A.1 Introduction

APPENDIX A

GRAM-CHARLIER COEFFICIENTS

In this section, the relations between the Gram-Charlier coefficients and the

moments and cumulants of joint random variables are given. These relations have been

obtained from Beaman and Hedrick(1981) and Nigam(1983).

A.2 Gram-Charlier Coefficients

A multivariate Gram-Charlier expansion of a probability distribution p(x) for the n­

dimensional vector X is written as

().iT cp(x) (A.I)

where

(A.2)

The symbol cP(x) represents a multivariate gaussian distribution which can be written as

(A.3)

where the matrix S is the covariance matrix for the vector x. The Gram-Charlier coefficients, rjlj2'''jn' are related to the cumulants, Kjd2'''jn ' of

the joint random variables by the equation

1 jT= 0

0 0< jT< 3 rjd2'''jn =

(_l)h Kjlh'''jn . , . i ., h· J2' ... In' jT~ 3

The joint moments and cumulants are related to each other through the joint

characteristic function, MOO, which is defmed as

(A.4)

152

M(e) !E < exp(i e . x.) > = J .... J exp(i e . 3) p(3) d.x (A.5)

n

In this equation, the symbol (.) represents the scalar product of two vectors and i = {:l.

The joint moments of the vector ~ are related to the derivatives of the characteristic function

by the equation

(A.6)

The above moments are algebraically related to the joint central moments, J.l.hh'''jn ' which ~

are defmed as n .

lLIJ'Z"'J'n = < n (x·_<X·>)Ji > r-J i=1 1 1 (A.7)

The joint cumulants of the vector ~ are related to derivatives of the natural logarithm

of the characteristic function by the equation

(A.8)

B.1 Introduction

APPENDIX B

EVALUATION OF EXPECTATIONS

In. this appendix, details of the evaluation of two types of expectations are given.

The first kind involves the quadratic nonlinearity used throughout this study. Fonns of this

expectation are given by equations (2.66), (2.71), and (5.20). The second kind of

expectation involves central moments of random variables whose probability distribution is

approximated by a Gram-Charlier expansion. Recall that due to computational limitations,

only moments up to third order are included in the expansion. Thus, fourth and higher

order cumulants of the approximate probability distribution are zero. The fourth and higher

order central moments, however, are not zero and may need to be computed. In. particular,

the fourth order central moment is needed to compute the quadratization coefficients.

Higher order moments may also be needed depending on the nonlinearity. A simple

method for approximating higher order central moments based on the truncated probability

distribution is described.

B.2 Expectations Involving Quadratic Nonlinearity

Consider the following expectation involving a general nonlinearity g(x)

k = 0,1,2 (B.1)

The non-gaussian random variable x is a represented by its mean, J.I.x, and a zero-mean

component x by the equation

(B.2)

The expectation in equation (B.1) needs to be evaluated to detennine the mean response and

the quadfatization coefficients as described in Chapter 2. Using the defmition of

expectation, Ek is written as the following integral

k=O, 1,2 (B.3)

154

where p(i) is the probability distribution of i. This non-gaussian distribution is

approximated by the truncated Gram-Charlier expansion

(B.4)

where cp(i) is a gaussian distribution defmed by the equation

(B.5)

and ~im is the mth central moment ofx. Substituting equation (B.4) into equation (B.3)

leads to

(B.6)

(B.7)

(B.8)

where

n=O, ... ,5 (B.9)

For the quadratic nonlinearity used throughout this study, equation (B.9) can be

written in the following general form

In = J -yla + ~x + il(a + ~x + i) in cp(i) di n=O, ... ,5

This integral is evaluated by the following procedure. A new variable I; = i/{iiii is

introduced and substituted into equation (B.IO). This leads to

n+2 00

In = )'(~2)T J I it + I; I( it + I; )I;n cp!;(I;) dl; """

where

(B.IO)

(B.ll)

_ a+J.l.x a ::;:--..fIii2

155

The integration in equation (B.ll) is split over several regions as

n+2 -li

In ::;: y(~2r2 J -( a + ~ )2 ~n cp~(~) d~ """

n+2 0 + y(J.LX2)"'2 J ( a + ~ )2 ~n cp~(~) d~

-a n+2 co

+ y(J.Lx2f2 J ( a + ~ )2 ~n cp~(~) d~

Making several changes of variables, equation (B.13) is rewritten as

(B.12)

(B.13)

(B.14)

These integrals can be evaluated using standard integral tables and integration by parts.

Evaluating In for n up to 5 yields

where

10 ::;: 2y (J.Lx2){ (a2 + l)ql + it q2)

11 ::;: 4Y(J.Lx2)3/2{a ql + q2}

12 ::;: 2y (~2)2{ (a2 + 3)ql + a q2)

13 ::;: 4y (J.LX2)5/2{ 3a ql + 4 q2)

14 ::;: 2Y(J.Lx2)3{3(a2 + 5)ql + aq2}

Is ::;: 4y (J.Lx2)7/2{ 15a ql + (a2 + 24)q2)

q2 ::;: _1_ exp( - ! a2) {2rt 2

(B.15)

(B.16)

(B.17)

(B.18)

(B.19)

(B.20)

(B.2l)

(B.22)

156

B.3 High Order Central Moments By defmition, the mth central moment, ~im , of a random variable, x, is computed

by the equation

~im = J (x - ~x)m p(x) dx

where p(x) is the probability distribution of x. It is assumed that the probability

distribution is approximated by a Gram-Charlier expansion of the form

00 aj",(x) p(x) = 1 r· ~

j=O J axJ

(B.23)

(B.24)

where C\l(x) is a gaussian distribution and rj is related to the jth moment of the random

variable x as described in Appendix A. If the distribution is truncated at order j=jv the jt+ 1

and higher order cumulants of the approximate probability distribution are zero. The jt+ 1

and higher order central moments, however, are not zero and may need to be computed. In

this study, the expansion is truncated at the third order. However, the fourth order central

moment is needed to compute the quadratization coefficients. A simple method for

approximating these higher order central moments is described.

If p(x) is given by the truncated Gram-Charlier expansion, evaluation of the integral

in equation (B.23) yields integrals of the following form

(B.25)

To evaluate this integral the relations

(B.26)

(B.27)

(B.28)

are used. The first equation reflects a property of gaussian distributions. The second

equation defmes the Hermite polynomial, Hi~). The third relation states simply that the

157

quantity ~m Hj(~) can be written as an expansion of Hennite polynomials where ~'s are the

coefficients. To initiate the evaluation of I.nj. equation (B.26) is substituted into equation

(B.25) and the variable change ~ = (x - 11x)/~ is made. Then. applying equation (B.27)

yields

.!!!i 00 •

Imj = (~2) 2 J ~m (-l)l H/~) q,t;(~) d~ (B.29) """

where q,t;(~) is defined in equation (B.12). Substituting equation (B.28) into (B.29) and

recalling the variable change gives

.!!!i 00 • l+j ~akq,E.(~) I.nj = (~2)2 J (-I)l~ek(~X2)2 a k d~

""" k=O ~x (B.30)

Interchanging the order of summation and integration yields

(B.31)

As an example. the fourth order and fIfth order central moments of a random

variable whose distribution is approximated by a third order Gram-Charlier expansion can

be computed. Specifically. it is found that

(B.32)

(B.33)

APPENDIX C PIERSON-MOSKOWITZ WAVE SPECTRUM

In offshore engineering, the wave elevation spectrum is frequently defmed by the

Pierson-Moskowitz spectrum. The analytical expression for the two-sided fonn of this

spectrum, which is used throughout this investigation, is given by the equation

(C.I)

where Uw is the wind speed, g is the acceleration of gravity, and Ol is the frequency. The

constant parameters (X and p are commonly assumed to have the following values

(X = .0081

P = .74

(C.2)

(C.3)

For given (X and p, the wave spectrum given by equation (C.l) is completely

defmed by the wind speed. In offshore practice, however, it is more common to specify

the wave spectrum in tenns of the significant wave height, Hs' and the mean period, Tavg.

The significant wave height is the average height of the highest one third of the waves in a

particular sea state. The mean period is related to the spectral moments. These two

parameters are related to the wind speed by the following equations

(C.4)

Olavg = 21t = .920 (1tP) 1/4 LU Tavg w

(C.5)

More details on these relationships can be found in references such as Chakrabarti(1987).

Using these equations, an alternative fonn of the Pierson-Moskowitz spectrum can be

159

produced in tenns of the significant wave height and the mean frequency, coavg•

Specifically,

4

Sllll(CO) = .111H; ~:I~ eXP[_.444(CO!Vgj] (C.6)

As an example, Figure C.1 shows a one-sided fonn of the P-M spectrum for

Uw = 30 m/sec. Further, the significant wave height, mean period, and mean frequency

for this and other wind speeds based on the values of a and ~ given in equations (C.2) and

(C.3) are given in Table C.l.

Table C.1

Jlw mlsec Hli: m Tru sec rom radlsec

20 8.5 11.3 .557

22 10.3 12.4 .506

24 12.3 13.5 .464

26 14.4 14.7 .429

28 16.7 15.8 .398

30 19.2 16.9 .371

32 21.8 18.0 .348

34 24.6 19.2 .328

36 27.6 20.3 .310

120

100

~ U)

80 . N

E

60

:§:

cl 40

20

0

0.0

160

0.2 0.4 0.6 0.8 CD radlsec

Figure C.l Pierson-Moskowitz Wave Spectrum (Uw = 3OmJsec. H. = 19.2m. T""II = 16.9sec)

1.0

D.1 Introduction

APPENDIX D

SIMULATION METHODS

The simulation methods used to generate random wave force time histories are

reviewed in this section. The force time histories are used in a Newmark numerical

integration procedure to obtain simulated responses of an idealized TLP.

D.2 Linear Wave Simulation

In offshore engineering, random wave force time histories are typically computed

by transforming random wave elevation time histories based on suitable physical relations.

The sum of harmonics method described by Borgman( 1969) is a common method used to

simulate linear waves. Wave elevation time histories are generated from a two-sided target

spectrum, Sl1l1(OO), by the following summation

N

11 (x,t) = ~ 11j COS(lCjX - OOjt + £j) )=1

where

(D.I)

(D.2)

(D.3)

The harmonic component with frequency OOj has an amplitude, 11j' which is related to the

area of a slice of the target wave spectrum at OOj. In this study, the frequency width of each

slice, L\oo, is the same, although this is not necessary. In addition, the phase angle, £j' is a

random variable with a uniform probability distribution from 0 to 2n. For a constant x

value, it follows that the random process, l1(X,t), is zero mean. Also by virtue of the

central limit theorem it approaches a gaussian probability distribution as the number of

harmonic components tends to infmity.

D.3 Linear Wave Force Simulation

A linear force, rl)(t), is obtained from the wave elevation by the following linear

transform written in the frequency domain as

162

00

fl>(t) = I H~>(ro) 'i1(X,ro) exp(irot) dro (0.4)

where H~I>(ro) is the linear force transfer function and 'i1(x,ro) is the Fourier transfonn of the

wave elevation. Force time histories are generated by taking the Fourier transfonn of the

hannonic summation in equation (D. 1) and substituting into equation (0.4) which yields

the following hannonic summation for the linear force

It is noted that the linear force depends on the horizontal position x. If x is time dependent,

as is the case in a compliant platfonn, the linear force actually has higher order behavior.

This is the so called body motion effect. By accounting for the time dependent position,

the force is no longer zero mean and is no longer gaussian, even though it is obtained by a

linear transfonnation. This equation is used to simulate the linear force in the numerical

integration computer program. As an option, the displaced position effect can be neglected

to isolate its contribution to the total response.

D.4 Drag Force Simulation

The nonlinear Morison equation with relative velocity effects is used to model the

drag force. The total drag force on a vertical cylinder is obtained by the following

integration over the submerged length of the cylinder

ll(X,t)

fD(t) = tpocCD I I U(z) + U(X,z,t) - xl(t)1 (U(z) + U(X,z,t) - Xl(t») dz (0.6) -ds

The variables have been defmed previously in Chapter 5. The wave field water velocity,

u(x,z,t), is a random process which is simulated by an equation similar to equation (D.5).

Accounting for the displaced position of the TI..P and the integration to the free surface as

opposed to the mean free surface can readily be accomplished in the simulation. These

effects, however, are not readily modeled in the analytical procedure presented in this

study. They are neglected in the simul~tion to prevent obscuring the validity of the

163

analytical procedure. Thus, the simulated drag force is computed for a TLP in the

undisplaced position, and the integration is carried out only to the mean free surface. The

integration in equation (D.6) is performed by the trapezoidal rule.

D.S Quadratic Potential Force Simulation

The wave elevation and velocity head drift forces are simulated directly from the

second order Volterra series functional. The force is written in the frequency domain as

00

£<2)(t) = If H~)(COl,C02) fi(x,col) fi(x,co2) exp(ico1t) exp(ico2t) dcoldco2 (D.7)

where Hi2)(COl,C02) is the quadratic force transfer function.

A straightforward procedure to simulate the quadratic order force is to take the

Fourier transform of the harmonic summation in equation (D.I) and substitute it into

equation (D.7). This yields the following double summation for the quadratic force

N N £<2)(t) = ~1 ~ t11m 11n {Re[Hi)(com,con)] COS[(lCm+lCn)X - (com+con)t + (Em+en)]

At a particular time, t, 4N2 harmonic computations are performed to evaluate the

force. IfNg is the number of steps in the time history, the total number of harmonic

computations is 4Ns N2. Since Ng is in general several times greater than N, the number of

harmonic computations for one time history is of order N3. This proves to be too costly

for practical use. A much more efficient method makes use of digital FFf algorithms. In

order to use this more efficient method, a tradeoff must be made in that the displaced

position cannot be included in the quadratic force calculation. However, as with the drag

force, the displaced position effect in the quadratic force causes higher order effects which

are neglected in the analytical method anyway.

164

The digital procedure makes use of the Fourier series transform which along with

its inverse is defmed by the following equations

T

fk = t J f(t) exp( -ikL\rot) dt (D.9)

-f(t) = L fk exp(ikL\rot) k~

(0.10)

where T is the period over which f(t) repeats itself and fk is the Fourier series transform of

f(t).

The Fourier series transform of the quadratic force is obtained by substituting the

time domain form of the second order Volterra series which is written as

00

£<2}(t) = II h7}('tl,'t2) 11(t-'tl) 11(t-'t2) d'tld't2 (0.11)

into equation (0.9). Changing the order of integration and summation yields

~} = i H7}(m.(\ro,(k-m).(\ro) 11m 11k-m (0.12) m=-oo

where 11m is the Fourier series transform of 11(t). Using the sum of harmonics

representation for the wave elevation and using the same frequency increment for the

harmonic summation and the Fourier series it is readily shown that

r t11m exp( -im) m~O

11m = ~ * m ~ 0

(0.13)

l 11_m

Since the wave elevation is band limited at the frequency NL\CI) the summation in equation

(0.12) need only be carried out over a fmite number of points. The summation can be

rewritten as

k (2) N·k (2) __

L H f (m,k-m) 11m 11k-m + 2 L H f (k+m,-m) 11-m 11k+m m=O m=l

(0.14)

165

where the symmetry properties of the quadratic transfer function have been utilized. Also

the presence of the frequency increment, Aco, in the arguments of the transfer function is

implied. The first tenn on the right hand side of equation (0.14) is the Fourier series of the

high frequency part of the quadratic force, the other tenn is the low frequency part.

The quadratic force time history is obtained from Ji> by applying the inverse Fourier

series transfonn given by equation (0.10). An FFr algorithm can be used to do this

efficiently. The time savings of this method compared to the straightforward method is

obvious. The number of operations in equation (0.13) is of order N, in equation (D.14) of order N2, and for the FFr algorithm of order Ns 10g(Ns). Since the total number of

operations for one time history is obtained by addition of all these operations, this

procedure requires an order N2 number of operations. This is a substantial savings over

the direct method.

This digital procedure has been verified by averaging the quadratic force power

spectral density from 100 realizations and comparing it to the analytical power spectral

density. This has been done for quadratic surge force and pitch moment on the idealized TLP for a P-M spectrum with Hs=1.36 m and Tavg= 4.51 sec. The results are shown in

Figures D.1 and D.2 and show good agreement in both the low frequency and high frequency regions.

&! ~

OJ

Z ~

g g =:

(/)

~ "'.-. ~ Z ~

~ N'''';: -rn-

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.0

0

0.20

0.15

0.10

0.05

166

-- Analytical

••• •••• . Simulation

2 3 ro rad/sec

Figure 0.1 Spectral Density of Quadratic Wave ElevationNelocity Head Force on a TLP (llw = 8 mlsec. Ii. = 1.36 m. Ts..g" 4.51 sec)

-- Analytical

•....... Simulation

4

0.0 E::::::~:::"""":==========--~..I...-___ --1. __ -==::J o 2 3

ro rad/sec

Figure 0.2 Spectral Density of Quadratic Wave ElevationNelocity Head Moment on a TLP

(llw .. 8 mlsec . Ii.. 1.36 m. TB"9" 4.51 sec)

4

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Moe, G., and Verley, RL.P., 1980, "Hydrodynamic damping of offshore structures in waves and currents", Proceedings of the 12th Annual Offshore Technology Conference, Houston, Texas, OTC 3798, pp. 37-44

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Nielsen, F.G., and Herfjord K., 1985, "The importance of non-linear wave forces to fatigue of deepwater structures", Proceedings o/the 17th Annual Offshore Technology Conference, Houston, Texas, OTC 4952, pp. 493-501

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