Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 1, pp. 61-83, Warsaw 2012 50th anniversary of JTAM MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY OF A THIN-WALLED BEAM SUBJECTED TO ECCENTRIC AXIAL LOADS Goran Janevski, Predrag Kozić, Ratko Pavlović University of Nǐs, Department of Mechanical Engineering, Nǐs, Serbia e-mail: gocky.jane@gmail.com 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 andmo- ment Lyapunov exponents are important characteristics for determining the almost-sure and moment stability of a stochastic dynamic system. As an example,we study the almost-sure andmoment stability of a thin- walled beam subjected to an eccentric stochastic axial load.The validity of the approximateresults formomentLyapunovexponents is checkedby thenumericalMonteCarlo simulationmethod for this stochastic system. Key words: eigenvalues, perturbation, stochastic stability, thin-walled beam, mechanics of solids and structures 1. Introduction In the recent years, there has been considerable interest in the study of the dynamic stability of non-gyroscopic conservative elastic systems whose para- meters 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, defined by λq = lim t→∞ 1 t log‖q(t;q0)‖ (1.1) 62 G. Janevski et al. where q(t;q0) is the solution process of a linear dynamic system. The almost- sure stability dependsupon the sign of themaximal Lyapunov exponentwhich is an exponential growth rate of the solution of the randomly perturbed dyna- mic system. The negative sign of themaximal Lyapunov exponent implies the almost-sure stability, whereas a non-negative value indicates instability. The exponential growth rate E[‖q(t;q0, q̇0)‖p] is providedby themomentLyapunov exponent defined as Λq(p)= lim t→∞ 1 t logE[‖q(t;q0)‖p] (1.2) where E[·] denotes the expectation. If Λq(p) < 0, then, by definition E[‖q(t;q0, q̇0)‖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 stochastic systems, the actual evaluations of the moment Lyapunov exponents are very difficult. Arnold et al. (1997) constructed an approximation for themoment Lyapu- nov exponents, the asymptotic growth rate of the moments of the response of a two-dimensional linear system driven by real or white noise. A pertur- bation approach was used to obtain explicit expressions for these exponents in the presence of small intensity noises. Khasminskii and Moshchuk (1998) obtained an asymptotic expansion of the moment Lyapunov exponents of a two-dimensional system under white noise parametric excitation in terms of the small fluctuationparameter ε, fromwhich the stability indexwasobtained. SriNamachchivaya et al. (1994) used a perturbation approach to calculate the asymptotic growth rate of a stochastically coupled two-degrees-of-freedom sys- tem. The noise was assumed to bewhite and of small intensity in order to cal- culate the explicit asymptotic formulas for themaximumLyapunov exponent. Sri Namachchivaya and Van Roessel (2004) used a perturbation approach to obtain an approximation for themoment Lyapunov exponents of two coupled oscillators with commensurable frequencies driven by small intensity real noise with dissipation.The generator for the eigenvalue problemassociatedwith the moment Lyapunov exponents was derived without any restriction on the size of pthmoment. Kozić et al. (2009, 2010) investigated the Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to a white noise parametric excitation. In the first, almost-sure and moment stability of the flexural-torsion stability of a thin elastic beam subjec- ted to a stochastically fluctuating follower force were studied. In the second, moment Lyapunov exponents and stability boundary of the double-beam sys- tem under stochastic compressive axial loading were obtained. Pavlović et al. (2007) investigated the dynamic stability of thin-walled beams subjected to Moment Lyapunov exponents and stochastic stability... 63 combined action of axial loads and endmoments. By using the direct Lyapu- nov method, the authors obtained the almost-sure stochastic boundary and uniform stochastic stability boundary as the function of characteristics of the stochastic process and geometric and physical parameters. 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 small intensity such that one can obtain an asymptotic growth rate. We apply the perturbation theoretical approach given in Khasminskii and Moshchuk (1998) to obtain second-order weak no- ise expansions of the moment Lyapunov exponents. The Lyapunov exponent is then obtained using the relationship between the moment Lyapunov expo- nents 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 eccentric stochastic axial loads. Themotion of such an elastic system is gover- ned by the partial differential equations in the paper byPavlović et al. (2007). The approximate analytical results of the moment Lyapunov exponents are compared with the numerical values obtained by theMonte 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 q̈1+ω 2 1q1+2εβ1q̇1− √ εξ(t)(K11q1+K12q2)= 0 q̈2+ω 2 2q2+2εβ2q̇2− √ εξ(t)(K21q1+K22q2)= 0 (2.1) where q1,q2 are generalized coordinates, ω1,ω2 are natural frequencies and 2εβ1,2εβ2 represent small viscous damping coefficients. The stochastic term√ εξ(t) is the white-noise process with small intensity with zero mean and autocorrelation functions Rξξ(t1, t2)=E[ξ(t1)ξ(t2)] = σ 2δ(t2− t1) (2.2) σ is the intensity of the random process ξ(t), and δ(·) is the Dirac delta. Using the transformation q1 = x1 q̇1 = ω1x2 q2 = x3 q̇2 = ω2x4 (2.3) and denoting 64 G. Janevski et al. pij = Kij ωi σ i,j =1,2 (2.4) the above Eqs. (2.1) can be represented in the first-order form by a set of Stratonovich differential equations dX =A0Xdt+εAXdt+ √ εBXdw(t) (2.5) where X = [x1,x2,x3,x4] ⊤ is the state vector of the system, w(t) is the standardWeiner process and A0,A and B are constant 4×4matrices given by A0 =      0 ω1 0 0 −ω1 0 0 0 0 0 0 ω2 0 0 −ω2 0      A=      0 0 0 0 0 −2β1 0 0 0 0 0 0 0 0 0 −2β2      B=      0 0 0 0 p11 0 p12 0 0 0 0 0 p21 0 p22 0      (2.6) Applying the transformation x1 = acosϕcosθ1 x2 =−acosϕsinθ1 x3 = asinϕcosθ2 x4 =−asinϕsinθ2 (2.7) P = ‖a‖p = √ (x21+x 2 2+x 2 3+x 2 4) p 0¬ θ1 ¬ 2π 0¬ θ2 ¬ 2π 0¬ ϕ ¬ π 2 −∞ < p < ∞ yields the following set of Stratonovich equations for the pth power of the norm of the response and phase variables (ϕ,θ1,θ2) d‖a‖p = εα∗1 dt+ √ εγ∗1 dw(t) dϕ = εα ∗ 2 dt+ √ εγ∗2 dw(t) (2.8) dθ1 =(ω1+εα ∗ 3)dt+ √ εγ∗3 dw(t) dθ2 =(ω2+εα ∗ 4)dt+ √ εγ∗4 dw(t) In the above transformations, a represents thenormof the response, θ1 and θ2 are the angles of the first and second oscillators, respectively, and ϕ describes the coupling or exchange of energy between the first and second oscillator. In the previous equation, we introduced the following marking Moment Lyapunov exponents and stochastic stability... 65 α∗1 =−2pP(β1 sin2θ1cos2ϕ+β2 sin2θ2 sin2ϕ) α∗2 = β1 sin 2θ1 sin2ϕ−β2 sin2θ2 sin2ϕ α∗3 =−β1 sin2θ1 α∗4 =−β2 sin2θ2 γ∗1 =− pP 2 (p11 sin2θ1cos 2ϕ+p22 sin2θ2 sin 2ϕ +p12 sinθ1cosθ2 sin2ϕ+p21cosθ1 sinθ2 sin2ϕ) γ∗2 = p11 4 sin2θ1 sin2ϕ− p22 4 sin2θ2 sin2ϕ+ p12 2 sinθ1cosθ2 sin 2ϕ − p21 2 cosθ1 sinθ2cos 2ϕ γ∗3 =−p11cos2θ1−p12cosθ1cosθ2 tanϕ γ∗4 =−p22cos2θ2−p21cosθ1cosθ2cotϕ (2.9) The Itô versions of Eqs. (2.8) have the following form d‖a‖p = εα1dt+ √ εγ1dw1(t) dϕ = εα2dt+ √ εγ2dw(t) (2.10) dθ1 =(ω1+εα3)dt+ √ εγ3dw(t) dθ2 =(ω2+εα4)dt+ √ εγ4dw(t) where αi are given in Appendix 1 and γi = γ ∗ i , (i =1,2,3,4). Following Wedig (1998), we perform the linear stochastic transformation S = T(ϕ,θ1,θ2)P P = T −1(ϕ,θ1,θ2)S (2.11) introducing the new norm process S by means of the scalar function T(ϕ,θ1,θ2) which is defined in the stationary phase processes θ1, θ2 and ϕ dS = P(ω1T ′ θ1 +ω2T ′ θ2 )dt+εP ( α1T +m0T ′ ϕ+m1T ′ θ1 +m2T ′ θ2 + 1 2 γ22T ′′ ϕϕ +γ2γ3T ′′ ϕθ1 +γ2γ4T ′′ ϕθ2 + 1 2 γ23T ′′ θ1θ1 +γ3γ4T ′′ θ1θ2 + 1 2 γ24T ′′ θ2θ2 ) dt (2.12) + √ εP(Tγ1+T ′ ϕγ2+T ′ θ1 γ3+T ′ θ2 γ4)dw(t) where m0 = α2+γ1γ2 m1 = α3+γ1γ3 m2 = α4+γ1γ4 (2.13) If the transformation function T(θ1,θ2,ϕ) is bounded and non-singular, both processes P and S possess the same stabilitybehavior.Therefore, the transfor- mation function T(θ1,θ2,ϕ) is chosen so that the drift term, of Itô differential Eq. (2.13), does not depend on the phase processes θ1, θ2 and ϕ, so that dS = Λ(p)S dt+ST−1(Tγ1+T ′ ϕγ2+T ′ θ1 γ3+T ′ θ2 γ4)dw(t) (2.14) 66 G. Janevski et al. BycomparingEqs. (2.12) and (2.14), it canbe seen that such a transformation function T(ϕ,θ1,θ2) is given by the following equation [L0+εL1]T(ϕ,θ1,θ2)= Λ(p)T(ϕ,θ1,θ2) (2.15) Here L0 and L1 are the following first and second-order differential operators L0 = ω1 ∂ ∂θ1 +ω2 ∂ ∂θ2 L1 = a1 ∂2 ∂ϕ2 +a2 ∂2 ∂θ21 +a3 ∂2 ∂θ22 +a4 ∂2 ∂ϕ∂θ1 +a5 ∂2 ∂ϕ∂θ2 +a6 ∂2 ∂θ1∂θ2 + b1 ∂ ∂ϕ + b2 ∂ ∂θ1 + b3 ∂ ∂θ2 + c (2.16) where ai = ai(ϕ,θ1,θ2), (i = 1,2, . . . ,6), bj = bj(ϕ,θ1,θ2), (j = 1,2,3), and c = c(ϕ,θ1,θ2) are given in Appendix 2. Equation (2.15) defines an eigenvalue problem for a second-order differen- tial operator of three independent variables, in which Λ(p) is the eigenvalue and T(ϕ,θ1,θ2) the associated eigenfunction. From Eq. (2.14), the eigenva- lue Λ(p) is seen to be the Lyapunov exponent of the pth moment of system (2.5), i.e., Λ(p) = Λx(t)(p). This approach was first applied by Wedig (1998) to derive the eigenvalue problem for themoment Lyapunov exponent of a two- dimensional linear Itô stochastic system. In the following section, themethod of regular perturbation is applied to eigenvalue problem (2.15) to obtain awe- ak 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 Λ(p) and the eigenfunction T(ϕ,θ1,θ2) are expanded in the power series of ε as Λ(p)= Λ0(p)+εΛ1(p)+ε 2Λ2(p)+ . . .+ε nΛn(p)+ . . . (ϕ,θ1,θ2)= T0(ϕ,θ1,θ2)+εT1(ϕ,θ1,θ2)+ε 2T2(ϕ,θ1,θ2) + . . .+εnTn(ϕ,θ1,θ2)+ . . . (3.1) Substituting perturbation series (3.1) into eigenvalue problem (2.15) and equ- ating the terms of equal powers of ε leads to the following equations Moment Lyapunov exponents and stochastic stability... 67 ε0 → L0T0 = Λ0(p)T0 ε1 → L0T1+L1T0 = Λ0(p)T1+Λ1(p)T0 ε2 → L0T2+L1T1 = Λ0(p)T2+Λ1(p)T1+Λ2(p)T0 ε3 → L0T3+L1T2 = Λ0(p)T3+Λ1(p)T2+Λ2(p)T1+Λ3(p)T0 ... εn → L0Tn+L1Tn−1 =Λ0(p)Tn+Λ1(p)Tn−1+Λ2(p)Tn−2 + . . .+Λn−1(p)T1+Λn(p)T0 (3.2) where each function Ti = Ti(ϕ,θ1,θ2), (i = 0,1,2, . . .) must be positive and periodic in the range 0¬ ϕ ¬π/2, 0¬ θ1 ¬ 2π and 0¬ θ2 ¬ 2π. 3.1. Zeroth order perturbation The zeroth order perturbation equation is L0T0 = Λ0(p)T0 or ω1 ∂T0 ∂θ1 +ω2 ∂T0 ∂θ2 = Λ0(p)T0 (3.3) From the property of the moment Lyapunov exponent, it is known that Λ(0)= Λ0(0)+εΛ1(0)+ε 2Λ2(0)+ . . .+ε nΛn(0)= 0 (3.4) which results in Λn(0) = 0 for n =0,1,2, . . .. Since eigenvalue problem (3.3) doesnot contain p, the eigenvalue Λ0(p) is independentof p.Hence, Λ0(0)= 0 leads to Λ0(p)= 0 (3.5) Now, partial differential Eqs. (3.3) have the form ω1 ∂T0 ∂θ1 +ω2 ∂T0 ∂θ2 =0 (3.6) The solution to Eq. (3.6) may be taken as T0(ϕ,θ1,θ2)= ψ0(ϕ) (3.7) where ψ0(ϕ) is an unknown function of ϕ which has yet to be determined. 3.2. First order perturbation The first order perturbation equation is L0T1 = Λ1(p)T0−L1T0 (3.8) 68 G. Janevski et al. SincehomogeneousEq. (3.6)hasanon-trivial solutionasgivenbyEq. (3.7), for Eq. (3.8) to have a solution, it is required that, from the Fredholm alternative (L0T1,T ∗ 0 )= (Λ1(p)T0−L1T0,T∗0 )= 0 (3.9) In thepreviousequation, T∗0 = ψ0(ϕ) is theunknownsolution to theassociated adjoint differential equation of (3.6), and (f,g) denotes the inner product of functions f(ϕ,θ1,θ2) and g(ϕ,θ1,θ2) defined by (f,g)= π/2 ∫ 0 2π ∫ 0 2π ∫ 0 f(ϕ,θ1,θ2)g(ϕ,θ1,θ2)dθ1dθ2dϕ (3.10) Considering (3.7), (3.8) and (3.10), expression (3.9) now has the form π/2 ∫ 0 2π ∫ 0 2π ∫ 0 (Λ1(p)ψ0−L1ψ0)ψψ0(ϕ)dθ1dθ2dϕ =0 (3.11) and will be satisfied if and only if 2π ∫ 0 2π ∫ 0 (Λ1(p)ψ0−L1ψ0)dθ1dθ2 =0 (3.12) After the integration of the previous expression, we have that L(ψ0)= A1(ϕ) d2ψ0 dϕ2 +B1(ϕ) dψ0 dϕ +C1(ϕ)ψ0−Λ1(p)ψ0 =0 (3.13) where A1(ϕ)= 2π ∫ 0 2π ∫ 0 a1(ϕ,θ1,θ2)dθ1dθ2 B1(ϕ) = 2π ∫ 0 2π ∫ 0 b1(ϕ,θ1,θ2)dθ1dθ2 (3.14) C1(ϕ)= 2π ∫ 0 2π ∫ 0 c(ϕ,θ1,θ2)dθ1dθ2 Finally, there A1(ϕ) =− 1 128 [p211+p 2 22−2(p212+p221)]cos4ϕ− p212−p221 16 (ω21 −ω22)cos2ϕ + 1 128 [p211+p 2 22+6(p 2 12+p 2 21)] Moment Lyapunov exponents and stochastic stability... 69 B1(ϕ) =− 1 64 (p−1)[p211+p222−2(p212+p221)]sin4ϕ − 1 8 (p212 sin 2ϕtanϕ−p221cos2ϕcotϕ) (3.15) + 1 32 {16β1 −16β2− [(p+2)(p211−p222)+2(p−1)(p212−p221)]}sin2ϕ C1(ϕ) = 1 128 p(p−2)[p211+p222−2(p212+p221)]cos4ϕ − 1 32 {16β1 −16β2− [(p+2)(p211−p222)−4(p212−p221)]}cos2ϕ + 1 128 p{−64β1−64β2+[(10+3p)(p211+p222)+2(6+p)(p212+p221)]} Since coefficients (3.15) ofEq.(3.13) areperiodic functionsof ϕ, a series expan- sion of the function ψ0(ϕ) may be taken in the form ψ0(ϕ)= N ∑ k=0 Kk cos2kϕ (3.16) Substituting (3.16) in (3.13), multiplying the resulting equation by cos2kϕ (k = 0,1,2, . . .) and integrating with respect to ϕ from 0 to π/2 leads to a set of 2N +1 homogeneous linear equations for the unknown coefficients K0,K1,K2, . . . N ∑ j=0 AjkKj = Λ1(p)Kk (3.17) where Ajk = π/2 ∫ 0 L(cos2jϕ)cos2kϕdϕ k =0,1,2, . . . ,N (3.18) When N tends to infinity, solution (3.16) to equations tends to the exact solution.The condition for systemhomogeneous linear equation (3.17) to have nontrivial solutions is that the determinant of system homogeneous linear equations (3.17) is equal to zero. The coefficients Ajk to order N = 3 are presented in Appendix 3. In the case when N = 0, we assume solution (3.16) in the form ψ0(ϕ) = K0, from conditions that A00 = 0, the moment Lyapunov exponent in the first perturbation is defined as Λ1(p)=− p 2 (β1+β2)+ p(10+3p) 128 (p211+p 2 22)+ p(6+p) 64 (p212+p 2 21) (3.19) 70 G. Janevski et al. In the casewhen N =1, solution (3.16) has the form ψ0(ϕ)= K0+K1cos2ϕ, themoment Lyapunov exponent in the first perturbation is the solution to the equation Λ21 +d (1) 1 Λ1 +d (1) 0 = 0 where coefficients d (1) 0 and d (1) 1 are presen- ted in Appendix 4. In the case when N = 2, solution (3.16) has the form ψ0(ϕ) = K0+K1cos2ϕ+K2cos4ϕ, the moment Lyapunov exponent in the first perturbation is the solution to the equation Λ31+d (2) 2 Λ 2 1+d (2) 1 Λ1+d (2) 0 =0 where coefficients d (2) 0 ,d (2) 1 and d (2) 2 are presented inAppendix5.However, for N =3, it is impossible to obtain explicit expressions of Λ1(p) and numerical results must be given. 4. Application to a thin-walled beam subjected to an eccentric stochastic axial load 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-torsinal vibration stability of a homogeneous, isotropic, thin-walled beam with two planes of summetry which is subjected to an eccentric axial load (Fig.1a), where R is the eccentricity. By transferring the eccentric load to the plane of symmetry of the cross-section of the beam, an axial load and a couple are obtained, which are shown in Fig.1b. The governing differential equations for the coupled flexural and torsional motion of the beam can be written as (Pavlović et al., 2007) ρA ∂2U ∂T2 +αu ∂U ∂t +EIy ∂4U ∂Z4 +M(T) ∂2φ ∂Z2 +F(T) ∂2U ∂Z2 =0 ρIp ∂2φ ∂T2 +αφ ∂φ ∂T − ( GJ −F(t)Ip A )∂2φ ∂Z2 +M(T) ∂2U ∂Z2 +EIs ∂4φ ∂Z4 =0 (4.1) where U is the flexural displacement in the x-direction, φ is the torsional displacement, ρ is the mass density, A is the area of the cross-section of beam, Iy, Ip, IS are the axial, polar and sectorial moment of inertia, J is Saint-Venant’s torsional constant, E is Young’s modulus of elasticity, G is the shear modulus, αU, αφ are viscous damping coefficients, T is time and Z is the axial coordinate. Using the following transformations U = u √ Ip A Z = zl R = rl Moment Lyapunov exponents and stochastic stability... 71 Fig. 1. Geometry of the thin-walled beam system T = ktt M(T)=RF(T) F(T)= FcrF(t) Fcr = π2EIy l2 k2t = ρAl4 EIy e = AIs IyIp εβ1 = 1 2 αU l2 √ ρAEIy εβ2 = 1 2 αφl 2 √ A ρEIyI2p S1 = l2r2A Ip S2 = GJAl2 π2EIyIp (4.2) where l is the length of the beam, Fcr is theEuler critical force for the simply supported narrow rectangular beam, S1 and S2 are slenderness parameters, β1 and β2 are reducedviscousdampingcoefficients,weget governing equations as ∂2u ∂t2 +2εβ1 ∂u ∂t + ∂4u ∂z4 +π2 √ S1F(t) ∂2φ ∂z2 +π2F(T) ∂2u ∂z2 =0 ∂2φ ∂t2 +2εβ2 ∂φ ∂T −π2[S2−F(t)] ∂2φ ∂z2 +π2 √ S1F(t) ∂2u ∂z2 +e ∂4φ ∂z4 =0 (4.3) 72 G. Janevski et al. Taking free warping displacement and zero angular displacements into ac- count, the boundary conditions for the simply supported beam are u(t,0)= u(t,1)= ∂2u ∂z2 ∣ ∣ ∣ ∣ (t,0) = ∂2u ∂z2 ∣ ∣ ∣ ∣ (t,1) =0 φ(t,0)= φ(t,1)= ∂2φ ∂z2 ∣ ∣ ∣ ∣ (t,0) = ∂2φ ∂z2 ∣ ∣ ∣ ∣ (t,1) =0 (4.4) Consider the shape function sinπz which satisfies the boundary conditions for the first mode vibration, the displacement u(t,z) and twist angle φ(t,z) can be described by u(t,z) = q1(t)sinπz φ(t,z) = ψ1(t)sinπz (4.5) Substituting u(t,z) and φ(t,z) from (4.5) into equations of motion (4.3) and employing Galerkin’s method, the unknown time functions can be expressed as ü1+ω 2 1u1+2β1εu̇1− √ ε(K11u1+K12ψ1)F(t) = 0 ψ̈1+ω 2 2ψ1+2β2εψ̇1− √ ε(K21u1+K22ψ1)F(t)= 0 (4.6) If we define the expressions ω21 = π 4 ω22 = π 4(S2+e) K11 = K22 = π 4 K12 = K21 = π 4 √ S1 (4.7) and assume that the compressive axial force is stochastic white-noise process (2.2) with small intensity F(t)= √ εξ(t) (4.8) then Eq. (4.4) is reduced to Eq. (2.1). Using the above result for the moment Lyapunov exponent Λ(p)= εΛ1(p)+O(ε 2) (4.9) with thedefinitionof themoment stability Λ(p) < 0,wedetermineanalytically (the case where N = 0, Λ1(p) is shown with Eq. (3.19)) the pth moment stability boundary of the oscillatory system in the first-order perturbation β1+β2 > σ 2π4 ( 1+ 1 S2+e )(10+3p 64 + 6+p 32 S1 ) (4.10) Moment Lyapunov exponents and stochastic stability... 73 It is known that oscillatory system (4.3) is asymptotically stable only if the Lyapunov exponent λ < 0. Then expression λ = ελ1+O(ε 2) (4.11) is employed to determine the almost-sure stability boundary of the oscillatory system in the first-order perturbation β1+β2 > σ 2π4 ( 1+ 1 S2+e )( 5 32 + 3 16 S1 ) (4.12) For the sake of simplicity, we assume, in what follows, that two viscous dam- ping coefficients are equal β1 = β2 = β (4.13) For this case, we determine the almost-sure stability boundary of the oscilla- tory system in the first-order perturbation β > 3π4σ2 32 ( 1+ 1 S2+e )(5 6 +S1 ) (4.14) and the pth moment stability boundary is β > σ2π4 64 ( 1+ 1 S2+e )[ 5+ 3 2 p+(6+p)S1 ] (4.15) With respect to standard I-section,we can approximatiely take h/ ≈ 2, b/δ1 ≈ 11, δ/δ1 ≈ 1.5, where h is depth, b is width, δ is thickness of the flanges and δ1 is thickness of the rib of I-section. These ratios yield S1 ≈ 6(R/h)2, S2 ≈ 0.01928(l/h)2 and e ≡ 1.276. Figure 2 shows the almost-sure stability boundaries with respect to the damping coefficient β and intensity of randomprocess σ. The stability regions are given in space for a constant geometrical ratio (l/h =10) of length of the beam and depth of the standard I-profil. They are enlarged when the axial force is closer to the axis of symmetry, with the greatest enlargement in the casewhen the force acts towards themain axis of symmetry.With the increase of ratio R/h, the stability regions are reduced. Figure 3 shows the almost-sure and pthmoment stability boundarieswith respect to the damping coefficient β and intensity of the random process σ. Note that the moment stability boundaries are more conservative than the almost-sure boundary. These boundaries become increasingly more conserva- tive as p increases, as shown in Fig.3. 74 G. Janevski et al. Fig. 2. Stability regions for almost-sure (a-s) stability for ε =0.1 and l/h =10 Fig. 3. Stability regions for almost-sure (a-s) and pth moment stability for ε =0.1, l/h =10 and R/h =0.2 5. Numerical determination of the pth moment Lyapunov exponent Numerical determination of the pth moment Lyapunov exponents is impor- tant in assessing the validity and ranges of applicability of the approximate analytical results. Inmany engineering applications, amplitudes of noise exci- tations are not small, and the approximate analytical methods, such as the method of perturbation of the method of stochastic averaging, cannot be ap- plied. Therefore, numerical approaches have to be employed to evaluate the moment Lyapunov exponents. The numerical approach is based on expan- ding the exact solution to the system of Ito stochastic differential equations in powers of the time increment h and the small parameter ε as proposed in Milstein and Tret’Yakov (1997). The state vector of system (2.5) is to be rewritten as a system of Ito stochastic differential equations with small noise in the form Moment Lyapunov exponents and stochastic stability... 75 dx1 = ω1x2dt dx2 =(−ω1x1−ε2β1x2)dt+ √ ε(p11x1+p12x3)dw(t) (5.1) dx3 = ω2x4dt dx4 =(−ω2x3−ε2β2x4)dt+ √ ε(p21x1+p22x3)dw(t) For the numerical solutions of the stochastic differential equations, theRunge- Kutta approximation may be applied, both with the error R = O(h4+ ε4h). The interval discretization is [t0,T ] : {tk : k = 0,1,2, . . . ,M; t0 < t1 < t2 < ... < tM = T} and the time increment is h = tj+1− tj. The following Runge-Kuttamethod is obtained for the (k+1)th iteration of the state vector X = [x1,x2,x3,x4] X (k+1) = {[ N1 0 0 N2 ] +ε [ β1M1 0 0 β2M2 ] + √ ε [ p11P11 p12P1 p21P2 p22P22 ]} X (k) (5.2) where Nk,Mk,Pk and Pkk (k =1,2) are 2×2matrices Nk = [ N1k N2k −N2k N1k ] Mk = [ M1k M3k M4k M2k ] Pkk = [ W1k W3k W4k W2k ] Pk = [ W1k W3k W5 Wk+5 ] (5.3) and themembers of previous matrices can be evaluated as follows N1k =1− h2ω2k 2 + h4ω4k 24 N2k = hωk ( 1− h2ω2k 6 ) M1k = h3 3 ω2k M2k =−2h ( 1− h2ω2k 3 + h4ω4k 36 ) M3k =−h2ωk ( 1− h2ω2k 9 ) M4k = h 2ωk ( 1− h2ω2k 6 ) W1k = h3/2 2 (ξ +2η)ωk W2k = h3/2 2 (ξ −2η)ωk W3k = h5/2 6 ξω2k W4k = √ h ( 1− h2ω2k 3 ) ξ W5 = √ h [ 1− h2 6 (ω21 +ω 2 2) ] ξ W6 = h3/2 2 (ξ−2η)ω2 W7 = h3/2 2 (ξ−2η)ω1 (5.4) Random variables ξi and ηi (i =1,2) are simulated as P(ξi =−1)= P(ξi =1)= 1 2 P ( ηi = −1√ 12 ) = P ( ηi = 1√ 12 ) = 1 2 (5.5) 76 G. Janevski et al. Having obtained L samples of the solutions to stochastic differential equations (5.1), the pth moment can be determined as follows E[‖X(tk+1)‖p] = 1 L L ∑ j=1 ‖Xj(tk+1)‖p ‖Xj(tk+1)‖= √ [X⊤j (tk+1)Xj(tk+1 (5.6) By the Monte-Carlo technique (Xie, 2005), we numerically calculate the pth moment Lyapunov exponent for all values of p of interest defined as Λ(p)= 1 T logE[‖X(T)‖p] (5.7) 6. Numerical results and conclusions In this paper, the moment Lyapunov exponents of a thin-walled beam sub- jected to eccentric stochastic axial loads are studied. The method of regular perturbation is applied to obtain a weak noise expansion of the moment Ly- apunov exponent in terms of the small fluctuation parameter. Theweak noise expansion of the Lyapunov exponent is also obtained. The slope of the mo- ment Lyapunov exponent curve at p = 0 is the Lyapunov exponent. When the Lyapunov exponent is negative, system (4.6) is stable with probability 1, otherwise it is unstable. For the purpose of illustration, in the numerical stu- dy we considere the set of system parameters β1 = β2 = β = 1, ε = 0.1, L =4000, h =0.0005, M =10000 and x1(0)= x2(0)= x3(0)= x4(0)= 1/2. Typical results of the moment Lyapunov exponents Λ(p) for system (4.6) given by Eq. (4.9) in the first perturbation are shown in Fig.4 for I-section, ε = 0.1, l/h = 10 and R/h = 0.2, the noise intensity σ = 0.2 and damping coefficient β = 1. The accuracy of the approximate analytical results is vali- dated and assessed by comparing them to the numerical results. The Monte Carlo simulation approach is usually more versatile, especially when the no- ise 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 pthmoment Lyapunov exponent is a randomnumber, with the mean being the true value of the pth moment Lyapunov exponent and stan- dard deviation equal to np/ √ L, where np is the sample standard deviation determined from L samples. It is evident that analytical results agree very well with the numerical results, except for N =0when the function ψ0(ϕ) does not depend on ϕ and Moment Lyapunov exponents and stochastic stability... 77 Fig. 4. Moment Lyapunov exponent Λ(p) for σ =0.2 and β =1 assumes the form ψ0(ϕ) = K0 = const. It is observed that the discrepancies between the approximate analytical andnumerical results decrease for a larger number N of series (3.16). Further increase of the N number of members does not make sense, because the curves merge into one. Further increase in the number of members in the supposed solution does not make sense also because the approximation of the exact solutions is worse. On the other side, the equation from which we can determine the value of the exponent of the moment Lyapunov exponent is of a higher order and the coefficients in them are of a more complex form. Acknowledgments Research supported by Ministry of Science and Environment Protection of Re- public Serbia, grant No. ON174011. The authors are grateful to the referees for the useful remarks which helped to improve this paper. Appendix 1 α1 =−2pP(β1 sin2θ1cos2ϕ+β2 sin2θ2 sin2ϕ) + p(p−2)P 16 (p11p22+p12p21)sin2θ1 sin2θ2 sin 22ϕ + pP 2 (p211 cos 2θ1cos 2ϕ+p212cos 2θ2 sin 2ϕ)[cos2θ1+(pcos 2ϕ−cos2ϕ)] + pP 2 (p222 cos 2θ2 sin 2ϕ+p221cos 2θ1cos 2ϕ)[cos2θ2+(psin 2ϕ+cos2ϕ)] + pP 8 cosθ1cosθ2 sin2ϕ{p22p21[p+2− (p−2)(cos2θ2+2sin2θ2cos2ϕ)] +p11p12[p+2− (p−2)(cos2θ1−2sin2 θ1cos2ϕ)]} + p(p−2)P 2 sin2ϕsinθ1 sinθ2(p11p21cos 2θ1cos 2ϕ+p22p12cos 2θ2 sin 2ϕ) 78 G. Janevski et al. α2 =(β1 sin 2θ1−β2 sin2θ2)sin2ϕ− 1 16 (p11p22+p12p21)sin2θ1 sin2θ2 sin4ϕ −(p11p21cos2θ1cos2ϕ+p22p12cos2θ2 sin2ϕ)sinθ1 sinθ2cos2ϕ −1 4 p211cos 2θ1 sin2ϕ(cos2θ1−cos2ϕsin2 θ1) + 1 4 p222cos 2θ2 sin2ϕ(cos2θ2+cos2ϕsin 2 θ2) + 1 2 p212cos 2θ2 sin 2ϕ(sin2θ1 sin2ϕ−cos2θ1 tanϕ) −1 2 p221cos 2θ1cos 2ϕ(sin2θ2 sin2ϕ−cos2θ2cotϕ) −p11p12cosθ1cosθ2 sin2ϕ(cos2θ1−cos2ϕsin2θ1) +p22p21cosθ1cosθ2cos 2 ϕ(cos2θ2+cos2ϕsin 2θ2) α3 =−β1 sin2θ1− 1 2 (p11cosθ1+p12cosθ2 tanϕ) 2 sin2θ1 α4 =−β2 sin2θ2− 1 2 (p22cosθ2+p21cosθ1cotϕ) 2 sin2θ2 Appendix 2 a1 = 1 32 (p11 sin2θ1−p22 sin2θ2)2 sin22ϕ + 1 2 (p12cosθ1 sinθ2cos 2ϕ−p21 sinθ1cosθ2 sin2ϕ)2 −1 4 (p11 sin2θ1−p22 sin2θ2)(p12 sinθ1cosθ2 sin2ϕ−p21cosθ1 sinθ2cos2ϕ)sin2ϕ a2 = 1 2 cos2θ1(p11cosθ1+p12cosθ2 tanϕ) 2 a3 = 1 2 cos2θ2(p22cosθ2+p21cosθ1cotϕ) 2 a4 =− 1 4 cos2θ1 sin2ϕ[p 2 11 sin2θ1− (p11p22−p12p21)sin2θ2] +p11p21cos 3θ1 sinθ2cos 2ϕ −p12cosθ1cosθ2 sin2 ϕ ( p11 sin2θ1− p22 2 sin2θ2+p12 sinθ1cosθ2 tanϕ ) a5 = 1 4 cos2θ2 sin2ϕ[p 2 22 sin2θ2− (p11p22+p12p21)sin2θ1] −p22p12 sinθ1cos3θ2 sin2ϕ −p21cosθ1cosθ2 cos2ϕ (p11 2 sin2θ1+p22 sin2θ2+p21cosθ1 sinθ2cotϕ ) a6 =(p11p22+p12p21)cos 2θ1cos 2θ2+cosθ1cosθ2(p11p21cos 2θ1cotϕ +p22p12cos 2θ2 tanϕ) b1 =(β1 sin 2θ1−β2 sin2θ2)sin2ϕ+ p−1 16 (p11p22+p12p21)sin2θ1 sin2θ2 sin4ϕ − (1 4 p211cosθ1 sin2ϕ+p11p12cosθ2 sin 2ϕ ) [cos2θ1+2(p−1)cos2ϕsin2θ1]cosθ1 Moment Lyapunov exponents and stochastic stability... 79 + (1 4 p222cosθ2 sin2ϕ+p22p21cosθ1cos 2ϕ ) [cos2θ2+2(p−1)sin2ϕsin2θ2]cosθ2 − 1 2 p212cos 2θ2 sin 2ϕ[(p−1)sin2θ1 sin2ϕ+cos2θ1 tanϕ] + 1 2 p221cos 2θ1cos 2ϕ[(p−1)sin2θ2 sin2ϕ+cos2θ2cotϕ] −(p−1)sinθ1 sinθ2cos2ϕ(p11p21cos2θ1cos2ϕ+p22p12cos2θ2 sin2ϕ) b2 =−β1 sin2θ1+ 1 2 (p211cosθ1+p11p12cosθ2 tanϕ)[(p−1)cos2ϕ −sin2ϕ]sin2θ1cosθ1+ p 2 (p11p22+p12p21)cos 2θ1 sin2θ2 sin 2ϕ + 1 2 p212 sin2θ1cos 2θ2[(p−1)cos2ϕ− sin2ϕ] tan2ϕ + p 2 p11p21cos 3θ1 sinθ2 sin2ϕ+ p 2 p22p12cosθ1cosθ2 sin2θ2 sin 2ϕtanϕ b3 =−β2 sin2θ2+ 1 2 (p222cosθ2+p22p21cosθ1cotϕ)[(p−1)sin2ϕ −cos2 ϕ]sin2θ2cosθ2+ p 2 (p11p22+p12p21)sin2θ1cos 2θ2cos 2ϕ + 1 2 p221 sin2θ2cos 2θ1[(p−1)sin2ϕ−cos2ϕ]cot2ϕ + p 2 p11p21 sin2θ1cosθ1cosθ2cos 2ϕcotϕ+ p 2 p22p12 sinθ1cos 3θ2 sin2ϕ c =−2p(β1 sin2θ1cos2 ϕ+β2 sin2θ2 sin2ϕ) + p(p−2) 16 (p11p22+p12p21)sin2θ1 sin2θ2 sin 22ϕ + p(p−2) 2 (p11p21cos 2θ1cos 2ϕ+p22p12cos 2θ2 sin 2 ϕ)sinθ1 sinθ2 sin2ϕ + p 2 (p11cosθ1cosϕ+p12cosθ2 sinϕ) 2{cos2θ1+[(p−1)cos2ϕ+sin2 ϕ]sin2θ1} + p 2 (p22cosθ2 sinϕ+p21cosθ1cosϕ) 2{cos2θ2+[(p−1)sin2 ϕ+cos2 ϕ]sin2θ2} Appendix 3 A00 =−Λ1(p)− p 2 (β1+β2)+ p(10+3p) 128 (p211+p 2 22)+ p(6+p) 64 (p212+p 2 21) A10 =− p+2 4 (β1−β2)+ 1 64 (p+2)2(p211−p222)+ 1 4 (p212−p221) A20 = (p+2)(p+4) 256 [p211+p 2 22−2(p212+p221)]− 17 32 (p212+p 2 21) A30 = 3 4 (p212−p221) A01 =− p 4 (β1−β2)+ p(p+2) 64 (p211−p222)− p 16 (p212−p221) A11=− 1 2 Λ1(p)− p 4 (β1+β2)+ 7p2+22p−8 512 (p211+p 2 22)+ p2+10p−56 256 (p212+p 2 21) 80 G. Janevski et al. A21 =− p+4 8 (β1−β2)+ p2+6p+8 128 (p211−p222)+ p+20 32 (p212−p221) A31 = p2+10p+24 512 (p211+p 2 22)− p2+10p+216 256 (p212+p 2 21) A02 = p(p−2) 256 [(p211+p 2 22)−2(p212+p221)] A12 =− p−2 8 (β1−β2)+ (p+2)(p−2) 128 (p211−p222)+ p−2 16 (p212−p221) A22=− 1 2 Λ1(p)− p 4 (β1+β2)+ 3p2+10p−16 256 (p211+p 2 22)+ p2+6p−80 128 (p212+p 2 21) A32 =− p+6 8 (β1−β2)+ 7p2+8p+12 512 (p211−p222)+ p+18 16 (p212−p221) A03 =0 A13 = p2−6p+8 512 [(p211+p 2 22)−2(p212+p221)] A23 =− p−4 8 [ (β1−β2)− 1 16 (p+2)(p211−p222)+ 3 4 (p212−p221) ] A33 =− 1 2 Λ1(p)− p 4 (β1+β2)+ 3p2+10p−36 256 (p211+p 2 22) + 2p2+12p−312 256 (p212+p 2 21) Appendix 4 d (1) 1 = p(β1+β2)+ ( 1 32 − 21p 28 − 13p 2 256 ) (p211+p 2 22)+ ( 7 16 − 11p 64 − 3p 2 128 ) (p212+p 2 21) d (1) 0 = 1 8 p(p−2)(β21 +β22)+ 1 4 p(2+3p)β1β2+ ( − 13p 2048 + p2 8192 + 5p3 4096 + 5p4 32768 ) (p411+p 4 22)+ ( − 5p 512 + p2 2048 + p3 512 + p4 8192 ) (p412+p 4 21)+ ( 3p 1024 + 97p2 4096 + 29p3 2048 + 37p4 16384 ) p211p 2 22+ ( −37p 256 + p2 1024 + p3 256 + p4 4096 ) p212p 2 21 + ( −23p 512 + 7p2 2048 + 17p3 2048 + 5p4 8192 ) (p211p 2 12+p 2 22p 2 21)+ ( −15p 512 + 7p2 2048 + 9p3 2048 + 5p4 8192 ) (p211p 2 21+p 2 22p 2 12)+ ( −3p 64 − 37p 2 256 − 21p 3 512 ) (β1p 2 22+β2p 2 11) + (5p 64 − 5p2 256 − 5p3 512 ) (β1p 2 11+β2p 2 22)+ (5p 32 − 7p2 128 − 3p3 256 ) (β1p 2 21+β2p 2 12) + (9p 32 − 15p2 128 − 3p3 256 ) (β1p 2 12+β2p 2 21) Appendix 5 d (2) 2 = 3p 2 (β1+β2)+ ( 5 32 − 31p 128 − 19p 2 256 ) (p211+p 2 22)+ (27 16 − 17p 64 − 5p 2 128 ) (p212+p 2 21) Moment Lyapunov exponents and stochastic stability... 81 d (2) 1 = (1 2 − 3p 8 − 9p 2 16 ) (β21 +β 2 2)− ( 1− 3p 4 − 15p 2 8 ) β1β2+ [( 3 256 − 47p 2048 + 41p2 32768 + 61p3 8192 + 35p4 32768 ) (p411+p 4 22)+ ( − 1 128 − 55p 1024 + 153p2 4096 + 133p3 4096 + 83p4 16384 ) p211p 2 22 ] + [(15 64 − 55p 512 − 39p 2 2048 + 13p3 2048 + 3p4 8192 ) (p412+p 4 21)+ (55 32 − 231p 256 − 71p 2 1024 + 13p3 1024 + 3p4 4096 ) p212p 2 21]+ [(1 8 + p 8 − 43p2 128 − 25p3 256 ) (β1p 2 22+β2p 2 11) + ( − 1 8 + 3p 16 − 19p2 128 − 13p3 256 ) (β1p 2 11+β2p 2 22) ] + [(3 8 + 45p 32 − 7p2 32 − 5p3 128 ) ·(β1p221+β2p212)+ ( −3 8 + 63p 32 − 5p 2 16 − 5p 3 128 ) (β1p 2 12+β2p 2 21) ] d (2) 0 = (p 4 − 3p 2 16 + p3 32 ) (β31 +β 3 2)+ ( −p 4 + 3p2 16 + 15p3 32 ) (β21β2+β1β 2 2) + ( −75p 216 + 9p2 216 + 277p3 220 + 7p4 221 − 49p 5 222 − 7p 6 223 ) (p611+p 6 22) + ( − p 216 + 227p2 216 + 63p3 220 − 3179p 4 221 − 2323p 5 222 − 469p 6 223 ) (p411p 2 22+p 2 11p 4 22) + ( −639p 215 + 93p2 215 + 2321p3 219 − 61p 4 220 − 413p 5 221 − 35p 6 222 ) (p411p 2 12+p 4 22p 2 21) + ( −339p 215 + 89p2 215 + 1137p3 219 − 29p 4 220 − 189p 5 221 − 35p 6 222 ) (p411p 2 21+p 4 22p 2 12) + ( −1407p 213 − 7p 2 213 + 1633p3 217 + 3p4 218 − 45p 5 219 − 3p 6 220 ) (p412p 2 21+p 4 21p 2 12) + ( −417p 214 + 55p2 214 + 1551p3 218 − 11p 4 219 − 291p 5 220 − 21p 6 221 ) (p211p 4 12+p 2 22p 4 21) + ( − 273p 214 + 59p2 214 + 815p3 218 − 43p4 219 − 131p5 220 − 21p6 221 ) (p211p 4 21+p 2 22p 4 12) + ( − 1465p 213 + 321p2 213 + 3807p3 217 + 229p4 218 − 211p5 219 − 21p6 220 ) (p211+p 2 22)p 2 12p 2 21 + ( 9p 214 + 6951p2 214 + 4545p3 218 − 621p 4 219 − 1133p 5 220 − 195p 6 221 ) (p212+p 2 21)p 2 11p 2 22 + (5p 28 − 25p 2 212 − 51p 3 214 + 7p4 214 + 7p5 216 ) (β1p 4 11+β2p 4 22) + ( − p 27 − 69p 2 212 + 133p3 214 + 115p4 214 + 63p5 216 ) (β2p 4 11+β1p 4 22) + (7p 26 + 17p2 28 − 41p 3 28 − 45p 4 210 ) β1β2(p 2 11+p 2 22)+ ( −15p 27 + 31p2 29 + 3p3 29 − 7p 4 211 ) ·(p211β21 +p222β22)+ ( p 27 + 15p2 29 − 45p 3 29 − 55p 4 211 ) (p211β 2 2 +p 2 22β 2 1) + ( − p 28 − 55p 2 211 + 153p3 213 + 133p4 213 + 83p5 215 ) p211p 2 22(β1+β2) 82 G. Janevski et al. + (117p 29 − 191p 2 211 − 109p 3 212 + 59p4 213 + 7p5 214 ) (p211p 2 12β1+p 2 22p 2 21β2) + ( −45p 29 − 469p 2 211 − 109p 3 212 + 129p4 213 + 27p5 214 ) (p211p 2 12β2+p 2 22p 2 21β1) + (63p 29 − 105p 2 211 − 49p 3 212 + 21p4 213 + 27p5 214 ) (p211p 2 21β1+p 2 22p 2 12β2) + ( − 39p 29 − 507p2 211 − 73p3 212 + 119p4 213 + 27p5 214 ) (p211p 2 21β2+p 2 22p 2 12β1) + ( − 21p 213 + 3p2 213 + 75p3 217 + p4 218 − 15p5 219 − p6 220 ) (p612+p 6 21) + (39p 28 − 9p 2 27 − 51p 3 212 + 9p4 211 + 3p5 214 ) (p412β1+p 4 21β2) + (21p 27 − 19p 2 29 − 27p 3 212 + p4 29 + 3p5 214 ) (p412β2+p 4 21β1) + ( −45p 26 + 125p2 28 − 15p 3 28 − 5p 4 210 ) (p212β 2 1 +p 2 21β 2 2) + ( −21p 26 + 53p2 28 − 3p 3 28 − 5p 4 210 ) (p212β 2 2 +p 2 21β 2 1) + (33p 25 + 127p2 27 − 25p 3 27 − 15p 4 29 ) (p212+p 2 21)β1β2 + (55p 26 − 231p 2 29 − 71p 3 211 + 13p4 211 + 3p5 213 ) p212p 2 21(β1+β2) References 1. ArnoldL.,DoyleM.M., SriNamachchivayaN., 1997,Small noise expan- sion of moment Lyapunov exponents for two-dimensional systems, Dynamics and Stability of Systems, 12, 3, 187-211 2. Khasminskii R., Moshchuk N., 1998,Moment Lyapunov exponent and sta- bility index for linear conservative system with small random perturbation, SIAM Journal of Applied Mathematics, 58, 1, 245-256 3. 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, 1689-1701 4. 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, 1435-1442 5. 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, 2142-2167 Moment Lyapunov exponents and stochastic stability... 83 6. PavlovićR., KozićP., 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, 690-700 7. Sri Namachchivaya N., Van Roessel H.J., 2004, Stochastic stability of coupled oscillators in resonance: A perturbation approach, ASME Journal of Applied Mechanics, 71, 759-767 8. Sri Namachchivaya N., Van Roessel H.J., Talwar S., 1994, Maximal Lyapunov exponent andalmost-sure stability for coupled two-degreeof freedom stochastic systems,ASME Journal of Applied Mechanics, 61, 446-452 9. Xie W.-C., 2005, Monte Carlo simulation of moment Lyapunov exponents, ASME Journal of Applied Mechanics, 72, 269-275 10. Wedig W., 1988, Lyapunov exponent of stochastic systems and related bi- furcation problems, [In:] Stochastic Structural Dynamics – Progress in Theory and Applications, AriaratnamT.S., SchuëllerG.I., Elishakoff I. (Eds.), Elsevier Applied Science, 315-327 Momentowe wykładniki Lapunowa i stateczność stochastyczna cienkościennej belki poddanej mimośrodowemu obciążeniu w kierunku osiowym Streszczenie W artykule zbadano wykładniki Lapunowa i momentowe wykładniki Lapunowa układów o dwóch stopniach swobody poddanych parametrycznemuwymuszeniu bia- łym szumem. Zastosowano regularnąmetodę perturbacyjną do wyznaczenia jawnych wyrażeń na te wykładniki w obecności szumów o małej intensywności. Wykładniki Lapunowa i momentowe wykładniki Lapunowa są ważnymi wielkościami w określa- niu prawie pewnej i momentowej stateczności stochastycznej układu dynamicznego. Jako przykład rozważono cienkościenną belkę poddaną mimośrodowemu obciążeniu osiowemu o charakterze losowym. Poprawność otrzymanych wyników przybliżenia momentowychwykładnikówLapunowa sprawdzonow drodze symulacji numerycznej przy wykorzystaniumetody Monte Carlo. Manuscript received February 7, 2011; accepted for print April 6, 2011