar func-
181 tion T h1; h2; h3; u1; u2ð Þ, which is defined on the stationary phase
182 process uj; the Itoˆ equation for the new pth norm process S can be
183 obtained from Itoˆ’s Lemma,
dS ¼ P
1
2
r2
11
@2
T
@h2
1
þ r2
21
@2
T
@h2
2
þ r2
31
@2
T
@h2
3
þ r2
41
@2
T
@u2
1
þ r2
51
@2
T
@u2
2
!
þ r11r21
@2
T
@h1@h2
þ r11r31
@2
T
@h1@h3
þ r11r41
@2
T
@h1@u1
þ r31r51
@2
T
@h3@u2
þ r41r51
@2
T
@u1@u2
þ m1
@T
@h1
þ m2
@T
@h2
þ m3
@T
@h3
þ m4
@T
@u1
þ m5
@T
@u2
þ m6T þ r11r61
@T
@h1
þ r21r61
@T
@h2
þ r31r61
@T
@h3
þ r41r61
@T
@u1
þ r51r61
@T
@u2
!
dT
þ P

r11
@T
@h1
þ r21
@T
@h2
þ r31
@T
@h3
þ r41
@T
@u1
þ r51
@T
@u2
r61T

Â dW tð Þ (22)
184 In the case that transformation function TðujÞ is bounded and non-
185 singular, both processes P and S possess the same stability behav-
186 ior. Therefore, transformation function TðujÞ is chosen so that the
187 drift term of the Itoˆ differential Eq. (43) does not depend on the
188 phase process uj, so that
dS ¼ K pð ÞSdt þ
S
T

r11
@T
@h1
þ r21
@T
@h2
þ r31
@T
@h3
þ r41
@T
@u1
þ r51
@T
@u2
þ r61T

dW tð Þ (23)
189 Comparing Eqs. (22) and (23), transformation function T h1; h2;ð
h3; u1; u2Þ is given by the following equation:
L0 þ eL1½ ŠT h1; h2; h3; u1; u2ð Þ ¼ K pð ÞT h1; h2; h3; u1; u2ð Þ (24)
190where L0 and L1 are the differential operators in the forms
L0 ¼ x1
@
@h1
þ x2
@
@h2
þ x3
@
@h3
;
L1 ¼ a1
@2
@h2
1
þ a2
@2
@h2
2
þ a3
@2
@h2
3
þ a4
@2
@u2
1
þ a5
@2
@u2
2
þ a6
@2
@h1@h2
þ a7
@2
@h1@h3
þ a8
@2
@h1@u1
þ a9
@2
@h1@u2
þ a10
@2
@h2@h3
þ a11
@2
@h2@u1
þ a12
@2
@h2@u2
þ a13
@2
@h3@u1
þ a14
@2
@h3@u2
þ a15
@2
@u1@u2
þ b1
@
@h1
þ b2
@
@h2
þ b3
@
@h3
þ b4
@
@u1
þ b5
@
@u2
þ c (25)
191where
ai ¼ ai h1; h2; h3; u1; u2ð Þ; i ¼ 1; 2; …15;
bj ¼ bj h1; h2; h3; u1; u2ð Þ; j ¼ 1; 2; …5 (26)
192and
c ¼ c h1; h2; h3; u1; u2ð Þ (27)
193Functional coefficients ai; bj, and c are available in the.nb
194Mathematica 8 file - (http://www.2shared.com/file/-pI4nFbZ/
195coeff_ai_bj_c_and_Lyapunov_1.html).
1963 Moment Lyapunov Exponents
197The eigenvalue problem for a differential operator of five
198independent variables is identified from Eq. (24), in which K pð Þ is
199the eigenvalue and T h1; h2; h3; u1; u2ð Þ is the associated eigen-
200function. Equation (23) defines that the eigenvalue K pð Þ is the
201Lyapunov exponent of the pth moment of system in Eq. AQ6(6). This
202approach was first applied by Wedig [17] to derive the eigenvalue
203problem for the moment Lyapunov exponent of a two-
204dimensional linear Itoˆ stochastic system. Applying the method of
205regular perturbation, both the moment Lyapunov exponent K pð Þ
206and the eigenfunction T h1; h2; h3; u1; u2ð Þ are expanded in power
207series of e to obtain a weak noise expansion of the Lyapunov

208 exponent of a system with stochastic excitation, described by six-
209 dimensional components as
K pð Þ ¼ K0 pð Þ þ eK1 pð Þ þ e2
K2 pð Þ þ ÁÁÁ
þ en
Kn pð Þ þ ÁÁÁ;
T h1;h2;h3;u1;u2ð Þ ¼ T0 h1;h2;h3;u1;u2ð Þ þ eT1 h1;h2;h3;u1;u2ð Þ
þ e2
T2 h1;h2;h3;u1;u2ð Þ þ ÁÁÁ
þ e2
T2 h1;h2;h3;u1;u2ð Þ þ ÁÁÁ
þ en
Tn h1;h2;h3;u1;u2ð Þ þ ÁÁÁ (28)
210 Equating terms of the equal powers of e from Eqs. (24) and (28),
211 the following equations may be written as
e0
: L0T0 h1; h2; h3; u1; u2ð Þ ¼ K0 pð ÞT0 h1; h2; h3; u1; u2ð Þ;
e1
: L0T1 h1; h2; h3; u1; u2ð Þ þ L1T0 h1; h2; h3; u1; u2ð Þ
¼ K0 pð ÞT1 h1; h2; h3; u1; u2ð Þ þ K1 pð ÞT0 h1; h2; h3; u1; u2ð Þ;
e2
¼ K0 pð ÞT2 h1; h2; h3; u1; u2ð Þ þ K1 pð ÞT1 h1; h2; h3; u1; u2ð Þ
þ K2 pð ÞT0 h1; h2; h3; u1; u2ð Þ;
e3
þ K2 pð ÞT1 h1; h2; h3; u1; u2ð Þ þ K3 pð ÞT0 h1; h2; h3; u1; u2ð Þ;
…
en
: L0Tn h1; h2; h3; u1; u2ð Þ þ L1TnÀ1 h1; h2; h3; u1; u2ð Þ
¼ K0 pð ÞTn h1; h2; h3; u1; u2ð Þ þ K1 pð ÞTnÀ1 h1; h2; h3; u1; u2ð Þ
þ Á Á Á þ Kn pð ÞT0 h1; h2; h3; u1; u2ð Þ (29)
212 where each function Ts h1; h2; h3; u1; u2ð Þ; s ¼ 1; 2; 3; … must be
213 positive and periodic in the range 0 hi 2p:
214 3.1 Zeroth Order Perturbation. From Eqs. (24) and (29),
215 the zeroth order perturbation equation is L0T0 ¼ K0 pð ÞT0 and can
216 be written as
x1
@T0 h1; h2; h3; u1; u2ð Þ
@h1
þ x2
@T0 h1; h2; h3; u1; u2ð Þ
@h2
þ x3
@T0 h1; h2; h3; u1; u2ð Þ
@h3
¼ K0 pð ÞT0 h1; h2; h3; u1; u2ð Þ
(30)
217 Equation (30) can be easily solved from the moment Lyapunov
218 exponent characteristic, which results in Kn 0ð Þ ¼ 0 for
n ¼ 0; 1; 2; 3; … because of K 0ð Þ ¼ K0 0ð Þ þ eK1 0ð Þ þ e2
K2 0ð Þ
þ Á Á Á þ en
Kn 0ð Þ ¼ 0. The eigenvalue K0 pð Þ is independent of p.
219 Hence, K0 0ð Þ ¼ 0 leads to K0 pð Þ ¼ 0. The solutions of Eq. (30)
220 have a periodic solution if and only if
K0 pð Þ ¼ 0; T0 h1; h2; h3; u1; u2ð Þ ¼ 1 (31)
3.2 First-Order Perturbation. The equation given from
221 first-order perturbation is
L0T1 h1; h2; h3; u1; u2ð Þ þ L1T0 h1; h2; h3; u1; u2ð Þ
(32)
222 and Eq. (32) has a periodic solution if and only if
ð2p
0
ð2p
0
ð2p
0
ðp=2
0
ðp=2
0
L1 Á 1 À K1 pð Þ½ Šdu1du2dh1dh2dh3 ¼ 0 (33)
223which yields
K1 pð Þ ¼
1
2p5
ð2p
0
ð2p
0
ð2p
0
ðp=2
0
ðp=2
0
c h1; h2; h3; u1; u2ð Þ½ Š
Â du1du2dh1dh2dh3
¼ r2 k2
1 þ k2
2
1024
p 9p þ 46ð Þ þ
k2
3
128
p 3p þ 10ð Þ
!
À
p
4
b1 þ b2 þ b3ð Þ (34)
224The first-order perturbation equation can be rewritten as
x1
@T1 h1; h2; h3; u1; u2ð Þ
@h1
þ x2
@T1 h1; h2; h3; u1; u2ð Þ
@h2
þ x3
@T1 h1; h2; h3; u1; u2ð Þ
@h3
þ c h1; h2; h3; u1; u2ð Þ ¼ K1 pð Þ
(35)
225It is important to take into consideration commensurable frequen-
226cies where exists a relation of the form of m1x1 ¼ m2x2 ¼ m3x3,
227where m1; m2, and m3 integers and expressions for second and
228third frequency are x2 ¼ l1x1 and x3 ¼ l2x1, respectively. The
229function c h1; h2; h3; u1; u2ð Þ in Eq. (35) can be written in the form
c h1; h2; h3; u1; u2ð Þ ¼ K1 pð Þ þ f0 h1; h2; h3ð Þ
þ f1 h1; h2; h3ð Þ cos 2u1 þ f2 h1; h2; h3ð Þ
Â cos 2u2 þ f3 h1; h2; h3ð Þ cos 4u1
þ f4 h1; h2; h3ð Þ cos 4u2 þ f5 h1; h2; h3ð Þ
Â sin 2u2 þ f6 h1; h2; h3ð Þ cos 2u1 cos 2u2
þ f7 h1; h2; h3ð Þ cos 2u1 sin 2u2
þ f8 h1; h2; h3ð Þ cos 2u1 cos 4u2
þ f11 h1; h2; h3ð Þ sin 2u1 sin u2 (36)
230Functions fr h1; h2; h3ð Þ are periodic in h1; h2, and h3 and given as
fr h1; h2; h3ð Þ ¼ U srf g; r ¼ 0; 1; …; 11 (37)
231where U½ Š is the vector in the form
Ub c ¼ h1 b1 h2 b2 h3 b3 r2
k2
1 r2
k2
2 r2
k2
3 r2
k1k2 r2
k1k3 r2
k2k3
Ä Å
232and the values of column vectors srf g are available in the.nb file –
233(http://www.2shared.com/file/-xPP9Snp/Sr_online.html). Equa-
234tion (35) cannot be obtained explicitly for T1 h1; h2; h3; u1; u2ð Þ:
235The combination of coefficients u1 and u2 in Eq. (36) suggests
236that function T1 h1; h2; h3; u1; u2ð Þ can be written as
T1 h1; h2; h3; u1; u2ð Þ ¼ T10 h1; h2; h3ð Þ þ T11 h1; h2; h3ð Þ cos 2u1
þ T12 h1; h2; h3ð Þ cos 2u2 þ T13 h1; h2; h3ð Þ
Â cos 4u1 þ T14 h1; h2; h3ð Þ cos 4u2
þ T15 h1; h2; h3ð Þ sin 2u2 þ T16 h1; h2; h3ð Þ
Â cos 2u1 cos 2u2 þ T17 h1; h2; h3ð Þ cos 2u1
Â sin 2u2 þ T18 h1; h2; h3ð Þ cos 2u1 cos 4u2
þ T19 h1; h2; h3ð Þ cos 4u1 cos 2u2
þ T110 h1; h2; h3ð Þ cos 4u1 cos 4u2
þ T111 h1; h2; h3ð Þ sin 2u1 sin u2 (38)
237Substituting Eq. (38) into Eq. (35) and equating terms of the equal
238trigonometry function to give a set of partial differential
239equations,

x1
@T1r h1; h2; h3ð Þ
@h1
þ x2
@T1r h1; h2; h3ð Þ
@h2
þ x3
@T1r h1; h2; h3ð Þ
@h3
þ fr h1; h2; h3ð Þ ¼ 0; r ¼ 0; 1; …; 11 (39)
240 The functions T1r h1; h2; h3ð Þ can be written as
T1r h1; h2; h3ð Þ ¼ U½ Š vrf g þ Cr h2 À l1h1; h3 À l2h1ð Þ;
r ¼ 0; 1; …; 11 (40)
241 where the calculated values of column vectors vrf g are available in
242 the.nb file – (http://www.2shared.com/file/7Eb8_wIV/Vr_onli-
243 ne.html). Here, Crðh2 À l1h1; h3 À l2h1Þ are the same arbitrary func-
244 tions of three variables, for which we make assumptions as follows:
Cr h2 À l1h1;h3 À l2h1ð Þ ¼ A1r þ B1r sin 2h2 À 2l1h1ð Þ
þ C1r sin 4h2 À 4l1h1ð Þ
þ D1r sin 2h3 À 2l2h1ð Þ
þ E1r sin 4h3 À 4l2h1ð Þ; r ¼ 0;1;…;11
(41)
245 As each function T1r h1; h2; h3ð Þ; r ¼ 0; 1; …; 11 must be positive
246 and periodic, unknown constants A1r; B1r; C1r; D1r; and E1r can
247 be determined using conditions
T1r 0; 0; 0ð Þ ¼ T1r 0; 0; 2pð Þ ¼ T1r 0; 2p; 0ð Þ ¼ T1r 2p; 0; 0ð Þ
¼ T1r 2p; 2p; 0ð Þ ¼ T1r 0; 2p; 2pð Þ
¼ T1r 2p; 0; 2pð Þ ¼ T1r 2p; 2p; 2pð Þ ¼ 0 (42)
@T1r 0; 0; 0ð Þ
@h1
¼
@T1r 2p; 0; 0ð Þ
@h1
;
@T1r 0; 0; 0ð Þ
@h2
¼
@T1r 0; 2p; 0ð Þ
@h2
;
@T1r 0; 0; 0ð Þ
@h3
¼
@T1r 0; 0; 2pð Þ
@h3
; r ¼ 0; 1; …; 11 (43)
248and give the following constants available in the.nb file – (http://
249www.2shared.com/file/247fMXuz/Coefficients_A1r_B1r_C1r_D1r_a.
250html).
2513.2 Second-Order Perturbation. The second-order pertur-
252bation equation must satisfy the condition of periodic function
T2 h1; h2; h3; u1; u2ð Þ in h1; h2, and h3, given as
L0T2 h1; h2; h3; u1; u2ð Þ þ L1T2 h1; h2; h3; u1; u2ð Þ
¼ K1 pð ÞT1 h1; h2; h3; u1; u2ð Þ þ K2 pð Þ (44)
253We have
K2 pð Þ ¼
1
2p5
ð2p
0
ð2p
0
ð2p
0
ðp=2
0
ðp=2
0
a1
@2
T1
@h2
1
(
þ a2
@2
T1
@h2
2
þ a3
@2
T1
@h2
3
þ a4
@2
T1
@u2
1
þ a5
@2
T1
@u2
2
þ a6
@2
T1
@h1@h2
þ a7
@2
T1
@h1@h3
þ a8
@2
T1
@h1@u1
þ a9
@2
T1
@h1@u2
þ a10
@2
T1
@h2@h3
þ a11
@2
T1
@h2@u1
þ a12
@2
T1
@h2@u2
þ a13
@2
T1
@h3@u1
þ a14
@2
T1
@h3@u2
þ a15
@2
T1
@u1@u2
þ b1
@T1
@h1
þ b2
@T1
@h2
b3
@T1
@h3
þ b4
@T1
@u1
þ b5
@T1 h1; h2; h3; u1; u2ð Þ
@u2
þ c À K1 pð ÞŠT1½
'
du1du2dh1dh2dh3 (45)
K2 pð Þ can be obtained symbolically. We made a program in software Mathematica 8 to pass the critical point of the work and to find
254 the solution of the integral given in Eq. (45). Procedure in detail is presented in Appendix A. After integrating, the solution has the fol-
255 lowing form:
K2 pð Þ ¼
K20 pð Þ þ K21 pð Þ sin 4l2p þ K22 pð Þ cos 4l2p þ K23 pð Þ cos 3l2p sin l2p þ K24ðpÞ cos l2p sin 3l2p
x1 l2
1 À 1
À Á
l2
2 À 1
À Á
l2
1 À l2
2
À Á
2 þ cos 4l2pð Þ
(46)
256 The values K20 pð Þ; K21 pð Þ; K22 pð Þ; K23 pð Þ, and K24 pð Þ are available in the.nb file – (http://www.2shared.com/file/SLB4GzXx/Lamb-
257 da_parts_second_perturbati.html). The weak noise expansion of the moment Lyapunov exponent in the second-order perturbation for
258 the stochastic system [5] is determined in the form
K pð Þ ¼ eK1 pð Þ þ e2
K2 pð Þ þ O e3
À Á
(47)
259 The Lyapunov exponent for the system in Eq. (26) can be obtained from Eq. (47) by using a property of the moment Lyapunov
260 exponent,
k ¼
dK pð Þ
dp

p¼0
¼ ek1 þ e2
k2 þ O e3
À Á
¼ e
23
512
k2
1 þ k2
2
À Á
þ
5
64
k2
3
!
r2
À
1
4

ðb1 þ b2 þ b3Þ
'
þ e2 k20 þ k21 sin 4l2p þ k22 cos 4l2p þ k23 cos 3l2p sin l2p þ k24 cos l2p sin 3l2p
x1 l2
1 À 1
À Á
l2
2 À 1
À Á
l2
1 À l2
2
À Á
2 þ cos 4l2pð Þ
þ O e3
À Á
(48)
261 The values k20; k21; k22; k23, and k24 are available in the.nb file –
262 (http://www.2shared.com/file/qXWvZYOS/Small_Lambda_20_21_
263 22_23_24.html).
2643.3 Stochastic Stability Conditions. The values used for the pa-
265rameters of the stochastic system in the calculations for determining
266moment Lyapunov exponent and Lyapunov exponent are given as follows:

E ¼ 1 Â 1010
NmÀ2
; K ¼ 2 Â 105
NmÀ2
; q ¼ 2 Â 103
kgmÀ3
;
l ¼ 10 m; A1 ¼ A2 ¼ A3 ¼ 5 Â 10À2
m2
; I1 ¼ 4 Â 10À4
m4
;
I2 ¼
25
4
Â 10À4
m4
; I3 ¼
35
4
Â 10À4
m4
(49)
267 where we assume, for simplicity, that
H1 ¼ H2 ¼ H3 ¼
K
qA
; K1 ¼ K2 ¼ K3 ¼
p2
l2qA
(50)
268 We determined analytically the pth moment stability boundary in
269 the first-order perturbation for various values p ¼ 1; 2; 4; respec-
270 tively, with the definition of the moment stability K pð Þ 0. Using
271 the results for the moment Lyapunov exponent from first-order
272 perturbation, K pð Þ ¼ eK1 pð Þ þ O e2
ð Þ, we obtained
K 1ð Þ 0 ¼)
pert:1
c0
55
768
ðk2
1 þ k2
2Þ þ
13
96
k2
3
!
r2
qA;
K 2ð Þ 0 ¼)
pert:1
c0
1
12
ðk2
1 þ k2
2 þ 2k2
3Þr2
qA;
K 4ð Þ 0 ¼)
pert:1
c0
41
384
ðk2
1 þ k2
2Þ þ
11
48
k2
3
!
r2
qA (51)
273 An almost-sure stability boundary of the oscillatory system can be
274 determined in the first-order perturbation from knowing that the
275 oscillatory stochastic system is asymptotically stable only if the
276 Lyapunov exponent k 0: From the expression k ¼ ek1 þ O e2
ð Þ,
277 we have
k 0 ¼)
pert:1
c0
23
384
ðk2
1 þ k2
2Þ þ
5
48
k2
3
!
r2
qA (52)
278 Following the same procedure, using the values of natural fre-
279 quencies x1 ¼ 19:739 sÀ1
; x2 ¼ 24:674 sÀ1
, and x3 ¼ 29:194 sÀ1
280 calculated for the parameters of the system, we determined the
281 moment stability boundary in the second-order perturbation in the
282 form of equations
K 1ð Þ 0 ¼)
pert:2
0:0004287 þ 0:02258090c0 þ 1:50619 Â 10À6
c2
0 0
K 2ð Þ 0 ¼)
pert:2
0:00099457 þ 0:0453544c0 þ 4:0165 Â 10À6
c2
0 0
K 4ð Þ 0 ¼)
pert:2
0:0025176 þ 0:0914795c0 þ 0:00001204c2
0 0
(53)
283 We determined the almost-sure stability boundary in the second-
284 order perturbation by the same procedure applied to k ¼ ek1
þ e2
k2 þ O e3
ð Þ, and we have the condition
k 0 ¼)
pert:2
0:0003586 þ 0:022485c0 þ 1:00412 Â 10À6
c2
0 0
(54)
285 The variation of moment Lyapunov exponent for double-beam
286 and three-beam systems on elastic foundation is presented in
287 Fig. 2. Values for the double-beam system in numerical experi-
288 ment are chosen from the present study Eq. (49) for the properties
289 of the beams 1 and 2 and compared with results for moment
290 Lyapunov exponent for the three-beam system on elastic founda-
291 tion. The main point of the presented results in Fig. 2 is that the
292 stochastic stability of the three-beam system on elastic foundation
293 is higher than for the double-beam system without foundation. We
294 compared stability when one more beam and elastic foundation is
295 included with the simple double-beam system. Our investigation
296 shows that, regardless, because we increased the number of free-
297 dom of the system with one more beam, elastic foundation has
298 significant influence on increasing the region of the stochastic
299stability, and the system considered in the present work is more
300stable because the values of K pð Þ are negative on the bigger
301region of the p. Negative values of the K pð Þ show when the sys-
302tem is stable. Figure 3 just shows moment stability boundaries for
303first perturbation with respect to the damping coefficients. We can
304conclude that the stability region is increasing with increasing of
305the damping coefficient, which was expected. AQ7
3064 Conclusions
307In this paper, the dynamic stability of a complex dynamic sys-
308tem of the class of six-dimensional under white noise excitations
309is studied through the determination of the moment Lyapunov
310exponents and the Lyapunov exponents. An eigenvalue problem
311for the moment Lyapunov exponent is established using the theory
312of stochastic dynamic systems. For weak noise excitations, a sin-
313gular perturbation method is employed to obtain second-order
314expansions of the moment Lyapunov exponent. The Lyapunov
315exponent is then obtained using the relationship between the
316moment Lyapunov exponent and the Lyapunov exponent. Expres-
317sions (51)–(54) show the almost-sure and pth moment stability
318boundaries in the first and second perturbation with respect to the
319damping coefficients c0: When the Lyapunov exponent is nega-
320tive, the system in Eq. (6) is almost-sure stable with probability 1;
321otherwise, it is unstable. Figure 2 shows the comparison in
Fig. 2 Moment Lyapunov exponent K pð Þ for r
5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2003103
p
; d1 5 d2 5 c0 5 0:01; thin lines - double beam [13],
thick lines – three beam system on elastic foundation, dashed
lines – second perturbation of the three beam system
Fig. 3 Stability regions for almost-sure (a-s) and pth moment
stability for e ¼ 0:002

322 transverse stochastic stability between a double-beam and three-
323 beam system on elastic foundation. It was concluded that the Win-
324 kler elastic layer has a significant influence on the stochastic sta-
325 bility of beams. This influence is reflected in the increase of
326 stochastic stability. Conditions of stability are determined for a
327 complex three-beam system, and, in future research, this study
328 can be extended to other structures, such as plates on viscoelastic
329 foundation. Transformations given in the present work for the
330 three-DOF system can be applied as a direct extension of the
331 work of Ariaratnam et al. [18] and Xie [19].
332Acknowledgment
333The research is supported by the Ministry of Science and Envi-
334ronment Protection of the Republic of Serbia, grant No. ON
335174011.
336Appendix A
337
338Second Order Perturbation – Integral Solving Procedure.
339See Figs. 4 and 5.
Fig. 4AQ8
Fig. 5

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