anh-zakovorotney a dual criterion of stochastic linearization method for multi-degree-of-freedom...

8/10/2019 Anh-Zakovorotney A dual criterion of stochastic linearization method for multi-degree-of-freedom systems subjected to random ex.pdf

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Acta MechDOI 10.1007/s00707-012-0738-5

N. D. Anh V. L. Zakovorotny N. N. Hieu D. V. Diep

A dual criterion of stochastic linearization methodfor multi-degree-of-freedom systems subjectedto random excitation

Received: 26 March 2012 / Revised: 10 July 2012 Springer-Verlag 2012

Abstract In this paper, a dual criterion of stochastic linearization method is developed for multi-degree-of-freedom systems subjected to random excitation. A closed system is obtained for determining coefcientmatrices of the linearized system. Two examples of two-degree-of-freedom systems subjected to randomexcitation are presented. The mean-square responses obtained by the dual approach are compared with thoseobtained by two other methods, namely, conventional linearization technique and energy method. The resultsshow that the dual criterion gives a good prediction on mean-square responses of the original nonlinear systemwhen the nonlinearity is increasing.

1 Introduction

Nonlinear stochastic dynamical systems have been investigated intensively by many approaches in the pastdecades. One of the most common approaches for solving nonlinear random vibration problems is the stochas-tic linearization method. The method has attracted many researchers due to its originality and capability forvarious applications in engineering. The rst studies about the stochastic linearization method were introducedsimultaneously by the three authors Booton [ 1], Kazakov [ 2], and Caughey [ 3,4]. They extended Krylov andBogoliubovs linearization technique [ 5] of deterministic problems to random problems. The fundamental ideaof the method lies in replacing an original nonlinear system under Gaussian random excitation by a linear oneunder the same excitation for which the coefcients of the equivalent system can be found from a speciedoptimization criterion, such as the mean-square criterion [ 3], energy criteria [6,7], spectral criteria [8,9], andprobability density criteria [ 10] in some probabilistic sense. The method then has been generalized to randomvibration of multi-degree-of-freedom (MDOF) systems by Foster [11 ], Iwan and Yang [ 12], and Atalik andUtku [ 13]. Later, Bruckner and Lin [14] generalized the method of stochastic linearization to a nonlinearsystem subjected to both parametric and external random white noise excitations. Some new ideas about the

N. D. Anh ( B ) N. N. HieuInstitute of Mechanics, 264, Doi Can, Ba Dinh, Hanoi, VietnamE-mail: [email protected]

N. N. HieuE-mail: [email protected]

V. L. ZakovorotnyDon State Technical University, 1 Gagarin Sq., Rostov-on-Don, RussiaE-mail: [email protected]

D. V. DiepInstitute of Military Vehicle, Hanoi, VietnamE-mail: [email protected]


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stochastic linearization method are presented in [1518] by several authors. Casciati and Faravelli [ 15] dis-cussed a new philosophy for stochastic equivalent linearization. Iyengar and Roy [ 16] studied responses of nonlinear systems under narrow-band inputs by a conditional linearization approach. Colajanni and Elishakoff [17] gave a new look at the stochastic linearization technique for hyperbolic tangent oscillators. Anh and

Schiehlen [ 18] proposed a mean-square criterion of error sample function for determining coefcients of thelinearized equivalent equation.The method of stochastic linearization was also reviewed in some survey papers by Spanos [ 19], Roberts

[20], Socha et al. [ 2123], Proppe et al. [ 24], Crandall [25], and in books by Robert and Spanos [ 26] and Socha[27]. These authors provided an overview in the area of theoretical and experimental aspects of the stochas-tic equivalent linearization method for analyzing responses of structural and mechanical nonlinear stochasticdynamic systems.

Recently, some approaches of the stochastic linearization have been proposed in Refs. [ 28,29]. A non-parametric linearization method for nonlinear random vibration analysis was presented by Fuijimura andKiureghian [ 28]. They employed a discrete representation of the stochastic excitation, the concept occurredfrom the rst-order reliability method for obtaining the rst-order approximation of the tail probability of thenonlinear system. In [ 29], Elishakoff et al. developed a new setting for the stochastic linearization methodsuggested by Anh and Di Paola. Their approach is applied to study mean-square responses of some nonlinearsystems under white noise excitations.

In [30,31], Anh et al. have proposed a dual criterion of stochastic linearization method for nonlinear single-degree-of-freedom systems subjected to white noise random excitations. The authors showed that the accuracyof the mean-square response is significantly improved when the nonlinearity is increasing.

Naturally, the dual criterion linearization approach needs to be generalized to MDOF systems subjectedto random forces. Therefore, in this study, our aim is to investigate the accuracy of an extended version of dual criterion in case of an MDOF system under random excitation. For this purpose, two examples of two-degree-of-freedom systems are considered. The results obtained in this way are compared with those obtainedby conventional linearization, energy method, and Monte-Carlo simulation.

2 A dual criterion for multi-degree-of-freedom systems

In this section, we are concerned with a multi-degree-of-freedom system in the following form:

M x +C x +Kx + (x, x) =Q (t ) , (1)where M = m i j nn , C = ci j nn , K = k i j nn are mass, damping, stiffness matrices of order n n,

= (x, x) = 1 2 . . . nT is a nonlinear nvector function of the generalized coordinate vector

x = x 1 x 2 . . . x nT and its derivative x = x 1 x 2 . . . x n

T . The symbol T denotes the transpose of amatrix. The external excitation Q (t ) is a zero-mean Gaussian vector process with the spectral density matrixSQ () = S i j () nn , where S i j () is the spectral density function of elements Q i and Q j .

The linearized form of Eq. ( 1) is given by

M q + C +C e q + K +K e q =Q (t ) , (2)where C e = cei j nn , K

e = k ei j nn are constant matrices which are determined by a specied criterion of linearization method. There are some criteria for determining the matrices C e and K e of the linearized system(2) (see [26,27]). For the case of a multi-degree-of-freedom system, however, it is seen that the minimummean-square error criterion is one of the most common criteria for nding these matrices. In the contextof conventional linearization method, as shown by Roberts and Spanos [26], two matrices Ce and K e aredetermined as follows:

E eT e mincei j ,k ei j , (3)

where E [] denotes the mathematical expectation operation ande = (q , q ) C e q K eq (4)


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is the error between the original system (1) and the linearized one (2). The criterion (3) leads to the followingsystem for determining the matrices C e and K e :

E qq T E qq T

E qqT

E q qT

K eTi

CeTi =

E i q

E i q, (5)

where C ei and Kei are the i th rows of the matrices C

e and K e , respectively.In the followingdevelopment of the conventional linearization, a dual criterion formulti-degree-of-freedom

systems under random excitation is proposed. As presented in [ 30,31], this dual criterion approach appearsfrom the idea that the nonlinear component of the original system is replaced by an equivalent linear compo-nent, in the rst step, and then the second step is made so that this linearized system is replaced by anothernonlinear system which belongs to the same class of the original system. That means, for the original system(1), the nonlinear vector function is replaced by the linear vector C e q +K eq and then this linear vector isreplaced by another nonlinear vector function which is considered as a product of a matrix D = d i j nn andthe original nonlinear vector function , where the matrices C e , K e and D are determined by the followingproposed criterion for the MDOF system ( 1):

E eT

e + E T

minC e ,K e ,D, (6)where

=C e q +K eq D (q , q ) (7)and is a detuning parameter taking two values 0 or 1. When the parameter is zero, the criterion ( 6) willbecome the criterion ( 3) of the conventional linearization method, and when is 1, the criterion (6) is the socalled dual one. The rst expectation of the criterion ( 6) is understood as a component of conventional replace-ment while the second expectation component describes a dual replacement of the linearization problem. Thecriterion ( 6) can be rewritten as follows:

n

r =1 r mincei j ,k ei j ,d i j , (8)

where

r = E r n

s=1cer s qs +k er s qs

2

+ E n

s=1cers qs +k ers qs

n

s=1d r s s

2

. (9)

The necessary condition for the criterion ( 8) is given by the following equations:

cei j

n

r =1 r = 0, ( i , j =1, 2, . . . , n ), (10)

k ei j

n

r =1 r = 0, ( i , j =1, 2, . . . , n ), (11)

d i j

n

r =1 r = 0, ( i , j =1, 2, . . . , n ). (12)

Equations ( 10), (11 ), and (12) can be rewritten in the following simpler forms (see [26] and Appendix A fordetails):

cei j

i = 0, ( i , j =1, 2, . . . , n ), (13)

k ei j i = 0, ( i , j =1, 2, . . . , n ), (14)

d i j

i = 0, ( i , j =1, 2, . . . , n ). (15)


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Equations (13), (14) and (15) lead to a system of equations with unknowns cei j , k ei j , d i j of C

e , K e , and D,respectively:

n

s=1ce

is E

qs

q j

+k e

is E qs

q j

1+n

s=1d is E s

q j

=1

1 + E i

q j , ( i , j

=1, 2, . . . , n), (16)

n

s=1ceis E qs q j +k eis E qs q j

1+

n

s=1d is E s q j =

11+

E i q j , ( i , j =1, 2, . . . , n ), (17)n

s=1ceis E qs j +k eis E qs j d is E s j =0, ( i , j =1, 2, . . . , n ). (18)

A combination of the three equations ( 16), (17) and (18) yields the system

E qq T E q q T 1+

E q T

E

qq T E

q

q T

1

+ E

q T

E q T E q T E T

K eTiC eTi

DTi =

11+

E i q1

1+ E i

q

0

, (19)

where Cei , Kei and Di are the i th rows of the matrices C

e , K e and D, respectively. By solving the nonlin-ear system ( 19) of 3n2 unknowns cei j , k

ei j and d i j , we obtain C

e , K e and D. However, these solutions alsodepend on the responses of the linearized system ( 2). Therefore, additional closed relationships of unknownelements cei j , k

ei j and d i j must be established. In the next section, these relationships are presented based on

the frequency-response function matrix of the linearized system ( 2).

3 Response of the linearized system via spectral density function matrix of input excitation

The frequency-response function matrix H () of the linearized system ( 2) takes the following form [ 26]

H () = 2M +i C +C e +K +K e 1

. (20)

The matrix H () is related to the unity impulse matrix h (t ) of the linearized system ( 2) by Fourier transformas follows:

H () =

h (t )ei t dt , (21)

h (t ) =1

2

H () ei t d. (22)

It is noted that the response q (t ) of the system (2) can be expressed by the Duhamel integral

q (t ) =

h (t 1) Q ( 1) d 1 . (23)

By using the integral ( 23), we have

E qq T =

h (t 1) E Q ( 1) Q T ( 2) h T (t 2 ) d 1d 2 . (24)


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The moment E Q ( 1) Q T ( 2 ) is related to the spectral density matrix SQ () of Q (t ) as follows:

E Q ( 1) QT ( 2 ) =

SQ () ei ( 2 1 ) d. (25)

Substituting ( 25) into the right hand side of ( 24) and using ( 21) yields

E qq T =

H () SQ () H T () d. (26)

Similar to the expression (26), we have the following expression forthe second-order moments matrix E q q T :

E q q T =

2H () SQ () H T () d. (27)

From ( 23), it is seen that the mean-response of q is equal to zero. Further, the two processes q and q are inde-pendent Gaussian ones. Thus, the moment E q q T is equal to zero, and all moments E q T , E q T and E T can be expressed via second-order moment elements E qi q j , E q i q j which depend on C e , K eas it is seen from ( 20), (26) and (27). Hence, substituting these moments into Eq. ( 19) yields a closed systemof unknowns C e , K e , and D. In general, the obtained system is a nonlinear algebraic one. It may be solvednumerically for determining C e , K e and D. An iterative scheme for nding unknown matrices C e , K e and Dcan be obtained as follows (see [ 11 13]):

(i) Assign initial estimations of the matrices Ce , K e , and D in order to obtain the frequency-response function matrix H () (Eq. 20), matrices E qq T , E q q T (see Eqs. ( 26), (27)) and then E q T , E q T , and E T .(ii) Substitute the obtained matrices into Eq. ( 19) to get new values of C e , K e , and D.

(iii) Use these values and return to step (i).(iv) Repeat step (i), (ii), and (iii) until the results from cycle to cycle are similar.

In fact, one can use either the linearization coefcient matrices or responses for the outset of the iterativeprocess. In the frame of this xed-point iteration, nonlinearities of the original system belonging to a stable typeare necessary for the convergence of the process. For example, hardening springs or hydraulic drag forces aresuch a type. For the case of conventional linearization ( =0), this point was shown and applied successfullyin Refs. [11 13] to some nonlinear systems. Those authors reported that the convergence of the above iterativescheme is quite fast. For the case of the dual criterion approach ( = 1), calculations in the iterative schemeappear more complicated than those of the conventional linearization due to the presence of the additionalmatrix D . However, the process for computations does not result in major difculties. In analytical aspects,because of the complicatedness of nonlinear functions containing elements of linearization coefcient matricesin the iterative scheme, one needs to investigate in more detail for further understanding this procedure.

In the next sections, in order to evaluate the accuracy of the above dual criterion of stochastic lineariza-tion for a multi-degree-of-freedom system, responses of two nonlinear systems under random excitation areinvestigated.

4 Two-degree-of-freedom oscillator with nonlinear stiffness

We consider the following nonlinear random two-degree-of-freedom system which was investigated by Jiaand Fang [ 32] and then by Anh and Hung [ 33]:

x 1 +h x 1 +21 x 1 +4 1 x 31 +2 3 x 1 x 22 = w 1 (t ) , x 2 +h x 2 +22 x 2 +4 5 x 32 +2 3 x 21 x 2 = w 2 (t ) ,

(28)


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where h , 1 , 2 , 1 , 3 , 5 are positive constants; w (t ) = w( t ) w( t )T is a zero-mean Gaussian white noise

stationary random vector process with the following correlation functions K i j ( ) ( i , j =1, 2)K i j ( ) = E w i ( t ) w j ( t + ) =2 S i i j ( ) , (29)

where () is Dirac delta function of time variable , i j is the Kronecker symbol, and the quantities S 1 , S 2are constant values of the spectral density of random excitations w1 (t ) , w 2 (t ) , respectively.

4.1 Exact solution

In the case of the same spectral density of random excitations S 1 = S 2 = S 0 , the Fokker-Planck equationcorresponding to the system ( 28) has an exact solution for the stationary probability density function [33] f ( x 1 , x 2 ) =C exp

h S 0

U ( x 1 , x 2) , (30)

where the potential energy U ( x 1 , x 2) and normalization constant C are determined as follows, respectively,

U ( x 1 , x 2) =12

21 x 21 +

12

22 x 22 + 1 x 41 + 3 x 21 x 22 + 5 x 42 , (31)

C 1 =

exp h

S 0U ( x 1 , x 2) d x 1d x 2 . (32)

The exact mean-square responses of x 1 and x 2 are given by

E x 2i =C

x 2i exp h

S 0U ( x 1 , x 2) d x 1d x 2 , ( i =1, 2). (33)

In the following calculations, we investigate approximate mean-square responses of the system ( 28) using threeapproaches, including the energy method and the linearization method with conventional and dual criteria.

4.2 Energy method

We now determine the equivalent stiffness matrix of the linearized system of the original nonlinear system(28) by employing the energy method approach [ 6,35]. For this purpose, the system (28) can be rewritten inthe matrix form as follows:

x 1 x 2 +

h 00 h

x 1

x 2 +21 00 22

x 1 x 2 +

4 1 x 31 +2 3 x 1 x 224 5 x 32 +2 3 x 21 x 2 =

w1 (t )w2 (t )

. (34)

Following Eq. (1) denote

M =1 00 1 , C =

h 00 h , K =

21 00 22

, =4 1 x 31 +2 3 x 1 x 224 5 x 32 +2 3 x 21 x 2

, x = x 1 x 2

. (35)

The linearized system of ( 28) is taken in the following form:

q1q2 +

h 00 h

q1

q2 +21 +k e11 k e12k e12 22 +k e22

q1q2 =

w1 (t )w2 (t )

, (36)

where k e11 , k e12 = k e21 , k e22 are linearization constants which are determined from an optimal criterion of thestochastic linearization method. The corresponding potential energy of the linearized system ( 36) is

U e =12

21 +k e11 q 21 +k e12 q1q2 +12

22 +k e22 q 22 . (37)


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The energy criterion states that the mean-square difference between the potential energy, associated with theoriginal nonlinear system ( 28) and its equivalent linear counterpart ( 36), be minimal [ 6]. That means

E (U

U e )2

min

k ei j

. (38)

From the energy criterion (38), we have the following system for determining the coefcients k ei j (i , j =1, 2):

k ei j E (U U e )2 =0, ( i , j =1, 2). (39)

Substituting expressions ( 31) and (37) into (39) yields the following system for k e11 , k e12 , k

e22 :

E q 41 E q31 q2 E q

21 q

22

E q 31 q2 E q21 q

22 E q1q

32

E q 21 q22 E q1q

32 E q

42

12 k

e11

k e121

2 k e22 =

E q 61 E q41 q

22 E q

21 q

42

E q 51 q2 E q31 q

32 E q1q

52

E q 41 q22 E q

21 q

42 E q

62

1 3 5

. (40)

The solution of the system ( 40) can be written as

12 k

e11

k e1212 k

e22

= E q 41 E q

31 q2 E q

21 q

22

E q 31 q2 E q21 q

22 E q1q

32

E q 21 q22 E q1q

32 E q

42

1 E q 61 E q 41 q 22 E q 21 q 42 E q 51 q2 E q

31 q

32 E q1q

52

E q 41 q22 E q

21 q

42 E q

62

1 3 5

. (41)

For the Gaussian random processes with zero-mean, we have a general expression for expectation as follows(see [ 34,35] for details)

E [ z1 z2 . . . z2m ] =all independent pairs j=k

E z j zk , (42)

where the number of independent pairs is equal to (2m)! (2m m!). In view of the representation of expression(42), the following terms will appear in the right hand side of Eq. ( 41):

E q 4i =3 E q 2i2

,

E q 3i q j =3 E q 2i E qi q j ,

E q2i q

2 j = E q

2i E q

2 j +2 E qi q j

2

, E q 6i =15 E q 2i

3,

E q 5i q j =15 E q 2i2

E q i q j ,

E q 4i q2 j =3 E q 2i

2 E q 2 j +12 E q 2i E qi q j

2 ,

E q 3i q3 j =6 E qi q j

3

+9 E q 2i E q 2 j E q i q j .

(43)

By substituting expressions (43) into Eq. ( 41), we obtain the solutions k ei j (i , j = 1, 2) . The solution (41)allows to analyze responses of the nonlinear system (28) via approximate responses of the linearized system(36) based on the frequency-response function matrix as presented in Sect. 3.


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4.3 Dual criterion approach

For the nonlinear system ( 34), it is seen that the nonlinear vector function only depends on the displacementx. Thus, by applying Eq. (19) to the system (34) and noting that the moments E q q T , E qq T , E q T , E q

Tare equal to zero for stationary processes q , q of the linearized equation ( 36), we get the followingsystem for determining the unknown matrices K e , C e , D:

E qq T 0 1+

E q T

0 E q q T 0 E q T 0 E T

K eTiC eTiDTi

=1

1+ E i q

00

. (44)

By solving Eq. (44), we obtain the following solutions for K eTi , CeTi , and D

Ti :

C eTi = 0 0T , (45)

KeTi =

11 + G

1 E i q , (46)

DTi =1

1 + E T

1 E q T G1 E i q , (47)

where the matrix G is given by

G = E qq T

1 + E q T E T

1 E q T , (48)

and the following nonsingular conditions are satised for the matrices G and E T

det (G )= 0, (49)det E T = 0. (50)

As shown in [13,26], one has the following property for Gaussian vector process q with zero-mean andsufciently smooth vector function (q ) having the rst partial derivative with respect to q:

E q T = E qq T E T , (51)

where

=

q1

q2. . .

qn

T

is the Nabla operator. By utilizing the property (51) for Eqs. (48) and (46),

the equivalent stiffness matrix K e can be expressed as

K eT =1

1 +In

1 +

E T E T 1

E TT

E qq T1

E T , (52)

where we assume that all inverse matrices in Eq. ( 52) exist, In is the unit matrix of size n . From Eq. ( 52), it iseasy to see that the matrix K e is a symmetric one; thus, the linearized system of the original nonlinear system(28) takes the form of Eq. ( 36).

In this subsection, the dual criterion is considered; thus, the value = 1 is taken for calculating approx-imate mean-square responses of the nonlinear system ( 28). The moments appearing in the right hand side of Eqs. (52) are determined by the expressions ( 43).


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4.4 Conventional linearization

In the case =0, the expression (52) gives the equivalent stiffness matrix K e of conventional linearization inthe following form:K eT = E T . (53)

By using ( 53), we get the following expressions for k ei j (i , j =1, 2):k e11 = E

1q1 =12 1 E q

21 +2 3 E q 22 ,

k e12 = k e21 = E 1q2 =4 3 E [q1q2] ,

k e22 = E 2q2 =2 3 E q

21 +12 5 E q 22 .

(54)

We employ these expressions for calculating approximate mean-square responses of the system (28).

4.5 Calculations of approximate mean-square responses

We now evaluate approximate mean-square responses of the nonlinear system ( 28) based on linearization coef-cient matrices of the linearized system (36) with three different approaches including the energy criterion(41), conventional linearization( 54), and dual criterion ( 52). For thatpurpose, we employa frequency-responsefunction matrix H () corresponding to the system ( 36)

H () =1

L ()2 +i h +2e22 2e12

2e12 2 +i h +2e11, (55)

where

2e11 =

21 +k

e11 ,

2e12 = k e12 ,2e22 = 22 +k e22 ,

(56)

and L () is dened as

L () = 2 +ih +2e11 2 +i h +2e22 4e12 . (57)Under the assumption ( 29, 30) of the random excitations w1 (t ) , w 2 (t ), the spectral density matrix Sw () of w (t ) is given by

Sw () =S 0 00 S 0

. (58)

Using expressions ( 26), (55) and (58), we obtain a matrix of second-order moment elements as follows:

E qq T =

H () Sw () H T () d =

1 L () L ()

r 11 () r 12 ()r 12 () r 22 ()

d, (59)

where

r 11 () = S 0 2 +2e222

+h22 +4e12 ,r 12 () = 2e12 S 0(220 2e11 2e22 ),r 22 () = S 0 2 +2e11

2

+h 22 +4e12 .(60)


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It is easy to see that the expression (57) can be rewritten as

L () = R (i ) = 4 (i )4 + 3 (i )3 + 2 (i )2 + 1 (i ) + 0 , (61)where R (i ) is the function of argument i with the coefcients

4 = 1, 3 = 2h , 2 = 2e11 +2e22 +h 2 , 1 = h 2e11 +h2e22 , 0 = 2e11 2e22 4e12 .

(62)

By applying a formula of integrals presented by Roberts and Spanos [ 26], the integrals in Eq. ( 59) can becomputed as follows:

E q 21 =

r 11 () d R (i ) R (i ) =

4

0 2 1 0

4 2 0 00 3 1 00 4 2 0

3 1 0 0 4 2 0 00 3 1 00 4 2 0

, (63)

E q 22 =

r 22 () d R (i ) R (i ) =

4

0 2 1 0

4 2 0 00 3 1 00 4 2 0 3 1 0 0 4 2 0 0

0 3 1 00 4 2 0

, (64)

E [q1q2] =

r 12 () d R (i ) R (i ) =

4

0 0 1 0

4 2 0 00 3 1 00 4 2 0 3 1 0 0 4 2 0 00 3 1 00 4 2 0

, (65)

where r i j , (i , j =1, 2) are determined from ( 60); here, they are rewritten as follows:

r 11 () = 24

+ 12

+ 0 ,r 22 () = 24 + 12 + 0 , 2 = S 0 , 1 = S 0 h 2 22e22 , 0 = 4e22 S 0 +4e12 S 0 , (66)

2 = S 0 , 1 = S 0 h 2 22e11 , 0 = 4e11 S 0 +4e12 S 0 ,1 = 22e12 S 0 ,0 = 2e12 S 0 2e11 +2e22 .


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Table 1 Mean-square response of x 1 of the system versus the parameter 1 ( 3 = 5 =0.2) 1 E x 21 ex E x

21 co Error (%) E x

21 en Error (%) E x

21 du Error (%)

0.1 1.2294 1.1457 6 .8094 1.2860 4.6004 1.2782 3.97010.2 0.9737 0.8907 8 .5213 0.9978 2.4835 1.0060 3.32380.5 0.6866 0.6166 10 .2020 0.6904 0.5462 0.7054 2.73461.0 0.5146 0.4573 11 .1469 0.5134 0.2392 0.5278 2.55252.0 0.3798 0.3343 11 .9881 0.3764 0.9022 0.3887 2.34845.0 0.2498 0.2176 12 .8658 0.2455 1.7275 0.2548 1.99807.0 0.2134 0.1854 13 .1180 0.2091 1.9866 0.2174 1.874010.0 0.1802 0.1562 13 .3457 0.1762 2.2251 0.1834 1.7578

Table 2 Mean-square response of x 2 of the system versus the parameter 1 ( 3 = 5 =0.2) 1 E x 22 ex E x

22 co Error (%) E x

22 en Error (%) E x

22 du Error (%)

0.1 0.9565 0.8735 8.6789 0.9716 1.5859 0.9867 3.15630.2 0.9737 0.8907 8.5213 0.9978 2.4836 1.0060 3.32380.5 0.9948 0.9119 8.3366 1.0313 3.6646 1.0319 3.72551.0 1.0084 0.9237 8.3999 1.0515 4.2710 1.0482 3.9453

2.0 1.0196 0.9321 8.5876 1.0655 4.4924 1.0602 3.97905.0 1.0309 0.9399 8.8329 1.0775 4.5219 1.0713 3.91667.0 1.0342 0.9420 8.9094 1.0808 4.5135 1.0744 3.891910.0 1.0371 0.9440 8.9813 1.0838 4.5031 1.0773 3.8689

By substituting E q 21 , E q22 , E [q1q2] from ( 63), (64) and (65) into the expressions appearing in ( 43) and

the results obtained into Eq. ( 52) and noting Eq. ( 56), we get a nonlinear equation system of three unknownsk e11 , k

e12 , k

e22 . It is difcult to nd exact solutions of this system; thus, one can use a numerical scheme to

determine the solutions k e11 , k e12 , k

e22 as presented in Sect. 3 for three approaches, namely, the conventional

linearization, energy method, and dual criterion.

4.6 Numerical results and discussions

In this section, we present a numerical example of mean-square responses x 1 , x 2 . The numerical results areobtained by using four methods including the exact solution method ( 33), conventional linearization ( 54),energy method (41), and dual criterion method ( 52) with a cycle procedure scheme as presented in the previ-ous section. These results for approximate mean-square responses E q 21 , E q

22 are compared with others

in Tables 1 and 2. where E x 2i ex , E x 2i co , E x

2i en , E x

2i du are mean-square responses of x i (i = 1, 2)obtained by the exact solution, conventional linearization, energy method, and dual criterion method, respec-

tively. The system parameters used are h = 1, 1 = 2 = 1, S 0 = 1. The error between the results of theexact and approximate solutions is dened as

Error = E x 2i approx E x 2i ex

E x 2i ex 100 % , ( i =1, 2). (67)

Table 1 shows a comparison between errors of mean-square responses E x 21 of the system for variousvalues of the parameter 1 whereas both parameters 3 and 5 are xed at 0 .2. It is seen that when the param-eter 1 increases from 0.1 to 10.0, the error of the conventional linearization method increases from 6.8094 to13.3457%, whereas the errors of the energy and dual criterion methods are quite small, about 4 %. The energymethod gives a good prediction for the mean-square response of the system. However, for large values of 1 ,for instance, 1 =7, 1 =10, the dual criterion leads to smaller errors than those of the energy method.Table 2 presents exact and approximate values of the mean-square response E x 22 of the system ( 28). It isalso seen that errors of the conventional linearization are larger than the ones of the energy and dual criterionmethods. In this table, the smallest error of the conventional linearization is 8.3366%, while largest error of thedual criterion method is about 2 times less, namely, 3.9790%. In general, there is a good agreement betweenresults of the energy and the dual criterion methods for mean-square values of both responses x 1 and x 2 of thesystem ( 28).


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N. D. Anh et al.

Fig. 1 A model of a two-degree-of-freedom system with nonlinearity of damper and stiffness

5 A model of a two-degree-of-freedom system

5.1 Equation of motion

Consider the two-degree-of-freedom system shown in Fig. 1 (see [26]). In this system, the absolute displace-ments y1 and y2 of the masses m 1 and m 2 are measured from the static equilibrium position, respectively.The mass m 1 is connected to the foundation by a linear damper, with coefcient c1 , and a nonlinear springof the linear-plus-cubic type, with linear coefcient k 1 , whereas m1 and m2 are connected by a linear springof stiffness k 2 and a nonlinear damper of the linear-plus-quadratic type, with linear coefcient c2 . The forceof the nonlinear spring is given by k 1 y1 1 +1 y21 , whereas the force in the nonlinear damper is given byc2 ( y2 y1) ( 1 +| y2 y1|). Assume that a random excitation F (t ) = m1 p(t ) acts on the mass m1 , where p(t ) is obtained by passing white noise through a linear rst-order lter. The spectral density function S p()of p(t ) is assumed to be a rst-order spectrum of the form

S p() =S 0

2 +2, (68)

where and S 0 are constants. In terms of the coordinates y1 , y2 , we have the following equations of motion:

m1 y1 +c1 y1 k 2 ( y2 y1) +k 1 y1 1 +1 y21 c2 ( y2 y1) ( 1 +2 | y2 y1|) =m1 p(t ),m2 y2 +k 2 ( y2 y1) +c2 ( y2 y1) ( 1 +2 | y2 y1|) =0.

(69)

Since the linear spring force and nonlinear damper element depend on the relative displacement y2 y1 andrelative velocity y2 y1 , respectively, it is convenient to introduce the transformation x 1 = y1 , x 2 = y2 y1 .Denote21 =

k 1m1

, 22 =k 2m2

, =m2m1

, 1 =1k 1m1

,

2 =2c2m2

, 1 =c1

2 k 1m1 , 2 =c2

2 k 2m2 .(70)

In terms of the coordinates x 1 and x 2 , the system ( 69) can be written as follows:

x 1

+2 11

x 1

+21 x 1

2 22

x 2

22 x 2

+ 1 x 31

2

x 2

| x 2

| = p (t ) ,

x 1 + x 2 +2 22 x 2 +22 x 2 + 2 x 2 | x 2| =0. (71)The system (71) is of the standard form given by Eq. ( 1). Here

M =1 01 1 , C =

2 11 2 220 2 22 , K =21 220 22

, = 1 x 31 2 x 2 | x 2| 2 x 2 | x 2|

,

Q (t ) = p (t )0 .

(72)

In the next calculations, we use the dual criterion stochastic linearization for nding approximate mean-squareresponses of the nonlinear system ( 71).


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5.2 Mean-square responses of the system

In order to simplify the calculations following [ 26], it is assumed that the responses q1 , q1 , q2 , q2 of the line-arized system corresponding to the nonlinear system (71) can be approximately taken as zero-mean Gaussianand independent distributions with the following probability density functions:

f ( z) =1

2 z exp 12

z2

2 z, ( z =q1, q1 , q2 , q2 ) , (73)

where 2 z is the variance of the response z ( z =q1 , q1 , q2 , q2). With these assumptions, it is easy to calculateexpectations appearing in Eq. (19) in order to get the following system for determining the three matricesC e , K e and D:

E q 21 0 0 0 1+

1 E q 41 00 E q 22 0 0 0 00 0 E q 21 0 0 00 0 0 E

q 22

1

+ 2 E

q 2

|q2

| 1

+ 2 E

q 2

|q2

| 1 E q 21 0 0 2 E q 2 |q2| 21 E q 61 2 22 E q 42 22 E q 420 0 0 2 E q 2 |q2| 22 E q 42 22 E q 42

k e11 k e21

k e12 k e22

ce11 ce21

ce12 ce22

d 11 d 21d 12 d 22

=1

1 +

1 E q 41 00 00 0

2 E q 22 |q2| 2 E q 22 |q2|0 00 0

. (74)

Using ( 73), we have the following expressions for the elements appearing in Eq. ( 74):

E z2n

= (2n

1)

!! 2n z , ( z

=q1 ,

q1 , q2 ,

q2) , ( n

=1, 2, . . . ) , (75)

E q 22 |q2| =4

2 3

q2 . (76)

In this study, we consider two particular cases of Eq. ( 74) corresponding to =0 (conventional linearization)and =1 (dual criterion).In the rst case, =0, we obtain the following results for the equivalent matrices C e and K e :

C e =0 4 2 2 q20 4 2 2 q2

, Ke =3 1 2q1 0

0 0. (77)

The expressions (77) are presented by Roberts and Spanos [ 26].In the second case, =1, by substituting ( 75) and (76) into Eq. (74) and solving the obtained system, we

get the following solutions for Ce, K

eand D:

C e =0

3 23 4

2 q2

03 23 4

2 q2, K e =

157

1 2q1 0

0 0, D =

37

17

9 403 4

04

3 4. (78)

The linearization of the nonlinear system (71) takes the form

1 01 1

q1

q2 +2 11 2 223

234

2 q20 2 22+3

234 2 q2

q1q2 +

21 +157 1 2q1 220 22

q1q2 =

p (t )0 .

(79)


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N. D. Anh et al.

Denote

2e = 2 +3 2

2 (3 4) 22

q2 ,

2e =21 +157 1 2q1 .

(80)

The frequency-response function matrix of Eq. (79) is

H () = 2 +2i 11 +2e 2i 2e2 22

2 2 +2i 2e2 +221

= H 11 () H 12 () H 21 () H 22 ()

, (81)

where the elements H i j () , (i, j =1, 2) are determined as follows: H 11 () =

2 +2i 2e2 +222 +2i 11 +2e 2 +2i 2e2 +22 2 2i 2e2 + 22

,

H 12 () =2i 2e2 + 22

2

+2i 11

+2e

2

+2i 2e2

+22

2 2i 2e2

+ 22

,

H 21 () =2

2 +2i 11 +2e 2 +2i 2e2 +22 2 2i 2e2 + 22,

H 22 () = 2 +2i 11 +2e

2 +2i 11 +2e 2 +2i 2e2 +22 2 2i 2e2 + 22.

(82)

The variances 2q1 and 2

q2 are determined by the following expressions (see Eqs. ( 26), (27)):

2q1 =

H 11 () S p () H 11 () d, (83)

2

q2 =

2 H

21 (

) S

p () H

21 () d. (84)

Substituting ( 68) into (83) yields

2q1 = S 0

| H 11 ()|2 2 +2

d = S 0

r 11 () R (i ) R (i )

d, (85)

where

r 11 () = 24 + 12 + 0 , (86) R (i ) = 5 (i )5 + 4 (i )4 + 3 (i )3 + 2 (i )2 + 1 (i ) + 0 , (87)

with the following coefcients of r 11 () and R (i ) :

2 = 1, 1 = 22 4 22e 2 , (88) 0 = 42 , 5 = 1, 4 = +2 11 +2 2e2 +2 2e2 , 3 = 22 +2e + 22 +4 1 2e12 + (2 2e2 +2 11 +2 2e2) , 2 = 2 1122 +2 2e22e + 22 +4 1 2e12 +2e + 22 , (89) 1 = 2e 22 + 2 1122 +2 2e22e , 0 = 2e 22 .


15/18


The integral ( 85) can be evaluated by the following formula:

2q1 = S 0 5

0 0 2 1 0

5 3 1 0 00

4 2

0 0

0 5 3 1 00 0 4 2 0 4 2 0 0 0 5 3 1 0 00 4 2 0 00 5 3 1 00 0 4 2 0

. (90)

The integral ( 84) can be expressed as

2q2 = S 0

| H 21 ()|2 2 +2

d = S 0

r 21 () R (i ) R (i )

d, (91)

where

r 21 () =6 , (92)and R(i ) is determined from the expression (87). The value of the integral (91) is given by

2q2 = S 0 5

0 1 0 0 0

5 3 1 0 00 4 2 0 00 5 3 1 00 0 4 2 0 4 2 0 0 0

5 3

1 0 0

0 4 2 0 00 5 3 1 00 0 4 2 0

. (93)

Similar to Sect. 3, a numerical procedure for calculating the responses 2q1 , 2

q2 is performed. In order toevaluate the mean-square response E q 22 , we can use the following expression:

2q2 = S 0 5

0 0 1 0 0

5 3 1 0 00 4 2 0 00 5 3 1 00 0 4 2 0 4

2 0 0 0

5 3 1 0 00 4 2 0 00 5 3 1 00 0 4 2 0

, (94)

where i , (i = 0, 5) are determined by ( 90). The numerical results of variances 2q1 and 2q2 are presented inthe following numerical results section.

5.3 Numerical simulation results and discussions

Results of calculations of the above considered example are performed to compute approximate mean-squareresponses by the conventional linearization (CL) method( =0) (using 77) and the dual criterion (DC) method


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N. D. Anh et al.

Table 3 The error of mean-square response of the system versus the parameter 1 for case 2 =0.05 1 E x 21 MC E x

21 CL Error (%) E x

21 DC Error (%) E x

22 MC E x

22 CL Error (%) E x

22 DC Error (%)

0.01 1.9258 2.0210 4 .9457 2.1101 9 .5723 0.8856 0.8870 0 .1602 0.9076 2.48220.05 1.4307 1.4336 0 .2010 1.5883 11 .0183 0.8288 0.8172 1 .3937 0.8512 2.70170.10 1.1421 1.1229 1 .6849 1.2785 11 .9429 0.7743 0.7627 1 .5031 0.8038 3.80600.20 0.8809 0.8360 5 .0997 0.9736 10 .5207 0.7104 0.6930 2 .4556 0.7404 4.21670.50 0.5875 0.5353 8 .8772 0.6358 8 .2278 0.6042 0.5822 3 .6359 0.6351 5.12251.00 0.4226 0.3733 11 .6576 0.4464 5 .6325 0.5166 0.4891 5 .3196 0.5438 5.25722.00 0.3017 0.2593 14 .0673 0.3101 2 .7735 0.4289 0.3924 8 .5070 0.4465 4.10415.00 0.1950 0.1619 16 .9842 0.1926 1 .2556 0.3114 0.2685 13 .7733 0.3171 1.8259

Table 4 The error of mean-square response of the system versus the parameter 1 for case 2 =2.0 1 E x 21 MC E x

21 CL Error (%) E x

21 DC Error (%) E x

22 MC E x

22 CL Error (%) E x

22 DC Error (%)

0.01 1.8728 2.0255 8 .1560 2.0394 8 .8969 0.2737 0.2428 11.2876 0.2671 2.41020.05 1.4442 1.5122 4 .7099 1.6089 11 .4014 0.2464 0.2160 12.3428 0.2437 1.10260.10 1.1931 1.2178 2 .0699 1.3320 11 .6413 0.2268 0.1970 13.1524 0.2256 0.54880.20 0.9381 0.9307 0 .7861 1.0436 11 .2479 0.2025 0.1746 13.7724 0.2030 0.26870.50 0.6360 0.6110 3 .9267 0.7032 10 .5719 0.1665 0.1427 14.2695 0.1692 1.63441.00 0.4581 0.4281 6 .5484 0.5001 9 .1649 0.1397 0.1189 14.8613 0.1429 2.27692.00 0.3230 0.2930 9 .2729 0.3463 7 .2215 0.1148 0.0967 15.7336 0.1176 2.43585.00 0.1988 0.1732 12 .8882 0.2071 4 .1827 0.0854 0.0711 16.7965 0.0876 2.5181

( =1) (using 78). Because the exact solution and solution obtained from the energy method are not availablefor the nonlinear system ( 71), therefore, in this section, we use results of the mean-square response obtainedfrom a Monte-Carlo simulation method in comparison with those of other approximate methods (CL and DCmethods). In the Monte-Carlo simulation, the number of realizations is taken to be 10 4 , time step is 0.1 (sec),duration of simulation is 100 (sec). It yields the error about 1%. In order to increase the accuracy, one needsto increase the number of simulation realizations. However, the computation time will increase considerably.In this simulation, the random excitation p(t ) is simulated as white noise passing through a linear rst-orderlter. The result of the Monte-Carlo simulation is denoted by E x 2

i MC (i

= 1, 2). The results E x 2

i CL of

the CL method and E x 2i DC of the DC method are presented in Tables 3 and 4 for various values of theparameters 1 and 2 . The error between the results of approximate methods (CL and DC methods) and theresult of the Monte-Carlo simulation (MC) is dened as

Error = 2 z, approx 2 z,MC

2 z,MC 100 % , ( z =q1 , q2 ). (95)

Table 3 presents the resultsof mean-square responses of q1 and q2 for the case of the parameters 1 =2 =1, 1 =0.05 , 2 =0.2, =1, =2, S 0 =1, theparameter 1 varies from 0.01 to 5.00 whereas the parameter 2 is xed at 0.05. It is seen that when 1 is small, for example, =0.01 , 1 =0.05 , 1 =0.10 , 1 =0.20,the errors of the CL method are smaller than the errors of the DC method. However, when 1 increases, theerrors of the CL method are greater than the ones of the CD method. For instance, for 1 = 5.0, the error of the DC method corresponding to q1 is 1.2556% while the error of the CL method is 16.9842%.

In Table 4, the parameter 2 is xed at 2.0 and the parameter 1 varies from 0 .01 to 5 .00, other parametersare the same as in Table 3. For the single-degree-of-freedom Dufng system, it is well-known that when theparameter 1 of the cubic term is increasing, the error obtained from the conventional linearization methodalso increases. However, for the two-degree-of-freedom system (71), due to the interaction between nonlinearterms, this observation is not seen in Table 4. For the DC method, it yields a good result when 1 is increasing.For instance, in case 1 = 5, the error of the mean-square response E x 21 DC is 4.1827% which is about 3times less than the error of E x 21 CL . Also, the error of E x

22 DC is 2.5181% which is about 6 times smaller

than the one of E x 22 CL .


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6 Conclusions

The method of stochastic linearization is an effective tool for solving random vibration problems of multi-degree-of-freedom system. Many researchers have suggested different linearization criteria in some probabi-

listic sense to obtain equivalent coefcient matrices of the linearization system for improving the accuracy of the method. In this study, a dual criterion of stochastic linearization method for MDOF systems under randomexcitations is suggested. A closed system for determining coefcient matrices of the linearization system isobtained. For illustration, it is shown that, in two considered nonlinear two-degree-of-freedom systems withlarge nonlinearity, the accuracy of the dual criterion method is significantly improved in comparison withthe conventional linearization method. For a two-degree-of-freedom oscillator with nonlinear stiffness, it isobserved that there is a good agreement between the results of the energy and dual criterion methods. This is anadvantage of the dual criterion method for the system possessing the potential energy. Also, for the consideredtwo-degree-of-freedom system with nonlinearity of damping and stiffness, the dual criterion method gives agood prediction on the mean-square responses of the system. Further investigations seem to be appropriate inorder to verify the advantages of this proposed dual criterion method.

Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development(NAFOSTED) under grant number: 107.04-2011.13 (09-Mechanics).

Appendix A

We prove expressions ( 13, 14, 15) as follows. By using (9), we have

cei j

n

r =1 r =

cei j

n

r =1 E r

n


2

+ E n

s

=1

cer s qs +k er s qs n

s

=1

d r s s

2

=n

r =1 E 2 r

n


n

s=1cer scei j qs

+ 2 n


n

s=1d r s s

n

s=1cer scei j qs

=n

r =1 E 2 r

n


n

s=1ir js qs (96)

+ 2 n


n

s=1d r s s

n

s=1ir js q

=n

r =1 E 2 r

n

j=1cer j q j +k er j q j q j ir

+ 2n

j=1cer j q j +k er j q j

n

j=1d r j s q j ir

= E 2 i n

j=1cei j q j +k ei j q j q j +2

n

j=1cei j q j +k ei j q j

n

j=1d i j j q j

=

cei j i ,


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N. D. Anh et al.

where the following property is used

x r s x i j =ir js , (97)

for all variables x i j (i, j =1, 2, . . . ).Similarly, we also obtain the expressions (14) and (15). The proofs are complete.

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anh-zakovorotney a dual criterion of stochastic linearization method for multi-degree-of-freedom...

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