Study on dynamic response of a moored structure subjected to compound external force

Subjected to Compound External Force

By:

Masanori Kobayashi*, KiyOshi Shimada*, TOlfru Fujihira*

Iñ order to estirnaíe motions aÑd moorïñg fOrces of a moored offshore structure subjected to the compound ex

terna! force of wave, wind and current, a computer slrnulàtión program was developed, in which slow drift oscillations

were taken into account

In this paper an outline and a calculation method of the program are described The results obtained by a model expertment for a Tèñsïôn Leg Platform (TLP) in irregular waves are compdred with the calculated oñés. the validity of the program is confirmed by good coincidence between them in the comparison Furthermore an example calcu lation for thé TLP under the one-leg-broken condition is shown.

1. Introduction

A floating offshore structure moored at sea is random-ly excited by external forces, namerandom-ly wave exciting forces, wind forces, current forces and so on furthermore, its mooring system sometimes shows strong nonlinearity So the structure produce complicated motions. Since a

safety policy and a rate of operation for the moored structure must be evaluated beforhand, it is very

im-portant to estimate such motions accurately"

The moored structur has the natural frequencies for the motiOns in the horizontal plane due to its mooring system in contrast to a free floating structure. When the frequency of a slowly oscillating drift force acting on the structure in irregular waves approaches the natural frequency, the structure produces large oscillations due to dynamic amplification, which affects mooring forces

intensively. Sinçe not only the external forces but the mooring characteristics are nonlinear in such a case, it is insufficeint to estimate the motions of the structure On the basis of conventional linear theory It is necessary to solve the nonliñear equations of motions by the time integral tçchnique.

In order to investigate the motions and the mooring forces the authors have been devélóped the simulatiOn

program which takes the slow drift oscillàtions into

consideration _{Furthermore a few kind of experiments} have been carried out to verify the validity of the pro-gram and the comparisons between their results and the calculated ones have been made. In this paper the fuñe-tiozi, the characteristics and the mathematical model of the program are briefly described. In addition the results of the experiment for a tension leg platform (TLP) in

* Ocean Engineering Research Sect., Akishima Laboratory,

Ship & Ocean Project Hq.

irregular waves are compared with the calculated ones. The sample simulation for the TLP under the one-leg-broken condition is also intródüced.

2. General discription of the program

The program computes the motions and the mooring foÑes of a moored offshore structure and a

conven-tional ship subjected to compound external forces in

time domain. The nonlinear equations of motions,

which take into account slowly oscillating drift forces, nonlinearity in the mooring system, visçous drag forces

and so on, are solved by the time integral technique The principal characteristics of the program are as

follows:.

(I) Hydro dynamic forces. of the structure, namely radiation forces, wave exciting forces etc., are

ob-tainéd by the calculations or the experiments as

input.

Irregular waves generated using an arbitrary wave spuctrum are considered. Thé Waves whose spec-trum is of ISSC, Bretsëhneider-Mitsuyasu or JONSWAP2 type can be generated with the inputs of a significant wave height, a mean wave period and so on. Furthermore, the calculation using as

input the time series of measured wave cañ be

performed.

Not only first order wave forces but nonlinear

slowly oscillating drift forces of second order are considered. The drift forces ate computed on the basis of Hsu3 or Pinkster's methOd.4>

Consideration of wind and current forces besides the wave exciting forces realizes the simulation in

the presence of the compoUnd external forces.

The wind Whose spectrum is of Davenport" type is generated with a few inputs.

Mooring forces are calculated quasi-statically 1

using a force-displacement characteristic curve: For a TLP, however, the tether force is calculated from the axial rigidity and the distance between a. moored point and an anchor.

(6) Motions of the structure a wave elevation, mooring forces and the displacement of the moored point are obtained by the computation. Their time series

Calculation Stagé Ç Start Data input

Hydodàmic Force Simulation Parameter etc.

'I, 15v. Matrix of Mass Wave Components Output Wave Mótion Tension etc; Stop Output Stage

(

Start Plotter Output Ç

SoP)

Fig. i Flow Chart of Program

Data Card

(>

Wave Data > Time Series Time Series Spectrum

Fig. 2 Coordinate System

Eqüáticús Of motions- are established to account for such a characteristic and the presence of wind and cur-rent forces. Practicality of the program at the design stage- is also taken into consideration;

3.1 Equations of motions

A set of coordinate axes G-xyz with -an origin fixed at

the center of gravity of the structure in still water, as shown in Fig. 2, is used to desciib the motions. The

motions is expressed as displacements (x0, y,zG) and

rotations (ça, O, sfr) of the center of gravity G. x,Yc, ZG and spectra . are dräwn- by a. plotter, and thèir statistics, namely a maximum, minimum, mean

value and so -on, are calculated.

(7) In the case of a conventional catenary moored

system the displacement of the moored point is

used as input of the program of mooring line

dynamics based on lumped mass method. It

enables the analysis of line dynamics in time do-maim

The program consists of two modules of the computa-tion and output stage. The former reads input data and

solves the equations of motions, the latter produces

drawings and statistics. The flow chart of the program is shown in Fig L

3. Mathematical thodél

When a floating offshore structure is moored, it has natural périods for the motions in the horizontal plane due to the horizontal reaction force caused by moonng

system. if the mooring system is loose, the natural

periods become longer and the amplification of the mo-tions may occur due tci slówly oscillating drift forces besides the motions in wave frequency region. Although the slowly oscillating drift forces are much smaller than the first order wave exciting forces, they produce a large displacement of the structure due to the small damping at the natural period.

Exciting Force Wave(lst & 2nd order)

Current. Wind

Drag Force

Mooring Force

Numerical Integration

.Spectrum Aña]ysià Analysis

Statistical Analysis Results

-denote surge, sway, heave andcp, O, 1r roll, pitch, yaw respectively.

The equations of motions, strictly speaking, must be described in the forth of convolution integral in time domain because of dependence of the radiation forces on a frequency.6 However, the exact solution by means

of a computer is so time-consuming that it does not

seem to be a practical method. So on the assumption

that the radiation force has the constant value at the

representative frequency, the equations of motions are described as the constant coefficient differential equa-tions. It is expressed as follows:

(M± A)X+ BX+ CX± D+ T

=FE±FDFw+Fc

(1)

Where

M: mass matrix of a structure

added mass matrix linear damping matrix hydrostatic stiffness matrix nonlinear damping force vector T: mooring force vector

FE: first order wave exciting force FD: slowly oscillating drift force vector

F:

wind force vector

F:

current force vector

displacement vector of a structure

Eq. (1) is a nonlinear equation because of the viscous,

mooring, drift force etc. k is solVed by the time step integral using the fourth order Runge-Kutta method

Descriptions of the terms in Eq. (1) follow. 3.2 Descriptions of terms in equations 3.2.1 Radiation force

The radiation force matrix at the peak or mean fre-quency of the wave spectrum is used. As.for the motions in horizontal plane, linear damping forces are neglected. 3.22 Wave and first order wave exciting force

The wave elevation whose spectrum is S() can be

written in the form

N

cos(ùt kxcoskysin+e1)....( 2)

where

= /2S(w)4o

&: circular frequency

k:

wave number

e:

uniform random number (O<.e<2ir)

X: angle between x axis and direction of wave propagation

4ú:

width of division of circular frequency

The width of division of the circular frequency,

4ù,

is determined randomly so that the time series does not repeat itself. When the wave elevation is

expressed as Eq. (2), first order wave exciting foÑes àre described by the following with an àmplitude response HFE(o) and a phase response eFE(0)).

FE(t)

₌

HEE((u) cos(0)t - k.x cos x

kjysinx+CFE(0)j)+eÍ) ( 3 )

3.2.3 Slowly oscillatiìig drift force

The slowly oscillating drift force which causes slow drift oscillations is a second order wave exciting force. The exact solution including a second order potential is necessitated to evaluate the force accurately. The exact solution is, however, supposed to be unsuitable to the design stage because of its time-consuming property

from the view point of a practical use. The present program perform an approximate calculation based on Hsu or Pinkster's method, which calculate the slowly oscillating drift forces from the steady drift forces in regular waves.

Hsu's method is summarized as follows: It is assumed that irregular waves can be characterized by a sequence

of a regular wave whose period and height change

every half wave length between zero crosses. Each wave

between zero crosses produces the steady drift force

corresponding to each regular wave. So the slowly oscillating drift forces are obtained in the form of a step function. On the other hand, according to Pinkster,

when the wave elevation is written as Eq. (2), the slowly oscillating drift forces are described as

FD(t)= HFD(

)j{cos(0)_O)f)

i=Jj=1 2

+ee1} ...(4)

with the response function of the steady drift forces in the horizontal plane, FIFD(04.

3.2.4 Viscous drag fOrce

In order to estimate the magnitude of the slow drift oscillation a viscOus drag force must be considered as a damping force. Since an offshóre structure genèrally consists of thin members Of a lower hull, a colUmn, a bracing etc, the viscOus drag force is calculated on the basis of Morison's formula". The force on the member is obtained by integrating the square of the relative ve-locity between a water particle and the member along its center. The sum over whole members results in the

viscoUs force on the structure. The viscous force in the normal direction on each thember is written in the form

D(t)=

where

p:

mass density of water a: radius of member' CD: drag force coefficient

wave orbital velocity in the normal direction v,: velocity of member in the nofmal direction If the member penetrates a free surface, the integration is made up to the free surface varying from time to time in order to consider the variation oía wetted surface of the member.

3.2.5 Mooring fórces

In order to apply an arbitrary mooring system to the present program static mooring characteristics are used as input. For a tension leg type mooring system,

how-ever, the mooring forces are obtained from the axial

rigidity 'and the elongation of the tether. Let rM and

(5)

r4 be the displacement of a moored point and the

coordinate of an anchor respectively, then mooring

forces on the moored pOint,. T1, is written as

r-r4

X {TO+(IrofrAI4Lo)XAE}

I

rfr4

L0

(6)

T0: initial tension of tether

L0 initial length of tether

AE:- axial rigidity

Since geornetical nonlinearity8' is taken into

considera-tion in Eq. (6), coupling between the horizóntal and

vertical motions are' considered. 3 2 6 Wind and current forces

Wind force

A wind force and a wind moment are written in the following form with wind force coefficients of the whole structure.

F(r)=J_pACDFV2 (wind fOrce)

M(t) =IpALCDMV2 (wind moment) where

p: mass density of air

A projected area in each mode

CDF, CDM: wind force and moment coefficients L: structure length

V: wind velocity at representative pòint of super-structure

Current force -- -

--

-Only a steady current is considered. The Current force force on -the structure is calculated by Eq. (5) with the current -velocity considered in the term of the relative velocity.

4. Experiment

In order to verify the validity of the program the

model tests for a TLP in irregular waves were carried out. Since the behavior of a TLP is relatively simple be-cause of restraint of vertical motions, heave, roll and pitch, the T-LP is suitable for verification of the program which takes the--slow drift oscillations into consideration.

The tests were carried out in Current Water Tank,

which has a water depth of 2.5 m, a breadth of 8.0 m

and a length of 55.0 m, at Akishima Laboratory of Mitsui Engineering & Shipbuilding Co, Ltd (MES)

The tank has a 2.5 rn-deep pit under the tank bottom. The pit was used in the model tests in oreder to get a water depth of 5.0 m

4.1 Model descriptiòn

A prototype- TLP was designed for operation at the sea of depth of 40Ó m and has a displaôement of 32 030 tons. A 1-: 80 scale model made of -aluminum is com-posed of square deck supported by six columns, which

are interconnected at their bottom by six rectangular

pontoons The load cells for measurement of tether

forces were mounted on the corner columns:

Each tether of the prototype is composed of a few

I

Tablè 1 Principal Dimensions

Unit : m

Fig. 3 ConfigUration of TLP

steel pipes. In the present tests, the model was moored at four corner columns by four simply modeled stain-less wires. An axial rigidities of the model tether was scaled down by third power of the scale ratio. Linear-spring property was verified by an elongation test within the displacement range in the tests. A lower end of the wire was connected to an anchor at. the bottom of the

Length - 77.8m Breàdth - 77.8rn Draft 26.7m Tether Span 64.6 m Displacement Volume 3 i 250m3 KG 343m Radius of Gyration -- 34.3m K 29.3 m K33 36-.5m

Tether Length from Anchor to

Column Bottom 358 m

Axial Rigidity of Tether (each) 3.52 x 10 if (34 500 MN} Intial Tension (each) 1 850 tf (18 .1 MN}

-

-r

i3.2 <12.8 i3.2

pit through a ball joint, and an upper end to load cell

on the deck through the corner column. The draft

and initial tensions of the tethers were adjusted by

moving load cells in the vertical direction. The difference of the initial tension among four tethers is 2.5% at the most.

Principal dimensions and a configuration of the TLP are shown in Table i and Fig. 3 respectively.

4.2 Measuring system

The mechanical friction to which a conventional

measuring system of carriage type is inevitably subjected, which is rolling friction caused in rollers of the carriage and so on, may affect the slow drift oscillations. In the present test the optical measuring system consisting of light emitting diodes was developed and used to avoid the mechanical friction.

The measuring system consists of two main devices. One is Position Sensor System (made by HAMAMATSU PHOTONICS K.K., Japan) which used light emitting

diodes (LEDs) and their sensor, the other is a

mini-computer system which analyses signals from the former

system. _{The Position Sensor System measures the}

locations of LEDs mounted on the model from time to time by the sensor like a camera placed away from the model. The signals from this device are fed to the mini-computer after analog-digital conversion. The

mini-Sensor Had Disk \__ LED Mini-Computer 1) Load Cell

Fig. 4 Measuring System

Wave Probe

V

Data Acquisition

computer converts the signals into the motions of the model in six modes by the technique of transformation of coordinate system. The cords of LEDs are selected and set so that their reaction forces do not affect motions of the model. The accuracy of the measuring system is, evaluating in resolution, about 2.0 mm for planar mo-tions and 0.6 degree for angular momo-tions.

The wire tensions were measured by four 25lb. load cells on the corner columns. Their cords were selected thin and flexible enough not to affect the motions of the model. A wave elevation was measured without the model by means of a servo-type wave probe (capacity of 200 mm) at the location where the model would be set, aiming to avoid mixing of the incident waves with the reflection from the model. The time record of the wave elevation was synchronized with those of motions and tensions after the measurement.

The scheme of the measuring system is shown in

Fig. 4.

4.3 Test conditions

Some ten cases of tests were carried out changing wave heights, mean wave periods and heading angles of ir-regular waves. In this paper the test under the following condition is described.

wave spectrum : ISSC type

significant wave height H113 : 10.0 m (in full scale) mean wave period T01 : 13.9 s ₍ do. ₎

heading angle _x : 225 deg

The appearance of the test is shown in Photo. 1.

Photo i Appearance of Experiment

5. Test and calculation results

Time history simulation were performed under the same condition as the test using measured wave data as input. The following numbers are all in full scale.

5.1 Calculation of hydrodynamic forces

The added mass matrix, the linear damping force

matrix, the response functions of the first order wave exciting forces and the steady drift forces used as input of the program were all calculated by the program based

on a three dimensional source distribution method.

This program had been developed by MES and is

called Dynamic Response Analysis of Marine and Offshore

5 Control Unit LED Driver

Position Sensor System Ampi. DC Ampi.

Fourier Analyser

Monitor Line Printer

Fig. 5 Element Division on Wetted Surface 3.0 2.0

o

o Wave o CaL 2.5 5.0 75 -Tv( s) Exp. 10.0 - 12.5

Fig. 7 Wave Drift Forces in Regular Waves

damping fOrce matrices. The drag coefficient of 1.0 was applied for all members. The slowly oscillating drift forces were calculáted by Pinkster's method.

5.2 Reproduction of irregular waves

In order to produce the same wave as measured in the test the time series fwave for 700 seconds (0t 700)

was sampled at intervals of 0.9 secOnd. The Fourier

series expansion of wave data gave amplitudes and

phases of wave components. The Fourier series expan sion was made after adding the 45 second data before and after the data for 700 seconds respectively to avoid the Gibbs' phenonienon.'" Added data are written in the form

(0):

45t0

(700):

700t745

The wave elevatiOn in the calculation was ¿/100 (0 t 100) tirnès as large as the measured one in the first 100 seconds in the time integral to restrain tránsient motions caused by abruptly induced forces.

5.3 ResUlts and comparison

The comparisons between the time history däta of the test and the calculation are shown n Fig. after remov-ing their initial data for 100 seconds. Since there is no measured data for heave, roll and pitch, only calculated

results for them are shown The tether tensions are

shown in the form of fluctuation components after sub-tracting the initial tension from actual one.

Coincidence between measured wave and computed

one shows that the same condition as the test were

produced in the calculation. Calculated responses of surge and sway agree well with the measured on s in both wave and low frequency regions. Although the

comparison for yaw cannot be made on acçount of

o

5 10 15 20 25

t,v ( s)

Fig. 6 Surge in Regular Waves

Stiuctures (DREAMS)."

In the calculation of the

hydrodynamic forces, the wetted surface of the TLP was divided into 372 elements on which sources were distn-buted. The view of the mesh division is shown in Fig. 5.

The results obtained from experiments for a few kind

of semisubmersibles have confirmed the validity of

DREAMS. For the present TLP the: experiments,

which were a wave exciting force test, a drift force test etc., were conducted and their results showed good agree-ment with thOse calculated by DREAMS. As examples of their comparison the response functions for a surge and drift force in head waves are shown in Figs 6 and 7 respectively.

The peak frequency of the wave spectrum was chosen representative frequency for the added mass and linear

WIVE (M) 3.00E02 -PITCH 5.00E02 -0. I

k;

5.00E02 -YPW (CEO) 2.50 -0.01 0.01 0..01 -600. SURGE (M)

;ÁAt

':\-A.

'Y!ro' "'VT

..wri!rv

_'yr

SwpY HERVE (M) 7.50E02 -(PEG) 2.50 -TENSION I (TON) 600. --600. TENSION 2 (TON) 500. --500. TENSION 3 (TON) 800. -800. TENSION 4 (TON) 600. '. ;i \ / TIME (S)

AAA

A

'

V " V

_V 5

Fig. 8 Time Series of Motion and Tènsioñ Fluctuation

- À ' TIME (S) '0 _/ 0: 7.50E02 0.150 -ROLL (CEO) 3.00E02 -TIME (S) Wave Ti c Ti

Ti 0=0=0

,T2 Exp. - Cal. 10.0 0.01 -10.0 4.00 0.l -4. 00 -8.00 4.00 -4.00 -8.00

80. 4 E. 8(F) (M*2,S 5(F) (M**2*5 5.0E-02 0.10 015. 0.20 F (HZ) -Exp. Cál.

Fig. 10 Spectrum of Surge

8(F) (M**2*5

Exp.

Cal.

Fig. 9 Spectrum of Wave

Exp.

Cál.

Fig. 11 Spectrum of Sway

inaccuracy of measurement, the calculated result and the measured one show a similar tendency. The calcu-láted tether tensions are .ïn close agreement with the measùred oñes.

By way of spectrum analysis the computed spectra of

Wave, surge, sway and yaw are shown in Figs. 9-12

Exp.

CaL

Fig. 12 Spectrum ôfYaw

compared with measured spectra. Although the wave spectrum indicates approximately zero around the fre-quency of 0.01 Hz, the spectra of surge, sway and yaw ae sharply peaked aroûnt it. It.próves that the amplifi-cation due to the slowly oscillating drift forces occured

From the above the validity of the program is confirmed by good coincidence between the calculation and the experiment.

Sample calculation

A sample simulatiofl for the TLP under the

one-leg-broken condition are introduced. On the assumption

that the Weather ide tether T1 is broken at the time of 300 second the simulation under the same wave condi-tion as mencondi-tioned before is shown in Fig 13

Remarkable snap loads in the tether T3 where locates opposite to T1 diagonally are shown, which is a charac-teristic of a TLP. The natural vibrations produeed by the sñap loads 1n the tether 7'3 cause extremely large tensions on the tethers T2 and T4 and the TLP is under the dangerous condition Roll and pitch show unique one-sided motions on account of restraint by the tether T3. On the other hand surge, sway and yaw do not change their characteristics. Their excursions are flot f _{süch magnittode that occured uuìder the} one-line-broken condition in the catenary moored system

Conclusions

The general discription and the mathematical model of the simulation program which calculates motions of

moored offshore structures subjected, to compound

external forces have been explained. The validity of

the program has been confirmed in comparison with the

experimental results for the TLP model in irregular

waves. Furthermore a sample simülation for the TLP has been introduced. Although the e uations of motions in the program are solved approximately, eg the

con-stant values are applied to the radiation forces, the program seems to be useful for the practical

calcula-tions.. 20. 5. 10. 5.0 0.9 5(F (DEC**2*S) -F (HZ) 5.0E-02 0.10 0.15 0.20

u. 5.00E03. -01 1.0.0 -SURGE (M) 5.00 --5.00 -10.0 5.00

°1

-5.00 -10. 0 HEPVE (M) 1.00 -SWY (M) 0.800 -TENSION 1 (TON) 1.00E+03 2.00E+03 -TENSION 2 (TON) 8. OOE+03 4. OOE+03 - 200

T3O=cJ

Ti

Fig. 13 Time Series of Motion and Tension Fluctuation (One-Leg-Broken Condition)

300 400 /Wave Ti_cJ=O=oTJ Ti

¡AA

'y'

ROLL (DEC) 12.0 6.00 -o n .

AAAA*&.

A

ALAAAAAAAA

4 P1(0 200 300 400. 500 600 6.00 -PITCH (DEC) 6.00 -0 200

30VVVVV0'vy

,yryvvyvrv

i 12.0 -YRW (DEG) 0.800

n A

L.

. & kM& AAM_

'T'VoT 5t!l

'1v'

PV

TiUi

W'IIiU-

-TIME (5) TIME (S) TIME (S) TIME (S) TIME (S) TIME (S) O 200 -1.00 300 400 d\f O 400 500 600 -1.00E+03 - TIME (S)

References

I) NIPPON KAlI! KYOKAI: Guide to Mooring System,

(1983) (in Japanese).

Y. Yarnanouchi, et al.: Ocean Wave Spectra, Bulletin of the Society of Naval Architects of Japan 609 (1980) p 160

(in Japanese).

F. H. Hsu, et al.: Analysis of Peak Mooring Force Caused by Slow Vessel Drift Oscillation in Random Seas, Proc. of Offshore Technology Conference, OTC 1159 (1970). J. A. Piiikster: Low Frequency Phenomena Associated

with Vessels Moóred at Sea, Society of Petroleúm Engineer-ing of AIME, SprEngineer-ing MeetEngineer-ing, SPE 4837 (1974).

A. G. Davenport: The spectrum of horizontal gustiness

near the ground in high winds, Qurt. J. Ray. Meteor. Soc.,

87 (1961), p. 194.

G. Van Oortmerssen: The Motions of a Moored Ship in

Waves, (1976), p. 67, Publication 510 NSMB.

J. P. Morison, et al.: The forces exerted by surface waves on piIes Petroleum, Trans., 189 (1950).

K. Yöshida, et äl.: Dynamic Response Characteristics of

taut Moored Platforms, JoUrnal of the Society of Naval

Architects Of Japan, 146 (1979), p. 195 (in Japanese). M. Kobayashi, et al.: Dynamic Response Analysis of

Marine & Offshore StrUcture (DREAMS), Technical Bul

letin of Mitsui Engineering & Shipbuilding Co., Ltd., TB

85-03, (1985).

A. Sommerfeld: Vorlesungen über Theoretische Physik (6), (1965).