Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators

Nội dung text: Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators

1. Vietnam Journal of Mechanics, VAST, Vol.41, No. 1 (2019), pp. 1 – 15 DOI: PERFORMANCE ANALYSIS OF GLOBAL-LOCAL MEAN SQUARE ERROR CRITERION OF STOCHASTIC LINEARIZATION FOR NONLINEAR OSCILLATORS Luu Xuan Hung1,2, Nguyen Cao Thang2,3,∗ 1Hanoi Metropolitan Rail Board, Vietnam 2Institute of Mechanics, VAST, Hanoi, Vietnam 3Graduate University of Science and Technology, VAST, Hanoi, Vietnam ∗E-mail: caothang2002us@yahoo.com Received: 22 March 2018 / Published online: 14 February 2019 Abstract. The paper presents a performance analysis of global-local mean square error criterion of stochastic linearization for some nonlinear oscillators. This criterion of sto- chastic linearization for nonlinear oscillators bases on dual conception to the local mean square error criterion (LOMSEC). The algorithm is generally built to multi-degree of free- dom (MDOF) nonlinear oscillators. Then, the performance analysis is carried out for two applications which comprise a rolling ship oscillation and two-degree of freedom one. The improvement on accuracy of the proposed criterion has been shown in comparison with the conventional Gaussian equivalent linearization (GEL). Keywords: probability; random; frequency response function; iteration method; mean square. 1. INTRODUCTION One popular class of methods for approximate solutions of nonlinear systems under random excitations is GEL techniques, which are most used in structural dynamics and in the engineering mechanics applications. This is partially due to its simplicity and appli- cability to systems with MDOF, and ones under various types of random excitations. The key idea of GEL is to replace the nonlinear system by a linear one such that the behav- ior of the equivalent linear system approximates that of the original nonlinear oscillator. The standard way is that the coefficients of linearization are to be found by the classical mean square error criterion [1,2]. Although the method is very efficient, but its accuracy decreases as the nonlinearity increases and in many cases it gives very larger errors due to the non-Gaussian property of the response. That is reason why many researches have been done in recent decades on improving GEL, for example [3–11]. One among them is LOMSEC that was first proposed by N. D. Anh and Di Paola [10], and then further developed by N. D. Anh and L. X. Hung [11]. The basic difference of LOMSEC from the c 2019 Vietnam Academy of Science and Technology
2. 2 Luu Xuan Hung, Nguyen Cao Thang classical GEL is that the integration domain for mean square of response taken over finite one (local one) instead of (−∞, ∞) in the classical GEL. As LOMSEC can give a good im- provement on accuracy, however, the local integration domain in question was unknown and it has resulted in the main disadvantage of LOMSEC. Recently a dual conception was proposed in the study of responses to nonlinear systems [12,13]. One remarkable advan- tage of the dual conception is its consideration of two different aspects of a problem in question allows the investigation to be more appropriate. Applying the dual approach to LOMSEC, a new criterion namely global-local mean square error criterion (GLOM- SEC) has been recently proposed L. X. Hung et al. [14, 15]for nonlinear systems under white noise excitation, in which new values of linearization coefficients are obtained as global averaged values of all local linearization coefficients. This paper is an additional research to aim at evaluating the improved performance of the proposed criterion; herein we analyse two more applications, which are a rolling ship oscillation and two-degree- of-freedom one. The results show a significant improvement on accuracy of solutions by the new criterion compared to the ones by the classical GEL. 2. FORMULATION Consider a MDOF nonlinear stochastic oscillator described by the following equa- tion Mq¨ + Cq˙ + Kq + Φ(q, q˙) = Q(t), (1) =   =   =   × where M mij n×n, C cij n×n, K kij n×n are n n constant matrices, defined as T the inertia, damping and stiffness matrices, respectively. Φ (q, q˙) = [Φ1, Φ2, , Φn] is a T nonlinear n-vector function of the generalized coordinate vector q = [q1, q2, , qn] and T its derivative q˙ = [q˙1, q˙2, , q˙n] . The symbol (T) denotes the transpose of a matrix. The excitation Q(t) is a zero mean stationary Gaussian random vector process with the spec- ( ) =  ( ) ( ) tral density matrix SQ ω Sij ω n×n where Sij ω is the spectral density function of elements Qi and Qj. An equivalent linear system to the original nonlinear system (1) can be defined as Mq¨ + (C + Ce) q˙ + (K + Ke) q = Q(t), (2) e h e i e h e i where C = cij , K = kij are deterministic matrices. They are to be determined n×n n×n T so that the n-vector difference ε = [ε1, ε2, , εn] between the original and the equiva- lent system is minimum. In the classical GEL shown in [16] by Roberts and Spanos, the matrices Ce, Ke are determined by the following criterion n T o E ε ε → min e e (i, j = 1, 2, . . . , n), (3) cij,kij e e where E{.} denotes the mathematical expectation operation and cij, kij are the (i, j) ele- ments of the matrices Ce, Ke and ε = Φ(q, q˙) − Ceq˙ − Keq. (4)
3. Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators 3 Using the linearity property of the expectation operator E{.}, criterion (3) can be written as  2 E ε → min e e (α = 1, 2, . . . , n). (5) α cij,kij The necessary conditions for the criterion (5) to be true are ∂  2 ∂  2 e E εα = 0, e E εα = 0, (i, j = 1, 2, . . . , n). (6) ∂cij ∂kij Combine (4) and (6), after some algebraic procedures, one gets the equivalent lin- earization coefficients as follows     e ∂Φi e ∂Φi cij = E , kij = E , (7) ∂q˙j ∂qj where Φi is the (i) element of Φ(q, q˙). The spectral density matrix of the response process q(t) is of the form Sq(ω) = [Sqiqj (ω)], (i, j = 1, 2, . . . , n), (8) where Sqiqj (ω) is the (i, j) element of Sq(ω). Using the matrix spectral input-output relationship to linear system (2), one gets T Sq(ω) = α(ω)SQ(ω)α (ω), (9) where α(ω) is the matrix of frequency response functions. It is known as −1 α(ω) = −ω2 M + iω(C + Ce) + (K + Ke) . (10) The mean values of the response can be calculated by the following equations Z∞ Z∞  n To T E qiqj = Sqiqj (ω)dω, E qq = α(−ω)SQ(ω)α (ω)dω, −∞ −∞ (11) Z∞ n To 2 T E q˙q˙ = ω α(−ω)SQ(ω)α (ω)dω −∞ A set of nonlinear algebraic equations (2), (7), (9)–(11) allows to find the mean values of response. Denote p(q) the stationary joint probability density function (PDF) of the T vector q = [q1, q2, , qn] . The criterion (5) can be written in the following form + + Z ∞ Z ∞  2 2 E ε = ε p(q)dq dq . . . dq → min e e (α, i, j = 1, 2, . . . , n). (12) α α 1 2 n cij,kij −∞ −∞ As the above-mentioned that the basic difference of LOMSEC from the classical GEL is that the integration domain for mean squares of response are taken over finite one (local one). Thus, LOMSEC requires +q +q Z 01 Z 0n  2  2 E ε = ε p(q)dq dq . . . dq → min e e (α, i, j = 1, 2, . . . , n), (13) α α 1 2 n cij,kij −q01 −q0n
4. 4 Luu Xuan Hung, Nguyen Cao Thang where q01, q02, , q0n are given positive values. The expected integrations in (13) can be transformed to non-dimensional variables by q01 = rσq1, q02 = rσq2, , q0n = rσqn with r a given positive value; σq1, σq2, , σqn are the normal deviations of random variables of q1, q2, , qn, respectively. Thus, criterion (13) become +rσq1 +rσqn Z Z  2  2 E ε = ε p(q)dq dq . . . dq → min e e (α, i, j = 1, 2, . . . , n), (14) α α 1 2 n cij,kij −rσq1 −rσqn where E[.] denotes the local mean values by LOMSEC. These values of random variables are taken as follows +rσq1 +rσqn +rσq1 +rσqn Z Z   Z Z E [.] = (.)p(q)dq1dq2 . . . dqn → E qiqj = qiqj p(q)dq1dq2 . . . dqn. For example −rσq1 −rσqn −rσq1 −rσqn (15) For zero-mean stationary Gaussian random variables, The classical GEL indecates that all odd-order means are null, all higher even-order means can be expressed in terms of second-order mean of the respective variable. These characteristics are also kept in LOMSEC and presented in the appendix. e e In GEL, the values σq1, σq2, , σqn are considered to be independent from cij, kij in e e the process of minimizing (14). Criterion (14) results in conditions for determining cij, kij as follows ∂  2  ∂  2  e E εα = 0, e E εα = 0, (α, i, j = 1, 2, . . . , n). (16) ∂cij ∂kij e e It is seen from (14) to (16) that the elements of cij, kij are functions depending on the e e e local mean values of random variables and also depending on r (i.e. cij = cij(r), kij = e kij(r)), which is not explicitly expressed here. Eqs. (2), (15) and (16) allow to determine e e the unknowns cij(r), kij(r) and the vector q(t) when r is given. However, is that the local domain of integration, namely in our case the value of r, is unknown and the open question is how to find it. Using the dual approach to LOMSEC, it is suggested that instead of finding a special value of r one may consider its variation in the entire global e e domain of integration. Thus, the linearization coefficients cij(r), kij(r) can be suggested as global mean values of all local linearization coefficients as follows s s D E 1 Z D E 1 Z ce = ce (r) = lim ce (r)dr, ke = ke (r) = lim ke (r)dr (17) ij ij s→∞ s ij ij ij s→∞ s ij 0 0 where h.i denotes conventionally the average of operators of deterministic functions. Ob- viously, Eqs. (2), (15), (16), (17) allow to determine the unknowns without specifying any value of r and the new criterion may be called global–local mean square error criterion (GLOMSEC).
5. Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators 5 3. APPLICATIONS 3.1. Rolling ship oscillation The rolling motion of a ship in random waves has been considered by Roberts [17], Roberts and Dacunha [18], David et al. [19]. The governing equation of motion, for ex- ample in [19], is √ ϕ¨ + βϕ˙ + αϕ˙ |ϕ˙ | + ω2 ϕ + δϕ3 = 2Dw(t), (18) where ϕ ≤ 35◦ is the roll angle from the vertical, ω is the undamped natural frequency of roll. The parameters β, α, δ are constant. The random waves is described√ as zero mean Gaussian white noise excitation, which is denoted by w(t), and 2D is the intensity of the white noise excitation. Note that equation (18) is only valid for ϕ ≤ 35◦. This, in turn, requires that δ and D are small such that the probability for the response trajectories to depart from the region of stability in the phase plane is extremely small. Under such con- ditions, for practical purpose, then it is reasonable to assume the existence of stationary random rolling motion. In order to obtain some simple analytical results, consider case with β = δ = 0 so that the rolling ship oscillator reduces to a quadratically damped linear stiffness oscillator as follows √ ϕ¨ + αϕ˙ |ϕ˙ | + ω2 ϕ = 2Dw(t). (19) The exact solution of the system (19) does not exist; however, an approximate prob- ability density function obtained by equivalent non-linearization (ENL) method follow- ing [19] or [20]. 2   3 3 3 8α − 8α (ω2 ϕ2+ϕ˙ 2) 2 P(ϕ, ϕ˙ ) = e 9πD , (20) 2 9πD 2πΓ 3 where Γ (.) is the Gamma function. Generally, ENL gives solutions with rather high accuracy and in many cases it agrees with Monte Carlo simulation (MCS) [20]. Thus, the solutions given by ENL can be used for evaluation of accuracy of ones obtained by other approximate methods, for example GEL. =  2  2 Consider the system (19) with ω 1. Denote E ϕ NL , E ϕ˙ NL the square mean responses of displacement and velocity determined from the probability density function =  2 =  2 (20), respectively. Additionally, when ω 1, we have E ϕ NL E ϕ˙ NL. Thus, the results are 2  D  E ϕ2 = E ϕ˙ 2 = 0.765 3 . (21) NL NL α For GEL, the nonlinear system (19) is replaced by a linear one as follows √ ϕ¨ + ce ϕ˙ + ϕ = 2Dw(t), (22)
6. 6 Luu Xuan Hung, Nguyen Cao Thang where ce is the linearization coefficient, for LOMSEC ce = ce(r) as known by (16) as follows ∂ ∂ h i E ε2 = E (αϕ˙ |ϕ˙ | − ce ϕ˙ )2 = 0. (23) ∂ce ∂ce Expand (23) and utilize (A.8)–(A.9), one gets  r r  q T 3 Z Z e 2 t ,r 3 2 c (r) = α E {ϕ˙ } , Tt3,r = t η(t)dt, T1,r = t η(t)dt . (24) T1,r 0 0 For the linear system (22), the mean square responses by LOMSEC are √ 2 2D D D  2 =  2 = = = E ϕ L E ϕ˙ L e e . (25) 2c (r) c (r) p Tt3,r α E {ϕ˙ 2} T1,r With r → ∞,(25) gives the solutions by the classical GEL as follows 2  D  E ϕ2 = E ϕ˙ 2 = 0.732 3 . (26) C C α Apply (17) for (24), one gets the linearization coefficient by GLOMSEC as follows  s   s  Z Z 1 1 Tt3,r ce = hce(r)i = lim  ce(r)dr = αE{ϕ˙ 2}1/2 lim  dr ≈ 1.49705αE{ϕ˙ 2}1/2. s→∞ s s→∞ s T1,r 0 0 (27) The limitation element in (27) can be approximately computed to be 1.49705. The solu- tions obtained by GLOMSEC are D D  D 2/3 E ϕ2 = E ϕ˙ 2 = = = 0.76415 . (28) GL GL ce 1.49705αE{ϕ˙ 2}1/2 α Denote Err(C), Err(GL) the relative errors of (26) and (28) to (21) respectively, one gets  2  2 E ϕ − E ϕ 0.732 − 0.765 Err = C NL ∗ = ∗ = (C) 2 100% 100% 4.314% E {ϕ }NL 0.765 (29)  2  2 E ϕ − E ϕ 0.764 − 0.765 Err = GL NL ∗ = ∗ = (GL) 2 100% 100% 0.130% E {ϕ }NL 0.765  D 2/3 Note that since (21), (26) and (28) all contain the same factor , so this factor α is reduced in the expression (29). The result in (29) shows that the solution by GLOMSEC agree with the one by ENL because of negligible differences between these solutions. In addition, these solutions contain the similar factor in their formulas. This means that GLOMSEC gives a significant improvement on accuracy of solution in comparison with the classical GEL.
7. Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators 7 3.2. Two-degree-of-freedom nonlinear oscillator Consider a two-degree- of-freedom nonlinear oscillator governed by the equation [20] 2
   2 3 x¨1 − λ1 − α1x˙1 x˙1 + ω1 x1 + ax2 + b (x1 − x2) = w1(t), (30) 2
   2 3 x¨2 − λ1 − λ2 − α2x˙2 x˙2 + ω2 x2 + ax1 + b (x2 − x1) = w2(t), where αi, a, b, λi, ωi(i = 1, 2) are constant. w1(t), w2(t) are zero mean Gaussian white noise and E {wi(t)wi(t + τ)} = 2πSiδ(τ) (i = 1, 2) where δ(τ) is Delta Dirac function, S1, S2 are constant values of the spectral density of w1(t), w2(t), respectively. The equa- tion (30) can be rewritten as follows 2 3 3 x¨1 − λ1x˙1 + ω1 x1 + ax2 + α1x˙1 + b (x1 − x2) = w1(t), (31) 2 3 3 x¨2 − (λ1 − λ2) x˙2 + ax1 + ω2 x2 + α2x˙2 + b (x2 − x1) = w2(t). Eq. (31) can be expressed in matrix form as follows " # 1 0 x¨  −λ 0  x˙  ω2 a  x  α x˙3 + b (x − x )3 w (t) 1 + 1 1 + 1 1 + 1 1 1 2 = 1 . − + 2 3 3 ( ) 0 1 x¨2 0 λ1 λ2 x˙2 a ω2 x2 α2x˙2 + b (x2 − x1) w2 t (32) Following Eq. (1), denote " # 1 0 −λ 0  ω2 a  α x˙3 + b (x − x )3 x  M = ; C = 1 ; K = 1 ; Φ = 1 1 1 2 ; x = 1 . − + 2 3 3 0 1 0 λ1 λ2 a ω2 α2x˙2 + b (x2 − x1) x2 (33) The linear equation to (32) is taken in the form of (2) as follows      e e     2 e e      1 0 x¨1 −λ1 + c11 c12 x˙1 ω1 + k11 a + k12 x1 w1(t) + e e + e 2 e = , 0 1 x¨2 c21 −λ1 + λ2 + c22 x˙2 a + k21 ω2 + k22 x2 w2(t) (34) e e where cij, kij(i, j = 1, 2) are the linearization coefficients. According to (4), the difference between (32) and (34) is ε = Φ(x, x˙) − CeX˙ − KeX (35)   " 3 3#  e e     e e  Φ1 α1x˙1 + b (x1 − x2) e c11 c12 x˙1 e k11 k12 Φ(x, x˙) = = 3 3 , C = e e ; x˙ = ; K = e e , Φ2 α2x˙2 + b (x2 − x1) c21 c22 x˙2 k21 k22 " # x  ε  α x˙3 + b (x − x )3 − ce x˙ − ce x˙ − ke x − ke x x = 1 ; ε = 1 = 1 1 1 2 11 1 12 2 11 1 12 2 . x 3 3 e e e e 2 ε2 α2x˙2 + b (x2 − x1) − c21x˙1 − c22x˙2 − k21x1 − k22x2 e e Use (16) for determining cij(r), kij(r)(i, j = 1, 2)   ( h i ) ∂E ε2 α E x˙4 + b(E x3x˙  + 3E x x2x˙  − 3E x2x x˙  1 = 2ce E x˙2 − 2 1 1 1 1 1 2 1 1 2 1 = 0, ∂ce 11 1  3  e e e 11 −E x2x˙1 ) − c12E [x˙1x˙2] − k11E [x1x˙1] − k12E [x2x˙1]  2    3   3   2   2   ∂E ε1 e  2 α1E x˙1x˙2 + b(E x1x˙2 + 3E x1x2x˙2 − 3E x1x2x˙2 e = 2c12E x˙2 − 2  3  e e e = 0, ∂c12 −E x2x˙2 ) − c11E [x˙1x˙2] − k11E [x1x˙2] − k12E [x2x˙2]
8. 8 Luu Xuan Hung, Nguyen Cao Thang  2    3   3   2   2   ∂E ε2 e  2 α2E x˙2x˙1 + b(E x2x˙1 + 3E x1x2x˙1 − 3E x1x2x˙1 e = 2c21E x˙1 − 2  3  e e e = 0, ∂c21 −E x1x˙1 ) − c22E [x˙1x˙2] − k21E [x1x˙1] − k22E [x2x˙1]   ( h i ) ∂E ε2 α E x˙4 + b(E x3x˙  + 3E x2x x˙  − 3E x x2x˙  2 = 2ce E x˙2 − 2 2 2 2 2 1 2 2 1 2 2 = 0, ∂ce 22 2  3  e e e 22 −E x1x˙2 ) − c21E [x˙1x˙2] − k21E [x1x˙2] − k22E [x2x˙2]   ( h i ) ∂E ε2 α E x x˙3 + b(E x4 + 3E x2x2 − 3E x3x  1 = 2ke E x2 − 2 1 1 1 1 1 2 1 2 = 0, ∂ke 11 1  3 e e e 11 −E x1x2 ) − c11E [x1x˙1] − c12E [x1x˙2] − k12E [x1x2]  2  (  3   3   3  2 2 ) ∂E ε α1E x˙1x2 + b(E x1x2 + 3E x1x2 − 3E x1x2 1 = 2ke E x2 − 2 h i = 0, e 12 2 − 4 ) − e [ ] − e [ ] − e [ ] ∂k12 E x2 c11E x˙1x2 c12E x2x˙2 k11E x1x2  2  (  3  3   3   2 2 ) ∂E ε α2E x1x˙2 + b(E x2x1 + 3E x1x2 − 3E x1x2 2 = 2ke E x2 − 2 h i = 0, e 21 1 − 4 ) − e [ ] − e [ ] − e [ ] ∂k21 E x1 c21E x1x˙1 c22E x1x˙2 k22E x1x2   ( h i ) ∂E ε2 α E x x˙3 + b(E x4 + 3E x2x2 − 3E x x3 2 = 2ke E x2 − 2 2 2 2 2 1 2 1 2 = 0. ∂ke 22 2  3  e e e 22 −E x1x2 ) − c21E [x˙1x2] − c22E [x2x˙2] − k21E [x1x2] (36) In order to simplify the calculation, assume that x1, x2 are independent from each other. As known that if is a stationary Gaussian random process with zero mean, so is x˙(t). Besides, a stationary random process is orthogonal to its derivative, so x1, x2 are independent from x˙1, x˙2, respectively. Use (A.3), (A.6) and (A.8) in the appendix to h 2n+1 2m+1i determine the local means in (36) and note that E xi xj = 0 (i 6= j). Thus, (36) gives the following result   E x˙4 T e ( ) = 1 =  2 2,r c11 r α1  2 α1E x˙1 , E x˙1 T1,r e e c12(r) = c21(r) = 0,   E x˙4 T e ( ) = 2 =  2 2,r c22 r α2  2 α2E x˙2 , E x˙2 T1,r       E x4 + 3E x2 E x2  T T  e ( ) = 1 1 2 =  2 2,r +  2 1,r k11 r b  2 b E x1 3E x2 , E x1 T1,r T0,r (37)       −E x4 − 3E x2 E x2  T T  e ( ) = 2 1 2 = −  2 2,r −  2 1,r k12 r b  2 b E x2 3E x1 , E x2 T1,r T0,r       −E x4 − 3E x2 E x2  T T  e ( ) = 1 1 2 = −  2 2,r −  2 1,r k21 r b  2 b E x1 3E x2 , E x1 T1,r T0,r       E x4 + 3E x2 E x2  T T  e ( ) = 2 1 2 =  2 2,r +  2 1,r k22 r b  2 b E x2 3E x1 . E x2 T1,r T0,r
9. Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators 9 In (37), let r → ∞, it gives the linearization coefficients by the classical GEL as follows e  2 e e e  2 c11 = 3α1E x˙1 , c12 = c21 = 0, c22 = 3α2E x˙2 , (38) e e  2  2
   e e  2  2
   k11 = k22 = 3b E x1 + E x2 , k12 = k21 = −3b E x1 + 3bE x2 . The following factors are defined and replaced in (38) Z∞ Z∞ t4η(t)dt t2η(t)dt T T 1 2 2,∞ 0 1,∞ 0 √ −t /2 = ∞ = 3, = ∞ = 1, η(t) = e . T1,∞ Z T0,∞ Z 2π t2η(t)dt η(t)dt 0 0 Apply (17) to (37), one obtains the linearization coefficients by GLOMSEC as follows  s  Z e e  2 1 T2,r c11 = hc11(r)i = α1E x˙1 lim  dr , s→∞ s T1,r 0  s  Z e e  2 1 T2,r e e c22 = hc22(r)i = α2E x˙2 lim  dr , c12 = c21 = 0, s→∞ s T1,r 0   s   s  Z Z e e  2 1 T2,r  2 1 T1,r k11 = hk11(r)i = b E x1 lim  dr + 3E x2 lim  dr , s→∞ s T1,r s→∞ s T0,r 0 0 (39)   s   s  Z Z e e  2 1 T2,r  2 1 T1,r k12 = hk12(r)i = −b E x2 lim  dr + 3E x1 lim  dr , s→∞ s T1,r s→∞ s T0,r 0 0   s   s  Z Z e e  2 1 T2,r  2 1 T1,r k21 = hk21(r)i = −b E x1 lim  dr + 3E x2 lim  dr , s→∞ s T1,r s→∞ s T0,r 0 0   s   s  Z Z e e  2 1 T2,r  2 1 T1,r k22 = hk22(r)i = b E x2 lim  dr + 3E x1 lim  dr , s→∞ s T1,r s→∞ s T0,r 0 0 where the limitation factors can be approximately computed to be  s   s  Z Z 1 T2,r 1 T1,r lim  dr ≈ 2.41189, lim  dr ≈ 0.83706. s→∞ s T1,r s→∞ s T0r 0 0 Consider the white noise spectral densities of w1(t), w2(t) respectively are S1 = S2 = S0, the spectral density matrix Sw(ω) of w(t) is defined by   S0 0 Sw(ω) = . (40) 0 S0
10. 10 Luu Xuan Hung, Nguyen Cao Thang The frequency response function to linear system (34) is −1 α(ω) = −ω2 M + iω(C + Ce) + (K + Ke) . (41) The matrices in (41) ware defined in (33) and (35) to be      2   e e   e e  1 0 −λ1 0 ω1 a e c11 c12 e k11 k12 M = , C = , K = 2 , C = e e , K = e e . 0 1 0 −λ1 + λ2 a ω2 c21 c22 k21 k22 After some matrix operations, the frequency response function (41) is defined as follows  2 e 2 e e e −1 −ω + iω(−λ1 + c11) + ω1 + k11 iωc12 + a + k12 α(ω) = e e 2 e 2 e . iωc21 + a + k21 −ω + iω(−λ1 + λ2 + c22) + ω2 + k22 (42)  2 In order to have a close equation system determining the unknowns, all the E xi ,  2 E x˙i , (i = 1, 2) must be defined. Use (11), (40) and after some matrix operations one gets + Z ∞  n To α11(ω)α11(−ω)+α12(ω)α12(−ω) α11(−ω)α21(ω)+α12(−ω)α22(ω) E xx = S0 dω, α11(ω)α21(−ω)+α12(ω)α22(−ω) α21(ω)α21(−ω)+α22(ω)α22(−ω) −∞ + + Z ∞ Z ∞  2  2 2  2  2 2 E x1 = S0 |α11(ω)| + |α12(ω)| dω, E x2 = S0 |α21(ω)| + |α22(ω)| dω, −∞ −∞ + Z ∞   n To 2 α11(ω)α11(−ω)+α12(ω)α12(−ω) α11(−ω)α21(ω)+α12(−ω)α22(ω) E x˙x˙ = S0 ω dω, α11(ω)α21(−ω)+α12(ω)α22(−ω) α21(ω)α21(−ω)+α22(ω)α22(−ω) −∞ + + Z ∞ Z ∞  2 2 2 2  2 2  2 2 E x˙1 = S0 ω |α11(ω)| +|α12(ω)| dω, E x˙2 = S0 ω |α21(ω)| +|α22(ω)| dω, −∞ −∞ (43) where the elements αij are defined from (42). Eq. (43) is solved either together with (38) or (39) to define the unknowns by the classical GEL or by GLOMSEC, respectively. In order to solve the above equations, it is needed to utilize computationally approximate meth- ods, for example, an iteration method is applied as follows: (i) Assign an initial value to the mean square responses of (43); (ii) Use (38) or (39) to determine the instantaneous linearization coefficients by the classical GEL or GLOMSEC, respectively; (iii) Use (42) and (43) to determine new instantaneous value of the responses; (iv) Repeat steps (ii) and (iii) until results from cycle to cycle have a difference to be less than 10−4. For purpose of evaluating the accuracy of solutions while the original nonlinear sys- tem (30) does not have the exact solution, one can use an approximate probability density function given by ENL method that was reported in [20] as follows. 1 h 9 1 2 1 2 2 1 1 2 1 2 i −( πS ) 32 (α1+α2)( 2 x˙1+ 2 x˙2+U) +( 2 λ2−λ1)( 2 x˙1+ 2 x˙2+U) p(x1, x˙1, x2, x˙2) = Ce i , (44)
11. Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators 11 where U(x1, x2) is the potential energy of the system. 1 1 b U(x , x ) = ω2x2 + ω2x2 + ax x + (x − x )4 , (45) 1 2 2 1 1 2 2 2 1 2 4 1 2 and C is the normalization constant defined by " #−1 Z ∞ Z ∞ 1 h 9 1 2 1 2 2 1 1 2 1 2 i 2 −( πS ) 32 (α1+α2)( 2 x˙1+ 2 x˙2+U) +( 2 λ2−λ1)( 2 x˙1+ 2 x˙2+U) C = e i ∏ dxidx˙i . (46) −∞ −∞ i=1  2 The mean square responses E xi NL obtained by ENL are Z ∞ Z ∞ 1 h 9 1 2 1 2 2 1 1 2 1 2 i 2  2 2 −( ) (α1+α2)( x˙ + x˙ +U) +( λ2−λ1)( x˙ + x˙ +U) = πSi 32 2 1 2 2 2 2 1 2 2 E xi NL C xi e ∏ dxidx˙i. −∞ −∞ i=1 (47) Consider two cases of the given parameters. Tabs.1 and2 show the mean square responses of x1, x2 as well as their relative errors to solutions by ENL method (see also Figs.1 and2). Table 1. The mean squares of x1, x2 versus α (α1 = α2 = α) while λ1 = λ2 = ω1 = ω2 = a = b = S0 = 1 n 2o n 2o ErrC n 2o ErrGL n 2o n 2o ErrC n 2o ErrGL α1, α2 E x1 E x1 E x1 E x2 E x2 E x2 NL C |%| GL |%| NL C |%| GL |%| 0.1 1.57273 1.21597 22.684 1.40692 10.543 1.57273 1.15079 26.829 1.32675 15.640 1 0.49622 0.42145 15.068 0.48835 1.586 0.49622 0.36966 25.505 0.41930 15.501 5 0.25327 0.21986 13.191 0.25395 0.268 0.25327 0.20466 19.193 0.23409 7.573 10 0.19437 0.17091 12.070 0.19735 1.533 0.19437 0.16233 16.484 0.18625 4.178 Table 2. The mean squares of x1, x2 versus b while λ1 = λ2 = ω1 = ω2 = a = α1 = α2 = S0 = 1 n 2o n 2o ErrC n 2o ErrGL n 2o n 2o ErrC n 2o ErrGL b E x1 E x1 E x1 E x2 E x2 E x2 NL C |%| GL |%| NL C |%| GL |%| 1 0.49622 0.42145 15.068 0.48835 1.586 0.49622 0.36966 25.505 0.41928 15.505 10 0.36492 0.29566 18.980 0.33460 8.309 0.36492 0.29040 20.421 0.32769 10.202 50 0.33076 0.28086 15.086 0.31644 4.329 0.33076 0.28048 15.201 0.31597 4.472 100 0.32340 0.27930 13.636 0.31453 2.743 0.32340 0.27920 13.667 0.31440 2.783 From the relative errors of the approximate solutions with respect to the ones by ENL, it can be seen that GLOMSEC gives a significant improvement on accuracy of solu- tion in comparison with the classical GEL, especially when the nonlinearity is strong.
12. PerformancePerformance analysis of Global LocalLocal MeanMean SquareSquare ErrorError CriterionCriterion of stochastic linearization forfor nonlinearnonlinear oscillatorsoscillators Table 2. The mean squares of xx, versus b while 1  2  1  2 aS 1 2 0 1 Table 2. The mean squares of xx1212, versus b while 1  2  1  2 aS 1 2 0 1 2 2 Err 2 Err 2 2 Err 2 Err Ex2 Ex2 ErrCC Ex2 ErrGL Ex2 Ex2 ErrCC Ex2 ErrGLGL b Ex 11NL Ex 11C Ex 1 GL Ex 22NL Ex 22C Ex 22GL b NL C % GL % NL C % GL % % % % % 11 0.496220.49622 0.421450.42145 15.06815.068 0.488350.48835 1.586 0.49622 0.369660.36966 25.50525.505 0.419280.41928 15.50515.505 1010 0.364920.36492 0.295660.29566 18.98018.980 0.334600.33460 8.309 0.36492 0.290400.29040 20.42120.421 0.0.3232769769 10.20210.202 5050 0.330760.33076 0.280860.28086 15.08615.086 0.316440.31644 4.329 0.33076 0.280480.28048 15.20115.201 0.315970.31597 4.4724.472 100100 0.323400.32340 0.279300.27930 13.63613.636 0.314530.31453 2.743 0.32340 0.279200.27920 13.66713.667 0.314400.31440 2.7832.783 12 Luu Xuan Hung, Nguyen Cao Thang Figure 1. The mean square of x versus FigureFig. 11 TheThe mean mean square square of xof1 versusx11 versusα FigureFig. 22 TheThe mean mean square square of ofx2 versusx2 versusα Figure 2. The mean square of x2 versus 4. CONCLUSION This paper presents the proposed criterion with its algorithm built to MDOF nonlin- ear oscillators under Gaussian white noise excitation. The mode of formulating algorithm is also mainly based on the classical GEL. However, a key problem is to determine the
13. Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators 13 matrix of equivalent linearization coefficients in which the constant linearization coeffi- cients are defined as global mean values of all local linearization coefficients. The paper is an additional research to our previous ones [14, 15] to aim at evaluating the improved performance of the proposed criterion; herein we analyse two applications, which are a rolling ship oscillation and two-degree-of-freedom one. The results show a significant improvement on accuracy of solutions by GLOMSEC in comparison with the ones by the classical GEL. ACKNOWLEDGMENTS The paper is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.04-2018.12. REFERENCES [1] T. K. Caughey. Equivalent linearization techniques. The Journal of the Acoustical Society of America, 35, (11), (1963), pp. 1706–1711. [2] T. K. Caughey. Response of a nonlinear string to random loading. Journal of Applied Mechanics, 26, (3), (1959), pp. 341–344. [3] L. Socha. Linearization methods for stochastic dynamic systems. Lecture Notes in Physics, Springer, Berlin, (2008). [4] X. Zhang, I. Elishakoff, and R. Zhang. A stochastic linearization technique based on mini- mum mean square deviation of potential energies. Stochastic Structural Dynamics, 1, (1991), pp. 327–338. 17. [5] F. Casciati, L. Faravelli, and A. M. Hasofer. A new philosophy for stochastic equiv- alent linearization. Probabilistic Engineering Mechanics, 8, (3-4), (1993), pp. 179–185. [6] N. D. Anh and W. Schiehlen. New criterion for Gaussian equivalent linearization. European Journal of Mechanics - A/Solids, 16, (6), (1997), pp. 1025–1039. [7] C. Proppe, H. J. Pradlwarter, and G. I. Schueller.¨ Equivalent linearization and Monte Carlo simulation in stochastic dynamics. Probabilistic Engineering Mechanics, 18, (1), (2003), pp. 1– 15. [8] I. Elishakoff, L. Andriamasy, and M. Dolley. Application and extension of the stochas- tic linearization by Anh and Di Paola. Acta Mechanica, 204, (1-2), (2009), pp. 89–98. [9] R. N. Iyengar. Higher order linearization in non-linear random vibration. International Jour- nal of Non-Linear Mechanics, 23, (5-6), (1988), pp. 385–391. 7462(88)90036-4. [10] N. D. Anh and M. Di Paola. Some extensions of Gaussian equivalent linearization. In Pro- ceedings of International Conferenceon Nonlinear Stochastic Dynamics, Hanoi, Vietnam, (1995). pp. 5–16. [11] N. D. Anh and L. X. Hung. An improved criterion of Gaussian equivalent linearization for analysis of non-linear stochastic systems. Journal of Sound and Vibration, 268, (1), (2003), pp. 177–200. [12] N. D. Anh. Duality in the analysis of responses to nonlinear systems. Vietnam Journal of Me- chanics, 32, (4), (2010), pp. 263–266. [13] N. D. Anh. Dual approach to averaged values of functions. Vietnam Journal of Mechanics, 34, (3), (2012), pp. 211–214.
14. 14 Luu Xuan Hung, Nguyen Cao Thang [14] N. D. Anh, L. X. Hung, and L. D. Viet. Dual approach to local mean square error crite- rion for stochastic equivalent linearization. Acta Mechanica, 224, (2), (2013), pp. 241–253. [15] N. D. Anh, L. X. Hung, L. D. Viet, and N. C. Thang. Global–local mean square error criterion for equivalent linearization of nonlinear systems under random excitation. Acta Mechanica, 226, (9), (2015), pp. 3011–3029. [16] J. B. Roberts and P. D. Spanos. Random vibration and statistical linearization. Wiley, New York, (1990). [17] J. B. Roberts. A stochastic-theory for non-linear ship rolling in irregular seas. Journal of Ship Research, 26, (4), (1982), pp. 229–245. [18] J. B. Roberts and N. M. C. Dacunha. Roll motion of a ship in random beam waves: Compari- son between theory and experiment. Journal of Ship Research, 29, (1985), pp. 112–126. [19] D. C. Polidori, J. L. Beck, and C. Papadimitriou. A new stationary PDF approximation for non-linear oscillators. International Journal of Non-Linear Mechanics, 35, (4), (2000), pp. 657– 673. [20] C. W. S. To. Nonlinear random vibration: Analytical techniques and applications. CRC Press, (2011). APPENDIX T Suppose that the components of the vector x = (x1, x2, , xn) are zero-mean sta- tionary Gaussian random variables. Denote E{.} global mean values of random variables taken as follows + + Z ∞ Z ∞ E {.} = (.) p (x)dx1dx2 . . . dxn, (A.1) −∞ −∞ where p(x) is the stationary joint probability density function. For the Gaussian random processes with zero mean, one has the following general expressions for expectations [2] !  E {x1x2 x2n+1} = 0, E {x1x2 x2n} = ∑ ∏ E xixj , (A.2) all dependent pairs i6=j where the number of independent pair is equal to (2n)!/(2nn!) For example, E {x1x2x3} = 0, E {x1x2x3x4} = E {x1x2} E {x3x4} + E {x2x3} E {x1x4} + E {x1x3} E {x2x4} , (A.3) E {x1x2x3x4x5} = 0. If xi and xj (i 6= j) are uncorrelated, i.e. independent, then E{xixj} = 0, and 2n+1 2m+1 E{xi xj } = 0. Besides, formula (A.2) results in the following consequences m n 2n 2mo  2n n 2mo  2
    n  n 2o E xi xj = E xi E xj = (2n − 1)!! E xi (2m − 1)!! E xj , (A.4) where n and m are natural numbers. Denote [.] the local mean values of random variables taken as follows. +rσ +rσ Z x1 Z xn E [.] = (.)p(x)dx1dx2 . . . dxn, (A.5) −rσx1 −rσxn
15. Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators 15 where σx1, σx2 , σxn are the normal deviations of random variables, respectively, and r is a given positive value. Due to the symmetry of the expected integrations in (A.5), hereby (A.2) are also applied to the local mean values. If xi and xj (i 6= j) are uncorrelated,   h 2n+1 2m+1i i.e. independent, then E xixj = 0, and E xi xj = 0. All higher even-order local h 2n 2mi  2 means E xi xj can be expressed in terms of second order global means E xi and n 2o E xj as follows [16]. m h 2n 2mi  2n h 2mi  2
    n  n 2o E xi xj = E xi E xj = 2Tn,r E xi 2Tm,r E xj , (A.6) where r r Z Z 2n 2m 1 −t2 2 Tn,r = t η(t)dt, Tm,r = t η(t)dt, η(t) = √ e / . (A.7) 2π 0 0 If n = 0, m 6= 0 or n 6= 0, m = 0, then (A.6) leads to the following results, respectively r m Z h 0 2mi  n 2o h 2n 0i  2
    n E xi xj = 2T0,r2Tm,r E xj , E xi xj = 2Tn,r E xi 2T0,r with T0,r = η(t)dt. 0 (A.8) If r → ∞,(A.5) and (A.7) will give the same result as (A.4) of the classical case. 2 A local mean of xi |xi| that arises in an application of the paper was presented in [15], the obtained result as follows +rσ +rσ Z xi Z xi  2  2 3 E xi |xi| = xi |xi| p(xi)dxi = 2 xi p(xi)dxi −rσxi 0 r r Z 1 2 2 2 Z 3 3 −t σxi/2σxi 3 3 = 2 t σxi √ e σxidt =2σxi t η(t)dt, (A.9) 2πσxi 0 0 r Z  2   2
    3/2 3 E xi |xi| = 2Tt3,r E xi , Tt3,r = t η(t)dt. 0 T T If x = (x1, x2, , xn) is the displacement vector, then x˙ = (x˙1, x˙2, , x˙n) is the ve- locity vector and we also obtain the same formulas, respectively, for the random variables of velocity.