Forces on a large vertical cylinder in multi-directional random waves

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24 JULi 19Th

ARCH1EF

OFFSHORE TECHNOLOGY CONFERENCE 6200 North Central Expressway

Dallas, Texas 75206

ABSTRACT

The linear diffraction theory used for calculating pressures, forces and moments on large tructures in regular long crested waves is extended to short crested multi.directionai random waves and applie4 to the case of a large surface piercing cylinder. Computer calculated theoretical results are compared with the rosults of measurements on a cylinder in laboratory generated short crested seas. Comparisons between theory and experiment were made in terms of transfer functions, calculated in the latter case from the measured spectra of the waves, pressures, forces and moments. The linearity of the experimental results and their agreement with theory even in severe sea states show that the theory can be used for design purposes.

Forces on a Large Vertical Cyt inder in

Multi-Directional Random Waves

By

S. M. Huntington and D. M. Thompson, Hydraulics Researçh Station

THISPAPER IS SUBJECT TO CORRECTION

Offshore Technology Conference on behalf of the American Institute of Mining, Metallurgical, and Petroleum Engineers, lnc. (Society of Mining Engineers, The Metallurgical Society and Society of Petroleum Engineers), American Association of Petroleum Geologists, American Institute of Chemical Engineers, American Society

-of Civil Engineers, American Society -of Mechanical Engineers, Institute -of Electrical and Electronics

En-gireers, Marine Technology Society, Society of Exploration Geophysicists, and Society of Naval Architects and Marine Engineers.

This paper was prepared fOr presentation at the Eighth Annual Offshore Technology Conference, Hóuston, Tex., May 3-6, 1976. Permission to copy is restricted to an abstract of not more than 300 words. Illusfratio9 may not be copied. Such use of an abstract should contain conspicuous acknowledgment óf where and by whom the paper is presented.

References and illustrations at end of paper

lab. v;Scheepsbauwkunti

TEhnische Hogeschool

i

Deift

PAPER

NUMBEROTC 2539

INTRODUCTION

Large water depths and severe environmental condition' have led offshore engineers to consider the use of monolithic concrete structures with members large enough to modify the wave field. This means that the wave-structure interactions are generally in the

diffraction regime(1).

Unear diffraction theory(2) is usually used for design in this situation. This theory, which neglects viscous effects, has been used to calculate the pressures, forces and moments on large bodies in

long crested (uni-directional) sinusoidal waves which were also used in the laboratory to test the theory.

-However real seas are both random and short crested (le,

multi-directional with wave energy propagating simultaneously over

a wide spread of directions). In this paper the theory is extended to short crested randómn seas and tested against experiments on a surface piercing cylinder in the Complex Sea Basin at the Hydraulics Research Station where seas with an angular spread of energy cañ be generated.

The theoretical and experirneñtal comparisons are made in terms of the transfer functions which relate the incident wave spectra to the spectra of the responses of interest, namely pressures, forces and mOments. These transfer. functions are of great importance in design because they describe the response of the structure and enable its response spectrum corresponding to any wave spectrum ta be calculated. From the moments of the

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where

in line and transverse moments (see equations (29) and (28)

respectively).

Evaluation of transfer functions for test conditions In the physical model used to test the applicability of this theoretical work the directional wave spectrum was made up of wave trains from ten discrete directions. In order to make an accurate comparison of the experimental and theoretical results, transfer functions must be computed for the particular configura-tion adopted in the physical model. Therefore the expressions for the transfer functions must be represented by sums rather than integrals, and hence equations (22), (24), (25), (26), (27), (28) and (29) become: lo Vi =

j1

G(!j) . T2(w.)J

TfXd(w) c1 - Tf(w)

Tfyd()

=

c2 . Tpo)

Td(c..)

=

c2.

Tmyd(W)= c1 _{. Tm(w)} IO =

[E cos2(j) . G(ct)]

lo

Vi C2 = [

sjn2(.) - G(1.)]

J=l J J

where the subscript d indicates a short crested sea made up from a finite number of discrete wave trains from directions ., where

are the angles of incidence of the trains and

j

is integ1er and

runs from i to 10. G(cIj) is the relative weighting of the energy in each wave train.

The ten wave paddles used in the physical model are symmetrically disposed about the x axis, each subtending an angle of 110, giving a total angular spread defined by the centre lines of the paddles of ±493°.

The actual distribution of the total energy from each direc-tion was related to the integral of the appropriate distribudirec-tion curve chosen over the subtended angle associated with the paddle. 1f the wave paddies are numbered i to 10 in sequence as above, then the relative energies G(j) are:

j+5 Vi°

G(cij) = A

f

005n d _j = 1,10 ....(37)

=

(-495

and A is a normalisation factor given by:

A =

1/(1

cos'

d)

and n is the power of the cosine curve assumed for the energy

distribution.

The theoretical transfer functions used for comparison with the experimental results are calculated for uni-directional waves from equations (13), (14) and (15) and for multi.directional

waves for cos2 and cos6Ø energy distributions from equations

(30)(39).

EXPERIMENTAL STUDY

The Complex Sea Basin

The laboratory tests were performed in the Complex Sea

Basin at the Hydraulics Research Station, Wallingford. This

facility is a concrete basin 18 m square with a water depth of 1.5 m. Along one side of the basin there are ten vedge type wave making machines, each disposed on an axis through the centre of the basin. The axes of the centre lines of the outer wave paddles subtend an angle of 100° at the central focus of the paddles. The sea around the focal point is the summation of the ten individual wave trains generated by the ten paddles. There are wave absorbers along the three sides of the basin facing the array of paddles (see Figure 3).

The wave machines are servo-controlled to follow independent input signals. 1f all the input signals are made the same the paddles operate in phase to generate long crested waves. These can be regular or random. If the input signals are random and uncor-related with respect to one another a short crested random sea is generated. The input signal for each machine is generated by its own pseudo random binary signal generator which has a variable digital filter so that signals with any spectral shape can be generated(8). Short crested random seas with known directional spectra are generated by adjustment of the signals and hence the energy spectra of the individual generators. Figure 4 shows the cylinder under test in short crested random waves.

Test cylinder

Forces, moments and pressures were measured on the test piece, a vertical cylinder 0.6 in in diameter and 2 m high, positioned at the focal point of the basin.

The cylinder had pressure tappings for semi-conductor type silicon diaphragm pressure transducers in twelve positions, four transducers being used at each of three levels. The three levels were 0.2 m, 0.5 m and 1.0 m below the still water level; the orientations of the transducers at each level were 00, 900, 1800 and 2700 relative to the principal wave direction. Pressures up to 2000 N/rn2 were measured.

The cylinder was mounted on a six component dynamometer set in a recess in the floor of the basin. The recess had a rigid cover flush with the basin floor with a flexible seal between it and the cylinder where the cylinder' emerged through the cover. The interior of the cylinder and the dynamometer in its recess beneath the cover were kept at atmospheric pressure to avoid uplift.

pressures.

Dynamometer

A six component balance (dynamometer) was desigted and constructed to measure the forces and mornnts on the test piece in short crested random waves. The dynamometer (Figure 5) consists of a lower plate which is connected rigidly to the floor of the recess, and an upper plate (the force plate) to which the cylinder is attached. The two plates are separated by four strain gauged weighbars which measure tension and shear forces. The three orthogonal forces and moments are obtained from the output of the gauges by a special computing network. The dynamometer was calibrated in situ with the test piece attached by applying appropriate point loads in various directions. The calibration frame can be seen in Figure 3 which shows the test piece in position on the dynamometer at the focal point of the wave basin. The dynamometer measures forces up to 1500N peakpeak and

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':esponsé ectrum the extreme values required for design can be

estimated).

THEORETICAL DEVELOPMENT

Linear wave theory is assumed in the following analysis of the pressures, forces and moments exerted by waves on a surface piercing cylinder in the diffraction regime.

The co-ordinate convention chosen for the mathematical development is shown on Figure 1.

Long crested regular waves

In the above co-ordinate convention the potential function for a long crested sine wave propagating in the positive x direction

is given by Wiegel(4) as:

and where g = acceleration due to gravity, H = crest to trough wave height (le, 2 x wave amplitude), w = wave radian frequency, k = wave number, d = water depth, t = time.

where _m Bessel function of the first kind of order m, e0 = i

and 6m = 2 for m I.

Assuming that the waves diffracted by the cylinder can be described by a similar expansion, then their potential function can be expressed iii terms of Hankel functions. Thus the total potential, ie, the potential due to the incident waves plus the potential due to the diffracted waves,can be written:

+ A H0)(kr))cos m8)e_(t = gH . cosh kz . _{; (emimJm(1)}

m m

2w cosh kd m=O

Note that Hankel functions of the first kind have been used in equation (4) because, for large r,

H 1)(kr) ¡

-

e1

r -

iT

N irkr

2m+l 4

which represents an outgoing wave satisfying the radiation boundary condition at infinity.

The Am in equation (4) can be evaluated by applying the boundary condition that the water particle velocity normal to the cylinder is zero, ie,

at r = a

- Em

m J'a)

which gives:

Am

Hg) (ka)

where and H) are first derivatives with respect to r. Neglecting second order terms, the amplitude of the dynìamic pressure due to waves at the surface of the cylinder, r = a, is given by:

where p is the water density. Using equations (4), (7) and (8) we

have:

_Pall.coshkz.

m cos,mO)).e_iwt

_....(9)

iTka coslì kd

'rnO'

The amplitude of the total horizontal force F on the cylinder is obtained by integrating equation 9 over the whole of the surface of the cylinder (all 8, z) so that:

F 2ipgH - tanh kd

. e_t

k2

M=2

kdtanhkd i +sechkde"°t

....(li)

k _HÇ')(ka)

Long crested random waves

Bendat and Piersol (6) show that if S(w) is a wave spectrum,

and S(w) a pressure spectrum associated with S(w) then:

S(w) = T2(w) S(w)

..(l2)

where T(w) is the pressure transfer function and is merely the ratio of pressure per unit wave amplitude calculated for linear sinusoidal waves at frequency w. Thus, in the case of a surface piercing cylinder in the diffraction regime T(w) is, (from

equation (9)):

,

cos mO I cosh kz . _(emim Iirka

cosh kd m0

I

Similarly, from equation (IO), the force transfer functionT1 is:

q.

T1(w) =tp&.

tanh kd I

I _k

_{(l)' (ka)'}

and from equation (Il), the moment transfer function Tm is:

Tm(0') =

_a

-

kd tanh kd - 1 + seth kd

H1)(ka)

/

Consequently, the evaluation of the responses of'a structure in long crested random waves is a simple extension of the results

for long crested regular waves, since spectra of the pressures, forces and moments (S , S- and Sm respectively) can be calculated from equations (I 2)ifs) if the wave spectrum S(w) is known. In this study the transfer functions T

_{, I}

and Tm are calculated and compared with their experimenta?versions obtained by measuring Sp Sf Sm and S, and using them in equations of the

form: where

- cosh _{eC_Cfht)}

....(2)

Similarly the amplitude of the total overturning moment M about the sea bed (z O) is obtained by integrating the moment per unit length of cylinder derived from equation (9) over the whole depth so that:

1

2w cosh kd

w2 =gktanhkd

This can be written in polar co-ordinates as: The above expressions for pressure, force and moment amplitude

are similar to those of Chakrabarti and Tam(5) for long crested regular

= . cosh kz . m

_{(kr) cos n18)e_t}

waves of wave height H measured peak to peak.

2w cosh kd m=O rne m

FORCES ON A LARGE VERTICAL CYLINDER IN MULTI-DIRECTIONAL RANDOM WAVES OTC 2539

170 _'I

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Comparison between the short and long crested results shows that the in line forces and moments (and their related maxima) in short crested waves with a cos2Ø angular distribution are about 90% of those in long crested waves with the same total energy. The transverse forces and moments are about 50% of the in line forces, je, about 45% of the forces in long crested waves with the same total energy.

In long crested rändom waves the maximum value of the force or moment can be estimated from the root mean square value(3). However, in short crested waves the maximum force or moment can occur in any direction and its statistics are not known in terms of the rms values of the x and y components.

CONCLUSIONS

Linear diffraction theory has been extended to short crested random seas. In a sea with a os2 continuous distribution

of energy between ±7T the in line forces and moments on a vertical cylinder are 87% of those in a long crested sea with the same total energy. The transverse forces and moments are 50% of those in a long crested sea with the same total energy.

The experimental results lie on one curve indépendently of wave height and frequency spectral shape. This shows that the interaction between the structure and the waves is linear even in

severe sea states.

There is good agreement between theory and experiment for pressures in long and short crested random seas.

There is good agreement between theory and experiment for forces and moments in long and short crested random seas, with the theory over-estimating at the peak of the response for in line forces and moments, and underestimating in the transverse case.

The agreement between theory and experiment in severe sea states, and the over-estimation of the larger in line forces and moments shows that the linear diffraction theory can be used for

design in real seas.

-NOMENCLATURE

A - normalisation fâctor Am - complex constañts

F. - overall force amplitude on cylinder

G(c7) angular, weighting function - discrete case

H - wave height (crest to trough)

Hg)-

- Rankel function of fIrst kind, order m - Bessel function of first kind, order m

M - overall moment amplitude on cylinder _REFERENCES

spectrum

- transfer function

HOGBEN N Finid Loading on Offshore Structres, A State

of Art Appraisali Wave Loads. Maritime Technology

- cylinder radius

Monograph No 1, Royal Ins titutiön 'of Naval Architects, November 1974. - .

- constants 2. HAVELOCK T H The Pressure of Water Waves on a Fixed

- water depth Obstacle. Proc Royal Society,'A,Vol'175, 1940, pp -409-421. - acceleration due to gravity

- angular weighting function - continuous case

3. CARTWRIGHT D E On estimating the mean energy 'of sea waves from the highest wave in à record. Proc Royal Society, 'A, Vol -247, 1958, pp 24.48.

174

FORCES ON A LARGE VERTICAL CYLINDER IN MULTI-DIRECTIONAL RANDOM WAVES OTC 2539 j,m,n - integer indices k - wave number k1,k2 - constants p - pressure amplitude r - radial co-ordinate t

time

x,y,z - Cartesian co-ordinates

- velocity potential - incident waves - velocity potential - total wave field - general angular measure of incident waves - angle of wave incidence - discrete sources

Em constant

o - angular cO-ordinate p . - water density

- radian frequency

Subscripts

c - continüous energy distribution

d - discrete energy distribution

f - related to forces

m - related to moments --.

p - related to pressures w - related tó incident waves x - related to x axis y - related to y axis ACKNOWLEDGEMENTS

Thanks are due tO the Director of the Hydraulics Research Station, Wallingford, for permission to publish this paper, to Mr C J Malinowski for his help in the experimental work and to the members of the Electrical, Design and Field Studies Section of HRS who made the experiments possible The work was part of a programme of research authorized by the Ship and Marine Technology Requirements Board on behalf of the Department of

Energy. s -S a c1,c2 d g

g()

(5)

T(w

S(w)

)½

S()

.(l6)

(Note that while T above is written as T _(o.,) for simplicity, it is

also a function of ?e, z); the position of &e point of interest on the cylinder.)

MuIt-directionaI random waves

Ìi'lulti-directional (short crested) random seas can be

considered as linear sums of trains of long crested random waves from all directions; the wave trains from different directions being uncorrelated. The pressures, forces and moments on the cylinder in short crested waves can be derived from the sum of all the contributions from long crested waves from all angles. Thus the instantaneous total pressure at a point on the cylinder is the linear sum of the instantaneous pressures from the long crested waves incident from every angle in the directional spectrum. The instantaneous total forces and moments result from integrating this total pressure arid consequently they act in any direction.

Hence it is convenient to resolve the'forces and moments in the x and y directions, (see Figure 2), forces along the x axis and moments about the x axis being denoted by a subscript x. Similarly the subscript y is associated with the corresponding forces and moments with reference to the y axis.

Consider the pressure on the cylinder surface at angle O and depth z due to a train of long crested random waves incident in the direction Ø. This pressure is exactly the same as the pressure on the cylinder surface at angle (eo) if the waves had been incident along the x axis. Hence we can write:

where T(iø) is the pressure transfer function for the pressure profile around the cylinder due to long crested waves incident

from angle .

The spectrum of the total pressure which results from

-uñcorrelated long crested wave trains from all the angles in the directional spectrum is then:

Spc(W)

=

S()

-

T2 ()dØ

....(l

where the subscript c denotes a short crested sea with a wave energy distribution that is continuous with angle, and S(ul,) is the directional wave spectrum.

lt is usual to assume that the angular distribution of energy in a directional wave spectrum is the same at all frequencies so that

S(w,) = g(Ø)

. S(w)

Ir

where

_f

g()dØ =

ir

and S(w) is the frequency wave spectrum at any point in the short crested sea (je, the spectrum that would be obtained from

an ordinary field wave recorder in the sea). From equations (18) and (19):

S(w) = S(w) . f g() . T,2 (ci,)d

...(2 1)

Therefore, the transfer function relating the frequency wave spectrum to the pressure in short crested seas is:

T(w)

= E f7T g(Ø) . T,2 (w,)d I ½ ....(22)

The force from a train of long crested waves incident at

angle is given by equation (14), so that the spectrum of the total force in the x direction due to uncorrelated trains of waves from all angles in the directional spectrum is:

Sçxc()

. cas2 p . T2(w)d ....(23)

the cos term arising from the resolution of the force in the direction into the x direction.

Using equation (19) the transfer function relating the frequency wave spectrum to the spectrum of forces in the x

direction is: -

-ir

where

The equivalent rèlationship for forces in the y direction can be shown to be

TfyC() = k2 .- Tf(w)

-ir - ½

= [f g()

sin2 .

dj

-ir

The corresponding relationships for the moments aie

--

T(w) = k2

-. Tm(w)

-

---

Tmyc()

k1 . tm(C - - -

-The above relationships (equations (24), (26), (28) and (29)) show that where the angular distribution of energy is the same at all frequencies the transfer functions for forces and moments in long crested seas can easily be modified to give transfer functions for a one-dimensional frequency spectrum which takes account of short crested waves. As an illustration consider two examples of short crested seas, namely a cos2 and a.cos6 angulâr

distribution of wave energy between +ir/2 ànd ir/21

Hasselmann(7) uses the cos2 distribution as beingtypical of real

-seas, the cos' distribution is arbitrarily taken as an example of a narrower spread of energy. The factors and k2 have been calculated for both of these energy distributions, with the angular distribution symmetrical about the x axis for convenience. These factors k1 and k2 are 0.87 and 0.5 for the cos2 distribution, and 0.94 and 035 for the cos6 distribution. This means that for a

cos2 short crested sea, the force along the x axis, which is in

-line with the principal wave- direction, is 13% smaller than -the

-force estimated using long crested waves with the same total energy. The transverse force (ie, the force along the y axis) is 57% of the reduced in line force. Similar relationships hold between the To calculate the transfer functions for multi-directional seas

it is necessary to define another angular variable which is

measured in the same convention as e and refers to the angle of

wave incidence (see also Figure 2). _where

Tfxc(w k1, Sfxc(C)) ½ cos2* ½

. dJ

....(24) ...125)

s(w)

-Ir

[f

g() .

T (ci,ø) =

2z..

cosh kz . m cas m(O_)) ....(17)

irka

cosh kd m0 m

_{) (ka)}

(6)

moments up to 2000Nm peakpeak.

The likely errors in the dynamometer have been estimated to be about ±3% in calibration together with an absolute error of ±0.3% of the full scale reading in each output channel. Further errors arise from cross-talk effects, je, a signal in one channel due

to an output in another, and this has been estimated at up to 3% per channel, but experience in measuring forces and moments in the transverse direction in long crested random waves indicates that a total error of less than 2% is more realistic.

Test conditions

A Moskowitz spectrum and Spectrum 92 specified in NPL Ship Division Report I 50(o) were used as a basis for the wave conditions for the tests. The Moskowitz spectrum describes a fully arisen or equilibrium sea state(10), whereas Spectrum 92 was recorded in storm conditions in the North Sea and has the sharp narrow peak similar to the JONSWAP spectrum(7).

At a scale of 1:76 the former spectrum (referred to as spectrum A) has a significant wave height of 151 mm arid a mean zero crossing period of 1.51 s and the latter (referred to as spectrum B).has a significant wave height of 114 mm and a mean zero crossing period of 1.10 s. The wavelength to cylinder diameter ratio put the cylinder in the diffraction/inertia regime over the significant regions of both spectra.

Both spectra were generated with l9ng crested waves, ¡e with the total energy in one direction, and with two angular spreads,

namely cos2 and cos6 between either side of the principal direction. In the latter cases the directional spectra always had the same total energy and frequency spectral shape as the long crested waves, the total energy being distributed between the ten paddles as described previously. Each combination of spectrum and angular distribution was used at three different wave heights, one equal to the storm sea states specified above, and the others to give wave heights of two thirds and one thirdof those specified values. The

purpose of this was to check the linearity of the results with wave

height.

The tests

Since the diffracted wave field around the cylinder is significant, the incident wave field was calibrated before the cylinder was put into the basin. 'Fhis was possible because of the repeatability of the random signals applied to the wave

generators. The frequency spectrum of the incident waves (referred to as S(w) in the theoretical development) was measured at the focal point of the basin for each of the test conditions (ie, 2 spectra x 3 angular spreads x 3 wave heights)

and compared with the appropriate theoretical curve. Figure 6 shows comparisons of the theoretical and measured frequency spectra for two of these conditions, and show agreement which is typical of all the others.

In each of the test sea states the pressures around the cylinder and the overall forces and moments on the cylinder were recorded on magnetic tape and ultra-violet paper. Data for spectral analysis was recorded over exactly one short repeating random sequence to remove statistical uncertainties. Figure 7 shows examples of computed pressure, force and moment spectra. The transfer functions were calculated by dividing the response spectra one frequency at a time by the incident frequency spectrum (cf equation (16)) measured during the calibration of the waves.

RESULTS

- The theoretical transfer functions for pressures, forces and

moments for the 18 test conditions were computed using equations (13), (14), (15) and (30)(39). Note that the linearity ofthe theory means that the theoretical transfer functions are independent of wave height and spectral shape and vary only with angular distribution. In the cases of forces and moments there is further simplification because the in line and transverse transfer functions only differ from one another and the long crested transfer function by the constant factors c1 and c2 given in equations (35) and (36).

The various theoretical transfer functions for the long crested waves and the short crested waves with a cos2 angular distribution

of energy are plotted dimensionally on Figures 8-11. The transfer functions for long crested waves have been checked against those published by Hogben et ai(li).

Chakrabarti and Tam(s) show how the dimensional plots can be reduced to a dimensionless form.

The experimental transfer functions, calculated from pressure, force and moment response spectra and the measured wave spectra, are plotted on Figures 8-11 in groups for the different angular energy distributions. Each plot shows the results for 6 test condi-tions (le 2 frequency spectral shapes x 3 wave heights) with the

same angular distribution.

A striking result from the graphs in general is that the six sets of points on each graph define one curve in spite of some scatter on some of the transfer functions. This means that the experimental results are independent of wave height and frequency spectral shape as predicted by the theory even in high steep waves typical of storm conditions. This implies that a linear process relates the waves to the pressures, forces and moments. The results for the cos' angular distribution which are not included for reasons of space give exactly the same picture.

Pressures

Space does not permit the inclusion of the results for both O = 900 and O = 2700. However the experimental results for these two angles are the same, which is expected for a symmetrical body in a sea with a syitimettical angular distribution of energy. The experimental points (Figures 8 and 9) closely follow the theoretical curves down to the small inflexions which are evident in spite of the scatter at the pressure points nearest the free surface. (The increased scatter is probably due to free surface effects.) The curves also show the expected decay of pressure with

depth.

Forces and monents

The force and moment transfer functions (Figures 10 and 11) also show good agreement with theory except at the peak of the curves where the experimental points fall below the theoretical curve for in line forces and moments. This has also been observed with long crested sinusoidal waves0 1) Conversely the experimental results for the transverse forces and moments tend to lie above the theoretical curve near the peak.

The agreement between theory and experiment, particularly in the light of the over-estimation by the theory of the larger in line forces and moments, show that linear diffraction theory ca be extended to short crested random seas for use in design.

OTC '2539 S W HUNTINGTON AND D M THOMPSON

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POSITIVE WAVE DIRECTION a-H

K

AZ ,d,

K

J X d SWL

V. I

Fig.

i

Cyl Inder in unidirectional

waves.

*

OT 2539

S W HUNTINGTON AND D M THOMPSON ₁₇₅

4. WIEGEL R L Oceanographical Engineering, Prentice Hail Inc 8. FRYER D K, GILBERT G and WILKIE M J A Wave

5.

1964.

CHAKRABART! S K and TAM W A Gross and Local

Spectrum Synthesizer. Journal of Hydraulic Research, 11,3, 1973, pp 193-204.

Wave Loads on a Large Vertical Cylinder - Theory and 9. EWING J A and HOGBEN N Wave spectra from two Experiment. Offshore Technology Conference, OTC 1818,

1973. _{Ship Division, Report 150, 1971.}British Research Trawlers. National Physical Laboratory, 6. BENDAT J S and PIERSOL A G Measurement and 10. PIERSON W J Jr and MOSKOWITZ L A Proposed Spectral

Analysis of Random Data, Wiley and Sons, 1966. Form for Fully Developed Wind Seas based on the

7. HASSELMANN K et al Measurements of wind-wave

growth and swell decay during the Joint North Sea Wave

Similarity Theory of S A Kitaigorodskii. J. Geophysical Research, 69, No 24, 1964, pp 5181-5190.

Project (JONSWAP). Erganzungsheft zur Deutschen 11. HOGEEN N, OSBORNE J and STANDING R G Wave

Hydrographischen Zeitschrift Reihe, A(8°), Nr 12, 1973. _{Loading on Offshore Structures - Theory and Experiment.} Proc Synposium on Ocean Engineering, Royal Institution of

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Incident wave direction

y

Fig. 2

-

Cylinder in multidirectional waves.

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U C C L -c (r. a: ç-o .(1 L o U)

(10)

30

1.0

O

o 0.2 01. 06 08 70 1.2e 7.1.

FREQuENCY (Hz)

Fiq. 5

- Six component dynamometer.

LO 7.5 o 00 o 0.1. 08 12 16 20 FREQUENCY (Hz)

Fig.

6

- Examples of wave spectrum calibrations.

SPECTRUM B

SHORT CRESTED WAVES

COS2 DISTRIBUTION

FULL WAVEHEIGHT

THEORETICAL

O MEASURED

SPECTRUM A

LONG CRESTED WAVES

e FULL WAVEHEIGHT THEORETICAL O MEASURED

t

e s e

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o o o 0.4 08 1.2 16 2.0 FREQUENCY (Hz) o o

o

o o o IN LINE FORCE 0.6 0.8 1.2 16 2.0 FREQUENCY (Hz)

RESPONSE SPECTRA MEASURED IN SEA STATE SPECTRUM B WITH C0S2

ANGULAR ENERGY DISTRIBUTION ANO FULL WAVEHEIGHT

Fig.

7 - Examples of pressure, force and moment

spectra.

s. .-. 0.3 s- I-. 0.2 0 0.1

0

o o o IN LINE MOMENT

o

Q.4 0.8 1.2 1.6 2.0 FREQUENCY (Hz) 0.6 DEPTH 02m

9 0°

;i 0.5

o

.4 0.4 I-. Q- i, _tu 03 o 0.2 o 0.1 o

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E E

n

'o es 02 o-r p 7.0 7 f 76 0 0.2 ' 0 0- 7 7' 75 0 0 OREOUENCV (H. WEVE#7E0HT] '7 wSXEHE,GHr SPEC7RUM â 7 W7'E7EIGHT I WAVE77EIGHT i o WAVE7SE7OHT SPECTRUN 5 o 7 WAvE HElEE? J 7 OmOEPTH

Fig. 8

Pressure transfer functions for

cylinder in long crested random seas.

(13)

p 10 0-8 O-6 0-2 E E 0 E E 08 w 'n w 0.5 QL 0-2 'n A o 9' 9 0 O 90 A e

-'\g.

_A

N

0.5mDEPTH o o 1.OmOEPTH 0 0-2 Q-L 0.6 0.8 7-0 1-2 7-L 76 0 0-2 0-' 0-6 0-5 7-0 1-2 1-4 1-5 0 0.2 0-L 0-6 0-8 7-0 7-2 7-4 7.6 FREQUENCY (Hz)

Fig. 9

- Pressure transfer functions for

cyInder in short crested random seas with

cos 4

energy distribution.

(14)

O5 O.3 0.2 0.1 o E 0.5 o.'

2.

O _£ INLINE

t

FORCE

13 \.

eN

e IN LINE MOMENT NO TRANSVERSE FORCE

r .a

2 D2

*

0.3-

f

NO TRANSVERSE I £ MOMENT

I

'o

I

/

\

I

N

0.1-/

O 0

02 CL

06 08 1.0 1.2

iL

16 FREQUENCY (Hz/

Fig. 10 - Force and moment transfer functions

for cylinder in long crested random seas.

(15)

s E

&03

02 0.1 O 05 E

u

0.3 I- '4-02 ,_ 0.1 00 ₀₂ IN LINE MOMENT o Oh 06 08 1.0 1.2 1h 1.6 0 FREQUENCY (HzJ

Fig.

11 -

Force and

moment transfer functions

for cyIider in short crested random

_seas

with cos

energy distribution.

2 £ TRANSVERSE FORCE TRANSVERSE MOMENT / 2 D d.2 0.1. 05 08 1.0 1.2 1.1. 7'6 g