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)


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


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)


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


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


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)


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]


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


( )

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)


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.


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.


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)


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.


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.


REFERENCES

1. Arnold, L., Doyle, M.N., Sri Namachchivaya, N., 1997, Small noise expansion of moment Lyapunov exponents for two-dimensional systems, Dynamics and Stability of Systems, 12(3), pp. 187-211.

2. Khasminskii, R., Moshchuk, N., 1998, Moment Lyapunov exponent and stability index for linear conservative

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.

446-452. 4. Sri Namachchivaya, N., Van Roessel, H.J., 2004, Stochastic stability of coupled oscillators in resonance: A

perturbation approach, ASME Journal of Applied Mechanics, 71, pp. 759-767.

5. Kozić, P., Pavlović, R., Janevski, G., 2008, Moment Lyapunov exponents of the stochastic parametrical Hill΄s equation, International Journal of Solids and Structures, 45(24), pp. 6056-6066.

6. Kozić, P., Janevski, G., Pavlović, R., 2009, Moment Lyapunov exponents and stochastic stability for two coupled oscillators, The Journal of Mechanics of Materials and Structures, 4(10), pp. 1689-1701.

7. Kozić, P., Janevski, G., Pavlović, R., 2010, Moment Lyapunov exponents and stochastic stability of a double-

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

equations with small noise, SIAM Journal on Numerical Analysis, 34(6), pp. 2142-2167.

9. Pavlović, R., Kozić, P., Rajković, P., Pavlović I., 2007, Dynamic stability of a thin-walled beam subjected to axial loads and end moments, Journal of Sound and Vibration, 301, pp. 690-700.

10. Xie, W.C., 2005, Monte Carlo simulation of moment Lyapunov exponents, ASME Journal of Applied

Mechanics, 72, pp. 269-275. 11. Wedig, W., 1988, Lyapunov exponent of stochastic systems and related bifurcation problems, In: Ariaratnam,

T.S., Schuëller, G.I., Elishakoff, I. (Eds.), Stochastic Structural Dynamics–Progress in Theory and Applications,

Elsevier Applied Science, pp. 315 – 327. 12. Deng, J., Xie, W.C., Pandey M., 2014, Moment Lyapunov exponents and stochastic stability of coupled

viscoelastic systems driven by white noise, Journal of mechanics of materials and structures, 9, pp. 27-50.

13. Deng, J., 2018, Stochastic stability of coupled viscoelastic systems excited by real noise, Mathematical problems in Engineering, Article ID 4725148.

14. Deng, J., Zhong, Z., Li, A., 2019, Stochastic stability of viscoelastic plates under bounded noise excitation,

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 −

http://pjm.math.berkeley.edu/jomms/2009/4-10/p03.xhtml

http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6VJS-4YCWNN7-1-5X&_cdi=6102&_user=1793222&_pii=S0020768310000478&_orig=search&_coverDate=05%2F15%2F2010&_sk=999529989&view=c&wchp=dGLbVtz-zSkWb&md5=990c61b2ee24db395dc537dc4e637cc4&ie=/sdarticle.pdf


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 },+ − +


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 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 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


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


− += = = + − +

− + = − − − − + −

+ − + −= − − + + + + +


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

= − − + − − − +

+ − + + + + + − − + + + +

+ − −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


+ + + + − − + + +

+ − − + + + + − − + +

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+

moment lyapunov exponents and stochastic stability …

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