Response statistics of moored offshore structures

O/t1/772L/2/

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Response statistics of moored offshóre structures

Takeshi Kinoshita, Satoru Takase TECHNIScHE_UNIVEfl$jj

Lboratorumo,

Institute of Industrial Science, University of Tokyo

OOPthYdm,il

Archjef

Mekelweg_{2,262$ CO Defft} TeL 015.788873._Faz

Introduction

For the design of a mooring system, it is required to analyze a motion

of a floating structure moored in random sea by a statistical method.

Extreme values of the rnotioii should be investigated.

There are two components of a motion response of the moored floating

structure, a linear response and a quadratic response; the latter is called

slow-drift motion. The linear response is Gaussian, but tile quadratic

response is non-Gaussian. Total response is generally non-Gaussian al-though it is conventionally treated as Gaussiaii.

In the previous papers, the authors analyzed the extreme values in

order to verify the theory which they proposed1)4). The extreme values

in those papers, however, scatter significantly. In this paper, we study

about estimation of tile extreme values by _{an experimental way.}

First subject is to obtain the extreme values from experiments. We

dis-cuss the way to get reasonable estimation of the extreme values. Secondly

we compare several theoretical methods to predict the extreme values.. It

is hard to treat a total response, because it is neither linear nor Gaussian.

Naess5),6) tried _{to calculate a linear and a quadratic response separately,}

and proposed square root of sum of squares(SRSS) formula. On the other

hand, Kato et al.7) propQsed a formula assuming that response process

x and its time derivative process ± are statistically independent, utiliz-ing a probability density function of total response x. We discuss their

formulas using experiment data.

Outline of experiment and data analysis

Model used in the experiments is a semi-submersible shown in Fig.1 and its particulars are in Table 1. The model is moored b.y linear springs and subjected to head seas..

The experiments are carried out for 6 cases of combination of wave

conditions and mooring spring coefficients, which are indicated in Table

2 and examples of wave spectrum are shown in Fig.2. The surge

N=

i(I+

I)T

4ir N N 7122

(2)

sensor system. Measured displacement data is analyzed in a following

ni ann e r

As an extreme value we adopt the expected largest amplitude in time interval T in this study.

E[(T)] = E[max{x(t); t0 <t < t0 + T}]

(1)

In the previous paper, we used the expected largest amplitude in

N observations. The relation between the time interval T[sec] and the

number of maxima N is approximately expressed as8

j'°wS(w)dw

(3)

where rn,, is a spectral moment with order

_{n and S(w) is a spectral}

density function of motion.

3. Getting smooth extreme values

It recluires reasonably long data to get statistically stable data. How-ever duratIon of measurement in a wave basin is limited because of re-flection by the tank end.

/

We examine the effect of length of data to the extreme values. We

carry out three types of experiment as shown in Table 2: (1) Average of

four times of 10 minutes measurement (Each four _{waves have différent} phase.), (2) One time of 30 minutes measurement, (3) One time of 50

minutes measurement.

Next, we discuss the analytical method of measured data. (a) When

we calculate the expected largest amplitude in time interval T, depending

on definition, each time interval data does not overlap as Fig.3(i). But in this method, we can average only a few number time interval data,

where T(or N) is large. (b) Each time interval data overlap with lag T as Fig.3(ii), in order to make smooth the results. In this calculation, _time

lag t

10[sec].

The results are shown in Figs.4r9. Values from the long time

measure-ments by method (a) also do not reduce their fluctuation. The _{average of}

four trials, even if each trial is short, rather reduce the fluctuation than

Lines of the valuês by method (b). show smoother results than those by method (a). But, Figs.6 and 8 show that these two methods tend to diverge when T increases. This is because oniy a few time interval data are averaged for a large time interval T in method (a) Thus it includes large statistical uncertainty. It is obvious that the values from method

(b) also include statistical uncertainty particularly for a large T.

Figs.4 and 5 show that the length of data significantly affects the

ex-treme values. To discuss in detail of this point, we show the results of each trial of 10 minutes measurement in FigJO.. These four trials have same condition except phase lag, and their values ought to agree, if the trial duration is enough. However, in fact, there are large differences

among these. In other Words, effect of random phae is large, then it is

difficult to get reliable data. Comparing with these discrepancies, we can

see in FigsA9 a differences appearing in Fig,lO fairly decrease by both methods (a) and (b), and that the values from 30 minutes measurement

are not so much far from the values from 50 minutes one, as shown in Figs.5 and 8.

As a result, long time measurement is effective _{to get reliable data, and}

method (b) is efficient to obtain the smooth expected largest amplitude, although the values from method, (b) still include statistical uncertainty. Therefore we use the method (b) of 30 minutes measurement in following analyses.

4. SRSS formula

4.1 experimental verification

A linear and a quadratic _{response of motion are represented as follows,}

where ((t) is wave elevation which is a statioûary Gaussian random

vari-able with a zero mean, g1(r) is a linear impulse response function and

g2(r1, T2) is analogous to the linear impulse response function and is called

the quadratic impulse response function.

3

x(t) x2(t) + xi(t) (4)

xi(t) f .g1(r)((t - T)dT (5)

It is hard to theoretically analyze the extreme values of a linear and a quadratic response simultaneously, since they are not independent.

Naess5'6 introduced the SRSS formula, treating a linear and à quadratic

response separately.

E[(T)j = VEi(T)j2 + E[2(T)]2

(7)

He divided 2 to a mean value and a varying part with a zero mean, then

E[(T)] =x + \/E[l(T)]2T+ (E[52(T)]

(8)

We call this formula as modified SRSS in here.

Kato et al.7) showed that the peak probability density function of_{s can} be obtained using the probability density function of s itself by assuming

that response process s and its time derivative process ± is independent

as follows,

d{Px+)}

(9)

4

where P(x) is a probability distribution function of _s.

The results are shown in Figs.1116. The total responses are

underes-timated by the SRSS formula. The value by the modified SRSS formula

is close to the total response than SRSS, still it underestimates _about

317%. In Figs.11,12,14 and 16, the values by _{Eci.(9) is very close to the} total responses. But in Figs.13 and 15, it differs from the total responses

by method (b), rather is close to those by method (a).

In these two cases, the differences between values by methods (a) and (b) are large, and there might 1)e relatively large statistical uncertainty, _{as discussed in} the previous section.

where _{p is peak probability density function of}

_{s and}

_{is mean value of} s. And the expected largest amplitude in N observations expressed as

E[Xm(N)1 = [°XPrnp(X)dX (10) (11) Pmp(x)

NP(x)'p(x)

Pmp(X) = = (12) pp(X) =

-P(x)

Naess introduced the correction factor q t Eq.(8), then

= +

q\/E[î1(T)]2

_{+ (E[2(T)]}

_-

₍₁₃₎ His calculation showed that q 1.2, and our experimental results show

smaller q value than it. Assuming q = 1.1, the errors reduce to about 7%. But since there are differences _{among the errors of each case, there is a} question yet if the concept of correction factor q is a good approximation in other cases.

4.2 approximate theory

Approximate theory of SRSS formula is also introduced by Naess. Since

a linear response is Gaussian, its expected largest amplitude in time in-terval T is asymptotically approximated as

v'(ir+ L)

,

(T» 1)

(14)

q1 =

_{21n(±.J!Z-.T)}

₍₁₅₎

27r\jmo

where 'y is Euler's constant and m7,. is a spectral moment obtained by

Eci.(3).

Naess6) derived following approximation _{for a quadratic response.}

E[(T)

_-

2ln{_ {i

-

r2(_T )_6]} TSD TSD

6=1r

(17)

1+r

r = exp(ir)

(18)

where 2 is a standard deviation of x2, TSD is a slow-drift period and ,c

is a relative damping, ic O.O82 in this experiment.

The results are shown in Figs.17r'22. Figures except Fig.19 show that when T is small, this approximate theory underestimates, and when T is

large, it overestimates. In the range of T in these examples, this theory

is a good approximation, but it may not be utilized for the prediction of a longer period.

5. Conclusion

From the above ánalysis, it is concluded as follows:

(1) Long time measurement is effective

_{to get reliable data, and the}

5

method (b) expressed by Fig3(ii) is efficient tö obtain the smooth

ex-pected largest amplitude.

IViodified SRSS formula with correction factor q 1.1 is a good

esti-mation in this case. In other cases, however, it is not clear whether this formula is good or not.

Approximate theory of SRSS formula can be utilized at small time

interval T, but is invalid for the prediction of _{very long period.}

Equation (9) is a good approximation, when the values by _methods

(a) and (b) agree each other.

We discussed several ways of prediction of extreme values. From this

study, it is the most important in future to estimate the _{extreme values}

where statistical uncertainty is minimized, from experiments.

References

Kinoshita, T., Takase, S. and Kato, S.: Statistical Analysis of Total Second Order Responses of Moored. Floating Structures _{in Random} Seas, Proc. of OMAR 1989 The Hague, Vol.2, pp.449 456, 1989.

Kinoshita, T.., Takase, S. and Takaiwa, K.: Probability Density

Func-tion of a Slow Drift MoFunc-tion of a Moored Floating Structure in Random Seas, Proc. of OMAE 1990 Houston, Vol.2, pp.97 105, 1990.

Takase, S., Kinoshita, T. and Matsui, T.: On Extreme Values of the Slow Drift Motion of _{a Moored Vessel Effects of the Second Order}

Potential and Coupled Motions, Proc. of OMAE

_{1991 Stavanger,}

Vol.2, pp.7L--'78, 1991.

Takase, S., Iinoshita, T. and Matsui, T.: Effects of the _{Second Ordei}

Potential and Coupled Motion ofa Slow Drift Motion of a Moored

Ves-sel, Journal of Soc. Nay. Archit. of Japan Vol.170, _{1991 (in Japanese).}

Naess, A.: The Response Statistics of Non-Linear, Second-Order

Trans-formations to Gaussian Loads, Journal of Sound and Vibration, _pp.103

129, 1987.

Naess, A.: Prediction of Extremes of Combined First-Order _and Slow-Drift Motions of Offshore Structures, Applied Ocean _{Research, Vol.11,} No.2, pp.l00'-'i10, 1989.

Kato, S. Ando, S. and Kinoshita, T.: On the

_{Statistical Theory of}

Total Second-Order Responses of Moored Floating _{Structures, OTC,}

Vol.4, pp.243''257, 1987.

Och, M,K.: On Prediction of Extreme Values, Journal _{of Ship Research,}

1973.

9) Longuet-Higgins, M.S.: On the Statistical Distribution of the Height of Sea Waves, Journal of Marine Research, Vol.XI, No.3, 1952.

Table i Principal dimensions of semi-submersible

Table 2 Parameter of experiment

case - 1 2 3 4 5 6

peak wave frequency

w[rad/sec]

2.2ir 2ir ir 2.2ir 2ir ir

measured sign. wave height H113 [cm] 2.16 3.83 4.18 2.25 3.86 4.24 spring coef. of surge mode [N/rn] 78.45 10.83 measurement item o o o o o o o o o o o o o o 10 min. 4 times 30 min. i time 50 min. i time Length(Lower Hull) L 1.15 m Breadth

B

0.6 m Draft d 0.2 m Center of Gravity .Xg 0.0 Yg 0.0 m Zg 0.175 m

Metacentric Height BM

0.0288 ni BML 0.0237 m

Radius of Gyration Pitch 0.3558 rn

A o 240 240 240 720 o f1 j

pp

1150

o

_-J >

Fig.1 _{Model of a sethi-subniersible. (unit: mm)}

r

x105

4.00

3.00

2.00

1.00

0.00

0.00

_1.60

x101

0.40

0.80.

1.20

o [nid/sec]

Fig.2 Wave spectra of experiments.

O

T

2T

3T

T

T

T

method (a)

T

_'r+T

I I.

T1

I

T

'z T

T

method (b)

Fig.3 Setting U of time interval.

/1

t

5.00

-::4.00

3.00

2.00

1.00

0.00

10 min. 4 times

30 min. i time

}

method (a)

10 min. 4 times

30 min. i time

}

method (b)

Fig.4 Data smoothing, case

2.00

_4..00

x102

5.00

b

EA

3.00

2.00

1.00

0.00

10 min. 4 times '

30 min. i time

_{methòd (a)}

50 min. i time

10 min. 4 times

30 min. i time

method (b)

50 min. i time

4.0.0

x102

2.00.

T[sec]

Fig.5 Data smoothing, case 2.

r

5.00

b

4.00

3.00

2.00

1.00

0.00

- -

_{-. 30 min. i time}

10 min. 4 times

-

30 min. i time

J

method (b)

Fig.6 _{Data smoothing, case 3.}

2.00

_4.00

><102..

T ec

r

5.0e

k4

_0O

3.Oo

2.Oo

1.00

0.00

10 min. 4 times

method (a)

3Omin.J time:

lOmin.4tjmes

J

_method(b)

---.

3Oflhlfl.ltime

'r

2.0e

Fig.? Data

5.00

400

e

3.00

2.00

10 min. 4 times

30 min. 1 time

_{method (a)}

- 50 min. i time

--. lOmin.4tin-ies

-

_{30 min. i time}

_{method (b)}

5Omin.ltjn-ie

-Fig.8 Data smoothing, case 5.

/

1.00

0.00

_2.00

_4.0.0

5.00

4.00

3.00

2.00

1.00

0.00

10 min. 4 times

method (à)

30 min. 1 time

10 min 4 times

method (b)

- 3Omin.ltime

Id)

Fig.9 _{Data smoothing, case 6.}

2.00

_4.00

x102

r

5.O

7

T[secj

><102

Fig.lo Four trials f lo minutes

_{rneureneit, case 1.}

2.0ø

4.0e

Tfsecj Fig.1i Extjeiie Values froi _experj11ent

i

/7

1.00

0.00

2.00

_4.00

_6.00

T[ec]

_X102

1.00

0.00

_2.00

_4.00

T[sec]

Fig.13 Extreme values from experiment,._{case 3.}

6.00

5.00

4.0.0

3.00

2.00

1.00

0.00

2.00

_4.00

_6.00

T[sec]

x102

r

5.00

T

.4.00

3.00

2.00

1.00

0.00

2.00

Fig.15 Extreme values from experiment, case

2-5

4.00

T[sec]

6.00

5.00

4.00

3.00

2.QO

1.00

0.00

caL by eq. (9)

-. method (à)

2.00

_4.00

_6.00

Ï[sec]

_x102

5.00

4.00

3.00

2.00

1.00

0.00'

2.00

_4.00

T[sec]

Fig.17 Approximate theory of SRSS formula, case

6.00

r

1.00

0.00

2.00

_4.00.

T[sec]

Fig.18 Approxinite. theory of.SR.SS formula, case 2.

.21

6.00

5.00

E-4

3.00

2.00

1.00

0.00

2.. 00

Fig.19 Approximate theory of SR.SS formula, case 3..

27

4.00

T[sec]

6...PQ

r

5.00

4.00

3.00

2.00

1.00

0.00

2.00

Fig.20 Approximate theory of SRSS formula, case 4.

4.OQ

6.00

1.00

0.00

2.00

4.00

T[sec].

Fig.21 Approximate theory of SRSS formula, cse 5.

6.00

5.00.

4.00

3.00

2.00

1.00

0.00

total

SRSS from experiment

SRSS approx. theory

2.00

Fig.22 Approximate theory of SRSS formula, case 6.

4.00

_6.00