A stationary model for periodic excitation with uncorrelated random disturbances

ProbabilisticEngineeringMechanics11 (1996) 191-203 PII:

ELSEVIER

S0266-8920(96)00014-8

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0266-8920/96 $15.00

A stationary model for periodic excitation with uncorrelated random disturbances Z. Hou, Y. Zhou, M. F. Dimentberg & M. Noori Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USA

The paper presents a stationary model for periodic excitations with random amplitude and phase disturbances for linear and nonlinear random vibration analysis. The disturbances are modeled as uncorrelated stationary white noise processes. Application of the model is demonstrated by stationary moment response of a linear single-degree-of-freedom system subject to such excitations. To find moment responses, an equivalent augmented system subject to parametric white noise excitations under certain constraint conditions is studied. Numerical results for the second and fourth-order moment responses are presented. The probability density function of the response is calculated based on the cumulantneglect closure method. NonGaussiauity of the response is discussed in terms of the excess factor. The results show that the random amplitude disturbance can significantly increase system moment response. The random phase modulation may increase or reduce the system moment response, depending on the value of relative detuning between the system natural frequency and the mean excitation frequency. The response may become Gaussian in the sense of up to the fourthorder moment for sufficiently large random phase or relative detuning. Copyright © 1996 Elsevier Science Ltd.

INTRODUCTION

factor. Comparison of the exact solution with an approximate solution by a stochastic averaging method was presented. Effects of important system parameters on the stationary response were investigated. Recently the model has been extended to the nonstationary case where the nonstationary random phase disturbance was modeled as a modulated stationary process, s'9 Different types of envelope function are employed in the study in order to investigate the effects of build-up, decay rate and duration of the nonstationary phase modulation. The nonstationary second and fourth-order moments are computed by numerically solving the transient moment equations. The significance of nonstationarity and nonGaussianity is demonstrated by some numerical results. However, the model of periodic excitation with random phase modulation was originally developed to represent basically periodic phenomena with random deviations from perfect periodicity, where fluctuations of the excitation amplitude are of secondary importance as compared with those o f the excitation frequency and, therefore, are ignored. As a result the amplitude of the periodic excitation is a constant so that the excitation magnitude is limited to a certain range. It may not be

The model of periodic excitation with random phase modulation was used by Dimentberg 1 and Wedig 2 in the investigation of Mathieu-type stochastic systems. By appropriately describing spatial random inhomogenuities, which lead in the time domain to random timewise variations of the parametric excitation frequency, such a random process model was applied to the studies of parametric excitation of a straight pipe due to slug flow of a two-phase fluid, 3 stability and subcritical dynamics o f structures with spatially disordered traveling parametric excitation. 4 Another application of the model was in the stability analysis of bridges in turbulent flow, where the turbulent fluctuation in the wind flow was represented by phase m o d u l a t i o n : The form of periodic functions with uniformly distributed random phase disturbance was also employed in simulation of earthquake ground motions. 6 The nonGaussian nature of stationary response of SDOF systems under periodic excitation with random phase modulation was generally discussed in Ref. 7, where equations were derived for up to the fourth-order moment and nonGaussianity was studied in terms of the excess 191

192

Z. Hou et al. ~.=1 p.=l DI=0 D2=0.02 1.5 l

• "..

. . . . . . Periodic excitation Periodic excitation with small phase modulation

.°o°"•.

.° ",..

.°"

0.5

A.~

0

-0.5

-1

-1.5 t

Fig. 1. Periodic excitation with small phase modulation. satisfactory for many engineering problems which involve comparable fluctuations in both amplitude and phase, such as modeling of nonstationary seismic ground accelerations which may exhibit nonstationarity in both amplitude and frequency content, l° This paper attempts to further develop an excitation model which includes randomness in both amplitude and phase. As a preliminary study, the random amplitude and phase disturbances are treated as uncorrelated stationary white noise processes. Application of this model to a linear SDOF system is demonstrated by solving an equivalent augmented system subjected to uncorrelated stationary white noise processes, as parametric excitations, with certain constraint conditions• Moments of stationary response up to the fourth-order are presented. The probability density function of the response is computed by the cumulant-neglect closure method. The excess factor is used as an index of nonGaussianity. The significance of amplitude and phase modulation is illustrated by some numerical results.

FORMULATION The equation of motion of the system under consideration is

d2x

dt 2 + 2a

+ f~2x = y(t)

(1)

where x(t) is the displacement response, f~ and a are, respectively, the natural frequency and damping constant of the system, and y(t) is the external excitation. For a periodic excitation with amplitude and phase

modulation, y(t) has the form

y(t) = (A + ~1(t)) cos v dv d--~=/.t + ~2(t)

(2)

where A is the mean magnitude and # is the mean frequency of the periodic excitation. Both of them are constant here. ~l(t) and ~2(t), the amplitude and phase disturbances, are stationary random processes. In the present study, ~l(t) and ~2(t) are assumed to be uncorrelated Gaussian white noise processes with zero mean, the intensities D1 and D2, respectively, i.e. E[~l(t)] = E[¢2(/)] = 0 E[¢1 (t)~l (t + 7-)] = D16(T )

E[~2(t)~2(t + r)] = D26(r ) E[~l(t)~2(t + r)] = 0

(3)

In eqn (3), 6(-) is the Dirac's delta function and E[. ] is the mathematical expectation operator. An index of nonGaussianity of the response used here is the excess factor K, defined by g

:

3

E[x4]

(E[x2])2

(4)

Both second and fourth-order moments of the response are required in order to evaluate K. For a Gaussian process, the excess factor K should be zero. The original system (1) subjected to the periodic excitation with random amplitude and phase disturbances expressed by eqn (2) can be reformulated in the state space by introducing the following state

Stationary model for periodic excitation

193

k=l p.=l Ot=O.O01 D2=0 1.8 ......

Periodic excitation Periodic excitation with $rr~ll amplitude modulation

1.2

0.6

v

I

I

I

7

-0.6

-1.2

-1.8

t

Fig. 2. Periodic excitation with small ampfitude modulation.

variables:

Stratonovich stochastic differential equations: dz 1 Z2 dt dz2 d--'7= -f~2zl - 2a z2 + [A + ~l (t)] Z 3

Z1 ~ X

--=

dx

22

-~-~-

Z3 ~ COS V 2 4 -~-

dz__2=

sin v

dt

(5)

In view of eqn (2), the corresponding stochastic Cauchy problem for the state variables zi (i = 1,2, 3, 4), may be written as a set of the following four 'physical' or

- [ # + ~2(t)] z4

dz4 dt = [# + ~2(t)] z3

(6)

Recognizing the difficulties of solving the associated

8.

4.

....

0

"t"i

r~

Fig. 3. Periodic excitation with large amplitude and phase modulations.

194

Z. Hou et al.

:l/I t 1.11 ~0 Dt/a

It8

16

Fig. 4. Second-order moment as a function of D1Q/A2 and D2/a.

Fokker-Planck equation for the probability density function of the response, this SDE set is analyzed by the method of moments. Due to the autonomous representation of trigonometric functions in eqn (5), this set is supplemented by an obvious constraint equation

z~ + z,~ = 1

(7)

By appropriately applying the Ito differential rule and the mathematical expectation operator, the deterministic moment equations for various orders of the response can be derived. In this study, after dealing with the extra constraint equations caused by eqn (7),

nine independent differential equations for the secondorder moments and 25 independent differential equations for the fourth-order moments are fully obtained. These equations may be arranged in matrix form as d {MII} = JAIl]9x 9(gli} + (CII}

~ (M~v} = [~w]25 × 25(M~v} + { q v }

where {MII} and {Mr,,}, JAn]9×9 and [Aiv]25×25, {Gl} and {Clv} are unknown moment vectors, coefficient matrices, and constant column vectors for second and

4.8

~ltl.I 1.11 0

~

: I),ta

(8)

0.11

111

Fig. 5. Fourth-order moment as a function of DI~2/A2 and DJa.

Stationary model for periodic excitation

195

l}-! b.OJ~ ~do,Oalf:lbO.I

~g

!.11

r. 0

IMa

Fig. 6. Excess factor as a function of Dlf~/A 2 and D2/a. fourth-order moments, respectively. Detailed expressions for these vectors and matrices are presented in the Appendix. Since only a stationary response is considered in the present paper, the time derivatives of the moments in eqn (8) are set to be zero and linear algebraic equations for the moments are solved numerically. In a later part of this study, the following nondimensional parameters are used to characterize the system: a / ~ represents the critical damping ratio of the system; DIQ/A ~ is the normalized intensity of amplitude fluctuation; D2/a denotes the bandwidth ratio of the excitation to the system; and A / a = (#2 _ f~2)/2# a is defined to describe the relative detuning between the

natural frequency of the system and the expected frequency of the excitation. There are two extreme cases. As D]~/A 2---, 0 and D2/a ~ O, which is the case where random amplitude and phase modulation vanishes, the response of the system reduces to a perfect periodic process. As DlQ/A 2 --~ 0 and D2/a -* oc, which is the case where random phase modulation is dominant, the response should reduce to a Gaussian process due to the normalization effect. As results, we have lim

K=15

lim

D I Q / A 2 ---*0 D2o~ ---*O

K=0

D I Q / A 2 --~0 D2a --* ~

I~t=t ;~=0.25 a / 9 ~ 0 . 1 0 ~ w : 2

/

l...O,t~2=° 2-~13/1,t= 1.6

3.-0,~."-3.2 4.-.Dd~k~*4.8

!

4_

-t5

-12

4

41

3

-3

8

9

12

A/a

Fig. 7. Second-order moments as a function*of Aa with different Dlf~/A2.

15

(9)

196

Z. Hou et al. O=l )--0.25c~l-O.I HI=

D-,/~2

I--D~IV~:-O 2.--~OrkZ-l.6 1.5.

3--.o,orx~r~2

4-.0,0/).=-4.8

! -t5

-t2

-g

-4

3

-3

6

9

t2

15

Ala

Fig. 8. Fourth-order moment as a function of A/a with different D]f~/A2. which may be easily evaluated by moments of the response of up to the /th order. By assuming that cumulants higher than a certain order for nonGaussian processes vanish, an approximate characteristic function is obtained by using finite terms in eqn (10). Then, the PDF of the response can be calculated by taking the inverse Fourier transform of the characteristic function. In the present study, cumulants higher than fourthorder are neglected.

The above asymptotic properties may be used to verify the numerical results. Once the moments of the response are obtained, the probability density function (PDF) of the response can be calculated by using the so-called cumulant-neglect closure method, ll'12 The characteristic function of the response can be generally expressed as

where i is the imaginary unit and ~l(X) is the lth cumulant of I", defined as

1 dllnPx(f) f=o ~l(X) = (i2~r)! d f z

NUMERICAL RESULTS To illustrate applications of the proposed model, numerical results for stationary moment responses of

(11) Jl.l:

l.--O~O/kZ-O 2---010/k~,1.6 ~OJl"

3--0,0/~,:'3.2 4-.oln~Z=4.8

I

I

9

Fig. 9. Excess factor as a function of A/a with different DI£I/A2.

12

Stationary model for periodic excitation

197

f~l X=O.2Secf~O.l D~fkO.s-t.6 l-.-D~o~=0 2--D'~w4 ~ 1 2

-15

-12

-6

4)

-3

0 M~

3

6

9

12

15

Fig. 10. Second-order moment as a function of A/t~ with different D2/a.

linear SDOF systems subjected to a periodic excitation with random amplitude and phase disturbances are presented. As a preliminary study, these disturbances are assumed to be uncorrelated. The nonGaussian nature of the response is investigated in terms of the excess factor as well as the probability density function of the response. To help us visualize the excitation process y(t), samples of the proposed model are graphically presented in Figs 1-3 based on computer simulation. Figure 1 shows the result of periodic excitation with small phase modulation only. The phase shift caused by phase modulation can be clearly observed. The excitation is bounded between - 1 and 1. Figure 2 depicts the periodic excitation under small amplitude disturbance,

but no phase modulation. There are high frequency amplitude fluctuations and the excitation may go beyond the range [-1,1]. The periodic excitation subject to both large amplitude and phase modulations is shown in Fig. 3. The excitation looks like a general stochastic process and the periodicity is no longer visible. By adjusting the amount of random disturbances in both amplitude and phase, the excitation process y(t) may be flexibly used to model different types of loading in engineering applications. Figures 4-6 presents the second and fourth-order moments as well as the excess factor of the stationary response as functions of Dlf~/A 2 and D2/a. It is observed from Figs 4 and 5 that increasing the amplitude fluctuation level, D1, results in larger second

fl,,I 1.=0.25 a/Q-O.! Dd;I/£~,.I.6

2

~

~ 1 2

J

1

-15

-12

4)

-6

-3

0

3

6

0

A/a

Fig. 11. Fourth-order moment as a function of A/c~ with different D2/a.

12

15

Z. Hou et al.

198

G/=.).-0.25 i Wfit,=O.I~R~a,,.I.6 i 1

.8 . ~

a---og,~a

0.0.i/

~

f, 2

2

0.2. '

~a Fig. 11. Excess factor as a function of A/a with different D2/o~. that K converges to zero for sufficiently large D2/a in the case of phase modulation alone and decreases to negative values for very large D]f~/A2. The results in the present case suggest that the response may become Gaussian for sufficiently large random phase disturbances at least in the sense of up to the fourth-order moment. Although it seems that K is more sensitive to D1 than to D2, the roles of D1 and D2 may not be compared directly since they have different physical interpretation. The role of relative detuning between the system natural frequency f~ and the mean excitation frequency # can be seen in Figs 7-9. Curves of the second and the fourth moments and the excess factor vs the detuning

and fourth-order moment responses, while increasing the phase modulation level D2 causes smaller moment responses in this resonance case (A/o~ = 0). The latter may be attributed to the normalization effect. The role of D2 may change with detuning A / a , as we can see later. In comparison with the case of a perfect periodic excitation, i.e. Dl = D2 = 0, the numerical results show that the existence of the random disturbance in either amplitude or phase significantly changes the moment responses. As shown in Fig. 6, the excess factor, K, achieves the value of 1.5 (as expected) when DIQ/A 2 and D2/a are all equal to zero. K decreases with increasing D]f~/A 2 and/or D2/a. Further calculations indicate

Q-I X-0.25 Or=0.2 ~,~-0 I.B.

1.111.

1.4.

1.2,

i'

~ 0.8 0.6 0.4 0.2

0 0.051

I 0.061

I 0.111

I 0.141

I 0.171

I 0.201

I 0.231

I 0.28t

a/O

Fig. 13. Second-order moment as a function of a/f~ with different D1.

I 0,291

Stationary mode~for periodic excitation

199

0,,,I 1,,,,0.25 D+=0.2 A/o=o

8

1--O+=0 2---D1=0.1

7

!

6

4

3 2 1 0 0.051

0,061

0.111

0.141

0.171 o/n

0.~91

0.231

0,261 .

Fig. 14. Fourth-order moment as a function of c~/f~ with different D factor, A / a , for a fixed D2/a and four different levels of DIQ/A 2 are illustrated. There is a peak near the resonance (A/a = 0) for each individual curve. The

1.4 I 1.2 +

1--D~=O 2--01',0.1 3---0+=0.2 4--D~=0.3

0.~"

0.60.4

0.2

l

o

1.

some larger values of IAx/al, the response may increase with D2/a. As expected, increasing the damping coefficient, a, will reduce system moment responses, as seen in Figs 13-18. The excess factor K may become zero at some points. Note that a Gaussian process has zero K. Therefore, for some particular combinations of parameters, the response of the system under nonGaussian excitation may demonstrate some Gaussian nature. The probability distribution may be close to that for a Gaussian process as far as only the moments not higher than the fourth order are concerned. The probability density function of the system response with different levels of DIQ/A 2 and D2/a are

moment response drops rapidly as IA/a] goes away from peak point and approaches a limiting value as IA/a] becomes larger. Larger amplitude disturbance increases the system moment response, as expected. It is also observed that the excess factor, K, decays to zero for sufficiently large positive detuning (# >> f~). In other words, the system response will be closer to a Gaussian distribution for larger A / a . Similar results are plotted in Figs 10-12 w h e r e DI~/)~ 2 is fixed and four different levels of DE/a are used. For very small ]A/a], the moment response always decreases with D2/a , but for

1.

0.291

.

l

~ ~ .

-0,2 ¸

-0.4

6/tl Fig. 15. Excess factor as a function of a/f~ with different D1.

Z. Hou et al.

200

~ 1 ),,O.25tgop0131,,0.1

3

?..5, ~

2~.4 4--o~'~ .2

1.

'

2

0.~ ~ i1[ it,,

01.

0.061

,! 0.081

~ 0.1'11

i,

o

0.141

0.171 a/rJ

,,

'"'

! 0.20t

'

! o.~1

I

i

! o.2111

i

i

|

0.~

Fig. 16. Second-order moment as a function of a/f~ with different D2. cautious when it is applied for estimating exceedance probabilities of high threshold values, as indicated in the literature,In

plotted in Figs 19 and 20. For comparison, the corresponding Gaussian distribution curves are also obtained by assuming that moments higher than second order are zero. As Dtf~/~ 2 increases, the curve becomes flatter and closer to the corresponding normal distribution, which agrees with results on nonGaussianity in terms of excess factor K. For large D2/o~, the difference between the PDF of the response and the corresponding normal distribution is also small, but the PDF curve becomes sharp. It should be mentioned that the cumulant-neglect closure method yields some negative PDF values (in the order of 10-3 or less in this study) for relatively large x, and therefore one should be

CONCLUSIONS A stochastic model for periodic excitations with uncorrelated random amplitude and phase disturbances is presented. Application of this model is demonstrated by stationary moment responses of a linear SDOF system subject to such an excitation, Numerical results for the second and fourth-order moments are illustrated

i')-,l J~,,0.2.$A/cI,,O D1=0.1

20 ¸ 18 16

1--Or~0 4---01-I.2

0|"

0.051

I

•

0.08t

0.11t

I

•

0.t41

i

II

I~,

0,171

I m

0.20t

~

o.~1

' "

|

0.261

W~ Fig. 17. Fourth-order moment as a function of ~/f~ with different D2.

0.291

Stationary mode~for periodic excitation fl,-I ~

201

A/a-O D ~ 0 . 1

1.2

1--D~ 2--.D,p0.4

3--D~0.8 4--D~1.2 0.8

0.8

0.4

0.2

0.081

0.111

4

0.1ql

-n.*'rl

0.201

-0.2 W[I

Fig. 18. Excess factor as a function of t~/f~ with different D2.

to show the effects of the level of the random disturbances, detuning and damping on the system response. The nonGaussian nature of the response is investigated in terms of t ~ excess factor as well as the probability density function. It has been shown that large amplitude disturbance will increase the system moment response, while large random phase modulation may increase or decrease the response, depending on relative detuning values. For sufficiently large random phase disturbances or relative detuning, the response process becomes closer to a Gaussian process in the sense of up to the fourth-order moment. Larger

detuning and system damping may result in smaller system moment responses. For some specific combinations of parameters the excess factor may become zero, implying that the response may be approximately treated as a Gaussian process in the sense of up to the fourth-order moment. The results may be extended to the case of correlated disturbances.

ACKNOWLEDGMENT This research is partly supported by ONR-URI under

i ~ l b,,0.2$ h/or-0 ~ t l - O . I D d e p 2

i-'~--~ -°°~'=°

0.0 -I~ I~___~...~-,~-Dd;:~l/~

o.0 ~

~

'

3

.

1 I

. -1.6lGau~iarl I

2

(

~

J

1

r

0.4

0.3

0.2

°-I

I 0.26

I 0.5

I 0.76

I 1

I 1.25

I 1.5

X

Fig. 19. Probability density function with different Dzf~/A 2.

I -1.76

202

Z. Hou et al. t ' ~ i b . o . 2 5 Atot.,..o o/t'i..0.10,dt'l/k~-l.e 1.o6.

0.7,

Ik

2 0.35

0.2

0.4

0.6

0.8

1 X

1.2

1.11

t.4

1.11

Fig. 20. Probability density function with different D2/o~.

grant no. N00014-93-1-0917. This support is gratefully acknowledged by the authors.

REFERENCES 1. Dimentberg, M. F., Statistical Dynamics of Nonlinear and Time-Varying System. Research Studies Press, Taunton, 1988. 2. Wedig, W., Analysis and simulation of nonlinear stochastic systems. In Nonlinear Dynamics in Engineering Systems, ed. W. Schiehlen. Springer, Berlin, 1989, pp. 337-344. 3. Dimentberg, M. F., A stochastic model of parametric excitation of a straight pipe due to slug flow of a two-phase fluid. Proceedings of the 5th International Conference on Flow-Induced Vibrations, Brighton, 1991, pp. 207-209. 4. Dimentberg, M. F., Stability and subcritical dynamics of structures with spatially disordered traveling parametric excitation. Probabilistic Engineering Mechanics, 1992, 7, 131-134. 5. Lin, Y. K. and Li, Q. C., New stochastic theory for bridge stability in turbulent flow. Journal of Engineering Mechanics, 1993, 119, 113-127. 6. Shinozuka, M. and Deodatis, G., Stochastic process models for earthquake ground motion. Probabilistic Engineering Mechanics, 1988, 3(3), 114-123. 7. Dimentberg, M. F., Hou, Z. K., Noori, M.N. and Zhang, W., NonGaussian response of a single-degree-of-freedomsystem to a periodic excitation with random phase modulation.

ASME Recent Development in the Mechanics of Continua (special volume) ASME-AMO, 160, pp. 27-33, 1993. 8. Hou, Z. K., Zhou, Y. S., Dimentberg, M. F. and Noori, M., Nonstationary nonganssian response of linear SDOF system under randomly disordered periodic excitation. Proceedings of the International Conference on Vibration Engineering, Beijing, 1994, pp. 415-420. 9. Hou, Z. K., Zhou, Y. S., Dimentberg, M. F. and Noori, M., A nonstationary stochastic model for periodic

10. 11. 12.

13.

excitation with random phase modulation. Probabilistic Engineering Mechanics, 1995, 10, 73-81. Grigoriu, M., Ruiz, S. E. and Rosenblueth, E., Nonstationary models of seismic ground acceleration. Earthquake Spectra, 1988, 4(3), 551-568. Zeman, J. L., Zur Losung Nichtlinearer Stochastischer Probleme der Mechanik, Acta Mechanica, 1972, 14, 157169. Wu, W. F. and Lin, Y. K., Cumulant neglect closure for nonlinear oscillators under random parametric and external excitation, International Journal of Nonlinear Mechanics, 1984, 19, 349-362. Schueller, G. I. and Bucher, C. G., Nongaussian response of systems under dynamic excitation. In Stochastic

Structural Dynamics (Progress in Theory and Applications), eds S. T. Ariaratnam, G. I. Schueller and I. Elishakoff. Elsevier, New York, 1988, pp. 219-239.

APPENDIX The m o m e n t vectors, coefficient matrices and the constant vectors in eqn (10) are presented as follows. F o r the second-order m o m e n t equation, the m o m e n t vector {Mn} is {MII} t = {roll, m12, m13, m14, m22, m23, m24, m33, m34) where

mij=E[zizj],

i = 1,2,3; j = 1,2,3,4

The nonzero elements of coefficient matrix [An] are a12 = 2, a2] = _f~2, a22 = - 2 a , a23 = A, a24 = 0 1

a25 = 1, a23 = - ~ D 2 a34 = - # , a36 = 1, a43 = #, a ~ = -½D2, a47 a52 -------2~ 2, a55 = --4cz

-----

1

Stationary model for periodic excitation

203

as6 --- 2A, a57 = 0, a58 = D1, a63 = --{22

all,18 = 1, &12,7 = - 2 ~ 2 ,

ass --- - ( 2 a + I o 2 ) , a67 : - ~

a12,12 = - ( 4 a + ½D2), &12,13 = 0, a12a4 = 2~

a68 ~- A, a69 --- 0, a74 : _ ~ 2 , a77 --- - ( 2 a

+lD2)

a76 : #

&12,n = #

a~2,16 = D~, a12,19 = 1, a13,8 = _Q2

a13,13 = - 2 ( a + D2), his,14 = - 2 # , a13,1s = ,k

, a78 = 0

a79 = )% a88 --- - 2 D 2 , a89 = - - 2 # , a98 = 2#

h~3,16 = 0, a~3,20 = 1, h14,9 = -f22, &~4,13 ----2#

a99 = - 2 D 2

&14,14 =

The n o n z e r o elements o f constant c o l u m n { C n } are C7 = 0 ,

C8 --~D2, c 9 = - #

{ M I v } t = { m l l l l , m1112, m1113, mll14, m1122, m1123, m1124, m1133, m1134, m1222, m1223~ m1224~ m1233, m1234, m1333, m1334, m2222~ m2223, m2224, m2233, m2234, m2333 ~m2334 ~m3333,

where

t)15,16 = - 3 / z

&16,23 = 1, &17,10 = - 3 Q 2 ,

&17,17 = - 8 a , ,

&17,18 = 4A

a17,19 = 0, alT,19 = 6Dl a18,11 = --3Q2, &18,18 = --(6oe + 1 D 2 ) , &18,19 = - #

h18,20 = 3A, t~18,21 ~---0 a18,22 = 3D1, a19,12 = --3~22, a19,18 ---- #

tl19,21 = 3A, &19,23 = 3D1, &20,13 = - 2 Q 2 :

E[z i zj z k Zl]

i=1,2,3,

a20,20 = -(4c~ + 2D2), a20,21 = -2/z

j---1,2,3, k=1,2,3,

l=1,2,3,4

The n o n z e r o elements o f coefficient matrix [Aw] are

4, (12,1 :

_Q2

a2,2 :

-2a,

a2,3 = ,~, a2,4 = 0

&2,5 = 3, a3,3, = - 1 D 2 , a3,4 = - - # a3, 6 = 3, an, 3 = / z , aa,4 = - 1 D 2 , &5,2 = - 2 Q 2 ,

&5,5 = - 4 ~ ,

&4,7 = 3

&5,6 = 2)~

t~5,7 = 0, a5, 8 = D1, t~5,10 = 2, t~6,3 = _ Q 2 t~6,6, :

hi4,16 = A, a14,21 = 1, &15,15 = - ~ D 2 ,

hi9,19 -------(6ct + I D 2 ) , t~19,20 -----0

m3334}

al,2 :

, t~14,15 = 0

&15,22 = 1, &16,15 = 3#, &16,16 = - 9 D 2

For the fourth-order m o m e n t equation, the m o m e n t vector { M t v } is

mijkl

-2(a + D2)

-(2a

+ I D 2 ) , (16,7 = - ] z , t~6,8 : )~

a6,9 = 0, a6,11 -- 2, t~7,4 = _ ~ 2 , &7,6 ---/z aT, 7 = --(2ot q- ½D2) , aT, 8 = 3D12 , aT, 9 = )~

t~20,22 ~---2A, &20,23 = 0, &20,24 ~---D I , a21,14 ~-~ - 2 Q 2

h2L2o = 2# h2Lel = --(4a + 2D2) , a21,22 = 0, a2L23 = 2A &21,25 = Dl, a22,15 = _~t-~2 a22,22 = -(2c~ + 9D2), a22,2s = - 3 # , a22,24 -- k a22,25 = 0, a23,16 ~--- _ ~ 2 a23,22 =

3#, h23,23

=

-(2a

+ 9D2) , t~23,23 --~ 0

(123,25 = A, &24,24 = - 8 D 2

a24,25 = - 4 # , tl25,24 ~-- 4#, a25,25 ~- - 8 D 2 The n o n z e r o elements o f constant column {Civ} are

a7,12 = 2, &8,8 = -2/92, &8,9 = - 2 # 2 , &8,13 = 2

c7 = 0, ~8 --- D2mll, c9 = -/Zmll, c12 = 0, c13 = D2ml2

a9,8 = 2#2, a9,9 = - 2 D 2 , a934 = 2

cl4 = - # m l 2 , cl5 = 3D2m13, C16 = D2ml4 - 2>m13

alo,5 = - 3 ~ 2 , hlO3O = - 6 a , alO,H = 3,k, hlO, t2 = 0

C19

:

0

&1o,~3 = 3D1, am,iT - 1

C20

:

D2m22, c21 = -/zm22, C22 ~--- 2D22m23

alL6 = - 2 f~2, aH,H = --(4a + ½D2), a11,12 = -t*

023 ~-~ D2m24 - 2/zm23, ~'24 ~--- 6D2m33

an,13 = 2)~, hl~,~4 = -3D12, an,is = D~

C25 : 3D2m34 - 3#m33