MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY …
Transcript of MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY …
FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 19, No 2, 2021, pp. 209 - 228
https://doi.org/10.22190/FUME191127014J
© 2021 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND
Original scientific paper*
MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC
STABILITY OF A THIN-WALLED BEAM SUBJECTED
TO AXIAL LOADS AND END MOMENTS
Goran Janevski1 , Predrag Kozić1, Ratko Pavlović1, Strain Posavljak2
1University of Niš, Faculty of Mechanical Engineering, Serbia 2University of Banja Luka, Faculty of Mechanical Engineering, Bosnia and Herzegovina
Abstract. In this paper, the Lyapunov exponent and moment Lyapunov exponents of two
degrees-of-freedom linear systems subjected to white noise parametric excitation are
investigated. The method of regular perturbation is used to determine the explicit asymptotic
expressions for these exponents in the presence of small intensity noises. The Lyapunov
exponent and moment Lyapunov exponents are important characteristics for determining
both the almost-sure and the moment stability of a stochastic dynamic system. As an
example, we study the almost-sure and moment stability of a thin-walled beam subjected to
stochastic axial load and stochastically fluctuating end moments. The validity of the
approximate results for moment Lyapunov exponents is checked by numerical Monte Carlo
simulation method for this stochastic system.
Key words: Eigenvalues, Perturbation, Stochastic stability, Thin-walled beam,
Mechanics of solids and structures
1. INTRODUCTION
In recent years there has been considerable interest in the study of the dynamic
stability of non-gyroscopic conservative elastic systems whose parameters fluctuate in a
stochastic manner. To have a complete picture of the dynamic stability of a dynamic
system, it is important to study both the almost-sure and the moment stability and to
determine both the maximal Lyapunov exponent and the pth moment Lyapunov exponent.
The maximal Lyapunov exponent is defined by
0
1lim log ( ; )qt
t qt→
= q (1)
Received November 27, 2019 / Accepted March 08, 2020
Corresponding author: Goran Janevski University of Niš, Faculty of Mechanical Engineering in Niš, A. Medvedeva 14, 18000 Niš, Serbia
E-mail: [email protected]
210 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
where 0( ; )t qq is the solution process of a linear dynamic system. The almost-sure stability
depends upon the sign of the maximal Lyapunov exponent which is an exponential growth
rate of the solution of the randomly perturbed dynamic system. A negative sign of the
maximal Lyapunov exponent implies the almost-sure stability whereas a non-negative value
indicates instability. The exponential growth rate E [||q(t;q0, 0q ||p] is provided by the moment
Lyapunov exponent defined as
0
1( ) lim log [ ( ; ) ]
p
qt
p E t qt→
= q (2)
where E [ ] denotes the expectation. If q(p) < 0 then, by definition E [||q(t;q0, 0q ||p] → 0) as
t → and this is referred to as the pth moment stability. Although the moment Lyapunov
exponents are important in the study of the dynamic stability of the stochastic systems, the
actual evaluations of the moment Lyapunov exponents are very difficult.
Arnold et al. [1] constructed an approximation for the moment Lyapunov exponents,
the asymptotic growth rate of the moments of the response of a two-dimensional linear
system driven by real or white noise. A perturbation approach was used to obtain explicit
expressions for these exponents in the presence of small intensity noises. Khasminskii
and Moshchuk [2] obtained an asymptotic expansion of the moment Lyapunov exponents
of a two-dimensional system under white noise parametric excitation in terms of the small
fluctuation parameter , from which the stability index was obtained. Sri Namachchivaya et al.
[3] used a perturbation approach to calculate the asymptotic growth rate of a stochastically
coupled two-degrees-of-freedom system. The noise was assumed to be white and of small
intensity in order to calculate the explicit asymptotic formulas for the maximum Lyapunov
exponent. Sri Namachchivaya and Van Roessel [4] used a perturbation approach to obtain an
approximation for the moment Lyapunov exponents of two coupled oscillators with
commensurable frequencies driven by small intensity real noise with dissipation. The
generator for the eigenvalue problem associated with the moment Lyapunov exponents was
derived without any restriction on the size of pth moment. Kozić et al. [5] investigated the
Lyapunov exponent and moment Lyapunov exponents of a dynamic system that could be
described by Hill’s equation with frequency and damping coefficient fluctuated by white
noise. The procedure employed in Khasminskii and Moshchuk [2] was applied to obtain an
asymptotic expansion of the Lyapunov exponent and moment Lyapunov exponents of an
oscillatory system under two white-noise parametric excitations in terms of the small
fluctuation parameter. These results were used to obtain explicit expressions of an
asymptotic expansion of the moment and almost sure stability boundaries of the simply
supported beam which was subjected to the axial compressions and varying damping which
were two random processes. In [6, 7], Kozić et al. investigated the Lyapunov exponent and
moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white
noise parametric excitation. In [6], almost-sure and moment stability of the flexural-torsion
stability of a thin elastic beam subjected to a stochastically fluctuating follower force were
studied. In [7], moment Lyapunov exponents and stability boundary of the double-beam
system under stochastic compressive axial loading were obtained. In [9], Pavlović et al.
investigated the dynamic stability of thin-walled beams subjected to combined action of
stochastic axial loads and stochastically fluctuating end moments. By using the direct
Lyapunov method, the authors obtained the almost-sure stochastic boundary and uniform
Moment Lyapunov Exponents and Stochastic Stability of a Thin-Walled Beam Subjected to Axial ... 211
stochastic stability boundary as the function of characteristics of stochastic process and
geometric and physical parameters.
Deng et al. [12] investigated the Lyapunov exponent and moment Lyapunov exponents
of flexural-torsional viscoelastic beam, under parametric excitation of white noise. The
system of stochastic differential equations of motion is first decoupled by using the method
of stochastic averaging for dynamic systems with small damping and weak excitations. The
moment and almost-sure stability boundaries and critical excitation are obtained analytically
which are confirmed by numerical simulation. Also, Deng in [13] studied the moment
stochastic stability and almost-sure stochastic stability through the moment Lyapunov
exponents and the largest Lyapunov exponent of flexural-torsional viscoelastic beam, under
the parametric excitation of a real noise.
Stochastic stability of a viscoelastic plate in supersonic flow as well typical example
of a coupled non-gyroscopic system through Lyapunov exponent and moment Lyapunov
exponents and are investigated by Deng et al. [14]. The excitation is modelled as a
bounded noise process. By using the method of stochastic averaging, the equations of
motion are decoupled into Itô differential equations, from which moment Lyapunov
exponents are readily obtained. The Lyapunov exponents are obtained from the relation
with moment Lyapunov exponents.
The aim of this paper is to determine a weak noise expansion for the moment Lyapunov
exponents of the four-dimensional stochastic system. The noise is assumed to be white noise
of such small intensity that an asymptotic growth rate can be obtained. We apply the
perturbation theoretical approach given in Khasminskii and Moshchuk [2] to obtain second-
order weak noise expansions of the moment Lyapunov exponents. The Lyapunov exponent is
then obtained using the relationship between the moment Lyapunov exponents and the
Lyapunov exponent. These results are applied to study the pth moment stability and almost-
sure stability of a thin-walled beam subjected to stochastic axial loads and stochastically
fluctuating end moments. The motion of such an elastic system is governed by the partial
differential equations in [9] by Pavlović et al. The approximate analytical results of the
moment Lyapunov exponents are compared with the numerical values obtained by the Monte
Carlo simulation approach for these exponents of a four-dimensional stochastic system.
2. THEORETICAL FORMULATION
Consider linear oscillatory systems described by equations of motion of the form
2
1 1 1 1 1 11 1 1 12 2 2
2
2 2 2 2 2 21 1 1 22 2 2
2 ( ) ( ) 0,
2 ( ) ( ) 0,
q q q K t q K t q
q q q K t q K t q
+ + − − =
+ + − − = (3)
where q1, q2 are generalized coordinates, 1, 2 are natural frequencies and 21, 22
represent small viscous damping coefficients. The stochastic terms 1( )t and
2 ( )t
are white-noise processes with small intensity with zero mean and autocorrelation functions
1 1
2 2
2
1 2 1 1 1 2 1 2 1
2
1 2 2 1 2 2 2 2 1
( , ) [ ( ) ( )] ( ),
( , ) [ ( ) ( )] ( ),
R t t E t t t t
R t t E t t t t
= = −
= = − (4)
212 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
where 1, 2 are the intensity of the random process 1(t) and 2(t), and ( ) is the Dirac
delta.
Using the transformation
1 1 1 1 2 2 3 2 2 4, , ,q x q x q x q x= = = = (5)
and denoting
ij ij jp K= , (i, j=1,2), (6)
the above Eqs. (3) can be represented in the first-order form by a set of Stratonovich
differential equations
1 2 ( ) ( )d dt dt dw t dw t= + + + 0 1 2X A X AX B X B X , (7)
where X = (x1 x2 x3 x4)T is the state vector of the system, w1(t) and w2(t) are the
standard Weiner processes and A0, A, B1 and B2 are constant 44 matrices given by
1
1 1
2
2 2
11 12
1
22 21
0 0 0 0 0 0 0
0 0 0 0 2 0 0, ,
0 0 0 0 0 0 0
0 0 0 0 0 0 2
0 0 0 0 0 0 0 0
0 0 0 0 0 0, .
0 0 0 0 0 0 0 0
0 0 0 0 0 0
p p
p p
− − = =
− −
= =
0
2
A A
B B
, (8)
Applying the transformation
1 1 2 1 3 2 4 2
2 2 2 2 2
1 2 3 4
1 2
cos cos , cos sin , sin cos , sin sin ,
( ) ,
0 2 , 0 2 , 0 2, ,
p p
x a x a x a x a
P a x x x x
p
= = − = = −
= = + + +
−
(9)
and employing Itô’s differential rule, yields the following set of Itô equations for the pth
power of the norm of the response and phase variables 1 2, , :
* * *
1 11 1 12 2
* * *
2 21 1 22 2
* * *
1 1 3 31 1 32 2
* * *
2 2 4 41 1 42 2
( ) ( ),
( ) ( ),
( ) ( ) ( ),
( ) ( ) ( ).
pd a dt dw t dw t
d dt dw t dw t
d dt dw t dw t
d dt dw t dw t
= + +
= + +
= + + +
= + + +
(10)
In the previous transformations, a represents the norm of the response, 1 and 2 are
the angles of the first and second oscillators, respectively, and describes the coupling or
exchange of energy between the first and second oscillator
Moment Lyapunov Exponents and Stochastic Stability of a Thin-Walled Beam Subjected to Axial ... 213
In the previous equation we have introduced the following markings
* 2 2 2 2
1 1 1 2 22 ( sin cos sin sin )pP = − + , * 2 2
2 1 1 2 2( sin sin )sin2 = −
*
3 1 1sin 2 = − , *
4 2 2sin2 = − , * 2 2
11 11 1 22 2[ sin 2 cos sin 2 sin ]2
pPp p = − +
*
21 11 1 22 2
1[ sin 2 sin 2 ]sin 2
4p p = − , * 2
31 11 1cosp = − , * 2
41 22 2cosp = − , (11)
*
12 12 1 2 21 1 2[ sin cos cos sin ]sin 22
pPp p = − + ,
* 2 2
22 12 1 2 21 1 2
1[ sin cos sin cos sin cos ]
2p p = − ,
*
32 12 1 2cos cosp tg = − ,
*
42 21 1 2cos cos cotp = − .
The Itô version of Eqs.(10) have the following form
1 11 1 12 2
2 21 1 22 2
1 1 3 31 1 32 2
2 2 4 41 1 42 2
( ) ( ),
( ) ( ),
( ) ( ) ( ),
( ) ( ) ( ),
pd a dt dw t dw t
d dt dw t dw t
d dt dw t dw t
d dt dw t dw t
= + +
= + +
= + + +
= + + +
, (12)
where i are given in Appendix 1 and *
ij ij = , (i, j=1, 2, 3, 4).
Following Wedig [11], we perform the linear stochastic transformation
1
1 2 1 2( , , ) , ( , , )S T P P T S−= = , (13)
introducing the new norm process S by means of the scalar function T(,1,2) which is
defined on the stationary phase processes 1, 2 and
1 2 1 2
1 2 1 1 1 2 2 2
1 2
1 2
1 2 1 0 1 2
00 01 02 11 12 22
11 21 31 41 1
12 22 32 42 2
( ) (
)
( ) ( )
( ) ( ) ,
dS P T T dt P T m T m T m T
m T m T m T m T m T m T dt
P T T T T dw t
P T T T T dw t
= + + + + + +
+ + + + + + +
+ + + + +
+ + + +
, (14)
where
0 2 11 21 12 22 1 3 11 31 12 32 2 4 11 41 12 42
2 2
00 21 22 01 21 31 22 32 02 21 41 22 42
2 2 2 2
11 31 32 12 31 41 32 42 22 41 42
, , ,
1( ), m , m ,
2
1 1( ), m , m ( ) .
2 2
m m m
m
m
= + + = + + = + +
= + = + = +
= + = + = +
(15)
214 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
If the transformation function T(,1,2) is bounded and non-singular, both processes
P and S possess the same stability behavior. Therefore, transformation function T(,1,2)
is chosen so that the drift term, of the Itô differential Eq. (15), does not depend on the
phase processes 1, 2 and , so that
1 2
1 2
1
11 21 31 41 1
1
12 22 32 42 2
( ) ( )
( )
( )
( ) .
dS p S dt S T T T T T dw t
S T T T T T dw t
−
−
= + + + + +
+ + + + (16)
By comparing Eqs. (14) and (16), it can be seen that such a transformation function
1 2( , , )T is given by the following equation
0 1 1 2 1 2[ ] ( , , ) ( ) ( , , ).L L pT T+ = (17)
In (17) L0 and L1 are the following first and second-order differential operators
0 1 2
1 2
2 2 2 2 2 2
1 1 2 3 4 5 62 2 2
1 2 1 21 2
1 2 3
1 2
,
,
L
L a a a a a a
b b b c
= +
= + + + + + + +
+ + + +
(18)
where 1a , 2a , 3a , 4a , 5a , 6a , 1b , 2b , 3b and c are given in Appendix 2.
Eq. (17) defines an eigenvalue problem for a second-order differential operator of three
independent variables, in which (p) is the eigenvalue and T(,1,2) the associated
eigenfunction. From Eq. (16), the eigenvalue (p) is seen to be the Lyapunov exponent of the
pth moment of system (7), i. e., (p) = x(t)(p). This approach was first applied by Wedig [11] to
derive the eigenvalue problem for the moment Lyapunov exponent of a two-dimensional linear
Itô stochastic system. In the following section, the method of regular perturbation is applied to
the eigenvalue problem (17) to obtain a weak noise expansion of the moment Lyapunov
exponent of a four-dimensional stochastic linear system.
3. WEAK NOISE EXPANSION OF THE MOMENT LYAPUNOV EXPONENT
Applying the method of regular perturbation, both the moment Lyapunov exponent
(p) and the eigenfunction T(,1,2) are expanded in power series of ε as:
2
0 1 2
2
1 2 0 1 2 1 1 2 2 1 2 1 2
( ) ( ) ( ) ( ) ( ) ,
( , , ) ( , , ) ( , , ) ( , , ) ( , , ) .
n
n
n
n
p p p p p
T T T T T
= + + + + +
= + + + + + (19)
Substituting the perturbation series (19) into the eigenvalue problem (17) and equating
terms of the equal powers of ε leads to the following equations
Moment Lyapunov Exponents and Stochastic Stability of a Thin-Walled Beam Subjected to Axial ... 215
0
0 0 0 0
1
0 1 1 0 0 1 1 0
2
0 2 1 1 0 2 1 1 2 0
3
0 3 1 2 0 3 1 2 2 1 3 0
0 1 1 0 1 1 2 2 1 1 0
( ) ,
( ) ( ) ,
( ) ( ) ( ) ,
( ) ( ) ( ) ( ) ,
( ) ( ) ( ) ( ) ( )n
n n n n n n n
L T p T
L T L T p T p T
L T L T p T p T p T
L T L T p T p T p T p T
L T L T p T p T p T p T p T
− − − −
→ =
→ + = +
→ + = + +
→ + = + + +
→ + = + + + + + ,
(20)
where each function 1 2( , , ) , 0,1,2,i iT T i= = must be positive and periodic in the
range 0 2 , 10 2 and 20 2 .
3.1. Zeroth order perturbation
The zeroth order perturbation equation is 0 0 0 0( )L T p T= or
0 01 2 0 0
1 2
( )T T
p T
+ =
. (21)
From the property of the moment Lyapunov exponent, it is known that
2
0 1 2(0) (0) (0) (0) (0) 0n
n = + + + + = , (22)
which results in (0) 0n = for 0, 1, 2, 3,....n = Since the eigenvalue problem (21) does
not contain p, the eigenvalue 0 ( )p is independent of p. Hence, 0 (0) 0 = leads to
0 ( ) 0p = . (23)
Now, partial differential Eqs. (21) have the form
0 01 2
1 2
0T T
+ =
. (24)
Solution of Eq.(24) may be taken as
0 1 2 0( , , ) ( )T = , (25)
where 0 ( ) is an unknown function of which has yet to be determined.
3.2. First order perturbation
The first order perturbation equation is
0 1 1 0 1 0( )L T p T LT= − . (26)
Since the homogeneous Eq. (24) has a non-trivial solution given by Eq. (25), for Eq. (26)
to have a solution it is required, from the Fredholm alternative, that following is satisfied:
* *
0 1 0 1 0 1 0 0( , ) ( ( ) , ) 0L T T p T L T T= − = . (27)
In the previous equation, *
0 0 ( )T = is an unknown solution of the associated adjoint
differential equation of (24), and (f,g) denotes the inner product of functions f (,1,2)
and g(,1,2) defined by
216 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
2 2 2
1 2 1 2 1 2
0 0 0
( , ) f( , , )g( , , )d d df g
= . (28)
Taking onto account (25), (26) and (28), the expression (27) has the form
2 2 2
1 0 1 0 0 1 2
0 0 0
( ( ) ) ( ) d d d 0p L
− = , (29)
and will be satisfied if and only if
2 2
1 0 1 0 1 2
0 0
( ( ) ) d d 0p L
− = . (30)
After the integration of the previous expression we have that
2
0 00 1 1 1 0 1 02
( ) ( ) ( ) ( ) ( ) 0d d
L A B C pdd
= + + − =
, (31)
where
( ) ( )2 2 2 2
1 1 1 2 1 2 1 1 1 2 1 2
0 0 0 0
( , , ) d d , b ( , , ) d d ,A a B
= =
2 2
1 1 2 1 2
0 0
( ) ( , , ) d d .C c
= (32)
Finally, 1A , 1B and 1C are
2 2 2 2
1 11 22 12 21
2 22 212 211 2
2 2 2 2
11 22 12 21
2 2 2 2
1 11 22 12 21
2 2 2 2
12 21
1
1( ) [ 2( )]cos 4
128
( )cos 216
1[ 6( )] ,
128
1( ) ( 1)[ 2( )]sin 4
64
1( sin cos cot )
8
1 16 16
32
A p p p p
p p
p p p p
B p p p p p
p tg p
= − + − + −
−− − +
+ + + +
= − − + − + −
− − +
+ −
2 2 2 2
2 11 22 12 21
2 2 2 2
1 11 22 12 21
2 2 2 2
1 2 11 22 12 21
2
1 2 11
[( 2)( ) 2( 1)( )] sin 2 ,
1( ) ( 2)[ 2( )]cos 4
128
1 16 16 [( 2)( ) 4( )] cos 2
32
1 64 64 [(10 3 )(
128
p p p p p p
C p p p p p p
p p p p p
p p p
− + − + − −
= − + − + −
− − − + − − − +
+ − − + + 2 2 2
22 12 21) 2(6 )( )] p p p p+ + + +
(33)
Moment Lyapunov Exponents and Stochastic Stability of a Thin-Walled Beam Subjected to Axial ... 217
Since the coefficients (33) of the Eq.(31) are periodic functions of , a series
expansion of the function 0() may be taken in the form
0
0
( ) cos2N
kk
K k=
= . (34)
Substituting (34) in (31), multiplying the resulting equation by cos 2k (k = 0, 1, 2 ...) and
integrating with respect from 0 to /2 leads to a set of 2N+1 homogenuos linear
equations for the unknown coefficients K0, K1, K2...
k1
N
0j
jjk K)p(KA ==
, (35)
where
( )2
0
cos(2 ) cos(2 )jkA L j k d
= , k=0, 1, 2, 3, ....N. (36)
When N tends to infinity, the solution (34) tends to the exact solution. The condition
for system homogeneous linear equations (35) to have nontrivial solutions is that the
determinant of system homogeneous linear equations (35) is equal to zero. The
coefficients Ajk to order N=4 are presented in Appendix 3.
In the case when N=0, we assume a solution (34) in the form 0() = K0. From conditions
that A00 = 0, the moment Lyapunov exponent in the first perturbation is defined as
2 2 2 2
1 1 2 11 22 12 21
(10 3 ) (6 )( ) ( ) ( ) ( ).
2 128 64
p p p p pp p p p p
+ + = − + + + + + (37)
In the case when N=1, the solution (34) has the form 0 0 1( ) cos2K K = + , then
moment Lyapunov exponent in the first perturbation is the solution of the equation 2 (1) (1)
1 1 1 0 0d d + + = where coefficients (1)
0d and (1)
1d are presented in Appendix 4. In
the case when N=2, the solution (34) has the form 0 0 1 2( ) cos2 cos4K K K = + + ,
the moment Lyapunov exponent in the first perturbation is the solution of the equation 3 (2) 2 (2) (2)
1 2 1 1 1 0 0d d d + + + = where coefficients (2)
0d , (2)
1d and (2)
2d are presented in
Appendix 5. However, for N > 2, it is impossible to obtain the explicit expressions of
1 ( )p and the numerical results must be given, for N = 3 and 4.
4. APPLICATION TO A THIN-WALLED BEAM SUBJECTED TO AXIAL LOADS AND END MOMENTS
The purpose of this section is to present the general results of the above sections in
the context of real engineering applications and show how these results can be applied to
physical problems. To this end, we consider the flexural-torsional vibration stability of a
homogeneous, isotropic, thin walled beam with two planes of symmetry. The beam is
assumed to be loaded in the plane of greater bending rigidity by two equal couples and
stochastic axial loads and stochastically fluctuating end moments (Fig. 1).
The governing differential equations for the coupled flexural and torsional motion of
the beam can be written as given by Pavlović et al. in [9]
218 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
2 4 2 2
2 4 2 2
2 2 2 4
2 2 2 4
( ) ( ) 0,
( ) ( ) 0,
u y
p
p s
U U U UA EI M T F T
tT Z Z Z
I UI GJ F t M T EI
T AT Z Z Z
+ + + + =
+ − − + + =
(38)
where U is the flexural displacement in the x-direction, is the torsional displacement,
is mass density, A is area of the cross-section of beam, Iy, Ip, IS are axial, polar and
sectorial moments of inertia, J is Saint–Venant torsional constant, E is Young modulus of
elasticity, G is shear modulus, U, are viscous damping coefficients, T is time and Z is
axial coordinate.
Fig. 1 Geometry of a thin-walled beam system
Using the following transformations
( ) ( )
2 42
2
2 22
1 2 2 2
, , , ( ) , ( ) ,
, , , ,
1 1, , ,
2 2
p
t cr cr
y scr cr y t
y y p
U
y p y py
IU u Z zl T k t F T F F t M T M M t
A
EI AIAlF M EI GJ k e
l EI I Il
l A GJAll s
EI I EI IAEI
= = = = =
= = = =
= = =
(39)
where l is the length of the beam, Fcr is Euler critical force, Mcr is critical buckling
moment for the simply supported narrow rectangular beam, S is slenderness parameter,
1 and 2 are reduced viscous damping coefficients, we get governing equations as
Moment Lyapunov Exponents and Stochastic Stability of a Thin-Walled Beam Subjected to Axial ... 219
( )
2 4 2 22 2
12 4 2 2
2 2 2 42 2
22 2 2 4
2 ( ) ( ) 0,
2 ( ) ( ) 0.
u u u usM t F T
tt z z z
us F t sM t e
Tt z z z
+ + + + =
+ − − + + =
(40)
Taking free warping displacement and zero angular displacements into account,
boundary conditions for the simply supported beam are
( ) ( )
( ) ( )
2 2
2 2
( ,0) ( ,1)
2 2
2 2
( ,0) ( ,1)
,0 ,1 0,
,0 ,1 0.
t t
t t
u uu t u t
z z
t tz z
= = = =
= = = =
(41)
Consider the shape function sin(z) which satisfies the boundary conditions for the
first mode vibration, the displacement ( . )u t z and twist ( , )t z can be described by
1( , ) ( )sinu t z q t z= , 2( , ) ( )sint z q t z = . (42)
Substituting ( , )u t z and ( , )t z from (42) into the equations of motion (40) and
employing Galerkin method unknown time functions can be expressed as
2
1 1 1 1 1 11 1 12 2
2
2 2 2 2 2 21 1 22 2
2 ( ) ( ) 0,
2 ( ) ( ) 0.
q q q K F t q K M t q
q q q K M t q K F t q
+ + − − =
+ + − − = (43)
If we are defined the expressions
2 4
1 = , 2 4
2 ( )s e = + , 4
11 22K K= = , 4
12 21 ,K K s= = (44)
and assume that the compressive stochastic axial force and stochastically fluctuating end
moment are white-noise processes (4) with small intensity
1( ) ( )F t t= ,
2( ) ( )M t t= , (45)
then Eq. (43) is reduced to Eq. (3).
Using the above result for the moment Lyapunov exponent in the first-order perturbation,
2
1( ) ( ) ( )p p O = + , (46)
with the definition of the moment stability (p) < 0, we determine analytically (the case
where N = 0, 1(p) is shown with Eq.(37)) the pth moment stability boundary of the
oscillatory system as
4 2 2
1 2 1 2
1 10 3 6
64 32
s e p ps
s e
+ + + + + +
+ . (47)
220 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
It is known that the oscillatory system (40) is asymptotically stable only if the
Lyapunov exponent 0 . Then expression
)(O 2
1 += , (48)
is employed to determine the almost-sure stability boundary of the oscillatory system in
the first-order perturbation
+
+
+++ 2
2
2
1
4
21 s16
3
32
5
es
es1
. (49)
In [9], Pavlović et al. by using the direct Lyapunov method, investigated the almost
sure asymptotic stability boundary of an oscillatory system as the function of stochastic
process, damping coefficient and geometric and physical parameters of the beam. According
to the authors, the condition for almost sure stochastic stability may be expressed by the
following expression
8 2 2 2 4 2 21 2 1 2 1 2 1 2( ) 2 ( )[ ( )] 4 ( ) 0s s s e s e + − + + + + + . (50)
For the sake of simplicity in the comparison of results, in the following we assume
that two viscous damping coefficients are equal
== 21 , (51)
For this case, we determine the almost-sure stability boundary as
+
+
++ 2
2
2
1
4
s6
5
es
es1
32
3
, (52)
and the pth moment stability boundary of the oscillatory system in the first-order
perturbation as
4
2 21 2
1[(10 3 ) 2(6 ) ]
128
s ep p s
s e
+ + + + +
+. (53)
Starting from Eq. (50), derived by Pavlović et al. [9], the almost sure stability
boundary can be determined in the form
4
2 21 2( )
2s
+ . (54)
With respect to standard I-section we can approximately take that ratios h / b 2,
b / 1 11, / 1 1.5, where h is depth, b is width, is thickness of the flanges and 1 is
thickness of the rib of I-section. These ratios give us s 0.01928(l/h)2 and e 1.176. For the
narrow rectangular cross section, according to assumption /h < 0.1, for thin-walled cross
sections s 1.88(l/h)2 and e 0, which is obtained using the approximation 1 + (/h)2 1.
Moment Lyapunov Exponents and Stochastic Stability of a Thin-Walled Beam Subjected to Axial ... 221
a) I-section b) Narrow rectangular cross section
Fig. 2. Stability regions for almost-sure (a-s) and pth moment stability for 0.1 =
Almost-sure stability boundary and pth moment stability boundary in the first-order
perturbation for I-section are given in Fig. 2a, and for narrow rectangular cross section in
Fig. 2b. It is evident that stability regions in the present study are higher compared to the
results obtained by Pavlović et al. [9]. Also, the moment stability boundaries (53) are
more conservative than the almost-sure boundary (52). It is evident that end moment
variances are about ten times higher for I-section than for narrow rectangular section,
when stochastic axial force vary only a little.
5. NUMERICAL DETERMINATION OF THE PTH MOMENT LYAPUNOV EXPONENT
Numerical determination of the pth moment Lyapunov exponent is important in
assessing the validity and the ranges of applicability of the approximate analytical results.
In many engineering applications, the amplitudes of noise excitations are not small so
that the approximate analytical methods such as the method of perturbation or the method
of stochastic averaging cannot be applied. Therefore, numerical approaches have to be
employed to evaluate the moment Lyapunov exponents. The numerical approach is based
on expanding the exact solution of the system of Itô stochastic differential equations in
powers of the time increment h and the small parameter as proposed in Milstein and
Tret’Yakov [8]. The state vector of the system (7) is to be rewritten as a system of Itô
stochastic differential equations with small noise in the form
1 1 2
2 1 1 1 2 11 1 1 12 3 2
3 2 4
4 2 3 2 4 22 3 1 21 1 2
,
[ 2 ] ( ) ( ),
,
[ 2 ] ( ) ( ).
dx x dt
dx x x dt p x dw t p x dw t
dx x dt
dx x x dt p x dw t p x dw t
=
= − − + + +
=
= − − + + +
(55)
For the numerical solutions of the stochastic differential equations, the Runge-Kutta
approximation may be applied, with error R = O(h4 + 4h). The interval discretization is
[ 0t , T]: { kt : k=0,1,2,3, ....M; 0t < 1t < 2t .........< Mt =T} and the time increment is h = tj+1 − tj.
222 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
The following Runge-Kutta method used to obtain the (k+1)th iteration of the state vector
X = (x1,x2,x3,x4)
2 2 4 4 3 2 3 2( 1) ( )1 1 11 1 1 1 1 11 1
2 2 5 2 2 2 22 ( )1 11 1 1 1
1 1 1 2
3 2 5 2( ) ( )12 1 2 2 12 1 2 23 4
( 2 )1
2 24 2 3
1 16 6 9
( 2 ) ,
2 6
k k
k
k k
h h p h hx x
h p h hh h x
p h p hx x
+ +
= − + + + +
+ − + + − +
+ + +
2 2 2 2 2 2( 1) 1 2 2 ( )1 1 12 1 11 1 1 1 1
2 2 4 4 3 2 4 4 2 2( )1 1 11 1 1 1 1 1
1 2
2 2 2 21 2 (1 2
12 2 3
1 1 16 3 6
( 2 )1 2 1
2 24 2 36 3
16 6
k k
k
k
h h hx h p h h x
h h p h h hh x
h hp h x
+
= − − + − + − +
− + − + + + − + − +
+ − −
3 2) ( )12 2 2 2
4
( 2 ) ,
2
kp hx
− +
3 2 5 2( 1) ( ) ( )21 2 2 2 21 1 2 23 1 2
2 2 4 4 3 2 3 2( )2 2 22 2 1 1 2 23
2 2 5 2 2 2 22 ( )2 22 2 1 2
2 2 2 4
( 2 )
2 6
( 2 )1
2 24 2 3
1 1 ,6 6 9
k k k
k
k
p h p hx x x
h h p h hx
h p h hh h x
+ + = + +
+ + − + + + +
+ − + + −
(56)
2 2 2 2 3 2( 1) 1 2 ( ) ( )1 2 21 2 2 24 21 2 1 2
2 2 2 2 2 21 2 2 ( )2 2 2
2 22 1 2 2 3
2 2 4 4 3 2 4 4 2
2 2 22 1 1 1 22
( 2 )1
6 6 2
1 1 16 3 6
( 2 )1 2
2 24 2 36
k k k
k
h h p hx p h x x
h h hh p h h x
h h p h h hh
+ −
= − − + +
+ − − + − + − +
− + − + + + − +
2( )241 .
3
kx
−
Random variables i and i (i=1,2) are simulated as
1
( 1) ( 1)2
i iP P = − = = = , 1 1 1
212 12i iP P
− = = = =
. (57)
Moment Lyapunov Exponents and Stochastic Stability of a Thin-Walled Beam Subjected to Axial ... 223
Having obtained L samples of the solutions of the stochastic differential equations
(56), the pth moment can be determined as follows
1 11
1( ) ( )
L pp
k j kj
E X t X tL
+ +=
= , 1 1 1( ) [ ( )] [ ( )]T
j k j k j kX t X t X t+ + += . (58)
Using the Monte-Carlo technique by Xie [10], we numerically calculate the pth
moment Lyapunov exponent for all values of p of interest as
1
( ) log ( )p
p E X TT
=
. (59)
6. CONCLUSIONS
In this paper, the moment Lyapunov exponents of a thin-walled beam subjected to
stochastic axial loads and stochastically fluctuating end moments under both white noises
parametric excitations are studied. The method of regular perturbation is applied to obtain
a weak noise expansion of the moment Lyapunov exponent in terms of the small
fluctuation parameter. The weak noise expansion of the Lyapunov exponent is also
obtained. The slope of the moment Lyapunov exponent curve at p = 0 is the Lyapunov
exponent. When the Lyapunov exponent is negative, system (43) is stable with
probability 1, otherwise it is unstable. For the purpose of illustration, in the numerical
study we considered set system parameters 1 = 2 = = 1, = 0.1, L = 4000, h = 0.0005,
M = 10000 and x1(0) = x2(0) = x3(0) = 1/2.
Typical results of the moment Lyapunov exponents (p) for system (43) given by Eq.
(46) in the first perturbation are shown in Fig. 3 for I-section and the noise intensity
1 = 0.1 and 2 = 0.15. The accuracy of the approximate analytical results is validated
and assessed by comparing them to the numerical results. The Monte Carlo simulation
approach is usually more versatile, especially when the noise excitations cannot be
described in such a form that can be treated easily using analytical tools. From the
Central Limit Theorem, it is well known that the estimated pth moment Lyapunov
exponent is a random number, with the mean being the true value of the pth moment
Lyapunov exponent and standard deviation equal to np / L , where np is the sample
standard deviation determined from L samples. It is evident that the analytical result
agrees very well with the numerical results, even for N = 0 when the function 0() does
not depend on and assumes the form 0() = K0.
The moment Lyapunov exponents (p) in the first perturbation for narrow rectangular
cross section and the noise intensity (1 = 0.15 and 2 = 0.01 are shown in Fig. 4. Unlike
the previous example, it is observed that the discrepancies between the approximate
analytical and numerical results decrease for larger number N of series (34). Further
increase of N number of members does not make sense, because the curves merge into one.
224 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
Fig. 3 Moment Lyapunov exponent )p( for I-section (1 = 0.1, 2 = 0.15)
Fig. 4 Moment Lyapunov exponent )p( for narrow rectangular cross section
(1 = 0.15, 2 = 0.01)
If we consider the influence of cross-sectional area of stability boundary, generally
speaking, the narrow rectangular cross section has smaller stability regions than the I-
section. As for the influence of intensity of stochastic force, the end moment variances
are about ten times higher for I-section than for narrow rectangular section, while the
difference in axial force variances is small.
Acknowledgments: This research was supported by the research grant of the Serbian Ministry of
Science and Environmental Protection under the number OI 174011.
Moment Lyapunov Exponents and Stochastic Stability of a Thin-Walled Beam Subjected to Axial ... 225
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system with small random perturbation, SIAM Journal of Applied Mathematics, 58(1), pp. 245-256. 3. Sri Namachchivaya, N., Van Roessel, H.J., Talwar, S., 1994, Maximal Lyapunov exponent and almost-sure
stability for coupled two-degree of freedom stochastic systems, ASME Journal of Applied Mechanics, 61, pp.
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beam system under compressive axial load, International Journal of Solid and Structures, 47(10), pp. 1435-1442. 8. Milstein, N.G., Tret’Yakov, V.M., 1997, Numerical methods in the weak sense for stochastic differential
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European Journal of Mechanics / A Solids, 78, Article ID 103849.
APPENDIX 1
2 2 2 21 1 1 2 2
2 2 2 2 2 2 2 2 2 21 1 11 1 12 2
2 2 2 2 2 2 2 2 2 22 2 22 2 21 1
11 22 12
2 ( sin cos sin sin )
{cos [( 1)cos sin ]sin }( cos cos cos sin )2
{cos [( 1)sin cos ]sin }( cos sin cos cos )2
( 2)(
16
pP
pPp p p
pPp p p
p p Pp p p
= − + +
+ + − + + +
+ + − + + +
−+ + 2
21 1 2)sin 2 sin 2 sin 2 ,p
2 22 1 1 2 2 11 22 12 21 1 2
2 2 2 2 2 211 1 1 1 22 2 2 2
2 2 2 2 2 2 2 212 2 1 1 21 1
1( sin sin )sin 2 ( )sin 2 sin 2 sin 4
16
1 1cos sin 2 (cos 2 cos 2 sin ) cos sin 2 (cos 2 cos 2 sin )
4 4
1 1cos sin (sin sin 2 cos ) cos cos
2 2
p p p p
p p
p tg p
= − − + −
− − + + +
+ − − 2 22 2(sin sin 2 cos ot ),c −
226 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
2 22 2 211 12
3 1 1 1 2 1sin 2 cos cos sin 22 2
p ptg
= − − +
2 22 2 222 21
4 2 2 2 1 2sin 2 cos cos cot sin 22 2
p p = − − +
.
APPENDIX 2
2 2 2 2 21 11 1 22 2 12 1 2 21 1 2
1 1a (p sin 2 p sin 2 ) sin 2 (p cos sin cos p sin cos sin )
32 2= − + − ,
+= 22
21
2212
14
211
2 tgcoscos2
pcos
2
pa ,
+= 2
22
12
221
24
222
3 cotcoscos2
pcos
2
pa ,
( ) 2 2 2 211 124 11 1 22 2 1 12 2 1 21 1 2
p pa p sin 2 p sin 2 cos sin 2 (p cos sin 2 sin tg p cos sin 2 sin 2 )
4 4= − − − − ,
2 2 2 222 215 11 1 22 2 2 12 2 1 21 1 2
p pa (p sin 2 p sin 2 )cos sin 2 (p cos sin 2 sin 2 p cos sin 2 sin cot )
4 4= − − − − ,
22
12
22116 coscosppa = ,
2 2 21 1 1 2 2 2 2 11 22 12 21 1 2
2 2 2 2 2 2 2 2 2 211 1 1 1 22 2 2 2
2 2 212 2
p 1b ( sin sin ) sin sin 2 (p p p p )sin 2 sin 2 sin 4
16
1 1 p cos sin 2 [cos 2(p 1)cos sin ] p cos sin 2 [cos 2(p 1)sin sin ]
4 4
1 p cos sin [(p
2
−= − − + + −
− + − + + − −
− − 2 2 2 2 2 2 21 1 21 1 2 2
11)sin sin 2 cos tg ] p cos cos [(p 1)sin sin 2 cos cot ],
2 + + − +
2 3 2 2 2 22 1 1 11 1 1 11 22 1 2
2 2 2 2 212 1 2 12 21 1 2
pb sin 2 p sin cos [(p 1)cos sin ] p p cos sin 2 sin
2
1 p p sin 2 cos (p 2 pcos 2 )tg p p cos sin 2 sin ,
4 2
= − + − − + +
+ − + +
2 3 2 2 2 23 2 2 22 2 2 11 22 1 2
2 2 2 2 221 1 2 12 21 1 2
pb sin 2 p sin cos [(p 1)sin cos ] p p sin 2 cos cos
2
1 p p cos sin 2 (p 2 pcos 2 )cot p p sin 2 cos cos ,
4 2
= − + − − + +
+ − − +
2 2 2 2 21 1 2 2 11 22 12 21 1 2
2 2 2 2 2 2 2 2 2 211 1 12 2 1 1
2 2 2 2 2 2 222 2 21 1 2
p(p 2)c 2p( sin cos sin sin ) (p p p p )sin 2 sin 2 sin 2
16
p (p cos cos p cos sin ){cos [(p 1)cos sin ]sin }
2
p (p cos sin p cos cos ){cos
2
−= − + + + +
+ + + − + +
+ + 2 2 22[(p 1)sin cos ]sin },+ − +
Moment Lyapunov Exponents and Stochastic Stability of a Thin-Walled Beam Subjected to Axial ... 227
APPENDIX 3
2 2 2 200 1 1 2 11 22 12 21
2 2 2 2 210 1 2 11 22 12 21
2 2 2 2 2 220 11 22 12 21 12 21
2 230 12 21
(10 3 ) (6 )( ) ( ) ( ) ( ),
2 128 64
2 1 1( ) ( 2) ( ) ( ),
4 64 4
( 2)( 4) 172( ) ( ),
256 32
3( ),
4
p p p p pA p p p p p
pA p p p p p
p pA p p p p p p
A p p
+ += − − + + + + +
+= − − + + − + −
+ + = + − + − +
= − 2 240 12 21( ),A p p= − +
2 2 2 201 1 2 11 22 12 21
2 22 2 2 2
11 1 1 2 11 22 12 21
22 2 2 2
21 1 2 11 22 12 21
22
31 11
( 2)( ) ( ) ( ),
4 64 16
1 7 22 8 10 56( ) ( ) ( ) ( ),
2 4 512 256
4 6 8 20( ) ( ) ( ),
8 128 32
10 24(
512
p p p pA p p p p
p p p p pA p p p p p
p p p pA p p p p
p pA p
+= − − + − − −
+ − + −= − − + + + + +
+ + + += − − + − + −
+ += +
22 2 2 2 222 12 21 41 12 21
10 216) ( ), ( ),
256
p pp p p A p p
+ +− + = − −
2 2 2 202 11 22 12 21
2 2 2 212 1 2 11 22 12 21
2 22 2 2 2
22 1 1 2 11 22 12 21
2
32 1 2
( 2)[( ) 2( )],
256
2 ( 2)( 2) 2( ) ( ) ( ),
8 128 16
1 3 10 16 6 80( ) ( ) ( ) ( ),
2 4 256 128
6 7 8 12( ) (
8 512
p pA p p p p
p p p pA p p p p
p p p p pA p p p p p
p p pA
−= + − +
− + − −= − − + − + −
+ − + −= − − + + + + +
+ + += − − + 2 2 2 2
11 22 12 21
2 22 2 2 2
42 11 22 12 21
18) ( )
16
14 48 14 304( ) ( ),
512 256
pp p p p
p p p pA p p p p
+− + −
+ + + += + − +
22 2 2 2
03 13 11 22 12 21
2 2 2 223 1 2 11 22 12 21
2 22 2 2 2
33 1 1 2 11 22 12 21
43
6 80 , ( ) 2( ) ,
512
4 1 3( ) ( 2)( ) ( ) ,
8 16 4
1 3 10 36 2 12 312( ) ( ) ( ) ( ),
2 4 256 256
p pA A p p p p
pA p p p p p
p p p p pA p p p p p
pA
− + = = + − +
− = − − − + − + −
+ − + −= − − + + + + +
+= −
22 2 2 2
1 2 11 22 12 21
8 10 16 3 56( ) ( ) ( ),
8 128 32
p p pp p p p
+ + + − + − + −
22 2 2 2
04 14 24 11 22 12 21
2 2 2 234 1 2 11 22 12 21
2 22 2 2 2
44 1 1 2 11 22 12 21
10 240 , 0, [( ) 2( )] ,
512
6 2( ) ( ) ( ) ,
8 16
1 3 10 64 2 12 512( ) ( ) ( ) ( ),
2 4 256 256
p pA A A p p p p
p pA p p p p
p p p p pA p p p p p
− += = = + − +
− + = − − − − + −
+ − + −= − − + + + + +
228 G. JANEVSKI, P. KOZIĆ, R. PAVLOVIĆ, S. POSAVLJAK
APPENDIX 4
2 2(1) 2 2 2 2
1 2 11 22 12 211
1 21p 13p 7 11p 3pd p( ) (p p ) (p p ),
32 28 256 16 64 128
= + + − − + + − − +
( )( ) ( )(1) 2 20 1 2 1 2
2 3 4 2 3 44 4 4 411 22 12 21
2 3 4 2 32 211 22
1 12 2 3
8 4
13 5 5 5( ) ( )
2048 8192 4096 32768 512 2048 512 8192
3 97 29 37 37
1024 4096 2048 16384 256 1024 2
d p p p p
p p p p p p p pp p p p
p p p p p p pp p
= − + + + +
+ − + + + + + − + + + +
+ + + + + − + +
42 212 21
2 3 44 412 21
2 3 42 2 2 211 21 22 12
2 3 2 32 2
1 22 2 11
56 4096
23 7 17 5( )
512 2048 2048 8192
15 7 9 5( )
512 2048 2048 8192
3 37 21 5 5 5( )
64 256 512 64 256
pp p
p p p pp p
p p p pp p p p
p p p p p pp p
+ +
+ − + + + + +
+ − + + + + +
+ − − − + + − −
2 21 11 2 22
2 3 2 32 2 2 2
1 21 2 12 1 12 2 21
( )512
5 7 3 9 15 3( ) ( ).
32 128 256 32 128 256
p p
p p p p p pp p p p
+ +
+ − − + + − − +
APPENDIX 5
2 2(2) 2 2 2 2
1 2 11 22 12 212
3p 5 31p 19p 27 17p 5pd ( ) (p p ) (p p )
2 32 128 256 16 64 128
= + + − − + + − − +
( )2 2
(2) 2 21 1 2 1 2
2 3 4 2 3 44 4 2 211 22 11 22
2
1 3 9 3 151
2 8 16 4 8
3 47 41 61 35 1 55 153 133 83( )
256 2048 32768 8192 32768 128 1024 4096 4096 16384
15 55 39
64 512 204
p p p pd
p p p p p p p pp p p p
p p
= − − + − − − +
+ − + + + + + − − + + + +
+ − −3 4 2 3 4
4 4 2 212 21 12 21
2 3 4 2 3 42 2 2 211 12 22 21
13 3 55 231 71 13 3( )
8 2048 8192 32 256 1024 1024 4096
9 165 109 47 17 3 153 61 35 17( )
64 512 2048 2048 8192 32 256 1024 1024 4096
p p p p p pp p p p
p p p p p p p pp p p p
+ + + + − − + + +
+ − − + + + + − − + +
2 2 2 211 21 22 12
2 3 2 32 2 2 2
1 22 2 11 1 11 2 22
2 3 2 32 2
1 21 2 12 1
( )
1 43 25 1 3 19 13( ) ( )
8 8 128 256 8 16 128 256
3 45 7 5 3 63 5 5( ) (
8 32 32 128 8 32 16 128
p p p p
p p p p p pp p p p
p p p p p pp p
+ +
+ + − − + + − + − − + +
+ + − − + + − + − −
2 212 2 21).p p+