Second-order, sum-frequency response statistics of tethered platforms in random waves

Applied Ocean Research 14 (1992) 23-32

Second-order, sum-frequency response statistics of tethered platforms in random waves A. Naess & G.M. Ness Department of Civil Engineering, The Norwegian Institute of Technology, Rich. Birkelands vei la, N-7034 Trondheim, Norway

(Received 8 January 1991; accepted 17 July 1991) The second-order, sum-frequency forces and motions of tension leg platforms in random seas are studied. The work is based on the assumption that these forces and motions can be characterized by a quadratic transfer function. A representation theorem for the second-order, sum-frequency force and response process is derived. This is used for calculating the statistical properties, including the probability density functions, of the forces and motions.

recently that computer programs have become available that make it possible to estimate in an apparantly satisfactory manner the second-order, sum-frequency hydrodynamic forces on a TLP structure in waves. Hence, previous evaluations of TLP structures have been based mostly on model test results and simplified theoretical calculations. One of the first published papers dealing with the springing response of the TLPs was written by DeBoom et al. t They made comparisons between high-frequency tether forces obtained by theoretical calculations and by measurements on a four-column TLP in regular waves. Their results to a certain extent lend support to the assumptions underlying the modelling of the high-frequency hydrodynamic forces. Further partial support was provided by Teigen 2 and Petrauskas and Liu. 3 In the last paper are presented measured springing forces on a large scale (1 : 16) TLP model in both regular and random waves. As already stated, it is only recently that development has started of computer programs that make possible the numerical solution of the complete second-order potential problem for the sum-frequency forces on a TLP in bichromatic waves? -8 It will be demonstrated in this paper that the availibility of such information enables us, under certain assumptions, to calculate the response statistics of the heave, roll and pitch motions of a TLP in short-crested random seas. The theoretical method developed will be illustrated by application to a large TLP in long-crested random waves. The results are mainly presented as plots of probability density functions (PDF) of the sum-frequency vertical response. An efficient numerical method for calculating the PDF of a combined first-order and second-order, sum-frequency response is also described. This method can be used to calculate the

INTRODUCTION When the tension leg platform (TLP) concept was evaluated for application as an oil production factility offshore, one of the concerns was the possible excitation of heave, roll and pitch motions. Even if the tether system is usually designed so that the resonance periods for these motions are in the interval from about 2-4 s, where very little wave energy is present, they may still be excited due to higher order effects. This concern is related to a recognition of the fact that the overall damping (including mechanical and hydrodynamic) is very low in these motion modes, so that even small higher order excitation forces may excite a resonant response including significant loads in the tethers. These high-frequency tension oscillations in the vertical tethers are often referred to as springing (or ringing), and they may have important implications for the evaluation of fatigue damage. One of the contributions to the higher order hydrodynamic excitation forces is due to second-order, sumfrequency potential forces, i.e. forces ocurring at frequencies equal to the sum of the frequencies appearing in the wave spectrum. The importance of these force components relative to other possible higher order contributions does not seem to be well understood at present. It appears that the usual approach is to assume that the second-order, sum-frequency forces constitute the dominating components, at least on a large-volume TLP. The first TLP to be installed offshore has been in operation since the early 1980s, but it is only quite Applied Ocean Research 0141-1187/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 23

A. Naess & G.M. Ness

24

PDF of the pure second-order, sum-frequency component as a special case.

THE SUM-FREQUENCY RESPONSE PROCESS IN SHORT-CRESTED SEAS The sea surface elevation X(t) of a short-crested random sea-way at a fixed reference point, is assumed to be a zero-mean, stationary Gauss,an process. X(t) is generally considered as a linear superposition of unidirectional Gauss,an wave fields of the same type as above, evaluated at the reference point. Let Sx(09,/3) denote the one-sided directional spectrum of X(t). Then X(t) can be approximated by the expression 9

X(t)

=

~

Re

~ [2Sx(09i, /3k)AogA/3]112Bik ei'°''

i=1 k=l

(1) where 0 ~< 09t < 092 < . . . < 09a~ and 0 ~< /71 < /32 < - . . < /3N ~< 2rt are equidistant discretizations of pertinent parts of the positive frequency axis and (0, 2r0, respectively. A09 = 09,+, - 09,, k/3 = /3,+, - /3,. {B,k} is a set of independent complex Gaussian N(0, l)-variables with independent, identically distributed real and imaginary parts. The use of i to denote both the imaginary unit x/Z-i- and an index in e '~'' should cause no confusion. Let F2(t) denote the rapidly varying, second-order (nonlinear) wave excitation forces in a given direction, e.g. heave, on a large-volume offshore structure subjected to the irregular, stationary short-crested seas X(t). Then F2(t) is also stationary, and by invoking the representation given by Naess, 9 it can be shown that (09 i = - 09i)

F2(,) =

M

M

Zo

~,o ~ Z ½ [Sx(I09,l,/3k)Sx(109)[,/3,)]'/2

i=

M j=

N

M

k=l

N

l=l

x A09A/3K2(09i, -- 09j; ilk,/3,)BikB~ e i~`~' <°kt (2) The index zero on a summation sign signifies that the summation index omits zero. In eqn (2) the following extension is also introduced

B i.k

=

Bi*k,

i =

1. . . . .

M;

k =

1. . . . .

practical difficulties of estimating K2(09, 09', [L [J') are at present still substantial. Therefore, very few numerical results are available to date, especially for the multidirectional case. The representation expressed by eqn (2) is based on the sum-frequency approximation K2(09, - 09';/3,33

giving the relation

0for09"09' > 0

Z2(t)

= fo l ( z ) ~ ( t - r)dz

(6)

where/(r) denotes the impulse response function of the dynamic model. Substituting from eqn (2) into eqn (6), it can be shown that 9 M

&(t) =

M

N

N

E0 E0 Z E ~ [s,<(I09,1,/3,)s,<(l~k,/3,)] '/~ i-

M

/=-M

k=l

l=1

X A09A/3 ]~(09i -- 09])K2(09i,

-

09j; ~k, /3l)BikB~l eit ....... sit

(7)

E[Bik B~] = 6ij6k,, i,j = - - M , . . . ,

(5)

This condition implies that all difference frequency components of the second-order excitation forces are neglected, in agreement with the requirement, retaining only sum-frequency terms. In view of eqn (5), this fact is clearly displayed by the far right-hand side of eqn (2). To obtain the simplification necessary for expressing the response of a structure within the same framework of nonlinear, second-order theory as above, it is required to adopt the assumption of linearized, time-invariant equations of motion. It is hard to say at present to what extent this is an acceptable approximation for the heave, roll and pitch motions of a TLP. What makes it difficult to evaluate the approximation, is the lack of precise knowledge about the influence of tether dynamics on the motion characteristics of the TLP hull. In fact, it seems difficult at this point to irrefutably justify the choice of a specific set of nonlinear equations of motion to model these response modes. Hence, we shall be content at this stage to adopt the assumption of a linear dynamic model. However, it is emphasized that even though the equations of motion are linear, the nonlinear nature of the second-order excitation loads is taken into full account. For cases where nonlinear, time-invariant equations of motion should be used, it is worth noting that stochastic methods have been developed that can provide reasonably good estimates of even extreme responses. J°H Adopting a linear, time-invariant dynamic model for the response Z2(t) to the second-order forces F2(t ), it is obtained that

N

(3)

=

--1,

1. . . . .

M;

k,l

=

1.....

N

(4) The function /('2(09, 09'; /3, fl') in eqn (2) denotes the bidirectional quadratic transfer function (BQTF). It can be identified as the (complex) amplitude of the secondorder force at a frequency 09 4- 09' due to the simultaneous incidence of two unit amplitude regular waves of frequency and direction 09,/3 and to',/3', respectively. The

where £(o9) denotes the transfer function of the linear dynamic model and is given by £(09) =

fo l(z)ei~'~dr

(8)

Equation (7) shows that Z2(t) has the same type of representation as F2(t) and that the BQTF of Z2(t), denoted by/t2(09, 09';/3,/3'), is given by the relation /t2(09, co'; /3, fl') =

£(09 + 09')K2(09, 09'; /3, fl')

(9)

Response statistics of tethered platforms in random waves

25

Table 1. Particulars of the TLP

Heave

Roll

Pitch

Eigen period [s]

2.29

2'40

2.50

Total mass [kg]

1-39 x 10s

--

--

Total mass moment of inertia IN m s2]

--

2.37 × 1011

2.63 x 10u

defined in Appendix 1. The W,(t) are zero-mean, stationary Gaussian processes, and W_,(t) is the Hilbert transform o f W, (t). As shown in Appendix 1, this implies that the spectral density of W,(t) 2 - W , ( t ) 2 only contains the sum of frequencies occurring in the wave spectrum Sx(oo). From eqn (12) it is obtained that the first four cumulants, kj, j = 1. . . . . 4, are given by (2_, = - 2 , ) Fig. I. Sketch of TLP structure.

MN

~--- kl

g[z2(t)]

Introducing the notation

=

EO 2~ =

0

(13)

= - MN MN

ttijkl = ½ [Sx(ltoi[, /~k)Sx(Io~jl, /%)],/2

Var[Z2(t)]

X A(.,oA}~ /~'2((J)i, --(./~j; #k, ]~l)

=

kz = 4 E 2~

(14)

a=l

(10) MN

equation (7) can be rewritten as M M N N

Z2(t)

=

Eo

k3 =

Zo

=

o

(15)

ct= - - M N

Eo E E H, yktB,kB;

e i(~'-°'y)'

(ll)

k=l I=l The subsequent developments will be limited mainly to the second-order, sum-frequency response process Z2(t). It is, however, obvious that the obtained results apply equally well with the required changes, to the secondorder forces F2 (t). In Appendix 1 the following representation of Z 2(t) is derived i=-M

8

MN

=

j=-M

MN

Z2(t) = ~ 2~ {W,(t) 2 - W , ( t ) 2}

(12) ~t=l Here the 2~ are the positive eigenvalues of the matrix T

96

44

(16)

0t=l Clearly, ku_l = 0 for n = 1, 2 . . . . . which implies that the P D F (probability density function)fz2(Z) is symmetrical about its mean value z = 0. In some cases it may be desirable to know k 2 and k 4 without having to solve for the eigenvalues 2~ of the matrix T, or equivalently, the eigenvalues/z~ of the matrix S ns, see Appendix 1. The following results are then useful k2 =

4 Tr {S HS}

(17)

96 Tr {(S HS):}

(18)

and Y

k4 =

"

il

Wave direction

0

i

11.3

Fig. 2. Horizontal cross-section of TLP at pontoon level and definition of coordinate system. Dimensions are in metres.

where Tr {-} denotes the trace operator, i.e. Tr {(au) } = Z a,. In contrast to the difference-frequency case, 9'12 the P D F of the second-order, sum-frequency response process Z2(t) is not known in closed form. In Appendix 2 an efficient numerical method is developed for estimating the P D F of the combined first-order and second-order, sum-frequency response with the P D F of the pure sumfrequency response as a special case. NUMERICAL EXAMPLES

The numerical example studies are based on an extensive

A. Naess & G.M. Ness

26 Table 2. Sea state parameters Sea state

H, (m)

Tp (s)

1 2 3

5-4 13.6 21.2

5.0 10.0 15-0

It is realized that the response Y(t) can be represented in a manner completely analogous to the representation o f e q n (11), where the kernel Hii~t in this case assumes the form

0-168 0-066 0.032

3

3

H, jk, = ½ [Sx(Ico, I, /~k)Sx(I,ojl, /~,)I'~2A,~AI~ ~ Y. .l~-I v : : l

set of data of the quadratic sum-frequency transfer function (QTF) for a T L P structure as shown in Fig. 1. 7,8 A horizontal cross-sectional view of the platform at pontoon level is given in Fig. 2, while some of the main particulars are listed in Table i. The origin o f the coordinate system is located at the mean free surface with the positive z-axis pointing upwards. For the sake of simplicity it is assumed here that heave, roll and pitch are decoupled from the horizontal plane motions and that the equations of motion are linear and time-invariant. This results in a three-degrees-of-freedom system of the form M Z 2 q- CZ 2 -+- KZ 2 =

F2(/)

(19)

Here M = the total mass matrix, C = the damping matrix and K = the stiffness matrix. Z2 = (Z2.~, Z2.2, Z2,3)T, where the second subindex refers to heave, roll and pitch, respectively. In general, heave, roll and pitch are coupled, which implies that the 3 x 3 matrices in eqn (19) are nondiagonal. The linear transfer function matrix f~(~o) = (£,,~.(eo)) of the equations of motion is given by L(co) =

( - c o 2 M -k i~oC + K ) - '

3

~ a~Z2,(t) =

(21)

aXZ2(/)

tt=l

Table 3. Cumulants

Wave direction

Sea state

~

-

coj)R2.,(~,,

-

~i; l~k, fl,)

(22) Here /~2,v ( ' , ", ; ", ") refers to the BQTF for the sumfrequency force in direction v. In the present paper the analysis will be simplified by assuming long-crested waves. It is also assumed that the heave, roll and pitch motions decouple. This implies that the summation in eqn (22) only extends over three terms, since all cross-terms disappear. As has already been mentioned, the total damping in the considered response modes is very small. In fact, it seems that realistic values of damping ratios lie in the interval from 0.5-1%. Therefore, results from example calculations with damping ratios of 0.5, 0.75 and 1% will be presented. To completely specify the kernel Hijkt, it is seen that the spectral density o f the sea surface elevation also needs to be given. Design of the offshore structures for the Norwegian Continental Shelf is often based on the use of the J O N S W A P spectrum. ~3 For the long-crested seas case, the wave spectrum St@o) is then given by

(20)

Consider any linear combination of the vertical plane motions, e.g., the total vertical displacement, Y(t) say, of a point in a platform leg. Then, since the relevant motions are small compared to platform dimensions,

Y(t) =

x ~£,,,(~o,

k2

k4

k4

k~ 0°

1 l 1 2 3

0"005 0-0075 0"01 0'005 0"005

0-588 0"376 0'283 0-160 0"040

0-561 0'329 0-217 0"040 0'003

1-81 2'33 2'70 1"58 1'73

22.5 °

1 1 1 2 3

0'005 0"0075 O-O1 0-005 0-005

0.846 0"524 0'371 0"243 0'065

1"238 0-599 0-360 0-087 0'007

!'73 2-18 2-61 1.48 1.47

45 °

1 1 I 2 3

0"005 0-0075 0"01 0-005 0"005

2 " 1 9 6 10.310 2.13 1"420 5'887 2.92 1.045 3 " 9 9 9 3"66 0 " 5 1 2 0 - 5 5 8 2"13 0.136 0 " 0 4 0 2"13

+ In 7 e x p

-

~-

1 //2o"2

(23)

where a~p denotes the peak frequency in rad/s and ~, 7 and ~r are parameters related to the spectral shape, ~r = 0"07 when eo ~< ~op and a = 0.09 when co > ~op. The parameter 7 is chosen to be equal to 3.0. The parameter has been determined from the following empricat relationship ~3 c~ =

5.78 (Hs/T2) 2"°36 (1 - 0.298 lnT)

(24)

Here Hs = significant wave height and T o = 2~/O9p = peak wave period. Example calculations for three different sea states have been carried out. These are specified in Table 2, where the corresponding values of ce have also been listed. Due to an error, the sea states have become slightly unrealistic. Estimates of the Q T F for the second-order, sumfrequency forces have been calculated 7'8 for three different wave headings, viz. 0, 22.5 and 45 ° (cf. Fig 2). Calculated values are available for points in the bifrequency plane with a frequency separation of 0.1 rad/s. The implied discretization of the frequency axes is too sparse for estimation of response statistics. To ensure sufficient resolution over the resonance peaks, which are very narrow for the considered reponses, a simple recursive

Response statistics of tethered platforms in random waves

27

0.4

Imml

PDF

16-

0.35

1412-

b.3

10-

0.25

80.2

64-

0.15 0.1

0 degrees 1~

22.5 degrees

45

:grees

Max ei|.val., ~=.005 ~

STD, ~=.005

U l l Max eig.val. , ~=.0075

S'rD, ~=.007J

Max ~g.vml., ~--.01

~

~

0.05 0

-2

-4

2

0

4

STD, ~--.01 Displacement [era]

Fig. 3. Comparison of the standard deviation (STD) and the maximum eigenvalue of the total displacement response in sea state 1 for the three wave headings and three values of the damping ratio ~. interpolation scheme in the bifrequency domain was implemented. At each step linear interpolation between the four vertices of every basic frequency square was carried out, giving the value assigned to the center point of that square. The interpolation process was continued until the resulting frequency discretization had about ten frequencies within the half-value width of the most narrow linear transfer function used in the response calculations. A sensitivity study was carried out to check that stable results were obtained. However, to what extent the Q T F obtained by the interpolation procedure is a good estimate o f the actual Q T F can for obvious reasons not be evaluated until further data are available. To limit the amount o f information to be presented, only results for the combined vertical displacement response of the center point of one of the columns with coordinates ( - 38, - 38, 0) (cf. Fig. 2) due to heave, roll and pitch, will be given here. Table 3 lists the calculated cumulants k2, k4 and the associated coefficient of excess ~2 = ka/k~, which equals zero for a Gaussian variable. A positive 72 is supposed to indicate that the associated P D F is taller and slimmer in the vicinity of the mean value (for a symmetrical PDF) than the corresponding

10o

I

I

I

i

!

2

4

6

8

10

PDF 10-1

10-~

lff 3

10~

10-2

I 12

Displacement [cm]

Fig. 5. PDF of total displacement response for wave heading 45°, sea state 1 and damping ratio ( = 0.005; : numerically calculated PDF; : corresponding Gaussian PDF. Gaussian PDF. This fact will be apparent from plots of the PDFs, which will be given below. It is worth noting that 72, which is often used as a measure of deviation from a Gaussian probability law, seems to be determined to a large extent by the damping and much less by the sea state. As could have been anticipated, the response variance is also strongly dependent upon the damping. But more important is the dependence on the sea state. It is recognized that the crucial parameter is the peak period of the i00

Imml

111-

PDF

iffI

12 J I0~

108-

i

64104i

20

S m ~III I

Sll ~III I

8 U IWll l

&IOta2

h

lllm 3

10

2

4

6

I• 8

I

10

12

Displacement [crn]

Fig. 4. Comparison of the standard deviation and the m a x i m u m eigenvaluc of the total displacement response for a wave direction of 45 ° and for various damping ratios and sea states.

Fig. 6. PDF of total displacement response for wave heading 45°, sea state 1 and damping ratio ( = 0.0075. Figure key as in Fig. 5.

28

A. Naess & G.M. Ness l&

r

I

1

I

[

I

r

I

1Ga

1

PDF

i

i

r

I

I

r

--~

PDF

"

'

l 1

lift

10 a

10-2

10-3

10-3

10-4

10a

10

I 1

I 2

I 3

1 4

~

I 5

I 6

t 7

I" 8

,

I 9

lt) 0.5

1

Fig. 7. PDF of total displacement response for wave heading 45°, sea state 1 and damping ratio ( = 0.01. Figure key as in Fig. 5 sea wave spectrum. The maximum response occurs when this is about twice the resonance period, clearly demonstrating the second-order character of the theory. To further highlight the response characteristics, graphical illustrations of the standard deviation and the maximum eigenvalue for a set of sea state and damping conditions are given in Figs 3 and 4. Note that if the maximum eigenvalue is a substantial part of the standard deviation this indicates that the eigenvalues decrease rapidly (cf. eqn (14)). A consequence of this is that the corresponding P D F tends to be of exponential type rather than Gaussian. If there had been many eigenvalues of about equal size, then, by invoking the central limit theorem, one would expect to get a near-Gaussian distribution. It has already been mentioned that there does not seem to be available a closed-form expression for the P D F of the sec0nd-order, sum-frequency response, but the method described in Appendix 2 is an efficient and accurate numerical tool for estimating the PDF. In Figs. 5-9 are plotted the obtained PDFs for the 45 ° wave heading together with the corresponding Gaussian densities, i.e. with zero mean and the same variance. A conspicuous feature of these plots is the substantial

I

I

1

2

I

I

l

I

3

4

5

6

PDF 10-~

10-2

10.3

104

10

Displacement

2

Z5

3

3.5

,~

Displacement [cm]

Displacement [cm]

100

1.5

[cm]

Fig. 8. PDF of total displacement response for wave heading 45°, sea state 2 and damping ratio ( = 0"005. Figure key as in Fig. 5.

Fig. 9. PDF of total displacement response for wave heading 45°, sea state 3 and damping ratio ( = 0-005. Figure key as in Fig. 5. deviation from the Gaussian law in the tails. This may come as a surprise since the low damping might induce one to expect a more Gaussian behaviour. Confer, however, the discussion on Figs. 3 and 4 above. The implication for extreme value prediction is not immediately obvious since the low damping also has a marked effect on clumping of large response peaks, which will tend to reduce the extreme response. 14'j5 Nevertheless, preliminary estimates seem to indicate that one must expect significantly larger extreme responses than predicted by the Gaussian assumption, in full agreement with the picture conveyed by the calculated PDFs. This problem is presently under investigation for the sumfrequency response.

CONCLUSIONS A general second-order theory has been developed for analysis of the response statistics of the resonant heave, roll and pitch motions of a large-volume T L P structure in short-crested random seas. The second-order character of the theory is reflected in the assumptions that the hydrodynamic excitation forces can be described in terms of bidirectional quadratic transfer function (QTF), and that the equations of motion are linear. The assumption that the excitation forces can be described by a Q T F implies that these forces occur at frequencies that are the sum of the frequencies in the wave spectrum. Based on an extensive set of data for the QTF pertaining to the heave, roll and pitch excitation forces and moments on a TLP, several example calculations have been carried out to study the properties of the sumfrequency response. The results clearly demonstate the second-order, sumfrequency nature of the excitation forces and moments by the fact that the sea state with a peak period of 5 s, which is about twice the resonance period of the considered response modes, induces the largest motions. The other two sea states used in the calculations, had peak

29

Response statistics o f tethered platforms in random waves

periods at 10 and 15 s, and corresponding to heavier seas, but still induced smaller motions. An efficient and accurate numerical method for calculating the probability density function (PDF) of the combined first-order and second-order response is described. This method is used to calculate the P D F of the second-order, sum-frequency vertical m o t i o n s of a point located in the center of one of the platform legs. The obtained results clearly show that this motion response is distinctly non-Gaussian.

REFERENCES 1. DeBoom, D.W., Pinkster, J.A. & Tan, P.S.G., Motion and tether force prediction for TLP. J. Waterway, Port, Coastal and Ocean Eng, ASCE, 110 (4), (1984) 472-86. 2. Teigen, P.S., The response of a TLP in short-crested waves. Proc. 15th Offshore Technology Conf. Houston, Texas, Paper No. OTC 4642, 1983, pp. 525-32. 3. Petrauskas, C. & Liu, S.V., Springing force response of a tension leg platform. Proc. 19th Offshore Technology Conf. Houston, Texas, Paper No. OTC 5458, 1987, pp. 333-8. 4. Eatock-Taylor, R. & Hung, S.M., Second-order diffraction forces on a vertical cylinder in regular waves. AppL Ocean Res. 6 (2), (1987) 19-30. 5. Kim, M.H. & Yue, D.K.P., The nonlinear sum-frequency wave-excitation and response of a tension-leg platform. Proc. 5th Int. Conf. on the Behaviour of Offshore Structures (BOSS'88), Trondheim, Norway, 1988, pp. 687-703. 6. Williams, A.N., Abul-Azm, A.G. & Ghalayini, S.A., Complete and approximate solutions for second-order wave forces on cylinder arrays. Proc. 9th Int. Conf. on Offshore Mechanics and Arctic Engineering (OMAE'90), Houston, Texas, March 1990, pp. 181-8. 7. Molin, B. & Chen, X.B., Calculation of second-order, sum-frequency loads on TLP hulls, Interim report (restricted) FNS Project no. 24841 'Vertical Resonant Motions of Tension Leg Platforms', Paris, France, 1990. 8. Chen, X.B., Bolang--A numerical model to evaluate the second-order loading on three-dimensional structures. Etude No. B4463030, Ref. IFP No. 38 740, Institut Francais de Petrole, Paris, February 1991. 9. Naess, A., Statistical analysis of nonlinear, second-order forces and motions of offshore structures in short-crested random seas. Probabilistic Eng. Mechanics. 5 (4), (1990) 192-203 10. Naess, A., Galeazzi, F. & Dogliani, M., Stochastic linearization method for prediction of extreme response of offshore structures. Proc. 1st European Offshore Mechanics Symp. (EUROMS'90), Trondheim, Norway, 1990, p. 115-20. 11. Naess, A., Galeazzi, F. & Dogliani, M., Extreme response predictions of nonlinear compliant offshore structures by stochastic linearization. Appl. Ocean Res. (in press) 12. Naess, A., The statistical distribution of second-order slowly-varying forces and motions. Appl. Ocean Res. 8 (2), (1986), 110-18. 13. Houmb, O.G. & Overvik, T., Parametrization of wave spectra and long term joint distribution of wave height and period. Proc. 1st Int. Conf. on the Behaviour of Offshore Structures (BOSS'76), Trondheim, Norway, 1976, pp. 144-69. 14. Naess, A., Prediction of extremes of combined first-order and slow-drift motions of offshore structures, Appl. Ocean Res. 11 (2), (1989) 100-10.

15. Naess, A., Approximate first-passage and extremes of narrow-band Gaussian and non-Gaussian random vibration, J. Sound Vib. 138 (3), (1990) 365-80. 16. Golub, G.H. & Van Loan, C.F., Matrix Computations, North Oxford Academic Publishing Co., Oxford, 1983. 17. Papoulis, A., Probability, Random Variables and Stochastic Processes. McGraw-Hill, New-York, 1965. 18. Rice, S.O., Distribution of quadratic forms in normal variables--Evaluation by numerical integration. Siam J. Sci. Stat. Comput., ! (4), (1980) 438-48. 19. Bender, C.M. & Orszag, S.A., Advanced Mathematical Methods for Scientists. McGraw-Hill, New York, 1978. 20. Press, W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T., Numerical Recipies--The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986. 21. Naess, A. & Johnsen, J.M., An efficient numerical method for calculating the statistical distribution of combined firstorder and wave-drift response. Proc. lOth Int. Conf. on Offshore Mechanics and Arctic Engineering (OMAE'91), Stavanger, Norway, June, 1991.

A P P E N D I X 1: R E P R E S E N T A T I O N OF T H E RESPONSE PROCESS In this appendix the theoretical results derived by Naess 9 will be adjusted to deal with the second-order, sumfrequency case. The results will be written out in a form amenable to numerical analyses. Due to the properties of/-Tr2(e~, - o9'; r, fl'), it can be shown that //,jk,

=

Hj,~k,

i,j

=

-M,...,

-1,

1. . . . . M ; k , l

=

1 .....

N

(A1) The adopted sum-frequency approximation implies that /-/~jkt = 0 for i "j > 0. Hence by defining the matrix S = (S~,), m, n = 1. . . . . M N , as follows Stun

=

n_M+i_Kikl ,

(j-

1)N+ l

k,l

1,

,

it can be demonstrated that the aggregate of terms Hiju are represented by the 2 M N × 2 M N matrix T

=

SH 0

(A3)

where 0 is an M N x M N z e r o - m a t r i x and S H = (S*) r is the Hermitian conjugate of S. By invoking the representation theorem proved by Naess 9 and expressing it in discretized form, it follows that the eigenvalue problem that has to be solved to obtain the desired representation in the present case can be written as Tv =

2v

(A4)

It is assumed that S, and thereby T, is nonsingular. Let 2~ and v~ = (v~,_us . . . . . v~,_l, v~.l. . . . . v~.Ms)r = -MN ..... - l , 1. . . . . M N ) denote the eigenvalues and orthonormal eigenvectors of T. In accordance

30

A. Naess & G . M . Ness M

with the developments in Ref. 9, it is convenient to introduce the notation

/=

=

{

I)U+k,

I =

l,

M;

. , . ,

- k~(~oi, ilk) (A10)

This result leads to the following decomposition of H,jk~ (see Naess 9) MN

and consider the eigenvector as a function of the two variables frequency and direction. Equation (A4) can then be rewritten as

:t=l

- v ~(a;,, flk)V ~(~oj, ill)*}

(All)

N

EO j~-M

E

HijklV(('OJ' ill)

:

'~ I ~ ( O i ,

ilk)

(A6)

I=1

which is the discretized version of the eigenvalue problem studied by Naess. 9 To determine the eigenvalues of the matrix T, it is expedient to make the following observation. The matrix S n S is Hermitian, non-negative definite matrix, which implies that all its eigenvalues are non-negative. It can further be shown that # is an eigenvalue of S nS if and only if +_~ are eigenvalues of T. Implicit in this statement is the fact that if 2 is an eigenvalue of T, then 2 is real and - 2 is also an eigenvalue. Therefore assuming that #,, ~ = 1. . . . . M N are the eigenvalues of S n S, then 2~, ~ = - M N . . . . . -1, 1. . . . . M N defined by 2~-

x~l~l,

....

~ =-MN

,-1

(A7)

and )~ =

x/-~,

~ = 1. . . . .

(AS)

MN

and the eigenvalues of T. Let v = v(~o~,ilk), be the eigenvector corresponding to an eigenvalue 2 of eqn (A4). Then the vector ~ = ~(o~, ilk) = ~((Di)'V((l)i, ilk), where z(a~i) = i ( = x/L-]") for e9~ < 0, and z(a~) = - i for tsg > 0, is the eigenvector corresponding to the eigenvalue - 2. This can be shown as follows. For ~oi < 0, M

=

1;

, --

(A5)

M

MI,-I

= i2v(~oi, fl~)

Yx.iN+k-l, i = - - M . . . . Y~t.(i

,'v

= i Eo E

N

M

N

/=l

j=l

/=1

where v_~(., .) = ~ ( . , .). Substituting the expression given by eqn (Al l) for H~jkt into eqn (1 l) gives MN

Z2(t)

=

~

W~(t) 2}

(A12)

v;.(egi, fl,)B~, ei'°''

(A13)

2~{W~(t) 2 -

~=l

where

w(t)

M

=

N

~o ~ i--Mk=l

which is a zero-mean, stationary real Gaussian process. For t fixed, the W~. = W..(t), 7 = - M N . . . . . -1, 1..... M N constitute a set of independent N(0, 1) variables. Noting that X(~o) is the transfer function of the Hilbert transform, 17 it follows from the linearity of the Hilbert transform that W ,(t) is the Hilbert transform of W~(t). From this it can be concluded that the spectral density of W~(t) 2 - W_~(t) 2 only contains sums of frequencies occurring in the wave spectrum Sx(o3). This fact is easily demonstrated by writing the representation of W,(t) as follows (suppressing the directional dependence): M

W~(t)

=

~ ~ c o s ( ~ o i t + ®j)

(AI4)

j=l

where ~ and ®j, j = I, . . . , M are independent sets of independent random variables, Uj being Rayleigh distributed and ®j uniformly distributed. Then M

/=

M

W ~(t) =

~ ~ s i n ( o g j t + ®j)

(A15)

j=l M

=

-i

=

-i

which gives

i=l i - I M N

W~(t)2 - W_~(t) 2 =

Zo j=

=

N

y~ Z Hijkl V(09i' fit)

M

E H, jk, v(ooj, fit)

-- i)]x(fOi,

ilk)

M

N

Z0 Z j=--M l = l

I

H,j,,

N

= j-

M l=l I

N

j--M l=l

+ oOt + o , + o J (A16)

=

-- )~(¢.0i,

ilk)

(A9) For ~oi > 0,

G cos id= 1

M l=l

This immediately leads to the desired conclusion. In addition to the sum-frequency, second-order response Z2(t), in general the response process will also contain a first-order, linear component Zl(t). It is convenient to derive a representation of Z1 (t) in terms of the 'eigenprocesses' We(t ). To this end, it is noted that in analogy with the representation given by eqn (1 t) for the second order response, the following expression for Z~ (t)

Response statistics of tethered platforms in random waves is obtained. M Zl(t)

Eo

=

x exp

N 2 t~rl(gOi, J~k)

X {1 Sx([O)il ' flk)Af.oAfl}l/2BikekO, t

x

(A17)

Here/~l (09, fl) denotes the linear transfer function from the directional sea way to the considered first-order response. Since it has been assumed that the eigenvectors v, constitute an orthonormal set, it follows that MN

N ~ /~l((Oi' ]~k)

i = - M k=l

x {½ Sx(l¢ofl, flk)AoJAfl} l/z v,(og~, ilk)

(A19)

Substituting from eqn (A18) into equation (A17), it is seen that the following representation is obtained MN

Zi (t) =

Z 0 G W,(t)

(A20)

ct= MN

APPENDIX 2: NUMERICAL INTEGRATION BY THE SADDLE POINT METHOD

Having achieved the desired representation of the response process Z(t) = Zl(t) + Z2(t), our goal is to compute the probability density function (PDF) of the random variable Z = Z(t) (t fixed). This will be done by first calculating the characteristic function g z ( ' ) of Z. To this end it is appropriate to take into account that in practice a number of the eigenvalues 2j may be zero or close to zero, so that the corresponding second-order terms in the response process representation are neglected. For notational convenience, M N will be replaced by M in this appendix. Assume that there are M~ non-negligible second-order terms. Then Z can be represented as follows MI Z

=

~ {t~j ( W j 2 - j=i M

+

~

W2j)

7{- C j < -

c jWj}

{cjWj + c j W j }

(A21)

J=Ml+l

The characteristic function is the Fourier transform of the PDF, that is

gz(O) =

E[ei°Z] =

f~oof z ( z ) ei°z dz

From eqns (A21) and (A22) it is found that

gz(O)

j=,=

1 + 42~02

{ exp

2} (A23)

2

The PDF of the random variable Z can be obtained from the Fourier inversion of the characteristic function and generally this has to be done numerically. By making a variable substitution w -- i0, the integral to be evaluated is

fz(z)

-

From the orthonormality of the eigenvectors, it can be shown that the expansion coefficients c~ are given by the relations Cc¢ =

j=MI+I

2(1 + 2i2j0)

~ o c~v~(c°i, ilk) et= -- MN

(A18)

M 20

I-I

c2o2}1

2(1 - 2i2j0)

M

i=--M k=l

/-Tq(~o,,/~,) {½sx(l~o,I, flk)AogAfl} I/= =

{

31

(A22)

2rdl[+i~exp{ -wz+a-ioo

~ I - ln(1 - 42~w2)

j=l

jw)Al dw

+ 2(1 - 2, jw) + 2(1 +

(A24) In eqn (A24) the fact that ( 1 - 42~w2) - l = exp {-ln(1 - 4),2w2)} has been used. Rice 18proposed a method for numerical evaluation of a similar integral using the so-called saddle point method. The idea behind the saddle point method is that in the integration of an analytic function one is free to choose the path of integration between two points as long as it stays within the domain of analyticity. This method turns out to be ideally suited to calculate the integral in eqn (24). A brief description of the method follows. The integrand in eqn (A24) is written in the form exp {p(w)} =

exp {¢p(u, v) + i~b(u, v)}

=

exp {q~(u, v)} [cos~b(u, v) + i sin~b(u, v)

(A25)

where

p(w)

=

M ( --wz + ~ {--ln(1j=l k

c) w2

42~w2)

c2j w

+ 2(1 - 22jw) + 2(1 + 22jw)

} (A26)

q~(u, v) and ~(u v) are, respectively, the real and the imaginary part of p(w) (w = u + iv). A point where p'(w) = 0, that is where Vcp(u, v) = V~k(u, v) = 0, is said to be a saddle point. 19 Furthermore, a steepest contour is defined as a contour whose tangent is parallel to Vcp. In the direction of -Vcp, the modulus of the integrand, that is l exp{p(w)} [, is decreasing most rapidly and is therefore called a steepest descent contour. It can also be shown that steepest contours are constant-phase contours, i.e. where ~b(u, v) = constant. ~9When the path of integration is deformed into a steepest descent contour, the main contributions to the integral come from the area close to the local maxima of (p(u, v), i.e. the saddle points. Also, along this contour oscillations in the integrand are avoided since ¢(u, v) = constant here. It is these particular properties that made the saddle point

32

A. N a e s s & G . M . N e s s

method so efficient compared to a direct integration. This is still true even if the integration contours used in practical applications are usually composed of straight line segments, which do not necessarily follow the steepest descent curves except perhaps locally at the saddle point. Due to the symmetry property p ( w * ) = p(w)* and the continuity of derivatives, it is easily verified that the real axis, w = u, is a steepest contour containing the possible saddle points. Saddle points on the real axis are then found by requiring that c~q~(u, O)/¢3u = 0, i.e.

additional path segments arise, running from u~ + iv, v ~

~ , and f r o m , - i v to u, -

iv, v - - ,

iv to

~ . However,

the integral along these paths vanishes as v--,-s_. Equation (A24) may therefore be rewritten as fz(z)

=

Re ~2~i .... ix, e"l"~dw

=

Re

~

,

e p'''dw

- - Z +

/=,

+

1 -

42~u 2 +

(1 + 2-2,u-~ J = j

(1 - 2)~ju)2 0

(A27)

~

The solution of this equation has to be obtained numerically, and the search interval is restricted by the poles on the real axis, urj. = + (22j) ~ , i.e.

The second equality in eqn (A29) follows from the property p ( w * ) = p(w)*. The numerical integration of the last term of eqn (A29) is based on the same simple trapezoidal rule as used by Rice, TM that is fz(z) ~ Re {~[½eP~"~' + ~ e"~"~+'J"b']}

(A30)

j=l -

min (~2i) < u~ < rain (2~j)

(A28)

where a saddle point has been denoted by u~. For the specific problem in this paper, only one saddle point is found. Obviously, the steepest contour along the real axis cannot be used for the integration, but the fact that there are always two or more steepest descent (and ascent) curves emerging from saddle point u~, can be exploited. 19 A numerical investigation has shown that the steepest descent curves cross the real axis perpendicularly. Consequently, the path of integration in eqn (A24) is shifted to pass through the saddle point and the resulting integral can be calculated effectively. By shifting the path, two

where hb is the spacing with the scaling factor I c~2¢P(u'v) ..... } 1/2

(A31)

v -- 0

The scaling factor is based on the curvature of the integrand at u~ in the direction of the path of integration. As a first approximation of the integral, h is set equal to 1 and is then progressively halved. Even though this quadrature formula is very simple, it has proved to be powerful. 2° Naess and Johnsen 21 have derived an upper bound to the error IE(v) l made in truncating an integral similar to the one in eqn (A29).