*\

Aftbody Slamming and Whipping Loads

G.K. Kapsenberg (V)1, A.P. van 't Veer (V)1, J.P. Hackett (LM)2, and M.M.D.Levadou (V)1

1 Maritime Research Institute Netherlands,2 Northrop Grumman Ship Systems

ABSTRACT

This paper describes a method of measuring qftbody slamming loads on a ship model in the seakeepingbasin. The method uses a large number of small pressure transducers mounted on the flat counter of thestern. The impact force is obtained by integrating the pressures. Experiments have been carried out on ascaled model of a modern cruise ship. The model was split into two segments at midships to simulate asimplified 2-node vibration mode. A structural mathematical model was made of the model as used for thetests. The analysis shows that the measured pressures can be used to predict the whipping loads on thesegmented model. A fatigue analysis was made for the vertical bending moment midships. It appears thatfor a following sea case the potential for fatigue damage is dominated by the whipping contribution. Theextreme vertical hull bending moment is determined by sailing in high head seas. The results of theexperiments show that the whipping component only contributes a small percent increase to the extremevertical bending moment. It is concluded that the measured loads can be used with a finite element model ofthe ship's structure to predict whipping stresses in the actual ship structure.

SYMBOLS

AiAZBH

Cij

CH

FE

[m2][m/s2][Nm]

[Nm][Pa3]

FZS L A M [kN]HF

H s

lijkyy

MMYN

PTP

VWFXSLAM

Zi

K

[m][m4][m][kg][kNm][-]

[kPa][s][m/s]

[m]

[m][deg][-]

[Pa]

[rad/s]

AreaVertical accelerationDamping of springHydrostatic restoring coefficientSpring stiffnessFatigue effectVertical slam forceHigh frequent part (due towhipping) on a signalSignificant Wave HeightMoment of InertiaRadius of GyrationMassVertical Bending MomentNumber of classes of the rainflowcountThe number of load cycles in class iPressurePeak period of wave spectrumVelocityWave frequent part of a signalLongitudinal distance of center ofeffort of slam force to midshipsHeave motionPitch motionTuned non-dimensional dampingcoefficientThe mean stress of class i

Frequency of encounter

INTRODUCTION

The goal of every commercial ship owner is tomaximize the revenue producing portion of the vesseland minimize the non-revenue producing portion suchas the engine room. A "shoe box" with no propulsionplant, ballast tanks, etc. would be considered the idealsolution. As a result, over the years an increasingdemand was placed on design naval architects toproduce ship designs with full beam main decks alongalmost the entire length of the ship. For containershipsthis allows more containers to be stowed on the maindeck. Likewise, for cruise ships, a full width main deckallows a full width superstructure that contains morepassenger cabins. Competition is now so intense thatthis trend has expanded to include filling out the aftportions of the hull between the full load waterline andmain deck. This added hull volume means morecontainer stowage capacity for container ships, morespace for passengers on cruise ships, and a full breadthvehicle deck for more vehicle stowage and/or a vehicleramp on ferries. Owners rationalize that as long as the

filling out of the hull occurs above the full loadwaterline, there should be no negative impact on calmwater powering and fuel economy.

This design philosophy has produced hull formswith progressively flatter and flatter sterns. Some shipshave been built with zero or near zero deadrise andbuttock angles near the transom. Such shapes havehydrodynamic benefits in calm water as demonstratedby Hamalainen and van Heerd (1998); however, theyare a far cry from the traditional cruiser sterns seen onMariner hull forms and liners like the SS United States.

At the same time that this hull form developmentwas occurring, a new propulsion system was beingdeveloped. The idea was to move the electricpropulsion motor from inside the hull to an azimuthingpod located external to the hull. This allows theelimination of the propeller shaft and shaft struts, therudder, and stern tunnel thrusters. This conceptprovides greater flexibility for the design engineer tolocate elements of the propulsion system; hence, anability to provide additional internal volume to therevenue producing portion of the vessel. It alsoproduces 10 to 15% improvement in calm waterpowering and significant improvements in vessel harbormaneuverability. The azimuthing pods encourageddesign naval architects to create stern designs that wereeven flatter than their conventional screw propelledsister ships to simplify the pod installation. Onevessel's counter approached a flat horizontal surface inway of the pods.

The first ship equipped with podded propulsionwas the Seili, an ice going supply vessel built byKvErner Masa in 1991. The first application to a cruiseship was in 1997, the Elation. Since then, poddedpropulsion has become the industry standard on cruiseships. The French Navy has now ordered a large LHDtype amphibious assault ship outfitted with twin pods,the first naval application for podded propulsion.

SLAMMING AND VIBRATION PROBLEM

It was not long before a serious drawback of theflatter stern design surfaced. Some of the new cruiseship designs began to report slamming and subsequentvibration problems in following seas at zero and lowship speeds. Yet ferries did not report these problems.While the hull forms of cruise ships and ferries aremuch the same, the operating profiles for the ships arequite different. Ferries sail at low speeds inside theharbor, and then come up to cruising speed and remainthere until they reach their destination. Cruise shipsspend large amounts of time operating at low speeds,even in the open seas. The reported slamming andvibration problem occurred in very mild sea states whenthe waves had a length on the order of the length of the

ship. In the process of investigating the zero speedfollowing sea stern slamming and vibration problem,questions began to arise with respect to possible hullgirder fatigue life. Later issues of ultimate hull girderlongitudinal strength surfaced. These issues broughttogether the sciences of hydrodynamics and structuralengineering.

within acceptable limits. The purpose of the specialtests was to determine the global structural loading andresponse. Specifically, it was to confirm if sufficient aftshape was present and if sufficient structural rigidityand strength existed to avoid the vibration andstructural problems present in those ships thatencountered severe slamming.

SUBJECT VESSEL

The cruise ship reported on herein is a typical state-of-the-art design with a twin podded propulsion system.The particulars of this 1,900 passenger cruise ship arein Table 1. Its hull form is a variation of a provendesign. Figure 1 shows the body plan with pod.Although the stern of the subject vessel is somewhatflat, it has considerably more shape aft than othervessels that have reported stern slamming problems.Note that the body plan shows a very reasonablebuttock rise moving aft, and section curvature near theskeg. The minimum deadrise angle exceeds 3 degrees.

-—-_

Figure 1 Profile and body plan of the cruise ship.

Prior to performing the special stern slamming testprogram reported here, a conventional seakeepingmodel test program was conducted. This programincluded measuring motions, accelerations, deckwetness, bow and stern emergence, and slamming. Inorder to investigate slamming, the model wasinstrumented with three pressure panels on the bowflare, one on the stem and one on the flatter portion ofstern near the transom. The size of each panel was 50mm x 50 mm. The forces recorded from these panelswere representative of the local structural loadsimposed on the „ship. All of these test results were

Length between perp.BeamDraftDisplacementDesign speedGM

233.0032.20

8.0039000

212.10

mmmtonktsm

Table 1 Main dimensions of the cruise ship.

EXPERIMENTAL METHOD

In general, impacts due to slamming containenergy in high frequency bands (1-20 Hz). At very lowdeadrise angles (below 5 deg), the high frequencycontent increases. Also, hydro-elastic effects might beimportant for these cases, as reported by Faltinsen(1996) and Bereznitski (2002).

If the high frequency component of the slammingpressure is important, it needs to be accuratelymeasured. This means that the experimental set-upneeds to have a high resonance frequency, at least twiceas high as the frequency content of the slamming signal.In addition to this, underwater video recordings of theaftbody of the model slamming in waves, Figure 2,revealed a large number of air bubbles underneath thestern during the impact. This indicates that there arestrong local effects.

The conventional test methodology is to isolate theaftbody of the model from the rest of the model. Theaftbody is then connected to the main portion of themodel in such a way as to allow a system of straingauges to be installed along the joint. Such a systemwill have a relatively low resonance frequency and willhence be insensitive to high frequency contents in thepressure signal. The measured force will be the result ofthe dynamics of the aftbody connection to the hull.Deriving the impact forces on the stern is notstraightforward. Therefore, this testing method wasdisregarded.

A large array of 33 pressure sensors was used toallow adequate definition of the impact, capturing boththe high frequency content of the input and the localeffects. The array used is illustrated in Figures 3 and 4.This set-up allowed direct measurement of pressuresincluding the high frequency content in the signal,without the dynamics of a strain gauge system.

Figure 2 Sequence of pictures from a video showing one slamming event in following waves.

Figure 3 Stern of cruise ship model with poddedpropulsion and pressure gauges.

STARBDARD

CENTRELINE

PÜRT-SIDE

STAT, O STAT. 1

Figure 4 Location and identification numbers for thepressure transducers model.

It is important with this set-up that all componentsin the model are as rigid as possible. This essentiallyrules out measuring the longitudinal stresses atmidships by means of a strain gauge system, just as asegmented stern model measuring the total impact forceis not considered acceptable, as discussed before. Thewhipping component of the vertical bending moment isone of the important consequences of aftbodyslamming, so a way to measure it was developed. A twostep approach was taken:

2. Measure the whipping loads on a model cut in halfwhile the frequency of the 2-node deformation isscaled from model to ship.

The pressures at the stern were measured for boththe flexible and rigid models. The assumption was thatthe total pressure would be the sum of the impactpressure and the pressure due to the deformation. If thiswas the case, based on the results of the model withdifferent natural frequencies, it would mean that thehydro-elastic effects could be ignored, at least forpractical purposes.

The objective of step 2 was to demonstrate that thephysics of the problem were understood by being ableto calculate the response of the segmented model.Using the measured pressures on the model allows thecalculation of the impact forces on the stern. Theimpact forces are then applied to a mathematicalwhipping model of the tow tank model in the water.

BUM IS 10

SLAM 19 200

"*** 0

SLAM 20 200 I

200-j

500

200

a-ma 100

Regular waves. Ampl=2.5 m. T=8.0 s. Headino=0 degTEST NO 337001

•Vvv-%

/W~~—

«24 200-I'• „]_

SLAM 24kPi

400SLAM 15 2 0 f l i.

1.2 1.3 1.4 1,5 1.6 1.7 1.8SECONDS

Figure 5 Recordings of pressure gauges 18-25during a slam.

1. Measure the impact loads (pressures) on a veryrigid model.

The validation then consists of the comparison ofthe calculated whipping moment to the measuredvalues. Positive validation would yield a reliable tool,when used in conjunction with a detailed structuralmodel of the ship, for determining the full scalewhipping response of the ship.

ANALYSIS OF THE IMPACT PRESSURE

When a slam occurs during the model tests inwaves, short duration impulses are measured on each ofthe pressure gauges. Figure 5 shows a time history foreight of the pressure transducers. This figure shows thatthere are important differences in the peak of the impactpressure (from 200 to 1000 kPa) and in the timing ofthe peak as a function of the different locations of thegauges. Also, the shape of the pressure pulses as afunction of time can be quite different for the varioustransducers.

There appears to be a high-pressure ridge thattravels with a rather high velocity over the aftbody ofthe model. The velocity of the pressure ridge wasdetermined by using the time difference associated withthe pressure peak passing the different gauges. Figure 6shows the analysis of a slamming event. This event wasselected from tests in a sea state characterized by aJONSWAP spectrum, Hs = 4 m, TP = 8 s and apeakedness parameter y = 3.3. The figure shows the 100kPa isobars at different time steps. The time is shown asthe small number in each isobar. The pressure gaugesare indicated as small dots. The figure further showsthat the impact started on starboard side, close to the aftend of the skeg, 2 m forward of the aft perpendicular(AP). The distance between the isobars is used to derivethe velocity of the pressure ridge. These velocities areindicated on the figure with the associated arrowshowing the direction of travel. The initial velocity ofthe pressure ridge is 13-14 m/s forward and to thestarboard side, and 21 m/s aft. The pressure ridgevelocity moving aft increases to 30 m/s after 0.3 s;athwartship it decreases to 6 m/s. The high pressureridge covers the full length of the aftbody in 0.5 s afterthe first impact, then it expands mainly athwartship at avelocity of 15-20 m/s to starboard and about 45 m/s toport. The total impact takes about 1 s.

Figure 7 shows a pressure map of the second stageof this impact, 0.55 s after the initial impact, when thepressure ridge moves athwartship to both port andstarboard. The peak pressure is now on the order of 100kPa. Figure 8 shows the second impact occurring at twolocations simultaneously, one on the centerline about 2m aft of the skeg and the second close to the centerlineon the starboard side. These two spots join to one largearea only 0.15 s after the impact, which covers the fulllength of the aftbody. The high pressure area further

expands to athwartship at a speed of 20 - 40 m/s. Thetotal impact lasts about 0.5 s.

-5 0 5X-location w.r.t. APP[m]

Figure 6 Contour plot of moving pressure fields overthe aftbody.

x[m]10 20

Figure 7 Pressure map, at t = 0.55 s, of the impactshown in Figure 6.

Analysis of a large number of impacts in differentwave conditions shows that most impacts have aduration between 0.4 to 0.75 s. This is illustrated inFigure 9.

The typical impact was found to be a movingpressure front with peak pressures between 300 to 500kPa in the initial stage of the impact, diminishing to 100to 200 kPa in the second stage. The width of the highpressure ridge is about 2 - 3 m. The high pressure area

grows very quickly longitudinally (velocity 30 - 40 m/s)until it covers the full length of the stern. After this ittravels athwartships at a velocity of 20 - 30 m/s. Thismakes the total duration of the impact on the order of0.5 s for the subject vessel.

-5 0 5X-location w.r.t. APP[m]

Figure 8 Contour plot of pressure field showing twolocations of the initial impact.

n1 /j

\

VHli

\

^ FLEXIBLE MODEL_ _ RIGID MODEL

SLAM DURATION [s]

Figure 9 Statistics of the duration of the impact in a4 m Sea State.

PRESSURE INTEGRATION

Knowing the details of the pressure distribution onthe aftbody, the impact force on the aftbody as afunction of time can be determined. This requirespressure integration in space. The width of the pressureridge requires a density of pressure gauges which is far

greater than practical. With the current experimentalset-up it is possible that the high pressure ridge is atsome instant "lost" because it is in-between the pressuretransducers. Therefore, it is important to carefully selecta spatial integration method of the pressure signals. Asimple and robust pressure integration technique,denoted (S), was tested first. The method makes use ofa so-called Delaunay triangulation as illustrated inFigure 10. This triangulation is unique; no other datapoints fall within the circle drawn through the cornerpoints of each triangle. The triangles are used in thepressure integration.

S 10 15X location w.r.t. APP [m]

20

Figure 10 Delaunay triangulation of the pressuregauges.

The measured time trace of the pressure at a pointis multiplied by 1/3 of the area of all surroundingpanels. In formula:

FZSLAM(t)= I P(t)- I A,/3 0)pressurepick-ups

adjacentpanels i

This procedure uses the complete area enclosed byall the grid points (329.9 m2). A similar procedure wasused to calculate the center of effort of the impact forceas a function of time. For example, the pressuremeasured in P10, see Figure 10, is multiplied by V3 ofthe area of triangles 2-4-7-8 and 10.

The proposed integration method is robust andsimple, but it requires a density of pressure gauges thatis high relative to the width of the high pressure ridge.Since this is not the case with the current set-up, the

resulting force, FZSLAMW, will be very spiky due to anover-estimation of the force at the moment the ridgetravels over one of the pressure gauges, and an under-estimation when the ridge is just between the gauges.

To overcome the expected inaccuracy noted in thesimple integration method, a more advanced pressureintegration is applied. This technique, denoted asmethod (C), regards the pressure as a moving pressureridge over a panel. The shape of the pressure ridge p(t)is considered constant for each panel. Since the velocityV of the pressure ridge can be determined, the timesignal at each point can be transformed from the timedomain to the spatial domain:

p(s) = p ( t ) V (2)

The direction and velocity of the moving pressureridge is detennined by the pressure-time signals at thethree corner points. The velocity vector at each cornerpoint is known, but normally only one of them actuallycrosses the subject panel. The pressure signal at thispoint is used to calculate the force on the panel in astrip-wise manner for each time step. Figure 11illustrates this spatial integration process over a singlepanel.

This result was then applied to a mathematicalmodel of the structure of the model hull as it was in thetowing tank. The objective was to show that theresponse of the mathematical model was similar to theresponse of the real model in the tank. Verification ofthis demonstrates an "understanding" of the physics ofthe problem.

MO*'

3.5

"ff: :z

is

.-1

Estimatedtime.-:;diirarlon of slam

CeiculBted IMPULSE jhNs727.Ö k'Ns i t r r t * * method -7445 kNs Aifranced miKhctt

J1ME [s]

Figure 12 Result of the Simple (S) and Advanced (C)pressure integration methods for one slam.

3 pressure signals«ten m*d*urod tn t corner point

Figure 11 Impression of moving pressure front over apanel.

The results of both pressure integration methodsare illustrated in Figure 12. The time trace of the signalis different, the peak of the signal is very different, butthe total impulse appears to be almost identical. Theimpulse JFdt is 7278 Ns for the simplified integration

method (S) and 7445 Ns for the advanced method (C).The analysis of other slams with both methods showedthat the difference in impulse between the two methodsis no greater than 3%.

Figure 13 Dynamic model of the cruise ship in thetowing tank.

MATHEMATICAL MODEL TO CALCULATETHE WHIPPING LOADS

A schematic of the model in the tank is illustratedin Figure 13: The model is represented by two masses(forward and aft hull), connected by a hinge, a springand a damper. Both masses (hull halves) are supportedby a hydrostatic spring. This spring C;J consists of the

coefficients c33, c35, c53 and c55 to take into accountheave and pitch restoring forces as well as the crosscoupling. Hydrodynamic damping is ignored because ofthe high frequencies of interest.

The equations of motions of this two-mass systemare built using:

a) The pitch motion equation of the forebody (index1) around the CG of this part of the model, and ofthe aftbody (index 2) around its CG. The distancefrom the CG of part 2 to the impact location is"slam*

IyyG]

-e2)+BH(e,-e2)' xslam

(3)

b) The heave motion equation of the forebody andaftbody is:

MFz,+dj3z, +^58, =0MAz2+c£z2+cA.e2+FZs lam=0

(4)

c) The hinge keeps the two parts connected. Thedistance from the local CG to the hinge is definedby Xi and x2:

z2 - x 2 0 2 = z, - Xj0j (5)

Equation (5) is used in equation (4), after which thetwo equations of (3) are summed and written as onesingle equation. Similarly, equation (4) can besubstituted in equation (3), after which the system canbe written as a differential equation:

(6)dt

in which M, the 6x6 mass matrix, can be derived fromthe equations (3) through (6). The y(t) vector containsthe unknown displacements and rotations and their timederivatives:

= (z1lz1,e1,e1,G2,92)1

(7)

The right hand side vector force F(y, t) contains therestoring forces and the slamming force. Thedifferential equations are solved using an explicitRunge-Kutta 4th order scheme.

DYNAMIC CALIBRATION

The model system is calibrated by hitting themodel on the extreme aft end and calculating theresponse. The response in this respect is either themoment in the connecting spring or the angulardeformation. The angular deformation is thedeformation mode with the lowest eigen frequency. Theresponse of this system on a triangular load, force as afunction of time, was calculated, see Figure 14. Theduration of the load was varied while the impulse,JF d t , was kept constant for the different cases.

> •

10TIME [sec)

Figure 14 Whipping response of the model on atriangular impulsive load.

to

E

1.5 -,

1.0 -

0 5

no -

—i ,\\

— - ^ -

1

! •

Flexible [__ -

N. [Rigid |. 'ÉL

1 1

—i 1 1

Dyn. response

• model

Static response .

-» — —.i

0.0 0.5 1.0 1.5 2.0

T(impulse)/T(resonance) [-]

2.5

Figure 15 Response of the schematized model to animpulse at the AP.

The results of the calculations are shown in Figure15. The duration of the impulse is normalized by theperiod of the angular deformation mode of the system.

This figure shows that for very short impulses it is notthe peak value of the force that is important, but theimpulse. When the duration of the impulse is very long,the response of the system goes asymptotically to thestatic case. The response of the system to a constantforce is denoted as the static response in Figure 15.

The model of the cruise ship was subjected to animpulsive load when lying at zero speed in calm water.This test was performed on both the flexible model andthe rigid model to determine the effect of modelstiffness on response frequency. The frequency of thewet eigen mode of the model and the dampingdetermined from the tests, are presented in Table 2. Thelow damping of the flexible model was expected, seeBetts (1977) and Bishop (1979). The high damping ofthe rigid model was quite surprising. The most likelyexplanation is associated with the beams added to thesegmented model to make it whole again. Internalfriction of the screw connections of the beams to thewood model was creating frictional damping.

Rigid modelFlexible model

NaturalFrequency

THzl1.750.83

Damping

r-i0.0490.007

Table 2. Results of model hammer tests.

- 8 A Z X S

P28SJ-200

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Figure 16 Response of flexible model on a hammerimpact.

As was illustrated in Figure 9, the duration of theslamming impact is on the order of 0.5 s. This allowsthe points for the rigid and flexible models to be plottedin Figure 15. The figure shows that the response of therigid model is not quasi-static; in fact, the rigid modelwas not rigid enough. The response of the flexible

model was closer to a quasi-static response than that ofthe rigid model.

The response of the flexible model on the impulseis shown in Figure 16. This figure shows the pressuremeasured by gauge P28 and the local verticalacceleration AZ. The signals are shifted in the figure forclarity. High frequency local vibrations are dominant inthe initial stage of the impact; these are damped inabout 2 s. After this time the simplified two-nodedeformation mode is dominant. About 0.5 s after theinitial impact, the pressure at P28 is reasonably in phasewith the local vertical acceleration (times -8). Thismeans that the measured pressure is only due to theadded mass effect. A similar relation between the localvertical acceleration and the pressure was found for therigid model.

200

P28rl

"200

time [s]

Figure 17 Pressure at gauge 28 during an impact onthe flexible (P28f) and on the rigid (P28r)model. Top graph: as measured; bottom:corrected for local deformation.

This result is used in the analysis of the impacts inregular waves. The measured load is split into parts: thefirst due to the actual wave impact and the second due

10

to the local deformation. The pressure due to the actualwave impact is defined:

PlMPACT = PMEASURED ~ PLOCAL DEFORMATION (8)

For P28 the pressure due to the local deformation is:

PLOCAL DEFORMATION = -8AZX=P28 (9)

The results of this analysis are shown in Figure 17. Thedifference in the peak values is quite low, and it isreduced by correcting them for the local deformationsas illustrated in the lower of the two plots. Note that thelocal vibrations are dominant over the two-nodebending mode at the time of the impact. A comparisonof the integrated pressure for the rigid and the flexiblemodel is shown in Figure 18 (regular following waves,amplitude 2.6 m, period 8 s.)

From these results it is concluded that the hydro-structural interaction is low, and the pressure due to thewhipping of the model is an order of magnitude lowerthan the pressure resulting from the impact. Hydro-elastic effects are again lower, so it is concluded that,for practical purposes, they can be ignored for theproblem of aftbody slamming.

70

Figure 18 Slamming force for the flexible and rigidmodel.

TUNING OF THE WHIPPING MODEL

The mathematical model presented before requiresas input the properties of the tow tank model (atprototype scale) and the stiffness CH and damping BH ofthe connecting spring. The value of CH follows from thenatural period of the two-node vibration and by

assuming uncoupled motions between heave and pitch,thereby neglecting the z-displacements in equation (3):

0, +2KCOn0,

2BH

= 0

K = -c o n - I F+A

yy

. 4 C H + C 5F

5 + A

TF+Alyy

(10)

The damping BH follows from the whipping momentdecay curve, which was obtained by hitting the modelwith a hammer. This damping curve contains bothhydrodynamic and structural damping. The firstcomponent is normally considered negligible at thisfrequency. The data for the proposed mathematicalmodel are listed in Table 3.

Mass [ton]kyv [m]LCG [m]C33 [kN/m]C35 = C53 [kN]C55 [MNm]CH [MNm]K r-1

Aft2223740.87

-53.5336521

10457522040

Forward1663643.8144.3027521-774338620

42470.0077

Table 3. Data of whipping mathematical model. Thedata are relative to the CG of the segment.LCG is relative to the midships cut.

tlO.5

"110 "Ï15 "Ï2Ö™" 125 Ï3Ö~""'Ï35 140TIME [sac]

ito.s 110.4 m

' 145 150 Ï55

Figure 19 Decay of the response to an impulse of thephysical model and the mathematicalmodel.

Figure 19 presents the correlation between themathematical model and the decay curve of the physical

11

model. The plot shows the response to an impulse. Theinitial conditions for starting the curve are obtainedusing the measured vertical bending moment (MY) atthat time. The initial part of the simulation is shown inthe detailed graph in the figure. Obviously, in the decaycurve, the vertical slam force (FZ) is zero. The tunednon-dimensional damping coefficient was K = 0.0077.This is a low value in comparison to full-scale ships,which have damping on the order of 2 to 3% of thecritical value.

USE OF THE WHIPPING MODEL TO ANALYZESLAMMING EVENTS

The objective of the whipping model was toanalyze slamming events to check if the physicalprocess was understood. If the slamming force, asderived from the pressure gauge measurements, isapplied to the whipping model, the response mustcorrespond to the response of the model in the tank.When that is the case, the loads can be applied to aFinite Element (FE) structural model of the ship

A first check was carried out to compare theresponse of the model to the simple integration method(S) and the advanced integration method (C). Thiscomparison is shown in Figure 20. The calculationsstart at T = 381 s with initial conditions zero, thus withno displacement, rotations or whipping moment presentin the hull.

The slamming force shows some spikes when thesimple integration method is used, but the model doesnot respond to such short duration, low impulse spikes.The result of the simulation shows an identical responseof the model with respect to the vertical bendingmoment.

RESULTS OF EXPERIMENTS IN WAVES

Tests were carried out in following sea conditionsat zero speed. The wave conditions consisted of longcrested irregular waves with significant wave heights (2and 4 m), using the JONSWAP spectrum with apeakedness parameter 3.3, and a peak period of 8 s. Theenergy spectrum of the vertical bending momentmidships is shown in Figure 21. The figure clearlyshows two peaks, the first at a frequency of 0.8 rad/sdue to the waves and the second at 5.2 rad/s due towhipping.

The measured MY (in the 4 m Sea State) is plottedin Figure 22. The total signal (top time trace) is splitinto a whipping component (using a high-pass filterwith cut-off frequency 3 rad/s), and a wave component.Both components are also shown in Figure 22. Thevertical bending moment due to whipping is clearlydominant in this condition. The results of the whippingmodel will be compared to the whipping component ofthe vertical bending moment midships. The comparisonwill mainly be made for the 4 m Sea State as specifiedabove.

Earlier discussion showed the whipping responseof the vessel due to a hammer impact. The validationwas made using zero initial conditions in thecalculations. In waves the situation is different. Figure23 shows that whipping due to previous slammingevents continues when the vessel is hit by the nextwave.

The timing between the existing whippingdeformation and the next slam impact appears to becrucial for the resulting response. For example, arelatively low impact force is able to minimize thewhipping moment at t = 26 s. A similar event is foundaround t = 110 s. On the other hand, the impacts at

— simple method (S)complex method (C)

,10"

"§81 ™382 ' 383"" "384 385 386 387 388 389TIME [s]

Figure 20 Calculated whipping response due to asingle slam.

~ 300

SP

EC

1

WAVE FREQUENCE W RAO'S

Figure 21 Spectrum of vertical bending momentshowing wave frequency and whipping part.

12

Long-crested Js, Hs 4.0 m, TO 8.0 s, Head 0 deg, Speed 0 knTEST NO 315001x105

MY TOTkNm

MY WHIPPING"kNm

MY WAVEkNm

60 80SECONDS

Figure 22 Time traces of the vertical bending moment midships. Top: the total signal; middle: the whippingcomponent; bottom: wave frequent component.

Long-crested Js, Hs 4 0 m TO 8 0 s. Head 0 deg. Speed 0 knTEST NO 315001

FISIAMmm

0.5

20 40 60 80 100 120 140SECONDS

Figure 23 Whipping bending moment and slam forcefrom tank tests.

t = 34 and t = 41 s increase the whipping moment morethan could be expected due to a "favorable" timing.

The whipping load is calculated using the initialconditions (angle of the fore and aft segments, andangular velocities) from the model tests. The amplitudeof the whipping load is used to obtain the difference inthe angle of the fore and aft segments. The angularvelocities are obtained from two angular rotations at

16

Figure 24 Measured and calculated whippingresponse due to an impact force.

13

x105

215 217 219 221TIME [si

223

Figure 25 Example of a slamming event that causes adecrease of the whipping moment.

x10

382 384 386 388 390TIME [sj

Figure 26 Example of a slamming event that causesan increase of the whipping moment.

consecutive time steps. Figure 24 through Figure 26present the results of the simulations with the whippingmodel compared to the measured vertical bendingmoment for three cases. Figure 24 shows how arelatively small impact increases the whipping momentsome 20%.

Figure 25 shows how a slam that occurs out ofphase with the whipping motion reduces the moment bya factor of four. The calculations predict the trendcorrectly, but the phasing of the whipping moment afterthe impact is incorrect.

Figure 26 gives an example of how a favorabletuning increases the whipping moment by a factor offour. The prediction of the whipping model agrees verywell with the actual measured moment.

The general good agreement between thecalculations using the whipping model and the actualmeasured vertical bending moment shows that the

whipping phenomenon is "understood", and that theresponse of the model can be reliably predicted by thewhipping model for individual cases.

STATISTICS OF WHIPPING LOADS

The previous section showed that the amplitude ofthe whipping moment is dependent on the timing of theslam relative to the existing whipping motion. Whensimulations with the whipping model were carried out,each slamming event could be predicted (using themeasured slamming force) if the initial conditions justbefore the impact were tuned. If this is not done,inevitably some drift will occur. This in turn willchange the tuning of the slam with the whipping motionand will have a large effect on the resulting peakwhipping moment.

Now consider a long record of measured whippingmoments and compare the statistics of the measuredwhipping peaks to those from the whipping model. Theinitial conditions for the whipping model are not tunedto match the experiments; therefore, individualslamming events can be very different betweenexperiment and simulation.

1.0E+00

§ 1.0E-01n

8 1.0E-02o

« 1.0E-03

XMC

v

easuredalculated

oQ_

1.0E-040 500 1000 1500 2000

Vertical bending moment - whipping [MNm]

Figure 27 Extreme values in a 4 m Sea State. Results ofthe whipping model and test data.

The measured peaks of the whipping moment inthe 4 m Sea State are compared to the peaks calculatedwith the whipping model, Figure 27. The vessel was atzero speed in a following sea condition. This resultshows that the distribution of the peaks is similar, sothat the differences, which develop due to "drift" in thewhipping model, are inconsequential to the extremewhipping moment.

14

Figure 28 shows a similar plot for a 2 m Sea State.The differences between actual measurements and thewhipping model at low probabilities of exceedance areimportant, but the tails of the distributions are againsimilar. These results mean that a simulation with thewhipping model as presented in this paper and themeasured slamming force cannot be used for fatigueassessments. However, the results of the whippingmodel can be used for the prediction of extreme values.

1.0E+00

£ 1.0E-01es<p

88 1.0E-02

1.0E-03

1.0E-04O 200 400 600 800 1000

Vertical bending moment - whipping [MNm]

Figure 28 Extreme values in a 2 m Sea State. Resultsof the whipping model and test data.

EFFECT OF WHIPPING ON FATIGUE

In addition to the tests in following seas at low shipspeeds, a series of experiments in high head and bowquartering seas at low ship speeds was carried out. Afatigue analysis (rainflow count) has been made of themeasured vertical bending moment and of the momentfrom which the whipping part has been removed byusing a low-pass filter. This filtered moment isindicated as the wave vertical bending moment.

The fatigue effect (FE) is in general defined by:

M

C

easured

alculatedVertical bending moment range [MNm]

Figure 29 Fatigue in head seas, 9 m Sea State at 9 ktsspeed.

600

„ 500 -I

§ 400

o 300

-g 200

z 100

0

• Total MY

D Wave MY

• •-

Vertical bending moment range [MNm]

Figure 30 Fatigue in following waves, 4 m Sea State(Tp = 8 s) at zero speed.

Total MYWF part of MY

Head seas,H s = 9 m;V = 6 kts2.60 E+l l2.07 E+l l

Followingseas, Hs =4 m; V = 01.73 E+l l8.62 E+08

Table 4. Fatigue effect, (MNm)3, in one hour on theMY midships for the model in the tank.

Fb = 2JO i n; (11)i=i

Since, in the elastic regime, the stresses are proportionalto the bending moment and because the structure is notbeing considered, equation (12) is a basis for comparingthe FE in different wave conditions.

(12)

The result of an experiment in a high (9 m) SeaState is shown in Figure 29. This figure shows the totaland the wave MY. The whipping part has an effectmainly on the lower MY ranges. This means that thefatigue portion in one hour is 80% dominated by thefirst order wave forces and 20% by the whippingcomponent, see Table 4.

Figure 30 shows a similar plot of a test in a 4 mSea State in following waves conditions (the same testas shown in Figure 22). This figure shows that theresults are dominated by the whipping response of the

15

ship. There is a very large number of load cycles atmedium ranges of the MY due to the whippingresponse of the vessel. The fatigue effects due to thetotal signal is a factor 200 times higher than those dueto the first order wave loads only. The fatigue effect inthis condition is of the same order as for the 9 m headseas condition mentioned before.

The extreme slamming events that are reported inthis paper only exist for a narrow range of sea states,which means a narrow range of the peak period of thewave spectrum. The worst case is in a sea state with apeak period of 8 s.

Figure 31 and Figure 32 show test results in a 2 mSea State with a peak period of 8 s and 14 s,respectively. Figure 31 shows that for the Tp = 8 s case,the fatigue effect is dominated by the whippingcontribution. Figure 32 shows that for the Tp = 14 scase, the wave contribution is dominant. The number ofload cycles is also an order of magnitude lower in thiscase, which makes the fatigue effect a factor 10 lowerthan the Tp = 8 s case.

snts

ote

v<

<ö.O

z

800

700

600500

400

300200

1000 L

• Total VBM

D Wave VBM

o o o o o o o o ot \ i - o - ( o o o o c N M - < o

Vertical bending moment range [MNm]

Figure 31 Fatigue in following waves, 2 m Sea State(Tp = 8 s) at zero speed.

200

150£S

'S 100

I 50 i.

• Total VBM

D Wave VBM

o oCM

o o o o ow io io m io00 O CM ^ CD

Vertical bending moment range [MNm]

Figure 32 Fatigue in following waves, 2 m Sea State(Tp = 14 s) at zero speed.

EFFECT OF WHIPPING ON EXTREME LOADS

The extreme values of the sagging and hoggingcomponent of the MY have been collected for thementioned tests in the 9 m head seas and the 4 mfollowing seas conditions. This analysis has also beendone on the total MY signal and the filtered signals (thewave frequency and the whipping part).

1.0.E+00t

1.0.E-040 1000 2000 3000 4000

MY-sagging [MNm]

Figure 33 Extreme values (sag) in head waves, 9 mSea State.

Figure 33 shows the extreme values of the test datain head seas and a 3-parameter Weibull distributionfitted through the data. This fit describes the shape ofthe distribution of the maximum values quite well.From here on use has been made of the fitted Weibullcurve for comparisons.

Figure 34 shows the results for the saggingcomponent of the MY in the same 9 m Sea State. Theplot shows the extremes of the wave component (WF)and the whipping component (HF) separately. Thesetwo signals were used to create the line "WF + HF".This line is constructed using the rules:

= M Y W A VE FREQUENCY

+ MYWHIPPING

Pr{MYS A G G I N G} = Pr{ MYWAVEFREQUENCY}

where Pr is the probability.

This means that the probability of an extremesagging moment is equal to the joint probability of anextreme moment due to the first order wave forces andan extreme whipping moment. The occurrence ofextreme events of these two phenomena is thereforeconsidered independent. The timing of the whipping

16

event is always such that the first peak, which is theextreme, occurs at the same time as a sagging peak.

Because of this last aspect, the same rule does notapply to extreme hogging moments as is illustrated inFigure 35. For this particular ship, the rule appears to bethat the whipping contribution is reduced to 50% at thetime of peak of the hogging moment.

M Y H O G G I N G = M Y WAVE FREQUENCY

+ Vl MYwHIPPING

Pr{MYH0GGiNG} = Pr{ MYWAVEFREQUENCY} (14)*

Another interesting feature from thesemeasurements is the sag/hog ratio. This ratio is for mostships larger than 1 due to non-linearities in the hullform (flare at the bow and/or the stern). It isunderstandable that this ratio increases for increasingwave height as is illustrated in Figure 36. This figureshows that the sag/hog ratio is quite large, about 1.8, forlarge vertical bending moments.

Figure 37 and Figure 38 show similar plots for the4 m following seas case. The whipping component isnow very dominant. The peak of the whipping momentoccurs at the same time as the peak of the hoggingmoment (unlike the head seas case). Therefore, the twoadded together give a distribution of extreme valueswhich is comparable to the distribution of the total MY.Adding the two components together for the saggingpeaks gives an over-prediction of the actual MY, as isillustrated in Figure 37. In fact, the extreme MY due tothe whipping contribution is always larger than theextremes of the total signal.

1 .OE+00

€ 1.0E-01

'S 1.0E-02

2a.

1.0E-03

Total

WF

HF

WF+HF

WF+HF/2

1000 2000 3000 4000

Vertical bending moment [MNm]

5000

Figure 35 Extreme values (hog) in head waves, 9 mSea State.

1.85CD> 1.6 Jg>I-- 1.4 -g>to

1.2 •

1.0 -

—O— wave MY

- 3 K - total MY

1000 2000 3000 4000

Vertical bending moment [MNm]

5000

Figure 36 Sag/hog ratio as a function of the magnitudeof the MY in head waves, 9 m Sea State.

1.OE+00

1.0E-030 1000 2000 3000 4000 5000 6000

Vertical bending moment [MNm]

Figure 34 Extreme values (sag) in head waves, 9 mSea State.

1.OE+00

J> 1.0E-01

i1.0E-02

1.0E-03

t

\\i

—¥

—A

-e

-Tota l

_ W F

h-HF

- WF+HF

1000 2000 3000 4000

Vertical bending moment [MNm]

5000

Figure 37 Extreme values (sag), 4 m following SeaState.

17

1.0E+00

1 .OE-01

1.0E-02

\

\

wX

— 1 — Total

- A - W F

-B-HF

C

ïe0_

1.0E-03

0 500 1000 1500 2000 2500 3000 3500

Vertical bending moment [MNm]

Figure 38 Extreme values (hog) component, 4 mfollowing Sea State.

PREDICTING FULL SCALE WHIPPING LOADS

The foundation has now been laid to predict waveloads for the prototype ship. This paper demonstratesthat the impact load can be derived from the measuredpressures. Applying this load to a structural model ofthe segmented model, it is possible to reproduce thewhipping bending moment at midships. The next step isto apply the measured impulse to a full 3D structuralmodel of the ship and do a time domain simulation todetermine the whipping response and peak stresses inthe ships structure.

CONCLUSIONS

This paper has introduced a reliable method formeasuring the excitation of a ship's hull due to aftbodyslamming impacts. The measured impact force has beenderived from a large array of pressure gauges. Testswith two models with varying stiffness indicated thatthe pressure due to the whipping response of the modelcan be superimposed on the impact pressure. Thismeans that there is hydro-structural interaction, but noimportant hydro-elastic effects.

The impact force has been used in a simplestructural schematization of the segmented model thatwas tested in the towing tank. The predictions with thiswhipping model compared very well to the measuredvertical bending moment. It is therefore concluded thatthe impact force can be applied to a structural model ofthe ship to determine the whipping moment and peakstresses for the full scale vessel.

Further conclusions are:

• A vessel having a modern stern shape lying at zeroor low speeds in following wave conditions canexperience heavy stern slamming. This slammingstarts occurring in mild sea conditions if the wavelength is on the order of the ship length.

• Stern slamming can be considered as an impactthat starts at a certain point and then expands firstlongitudinally and later athwartship over the stern.The high pressure ridge passes a single point on thestern in about 0.05 s. Due to the size of the sternthe total impact takes about 0.5 s.

• Instrumenting the aftbody of the model with a largearray of pressure gauges allows an accuratemeasurement of the impulsive force.

• The measured impulsive force can be used on adynamic structural model of the ship to predict thewhipping loads.

• The fatigue effects due to stern slamming in a 4 mfollowing Sea State at zero ship speed is on thesame order as the effects in a 9 m head seas SeaState at a ship speed of 6 kts.

• The peak of the whipping vertical bending moment(MY) in head seas occurs together with the peak ofthe sagging MY. In following seas the timing isdifferent: then the peak of the whipping MY occursat the same time as the peak of the hogging MY.

• The design value of the extreme MY shouldaccount for a larger than normally used sag-hogratio.

ACKNOWLEDGEMENT

The authors acknowledge the permission ofNorthrop Grumman Ship Systems to publish the resultsin this paper.

REFERENCES

Bereznitski, A. and Kaminski, M.L., "PracticalImplications of Hydroelasticity in Ship Design",Proceedings Int. Offshore and Polar EngineeringConference, Kyushu, 2002.

Betts, C.V, Bishop, R.E.D. and Price, W.G., "A Surveyof Hull Damping", Transactions RINA, Vol. 119, 1977.

Bishop, R.E.D and Price, W.G., "Hydro-elasticity ofShips", Cambridge University Press, 1979.

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Hamalainen, R. and Heerd, J. van, "Hydro-dynamicdevelopment for a large fast monohull passengervessel", Transactions SNAME Vol. 106, 1998.

Haugen, E.M, Faltinsen, O.M. and Aarsnes, J.V.,"Application of Theoretical and Experimental Studiesof Wave Impact to Wet Deck Slamming", ProceedingsFAST-97 Conference, Sydney, 1999.

Faltinsen, O.M., "Slamming", Colloquium for Ship andOffshore Hydrodynamics, Hamburg, 1996.

Faltinsen, O.M, "Water Entry of a Wedge byHydroelastic Orthotropic Plate Theory", Journal ofShip Research, Vol. 43, No 2, 1999.

Faltinsen, O.M., "Hydroelastic Slamming", Journal ofMarine Science and Technology, Vol. 5, No. 2, 2000.

Kurimo R., "Sea Trial Experience of the FirstPassenger Cruiser with Podded Propulsion",Proceedings of the 7™ Int. Symposium on PracticalDesign of Ships and Mobile units (PRADS)1 The Hague,1998.

Kvalsvold, J. and Faltinsen, O.M, "Slamming Loads onWet Decks of Multihull Vessels", ProceedingsHydroelasticity in Marine Technology Conference,Trondheim, 1994.

19