computation of drift forces for dynamic …...~d s directional vector of each waterline element a x0...

Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic EngineeringOMAE 2014

June 8-13, 2014, San Francisco, California, USA

OMAE2014-23074

COMPUTATION OF DRIFT FORCES FOR DYNAMIC POSITIONING WITHIN THE VERY EARLYDESIGN STAGE OF OFFSHORE WIND FARM INSTALLATION VESSELS

Philip H. Augener∗Research Assistant

Institute of Ship Design and Ship SafetyHamburg University of Technology

Hamburg, GermanyEmail: [email protected]

Stefan KrugerHead of Institute

Institute of Ship Design and Ship SafetyHamburg University of Technology

Hamburg, GermanyEmail: [email protected]

ABSTRACTThe German government has decided upon the changeover

from fossil and nuclear based electrical power generation to re-newable energies. Following from this offshore wind farms areerected in the exclusive economic zones of Germany. For thetransportation and installation as well as the maintenance of thewind turbine generators very specialized vessels are needed. Thecapability of dynamic positioning even in very harsh weatherconditions is one of the major design tasks for these vessels. Forthis reason it is important to know the external loads on the shipsduring station keeping already in the very early design stage.This paper focuses on the computation of wave drift forces inregular and irregular waves as well as in natural seaway. Forvalidation the results of the introduced calculation procedure arecompared to measured drift force data from sea-keeping tests ofan Offshore Wind Farm Transport and Installation Vessel.

NOMENCLATUREL Length of the vesselT Draught of the vesselm Total mass of the vesselg Gravitational accelerationdt Time stepΘ 3x3 matrix of inertia with respect to COG of the vesselβ Wave propagation direction β = π − µ

∗Address all correspondence to this author.

γ Waterline angle regarding the longitudinal axis of thevessel

ω Circular wave frequencyρ Density of sea waterµ Wave encounter angleL1 Non-shadow part of the waterlineTp Peak periodT1 Average wave periodxg Longitudinal coordinate of the center of gravityyg Tranverse coordinate of the center of gravityx0 Longitudinal coordinate of a waterline elementy0 Transverse coordinate of a waterline elementpw Complex amplitude of the pressure at a point on the

mean waterline~x Position vector of a waterline element~α Complex amplitude of the vessel’s rotation vector~xG Vector of center of gravityζa Real wave amplitude of an elementary waveζ Complex wave amplitudeζ Real wave amplitudeµ0 Main wave propagation directionωe Encounter frequency between the ship and the wavesω0 Modal frequency of the specified seaway classε j Random phase angle of irregular wavesε jl Random phase angle of natural seawayYp Complex transfer function of the pressure at a point at

the mean waterlineY4 Complex transfer function of the roll motion

1 Copyright c© 2014 by ASME

~∆s Directional vector of each waterline elementAx0 Area of the wetted transom in calm waterH1/3 Significant wave heightzx0 Vertical coordinate of the center of area of the transomYzr Complex transfer function of the relative vertical mo-

tion between the ship and the water surface at the wa-terline of the ship

Y1s Complex transfer function of the surge motion of theship’s center of gravity

Y2s Complex transfer function of the sway motion of theship’s center of gravity

Y3s Complex transfer function of the heave motion of theship’s center of gravity

Y ∗5 Conjugate-complex transfer function of the pitch mo-tion

Y ∗6 Conjugate-complex transfer function of the yaw motiony+w Absolute value of the transverse coordinate of the wa-

terlineα2 Drift force coefficientα2

0 Mean drift force coefficientSzz Energy density spectrum of a seawayF2(t) Wave drift force or wave drift moment of yaw in the

time domainF2

ξ(t) Longitudinal wave drift force in the time domain

F2η (t) Transverse wave drift force in the time domain

M2ζ(t) Wave drift moment of yaw in the time domain

a2(t) Square of the envelope of the random seawaydy+wdx dx Inclination of the waterline against the longitudinal axis

Y ∗ζ Wy Conjugate complex amplitude of the transverse slope of

the water surfaceF2

ξTime averaged longitudinal wave drift force in the fre-quency domain

F2η Time averaged transverse wave drift force in the fre-

quency domainM2

ζTime averaged wave drift moment of yaw in the fre-quency domain

~α∗ Conjugate complex amplitude of the vessel’s rotationvector

INTRODUCTIONThe German politics have decided upon the energy

turnaround for sustainability, meaning the changeover from nu-clear and fossil based electrical power generation to the use ofrenewable energies. One major stakeholder for this switch areoffshore wind farms. The areas for the erection of these windfarms are the exclusive economic zones of Germany in the NorthSea as well as the Baltic Sea. Several thousands of wind turbinegenerators (WTG) are required to fulfill the ambitious goals ofthe German government. For the installation at first place and

the maintenance in the following years, specialized vessels areneeded. In order to keep downtimes during the installation pro-cess and during the power production in the following years aslow as possible, vessels for both tasks must have the capabilityof dynamic positioning even in very harsh weather conditions.Hence dynamic positioning is a major design task for these kindsof vessels and should already be considered in the early designstage, especially under the aspect of a holistic ship design. Forthis fast and reliable calculation methods are needed. In additionto wind and current loads, wave drift forces are the third majorenvironmental loads that have to be considered for the design ofDP-Systems. While wind and current loads can be determinedbased on lateral areas and drag coefficients, for wave drift forcesmore sophisticated tools are needed. The reason for this is thedependency of the wave drift forces on the ship’s hull form, thewave height and period as well as the encounter angle. In the fol-lowing the nature of wave forces is explained and a calculationmethod based on potential theory for the determination of wavedrift forces in regular and irregular waves as well as in naturalseaway is described. At first the wave drift forces are calculatedin the frequency domain using the unit wave amplitude. Basedon the results in the frequency domain, calculations in the timedomain can be executed, which lead to time series of the longi-tudinal and transverse wave drift forces as well as the wave driftmoment of yaw. These time series can be used as input data fordynamic positioning computations in the time domain. Finallythe paper closes with a validation of the calculation results withmeasured wave drift forces of the Offshore Wind Farm Installa-tion Vessel Sietas Type 187.

WAVE FORCESWhen talking about wave forces on offshore structures and

ships it is important to distinguish between first and second or-der forces. The first order wave forces are mostly relevant forthe construction of offshore structures as stated in [1]. With azero mean value averaged over one period, the first order forcesoscillate with the wave frequency. Wave drift forces, which aresecond order wave forces, are about one magnitude smaller in re-lation to the first order forces. With a non-zero mean value, wavedrift forces are proportional to the square of the wave amplitudein harmonic waves. This leads to a constant timely weighted av-erage resulting force in the propagation direction of the waves.Hence wave drift forces have to be considered for station keep-ing. In irregular waves drift forces are proportional to the squareof the envelope of the exciting waves. The reason for this is theslow change of the wave drift forces regarding the variation of thewave heights in irregular seas. With good approximation the lon-gitudinal wave drift forces are equivalent to the added resistancein waves for ships with forward speed, but normally defined withan opposite sign.


FIGURE 1. COORDINATE SYSTEM USED FOR THE DESCRIP-TION OF THE THEORY.

COMPUTATION OF WAVE DRIFT FORCES AND THEWAVE DRIFT MOMENT OF YAW

In this chapter a calculation procedure for the longitudinaland transverse wave drift forces as well as for the wave drift mo-ment of yaw is explained. The procedure is subdivided into twoparts. The first part is the computation of the drift forces in thefrequency domain in harmonic waves of unit wave amplitude. Inthe second part the computation in the time domain using irregu-lar waves and natural seaway is explained. The described calcu-lation procedure is integrated in the ship design environment E4,for which a detailed description can be found in [2] and [3].

Frequency Domain ComputationsIn 1970 Boese published his theory for the calculation of

the added resistance in waves in the frequency domain [4]. Inthis theory the second order forces are determined based onfirst order results. For this approach the required first ordermotions of a ship are calculated according to strip method asdescribed in [5] and the first order pressures on the hull surfaceare determined with the pressure strip method by Hachmann [6].While normally the calculation of pressures is omitted incommon strip methods, this is the focus of the latter method.In order to fit the usual strip theory the pressure strip methodhas been adapted in [7]. Based upon this the derivation of thefollowing equations for the longitudinal and transverse wavedrift forces can be found in [8], while the equation for the wavedrift moment of yaw is from an unpublished paper by Soding.

Figure 1 shows the inertial coordinate system used for thedescription of the theory. The origin is located at the intersec-tion of the the midship section, the midship plane and the keel.Furthermore it has to be mentioned, that the system is rotated by180◦ around the longitudinal axis for the output of the results.Following from this the output of the longitudinal forces is posi-tive forward, the transverse forces are positive to portside and theoutput of the yaw moment is positive turning counterclockwise

around the upward pointing axis. For the time-averaged longitu-dinal wave drift force Eqn. (1) is valid and for the time-averagedtransverse wave drift force Eqn. (2) is used. Equation (3) showshow the time averaged wave drift moment of yaw is calculated.In Eqn. (1), Eqn. (2) and Eqn. (3) the forces on the hull of the ves-sel beneath the average still-water line are accounted for by thefirst summand, while the second summand considers the forceson the ship due to the relative motion between the vessel and thewater surface. The third summand in Eqn. (1) is a correction termvalid for ships with a non-wetted transom beneath the waterlineat forward speed. In Eqn. (3) the index ζ indicates the relevantvector component for the moment of yaw and the last two termsare needed because the origin of the coordinate system is not thecenter of gravity.

F2ξ=

12

(mω

2e Re

(Y2s Y ∗6 − Y3s Y ∗5

)(1)

+12

ρ g ∑Stb,Port

∫L

|Yzr|2dy+wdx

dx

+12

ρ g(|Y4|2 + |Y5|2

)· [Ax0 (T + zx0)]Transom

)ζ

2a

F2η =

12

(−mω

2e Re

(Y1s Y ∗6

)(2)

+ ρ g ∑Stb,Port

∫L

y+w Re(

Yzr0 Y ∗ζ Wy

)dx)

ζ2a

M2ζ= − 1

2ω

2e Re

[~α∗ × Θ~α

]ζ

(3)

+ ∑WL−Elements

((~x − ~xG)

×[|pw|2

4ρg~∆s × (0,0,−1)T

])ζ

+ xg F2η − yg F2

ξ

Equation (1) is already used in the ship design environmentE4 for the determination of the added resistance in waves as de-scribed in [9]. However the determination of the relative mo-tion between the water surface and the ship is improved. Whilefor the added resistance only the Froude-Kriloff forces, which


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-150

-100

-50

0

50

100

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Drift

forc

e in

kN

/ m

**2

wa

ve

am

plit

ud

e

Circular frequency [rad/s]

Enc.ang [deg] 0.0Enc.ang [deg] 45.0Enc.ang [deg] 90.0

Enc.ang [deg] 135.0Enc.ang [deg] 180.0 Enc.ang [deg] 225.0

Enc.ang [deg] 270.0 Enc.ang [deg] 315.0 Enc.ang [deg] 360.0

FIGURE 2. TYPICAL FREQUENCY DOMAIN RESULTS FOR F2ξ

WITHOUT SHORT WAVE CORRECTION.

are caused by the undisturbed pressure field due to the inci-dent wave, are used, the new implementation additionally ac-counts for the diffraction and radiation forces. Correspondingto [8] this is achieved by using the above mentioned approach byHachmann [6] to compute the complex amplitudes of the pres-sures at the mean waterline. Based on this the complex transferfunctions of the pressures are determined and with these the com-plex transfer functions of the relative vertical motion are foundby Eqn. (4).

Yzr =Yp

ρ g(4)

For the aforesaid way to determine the wave drift forces typ-ical results are revealed in the following. Figure 2 shows typicalresults for the longitudinal and Fig. 3 for the transverse wavedrift force, while in Fig. 4 characteristic results for the wave driftmoment of yaw are displayed.

In the diagrams the encounter angles are counted counter-clockwise from µ = 0◦ for following seas over the starboardside to µ = 180◦ for head seas and back over the port side ofthe vessel. For the symmetric model at an un-heeled floating po-sition, symmetric results for both sides of the vessel have to beexpected as well.

Faltinsen explains in [10], that wave drift forces for longwaves originate from the motions of the vessel, while drift forcesin short waves are mainly due to the reflection of the incomingwaves from the ship’s hull. For the longer waves the above men-tioned equations can be used, whereas the calculation for the zerospeed case for short waves is done in accordance to [10] as ex-plained below. Figure 5 shows the definitions used for the fol-lowing equations. It has to be mentioned that as β = 0◦ for head

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-500

0

500

1000

1500

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Drift

forc

e in

kN

/ m

**2

wa

ve

am

plit

ud

e





FIGURE 3. TYPICAL FREQUENCY DOMAIN RESULTS FOR F2η


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-8000

-6000

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0

2000

4000

6000

8000

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Drift

mo

me

nt

in k

Nm

/ m

**2

wa

ve

am

plit

ud

e





FIGURE 4. TYPICAL FREQUENCY DOMAIN RESULTS FOR M2ζ


seas, it has to be transformed by β = π − µ for the used angleof encounter µ , which is zero in following seas. Equation (5)shows how the longitudinal wave drift forces are calculated inshort waves, while Eqn. (6) is valid for the transverse wave driftforces and Eqn. (7) for the wave drift moment of yaw.

F2ξ short

=ρgζ 2

a

2

∫L1

sin2 (γ + β ) · sin(γ) dl (5)

F2η short =

ρgζ 2a

2

∫L1

sin2 (γ + β ) · cos(γ) dl (6)


FIGURE 5. DEFINITIONS FOR THE SHORT WAVE CORREC-TION [10].

M2ζ short

=ρgζ 2

a

2

∫L1

sin2 (γ + β ) ·(x0 · cos(γ) (7)

− y0 · sin(γ))

dl

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0

50

100

150

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Drift

forc

e in

kN

/ m

**2

wa

ve

am

plit

ud

e





FIGURE 6. TYPICAL FREQUENCY DOMAIN RESULTS FOR F2ξ

WITH SHORT WAVE CORRECTION.

Figure 6 shows characteristic results for the longitudinalwave drift force with the short wave correction as explainedin [10] applied for waves shorter than the length of the vessel.

In Fig. 7 typical results for the transverse drift force withthe short wave correction applied for waves shorter than half thelength of the vessel are shown and in Fig. 8 for the wave driftmoment of yaw, where the short wave correction is applied forwaves shorter than the length of the vessel. As already mentioned

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-500

0

500

1000

1500

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Drift

forc

e in

kN

/ m

**2

wa

ve

am

plit

ud

e





FIGURE 7. TYPICAL FREQUENCY DOMAIN RESULTS FOR F2η


above the results have to be symmetric for a symmetric vesselfloating in an un-heeled loading condition.

-15000

-10000

-5000

0

5000

10000

15000

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Drift

mo

me

nt

in k

Nm

/ m

**2

wa

ve

am

plit

ud

e





FIGURE 8. TYPICAL FREQUENCY DOMAIN RESULTS FOR M2ζ


Time Domain ComputationsThe results from the frequency domain calculations for dif-

ferent encounter angles with the unit wave amplitude, as de-scribed in the previous section, will be used as input data forthe time domain computation described in this section. In thebeginning a time domain computation procedure for the calcula-tion of the wave drift forces and the wave drift moment of yawin unidirectional irregular waves according to [1] is presented.


Based on this an enhancement for natural seaway is introducedin the second subsection.

Irregular Waves. Using the wave drift forces deter-mined with the equations above as input values and combiningthem with the seaway data from energy density spectra, it is nowpossible to determine time dependent forces in irregular waves ofarbitrary heights and with different encounter angles. Dependingon the selected input data, Eqn. (8) is valid for the calculation oftime series for the longitudinal and transverse wave drift forcesas well as the wave drift moment of yaw as explained in [1].

F2(t) =ρ

2· g · L · α2

0 · a2(t) (8)

Equation (8) shows the dependency of the wave drift forceson the square of the envelope of the random seaway, with theenvelope of the random seaway corresponding to the amplitudefluctuation of the random seaway under consideration. Further-more the dependency on the mean drift force coefficient becomesclear, which is determined by Eqn. (9), which indicates that themean drift force coefficient is a mean value weighted with theseaway defined by the energy density spectrum.

α20 =

∞∫0

α2(ω) · Szz(ω) dω

∞∫0

Szz(ω) dω

(9)

Equation (10) represents the definition of the drift forcecoefficient, either for the longitudinal or transverse wave driftforces as well as the wave drift moment of yaw. The equationshows how the drift force coefficient is computed from the fre-quency domain solution with ζ 2

a being the square of the unit waveamplitude.

α2(ω) =

F2(ω)ρ

2 · g · L · ζ 2a

(10)

The typical generation of a time series of a random seawayfrom a seaway spectrum is presented in Eqn. (11). The wavespectrum is subdivided into J parts.

ζ (t) =J

∑j=1

√2 · Szz(ω j)∆ω j · Re

(ei(

ω j t+ε j

))(11)

FIGURE 9. TYPICAL RESULTS FOR F2ξ(t).

FIGURE 10. TYPICAL RESULTS FOR F2η (t).

Based on Eqn. (11) the square of the envelope of the randomseaway can be determined with Eqn. (12).

a2(t) = ζ2(t) +

(dζ/dt)2

ω20

(12)

Typical results for the aforesaid calculation procedure forthe longitudinal wave drift force are presented in Fig. 9 and forthe wave transverse drift force in Fig. 10. Both time series clearlyconstitutes the non-zero mean value. Fig. 11 shows characteristicresults for the wave drift moment of yaw, where the non-zeromean value can be noticed as well. All three presented examples


FIGURE 11. TYPICAL RESULTS FOR M2ζ(t).

are extracted from time series with 104 seconds simulation timewith time steps of 0.5 seconds.

Natural Seaway. Adapted from the above described cal-culation procedure for unidirectional irregular waves from [1],this subsection deals with the calculation of time series for wavedrift forces and the wave drift moment of yaw in natural sea-way. Keeping in mind Eqn. (13), as stated in [11], and combiningEqn. (13) with Eqn. (9) this leads to Eqn. (14).

∞∫0

Szz(ω) dω ≈J

∑j=1

ζ 2 (ω j)

2(13)

α20 =

J∑j=1

α2(ω j) · ζ 2 (ω j)

J∑j=1

ζ 2 (ω j)

(14)

As described in [12] the complex wave amplitude for shortcrested seaway can be determined with Eqn. (15) by usingEqn. (16).

ζ (ω j, µl) =√

2Szz(ω j,µl)∆ω j ∆µl eiε j,l (15)

Szz(ω,µ) = Szz(ω) · 2π

cos2 (µ − µ0) , (16)

for |µ − µ0| ≤π

2, else = 0.

In Eqn. (17) the relation between the real and the complexwave amplitude is shown. The real wave amplitude is equal tothe absolute value of the complex wave amplitude.

ζ (ω j, µl) = |ζ (ω j, µl)| (17)

Based on Eqn. (14) and using Eqn. (17) with Eqn. (15) andEqn. (16) the mean drift force coefficient for natural seaway isderived according to Eqn. (18).

α20 =

J∑j=1

L∑

l=1α2(ω j,µl) · ζ 2(ω j, µl)

J∑j=1

L∑

l=1ζ 2(ω j, µl)

(18)

Based on the frequency domain results the wave drift forcecoefficients are determined according to Eqn. (19).

α2(ω,µ) =

F2(ω,µ)ρ

2 · g · L · ζ 2a

(19)

To generate the required time signal of the wave amplitudeEqn. (20) from [12] can be adopted.

ζ (t) =J

∑j=1

L

∑l=1

√2Szz(ω j,µl)∆ω j ∆µl · (20)

Re(

ei(

ω j t+ε j,l

))

Finally applying Eqn. (18), Eqn. (19) and Eqn. (20) insteadof Eqn. (9), Eqn. (10) and Eqn. (11) the above described proce-dure for irregular waves can be used for natural seaway as well.

VALIDATIONFor the validation of the presented calculation procedure

sea-keeping test results for the Offshore Wind Farm Installation


FIGURE 12. SIDE VIEW OF SIETAS TYPE 187.

Vessel Sietas Type 187 in unidirectional irregular waves are com-pared to computed results based on a model of the vessel in theship design environment E4. The Sietas Type 187 is purposebuild for the transportation and erection of offshore wind turbinesand it is thus belonging to the third generation of this specific shiptype. This generation of jack-up ships is especially designed andconstructed according to the specific requirements of the offshorewind industry, while older jack-up vessels are mostly convertedother ship types. The main reason why so much effort is put intothese new vessels is the fact, that no offshore wind turbines shallbe visible from ashore in Germany. This directly leads to instal-lation sites far away from the coastline and hence to greater waterdepths and a very harsh environment for the operation of thesevessels. The Sietas Type 187 is presented in Fig. 12 and the maincharacteristics of the vessel can be found in Tab. 1.From this table the length/beam ratio can be identified as rathersmall and the beam/draught ratio as large in comparison to toconventional ships. There are several boundary conditions lead-ing to these unusual ratios: For the transportation of the windturbines and foundations a large deck area is required, leadingto a great beam of the vessel. Additionally the relatively smalllength and draught of the vessel are caused by the restricted wa-ter depth and berth length at the offshore terminals, where thevessel loads its cargo. Another reason for the rather small lengthis the longitudinal bending moment, which is critical because ofthe required jacking capability of the vessel, especially in combi-nation with a rather small side depth. Furthermore it is necessaryto enable a secure standing position for the vessel in its jacked-up

TABLE 1. MAIN DIMENSIONS OF SIETAS TYPE 187.

Length [m] 139.40

Beam [m] 38.00

Draught (design) [m] 5.70

Depth [m] 9.12

Engine [kW] 4 x 2,500

Deadweight (design) [t] 6,500

Speed (design) [kn] 12.00

Crane lift main hoist [t] 900

(30 m outreach)

Maximum water depth for jacking [m] 45.00

Length/Beam [ ] 3.67

Beam/Draught [ ] 6.67

position. This works best with a small length/beam ratio with thelegs located as far away from each other as possible.

A critical examination concerning the applicability of thestrip theory is necessary, because of the unusual main dimensionratios of the vessel and the fact that strip methods are based onslender body assumptions. However then the length/beam ratio


of the Sietas Type 187 is 3.67, being still significantly larger thanthe threshold value of 2.5 for which a remarkable effectivenessfor the prediction of ship’s motion by strip theory is testifiedin [10]. In supplement the use of the strip theory is approvedwith good results for wave length as short as one third of thevessel’s length in [13], which leads to critical wave frequencieshigher than approximately 1.1 rad/second for the vessel underconsideration. Hence if the discretization of the strips is doneproperly, the large beam/draught ratio of the vessel should notbe a problem.

TABLE 2. ANALYZED SEA STATES.

Name Spectrum H1/3 [m] Tp[s]

W4 Pierson-Moskowitz 1.8 7.5



The analysis described below is based on a loading conditionof the vessel floating with even keel on its design draught. Theutilized model in E4 consists of the buoyancy body and the ship’sdisplacement, considering the weight distribution in all three spa-tial directions for the correct representation of the mass momentsand products of inertia. The complex transfer functions for allsix degrees of motions and the pressures are computed based onequidistant discretized stripes.The sea-keeping tests have been performed with a model in scale1:27 for the three sea states listed in Tab. 2. And for the sim-ulation the modified Pierson-Moskowitz spectrum according toEqn. (21) from [12] is used.

Szz(ω) = 173H21/3 T 4

1 ω−5 e(−692T 4

1 ω−4) (21)

The relation between the peak period and the average periodfor a Pierson-Moskowitz spectrum can be found in [14] and it isdefined as shown Eqn. (22).

T1 = 0.771 · Tp (22)

The wave drift forces in this section are simulated for 104

seconds, which is the recommended magnitude in [1] to achievereasonable results. Fifty random wave components are used for

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0

50

100

180 225 270 315 360

Driftforce [k

N]

Encounter Angle [°]

fxi calculated feta calculated fxi measured feta measuredcalculated measuredcalculated measured

FIGURE 13. COMPARISON OF THE FORCES FOR SEAWAY W4CALCULATED WITH SHORT WAVE CORRECTION.

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0

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180 225 270 315 360

Driftforce [k

N]


fxi calculated feta calculated fxi measured feta measuredcalculated measuredcalculated measured


the composition of the irregular seaway. And since the sea-keeping tests have been performed with an unidirectional sea-way, no angular spreading is used. The short wave correctionis applied for wave lengths smaller than the length of the ves-sel for the longitudinal and for waves shorter than half of theship’s length for the transverse drift forces. The comparison be-tween the measured wave drift forces and the computed results isdone by the mean values of the time series of the forces, becausethese measurements were available only. No measurements forthe wave drift moment of yaw have been carried out.For the seaway specified in Tab. 2 the comparisons of the mea-sured data to the computed results are presented in Fig. 13 forseaway W4, in Fig. 14 for W5 and in Fig. 15 for W6.

The presented results are reasonable for all three sea states.


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Driftforce [k

N]


fxi calculated feta calculated fxi measured feta measuredcalculated measuredcalculated measuredmeasuredmeasured


Especially the accordance for the longitudinal wave drift force infollowing seas is good, while the mean value of this force in headseas is overestimated by the computation. The transverse wavedrift forces are also overestimated by the presented calculationprocedure. But remembering that wave drift forces are second or-der forces, which are computed based on first order results here,and assuming an accuracy of 10-15 % for the first order results,the accuracy of the second order results may not be better than20-30 % as it is mentioned in [15]. Additionally conservativeresults are rather favorable for the design of DP-Systems.

CONCLUSIONSIn this paper a calculation procedure for the computation of

time series for the longitudinal and transverse wave drift forcesas well as the wave drift moment of yaw is presented. The useof potential theory makes the procedure fast and therefore appli-cable as a first principle design tool in the very early ship de-sign stage. The comparison to sea-keeping test results leads toan acceptable accordance. Together with the simulation of windforces as formulated in [16] the described procedure can be usedfor the generation of input data for a dynamic positioning codeas introduced in [17].

ACKNOWLEDGMENTThe authors are deeply grateful for the support of the re-

search project DYPOS by the Federal Ministry of Economics andTechnology of Germany. Furthermore the authors would like tothank the German shipyard J.J.Sietas KG for the permission touse the test results for validation.

REFERENCES[1] Clauss, G., Lehmann, E., and Ostergaard, C., 1994. Off-

shore Structures Volume II: Strength and Safety for Struc-tural Design. Springer-Verlag, London, United Kingdom.

[2] Buhr, W., Kruger, S., and Wanot, J., 1994. Offenes CAD-Methodenbank-System fur den schiffbaulichen Entwurf.Fortbildungskurs 28., Institut fur Schiffbau der UniversitatHamburg, Hamburg, Germany.

[3] Todd, T., 2002. “Unwinding the Design Spiral”. ShipPaxSTATISTICS 02, pp. 10–16.

[4] Boese, P., 1970. “Eine einfache Methode zur Berech-nung der Widerstandserhohung eines Schiffes im Seegang”.Schiffstechnik, 17, pp. 29–32.

[5] Soding, H., 1969. “Eine Modifikation der Streifenmeth-ode”. Ship Technology Research, 16, pp. 15–18.

[6] Hachmann, D., 1991. “Calculation of Pressures on a Ship’sHull in Waves”. Ship Technology Research, 38, pp. 111–133.

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