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24TH Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Whipping loads due to aft body slamming G.K. Kapsenbergl, A.P. van 't Veerl, J.P. Hackett and M.M.D. Levadou1 (1 Maritime Research Institute Netheriands, 2 Northrop Grumman Ship Systems) ABSTRACT This paper describes a method to measure slamming loads on the aft body of a ship model in the towing tank. The method uses a large number of pressure gauges in the flat part of the stern. The impact force is derived from the pressures by integration. Experiments have been carried out on a scaled model of a modern cruise vessel. The model consisted of two segments so that a simplified 2- node vibration mode is simulated. A structural mathematical model was made of the model as used for the tests. The analysis shows that the measured pressures can be used to predict the whipping loads on the segmented model. It is concluded that the measured loads can also be used on a finite element model of the ships construction to predict whipping stresses. Whipping stresses are important for fatigue loads and the extreme vertical hull bending moment. INTRODUCTION The goal of every commercial ship owner is to maximize the revenue producing portion of the vessel and minimize the non-revenue producing portion such as the engine room. A "shoe box" with no propulsion plant, ballast tanks, etc. would be considered the ideal solution. As a result, over the years an increasing demand was placed on design naval architects to produce ship designs with full beam main decks along almost the entire length of the ship. For containerships this allows more containers to be stowed on main deck. Likewise, for cruise ships, a full width main deck allows a full width superstructure that contains more passenger cabins. Competition is now so intense that this trend has extended to include filling out the aft portions of the hull above the full load waterline and below main deck. This added hull volume means more container stowage capacity for container ships, more space for passengers on cruise ships, and a full breadth vehicle deck for more vehicle stowage on ferries. Owners rationalize that as long as the filling out of the hull occurs above the full load waterline, there would be no negative impact on calm water powering and fuel economy. This design philosophy has produced hull forms with progressively flatter and flatter sterns. Some ships have been built with zero or near zero deadrise angles and buttock angles near the transom. Such shapes have hydrodynamic benefits in calm water as demonstrated by Hamalainen and van Heerd (1998~; however, they are a far cry from the traditional cruiser sterns seen on the Mariner hull form. At the same time that this hull form development was occurring, a new propulsion system was being developed. The idea was to move the electric propulsion motor from inside the hull to an azimuthing pod located external to the hull. This allows the elimination of the propeller shaft and shaft struts, the rudder, and stern tunnel thrusters. This concept provides greater flexibility for the design engineer to locate elements of the propulsion system; hence, an ability to provide additional internal volume to the revenue producing portion of the vessel. It also produces 10 to 15% improvement in calm water powering and significant improvements in vessel harbor maneuverability. The first ship equipped with podded propulsion was the 'Seili', an ice going supply vessel built by Kv~erner Masa in 1991; the first application to a cruise vessel was in 1997, the 'Elation'. Since then podded propulsion has become the industry standard on cruise ships. The combination of the pod with the flat stern has resulted in gains in propulsion efficiency. The azimuthing pods encouraged design naval architects to create stern designs that were even
flatter than their conventional screw propelled sister ships to simplify the pod installation. THE SLAMMING PROBLEM It was not long before a serious drawback of the flatter stern design surfaced. Some of the new cruise ship designs began to report slamming and subsequent vibration problems in following seas at zero and low ship speeds. Yet ferries did not report these problems. While the hull forms of cruise ships and ferries are much the same, the operating profiles for the ships are quite different. Ferries sail at low speeds inside the harbor, and then come up to cruising speed and remain there until they reach their destination. Cruise ships spend large amounts of time operating at low speeds even in the open seas. The reported slamming and vibration problem occurred in very mild sea states when the waves had a length in the order of the length of the ship. In the process of investigating the zero speed following sea stern slamming and vibration problem, questions began to arise with respect to a possible hull girder fatigue life problem. Later issues of ultimate hull girder longitudinal strength surfaced. These issues brought together the sciences of hydrodynamics and structural . . engineering. THE: 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 are in Table 1. Its hull form is a variation of a proven design. Figure 1 shows the aft body plan with pod. Although the stern of the subject vessel is somewhat flat, it has considerably more shape aft than other vessels that have reported stern slamming problems. Note that the body plan shows a very reasonable buttock rise moving aft, and section curvature near the skeg. The minimum deadrise angle exceeds 3 degrees. Prior to performing the special stern slamming test program reported here, a conventional seakeeping model test program was conducted which included three pressure panels on the bow flare and one on the flatter portion of stern near the transom. All of these test results were within acceptable limits. The purpose of the stern slamming seakeeping model tests and hull girder structural finite element investigation was to determine if sufficient aft shape was present and if sufficient structural rigidity and strength existed to avoid the vibration and structural problems present in those ships that encountered severe slamming. Length between perp. 233.00 m Beam 32.20 m Draft 8.00 m Displacement 39000 ton Speed 21 kn C 14 2.10 m Table 1 Main dimensions of the cruise vessel. DWL 6/ ~ 'POD Figure 1 Aft body of the cruise vessel. CONSIDERATIONS FOR SLAMMING MEASUREMENTS In general, impacts due to slamming contain energy in high frequency bands (1-20 Hz). At very low deadrise angles (below 5 de"), the high frequency contents increase with decreasing deadrise angle. Also hydroelastic effects might be important, as reported by Faltinsen [1996] and Bereznitski and Kaminski [20021. If the high frequency component of the slamming pressure is important, it needs to be accurately measured. This means that the set-up needs to have a high resonance frequency, at least twice as high as the frequency content of the slamming signal. In addition to this, under water video recordings of an aft body of a ship slamming in waves revealed a large number of air bubbles underneath the stern during the impact. This indicates that there are strong local effects.
The conventional test methodology is to isolate the aft body of the model from the rest of the model. The aft body is then connected to the main portion of the model in such a way as to allow a system of strain gages to be installed along the joint. Such a system will have a relatively low resonance frequency and will hence be insensitive to high frequency contents in the pressure signal. The measured force will be a result of the dynamics of the aft body connection to the hull. Deriving the impact forces on the stern is not straightforward. Therefore, this testing method was disregarded. PROPOSED METHODOLOGY Both aspects mentioned, the high frequency content of the input and the local effects, made us decide to use a large array of pressure sensors to define the impact. The array used is illustrated in Figure 2. This set-up allows direct measurement of pressures including the high frequency content in the signal, without the dynamics of a strain cause system. It is important in this set-up that all components in the model are as rigid as possible. This essentially rules out measuring the longitudinal stresses midships by means of a strain gauge system, just as a segmented stern model measuring the total impact force was not considered acceptable, as discussed earlier. However, the whipping component on the vertical bending moment is one of the important consequences of aft body slamming, so a way to measure it must be developed. The approach taken was, to solve the problem in two steps: 1. Measure the impact loads (pressures) on a very rigid model. 2. Measure the whipping loads on a model cut in two while the frequency of the 2-node deformation is scaled from ship to model. The pressures at the stern were measured for both the flexible and rigid model. The assumption was that the total pressure would be the sum of the impact pressure and the pressure due to the deformation. If this was the case, based on the results of the model with different natural frequencies, it would mean that the hydroelastic effects could be ignored; at least for practical purposes. The objective of step 2 was to demonstrate that the physics of the problem were understood by being able to calculate the response of the segmented model. Using the measured pressures on the stern of the model allows the calculation of the impact forces on the stern on the model. The impact forces are then applied to a mathematical whipping model of the tow tank model in the water. The validation then consists of the comparison of the calculated whipping moment to the measured values. Positive validation would yield a reliable tool, when used in conjunction with a detailed structural model of the ship, for determining the full scale whinnying resr,on.ce of the shin Figure 2 Aft body of cruise vessel with podded propulsion and instrumented with 33 pressure gauges. MODEL AND INSTRUMENTATION For the model tests reported here, a wooden model with a geometric scale ratio of 1:49, equipped with bilge keels, active stabilizer fins and podded propulsors was built. The model was built in two segments with the cut at Station 10 (midships). The connection was made via a steel plate that was instrumented to measure the vertical bending moment (VBM). This steel plate is illustrated in Figure 3. The gap between the two halves of the hull was closed with a thin rubber membrane to make it watertight. The weight and inertia of both segments were calibrated separately. A rigid model could be obtained again by fitting of longitudinal beams along the gunwales of the model. Accelerometers were fitted on the fore and aft end of each hull segment so that also the deformation of the vessel could be measured. The stern of the vessel was instrumented with 33 pressure pick-ups as illustrated in the photo in Figure 3 and the drawing in Figure 4. The sampling rate of the measurements was 286 Hz (full scale). Anti-aliasing filtering was applied and the
measured signals were run through an LP filter with a cut-off frequency of 95 Hz. : Regular waves, Arnpl=2~5 m, T=8.0 s, Heading=O:deg TEST NO 337001 ANALYSIS OF THE IMPACT PRESSURE ° When a slam occurs during the tests in waves, short duration impulses are measured on each of the pressure gauges. Figure 5 shows such a recording. This figure shows that there are important differences in the peak of the impact pressure (from 200 to 1000 kPa) and in the timing of the peak as a function of the different locations of the gauges. Also, the shape of the pressure pulses as a function of time can be quite different from the various sensors. SLAM 19 200 ~ kPa | O CLAM 20 200 ~ kPa o 1 __ 200 - l SLApMe21 1 00 ~ _ _ _ !500 SUM 22 : kPa O- 200 - SLAM 23 .~00 Jew _ .1~, a: 400 - SLAM 25 200 . . . . 1.5 1.6 1.7 1.8 SECONDS Figure 5 Recordings of pressure gauges 18-25 during a slam. Figure 3 Instrumented steel plate to measure vertical bending moment. STARBOARD 15 ~ O O 1 9 0 1° 110 o 3 120 _ _ _~ 0 13 i 10 0 1° O O ~ 17 25 NO z7o 190 280 20O Z9O Z1~ -I= 22 O 230 40 310 32O 33O STAT. 0 STAT. 1 CENTRELINE _ PORT-SIDE Figure 4 Location of the 33 pressure pick-ups in the stern of the model. There appears to be a high-pressure ridge that travels with a rather high velocity over the aft body of the vessel. The time difference of the pressure peak passing the different gauges was used to derive the velocity of the pressure ridge. Figure 6 shows the analysis of a slamming event; this event was selected from tests in a sea state characterized by a JONSWAP spectrum, Hs = 4.0 m, Tp = 8.0 s and a peakedness parameter ? = 3.3. The figure shows isobars at different time steps, the isobar shown is the 100 kPa level of the pressure peak during the rise of the pressure. Figure 6 shows that the impact started on starboard side, close to the aft end of the skeg, 2 meters forward of the aft perpendicular (APP). The distance between the isobars is used to derive the velocity of the pressure ridge; these velocities are indicated on the figure. The initial velocity of the pressure ridge is 13-14 m/s forward and to the starboard side and 21 m/saft. The velocity aft increases to 30 m/s after 0.3 s; side ways it decreases to 6 m/s. The high pressure ridge covers the full length of the aft body in 0.5 s after the initial impact; then it expands mainly sideways, at a velocity of 15-20 m/s to
starboard and about 45 m/s to port. The duration of this total impact is a bout 1 s. Figure 7 shows a pressure map of the second stage of this impact, 0.55 s after the initial impact, when the pressure ridge moves sideways to both port and starboard. The peak pressure is now on the order of 100 kPa. Figure 8, shows the second impact occurring at two locations simultaneously; one on the center line about 2 meters aft of the skeg and the second close to the center line on the starboard side. These two spots join to one large area only 0.15 s after the impact; which covers the full length of the aft body. The high pressure area further expands to the sides at a speed of 20 - 40 m/s. The total impact lasts about 0.5 s. :E s c' ~ 0 ~ ~ . = ..... b~ ., ~ A: soI ~~ °~ ~ 1- o.6 ~ O'6 1" -10~ - Analysis of a large number of impacts in different wave conditions shows that most impacts have a duration in the range of 0.4 to 0.75 s. This is illustrated in Figure 9 The impact is shown to be a moving pressure front with peak pressures of about 300 to 500 kPa in the initial stage of the impact, reducing to 100 - 200 kPa in the second stage. The width of the high pressure ridge is about 2.0- 3.0 m. The high pressure area grows very quickly (velocity 30 to 40 m/s) length-wise until it covers the full length of the stern. After this it travels sideways at a velocity of 20 - 30 mls. This makes the total duration of the impact in the order of 0.5 s for the subject vessel. . 0 t~ ,°~ a,~ ~ - 5 O o ~ IC ~~- ~ ~ e~/~wf~x,~=,~~ X-location w.r.t. APP[m1 Figure 6 Contour plot of moving pressure fields over the aft-body. The plot gives the 100 kPa isobar lines at different times and the velocities of the pressure peak. in Pressure [kPa] 200 1 190 180 170 1~ 1~ 140 130 ~ 1~ ~ 110 1 100 90 80 70 60 50 40 30 20 10 J o ' 1 1 1 1 1 , , , ~ 1 , 1 1 1 1 0 10 20 x [m] Figure 7 Pressure map of the impact shown in Figure 6. Time is 0.55 s. after the initial impact. X-location w.r.t. APPlmJ Figure 8 Contour plot of moving pressure field over the aft body after the second impact showing that the initial impact takes place at two locations. Hi, - .? 0 0.5 1 1.5 SLAM DURATION [s] Figure 9 Statistics of the duration of the impact in a 4.0 meter Sea State.
PRESSURE INTEGRATION The next step is to derive the impact force on the aft body as a function of time. This requires a pressure integration in space. The width of the pressure ridge requires a density of pressure gauges which is far greater than practically possible. With the current experimental set-up it is possible that the high pressure ridge is at some instant 'lost' because it is in between the pressure pick-ups. Therefore, a proper spatial integration of the pressure signals requires a careful selection of the integration method. A simple and robust pressure integration technique, denoted (S), was tested first. The method makes use of a so-called Delaunay triangulation as illustrated in Figure 10. This triangulation is unique; no other data points fall within the circle drawn through the corner points of each triangle. The triangles are used in the pressure integration. The measured time trace of the pressure in a point is multiplied with one third of the area of all surrounding panels. In formula: FZSLAM(t) = ~ P(t) pressure pick-ups ~ Al /3 (l) adjacent panels i This procedure uses the complete area enclosed by all the grid points (329.9 mid. A similar procedure was used to calculate the center of effort of the impact force as a function of time. 15~ 10t - .Q - . -10 - ~ = -15 2 4 6 8 10 12 X location w.r.t. APP [m] Figure 10 Delaunay triangulation using all pressure pick-ups in the aft-body of the model. 1 14 16 18 For example, the pressure measured in Pie, see Figure 10, is multiplied by i/3 of the area of triangles 2-4-7-8 and 10. The proposed integration method is robust and simple but it requires a density of pressure gauges which is high relative to the width of the high pressure ridge. Since this is not the case with the current set-up, the resulting force, FZS~AM(t), will be very spiky due to an over-estimation of the force at the moment the ridge travels over one of the pressure gauges, and an under-estimation when the ridge is just in-between the gauges. To overcome the expected inaccuracy noted in the simple integration method, a more advanced pressure integration is applied. This technique is denoted (C). This method regards the pressure as a moving pressure ridge over a panel. The shape of the pressure ridge ptt) is considered constant for each panel. Since the velocity V of the pressure ridge can be determined, the time signal at each point can be transformed from the time domain to the spatial domain: pts)=p~t) V :~\ ! ~ | ~~( Activ~nel I \~\ 100 So \ 3 pressure signals L AIL I ^^~ a corner point _- (2) - -5 Leo 160 140 5100 ~0 60 140 ~ _^v Figure 11 Impression of moving pressure front over a panel.
i Est mated time- duration of stem A: C 2.! In i Calculated IMPULSE {kNSl:' 7278 kNs serge method ~ 7445 kNs advanced my 0.5 nME [S1 ~9 1.S Figure 12 Result of the Simple and Advanced pressure integration methods for one slam. The direction and velocity of the moving pressure ridge is determined by the pressure-time signals at the three corner points. The velocity vector at each corner point is known, but normally only one of them actually crosses the subject panel. The pressure signal at this point is used to calculate the force on the panel in a strip-wise manner for each time step. Figure 11 illustrates this spatial integration process over a single panel. The result of both pressure integrations is illustrated in Figure 12. The time trace of the signal is different; the peak of the signal is very different, but the total impulse appeared to be almost identical. The impulse ~Fdtis 7278 Ns for the simplified integration method and 7445 Ns for the advanced method. The analysis of other slams with both methods showed that the difference in impulse between the two methods is limited to 3%. MATHEMATICAL MODEL TO CALCULATE THE WHIPPING LOADS A schematic of the model in the tank is illustrated in Figure 13. The model is represented by two masses, connected by a hinge, a spring and a damper. Both masses are supported by a hydrostatic spring. This spring cij consists of the coefficients C33, C35, C53 and CS5 to take into account heave and pitch restoring as well as the cross coupling. Hydrodynamic damping is ignored because of the high frequencies of interest. The equations of motions of this two mass system are built using: a) The pitch motion equation of the forward part (index 1) around the CG of this part of the model, and of the aft part (index 2) around its CG. The distance from the CG of part 2 to the impact location is Xs~am: F.. ~ ~ . IYYO1 + CS3Z1 + CSSD1 =CH (0102 ~BH (0102 IA~,82 + CS3Z2 + CSSD2 = CH (0102 ~ + BH (0102 ) + FZSlam Xslam (3) b) The heave motion equation of the forward and aft part: MFz1 + C33z1 + C3sOl = 0 <4' M Z2 + C33Z2 + C3s82 + FZslam = 0 c) The fact that the hinge keeps the two parts connected. The distance from the local CoG to the hinge is defined by x~ and x2: Z2 - X202 = Z1 - Xl81 (5) Equation (5) is used in equation (4) after which the two equations of (3) are summed and written as 1 single equation. Similarly equation (4) can be substituted in equation (3) after which the system can be written as a differential equation: M Yt ~ =F(y,t) (6) in which, M, the 6x6 mass matrix can be derived from the equations (3) through (6~. The yet) vector contains the unknown displacements and rotations and their time derivatives: ytt) = (Z., Z1,0,, 8,, 82, 02) (7) The right hand side vector force F(y, t) contains the restoring forces and the slamming force. The differential equations are solved using an explicit Runge-Kutta 4th order scheme.
MA , , , ,1, . . . cH BH MF ~ C hi , ,,,1,,,, Figure 13 Dynamic model of the cruise vessel as it was in the towing tank. DYNAMIC CALIBRATION The model system is calibrated by hitting the model on the extreme aft end and calculating the response. The response in this respect is either the moment in the connecting spring or the angular deformation. The angular deformation is the deformation mode with the lowest eigen frequency. The response of this system on a triangular load, force as a function of time, was calculated, see Figure 14. The duration of the load was varied while the impulse, .iFdt, was kept constant for the different cases. The results of the calculations are shown in Figure 15. The duration of the impulse is normalized by the period of the angular deformation mode of the system. This figure shows that for very short impulses it is not the peak value of the force that is important, but the impulse. When the duration of the impulse is very long, the response of the system goes asymptotically to the static case. The response of the system to a constant force is also shown in Figure 15; it is denoted the static response. The model of the cruise vessel was subjected to an impulsive load when lying at zero speed in calm water. The results of these tests, the frequency of the wet eigen mode of the model and the damping, are presented in Table 2. The low damping of the flexible model is a value that could be expected, see Betts et. al. (1977) and Bishop and Price (1979~. The high damping of the rigid model is quite surprising. Most likely there was internal frictional damping in the fixation of the additional beams as a result of the high accelerations. :2000 Q -2000 . 100 z Y 50 ~ 25 -4000 · , ~ . . n 0 2 ~ ~ ~ 10 12 14 16 18 2 TIME [see] Figure 14 Whipping response of the model on a triangular impulsive load. 1.4 - .2 - u' 1.0- o u' 0.8- E 0.6- ~ 0.4- ._ x no v. 0.0 - 0.0 \ 1' ~ \ |Flexible I ~ T Dynamic solution Model Static solution . - _ 0.5 1.0 T impulse / T resonance [-] Figure 15 Response of the schematized model to an impulse of varying duration at the extreme aft end. 2.0 As was illustrated in Figure 9, the duration of the slamming impact is on the order of 0.5 s. This allows the points for the rigid and flexible model to be plotted in Figure 15. The figure shows that the response of the rigid model is not quasi-static, in fact the rigid model was not rigid enough. The response of the flexible model will be closer to a quasi-static response than that of the rigid model. The objective of comparing the response of the two models is to compare the response of two models with varying flexibility. The response of the flexible model on the impulse is shown in Figure 16. This figure shows the pressure measured by gauge P28 and the local vertical acceleration. High frequency local vibrations are dominant in the initial stage of the impact; these are damped in about 2 seconds. After this time the simplified 2-node deformation mode is dominant. About 0.5 s after the initial impact, the pressure P28 is reasonably in phase with the local
vertical acceleration (times -8~. This means that the measured pressure is only due to the added mass effect. A similar relation between the local vertical acceleration and the pressure was found for the 'rigid' model. rigid model flexible model frequency damping [Hz] [-] 1.75 0.049 0.83 0.0069 Table 2. Results of hammer tests model. 20Q 160 120 80 -120 -160 ~ 200 0 Figure 16 Response of flexible model on a hammer impact. The figure shows the P28 pressure and the local vertical acceleration. This result is used in the analysis of the impacts in regular waves. The measured load is split into parts, the first due to the actual wave impact and a second part due to the local deformation. The pressure due to the actual wave impact is defined: PIMPACT = PMEASURED PLOCAL DEFORMATION (8) For P28 the pressure due to the local deformation Is: PLOCALDEFORMATION = 8 AZX=P28 (9) The results of this analysis are shown in Figure 17. The difference in the peak values is quite low; and it is reduced by correcting them for the local deformations as illustrated in the lower of the two plots. Note that the local vibrations are dominant over the 2-node bending mode at the time of the impact. A comparison of the integrated pressure for the rigid and the flexible model is shown in Figure 18. From these results it is concluded that the hydro-structural interaction is low; the pressure due to the whipping of the model is an order of magnitude lower than the pressure resulting from the impact. Hydro-elastic effects are again lower, so it is concluded that, for practical purposes, they can be ignored in the problem of aft body slamming. P28rl P28fl P28rc P28fg O -100 D tr 2 3 4 Figure 17 Impact of a regular wave on the flexible and the rigid model. Top graph: measured pressures at P28. Bottom graph: pressures corrected for local deformation.
X10 REGULAR WAVES ~Flexible model | ~ `; m AAAP! IT' OF I Rioid model I ad ~2 Cal in ~ ~ v~ Figure 18 Slamming force for the flexible and rigid model. Regular following waves, amplitude 2.6 m, period 8 sec. TUNING OF THE WHIPPING MODEL The mathematical model requires as input the properties of the model and the stiffness CH and damping BH of the connecting spring. The value of CH follows from the natural period of the 2-node vibration and by assuming uncoupled motions between heave and pitch, so by neglecting the z- displacements in equation (3~: .. . ~ 01 + 2KC)n (31 + COn 01 = 0 _ 2BH 2 4CH + C55 A (1 0) C,3 . IF+A C) = n yy YE The damping BH follows from the whipping moment decay curve, which was obtained by hitting the model with a hammer. This damping curve contains both hydrodynamic and structural damping. The first is normally considered negligible at this frequency. The full data of the proposed mathematical model are listed in Table 3. Figure 19 presents the correlation between the mathematical model and the decay curve of the physical model. The initial conditions for starting the curve are obtained using the measured vertical bending moment (MY) at that time. The initial part of the simulation is shown in the detailed graph in the figure. Obviously, in the decay curve the vertical slam force (FZ) is zero. The tuned non- dimensional damping coefficient was ? = 0.0077; this is a low value in comparison to full scale ships. Full scale ships have a damping on the order of 2 to 3% of the critical value. LINE = MEASURED CIRCLE = CALCU~TED 110 115 120 125 130 135 140 ~ 145 150 155 TIME [see] Figure 19 Decay curve of the physical model and the mathematical model for whipping due to an impulse (hammer blow). mass [ton] kyy [m] LCG [m] C33 [kN/m] C35 = C53 [kN] C55 [MNm] CH [MNm] ? [-] . Aft Forward . 22237 16636 40.87 43.81 -53.53 44.30 36521 27521 lW575 -7743 22040 38620 . 4247 0.0077 Table 3. Data of whipping mathematical model. The data is relative to the CG of the segment. LCG is relative to the cut. USE OF THE WHIPPING MODEL TO ANALYZE SLAMMING EVENTS The objective of the application of the whipping model to analyze slamming events was to check if the physical process was understood. If the slamming force, as derived from the pressure gauge measurements, is applied to the whipping model, the response of this model must correspond to the response of the model in the tank. When that is the case, the loads can be applied to a Finite Element (FE) structural model of the ship as built. A first check was carried out to compare the response of the model to the simple integration method (S) and the advanced integration method (C). This comparison is made in Figure 21. The calculations start at T = 381 seconds with initial conditions zero, thus with no displacement, rotations or whipping moment present in the hull.
::L:ong-cre:sted ~Js, Hs 4.:0 my, TO 8.0 s':Head 0 deg. Speed 0 kn TEST NO 3:1:5001 x10 MY TO:T klKlm - ::M~y WHIPPINGS ~ kNm to a| L .1 . 11 U ,5~ X o ::2- ~ iN MY WAVE 1 ~~L,~,7 N; ~~l a! , ~ T J~ \ SECONDS Figure 20 Time traces of the Vertical Bending Moment midships. Top: the total signal; Middle: the whipping component; bottom: wave frequent component. x 10 10 5 , , . , . . . simple method (S) complex method (C) At .e CL i" , ;: , ~ ~ . - . ~ · : .. Y . . , . ~ . -. . ~_c'~ . . . . . : : i., 381 382 ; Or O _ 385 386 387 S88 389 TIME [sl Figure 21 Calculated whipping response due to a single slam. The slamming force shows some spikes when the simple integration method is used, but the model does not respond to such short duration, low impulse spikes. The result of the simulation shows an identical response of the model in the sense of the vertical bending moment RESULTS OF EXPERIMENTS IN WAVES Tests were carried out in following sea conditions at zero speed. The wave conditions consisted of long crested seas, significant wave height 2.0 and 4.0 meters, irregular waves using the JONSWAP spectrum with a peakedness parameter 3.3, and a peak period of 8.0 s. The energy spectrum of the vertical bending moment midships is shown in Figure 22. The figure clearly shows two peaks, the one at a frequency of 0.8 rad/s due to the waves and the peak at 5.2 rad/s due to whipping. The measured MY (in de 4.0 m Sea State) is plotted in Figure 20. The total signal (top time trace) is split into a whipping component (using a high-pass filter with cut-off frequency 3 rad/s) and in a wave component. Both components are also shown in Figure 20. The vertical bending moment due to whipping is clearly dominant in this condition. The results of the whipping model will be compared to the whipping component of the vertical bending moment midships; the comparison will mainly be made for the 4.0 meter Sea State as specified above. Earlier discussion showed the whipping response of the vessel due to a hammer impact. The validation was made using zero initial conditions in the calculations. In waves the situation is different;
Figure 23 shows that whipping due to previous slamming events still occurs when the vessel is hit by the next wave. 500. 400 - .. in E it : ~ ~ ?°°- .~¢ 105 0:_ ~ 4 6 WAVE FREQUENT U. RADtS Figure 22 Spectrum of vertical bending moment ° showing distinct wave frequency part and whipping part. JONSWAP spectrum, Hs = 4.0 m, Tp = 8.0 s, Vs = 0, following waves. 6~11 Jr,WHPP09G 10Mm 2 L _ FZ SUb. _, kNm a_" . - ~!1~' x104 The timing between the existing whipping deformation and the next slam impact appears to be crucial for the resulting response. For example, a relatively low impact force is able to minimize the whipping moment at t=26 s. A similar event is found around t=110 seconds. On the other hand, the impacts at t=34 and t=41 s. increase the whipping moment more than could be expected due to a 'favorable' timing. The whipping load is calculated using the initial conditions (angles of the fore and aft segment, angular velocities) from the model tests. The amplitude of the whipping load is used to obtain the difference in the angle of the fore and aft segment. The angular velocities are obtained from two angular rotations at consecutive time steps. Figure 24 through Figure 26 present the results of the simulations with the whipping model compared to the measured vertical bending moment for three cases. Figure 24 shows how a relatively small impact increases the whipping moment some 20%. Figure 25 shows how a slam that occurs out of phase with the whipping motion reduces the moment by a factor of four. The calculations predict the trend correctly, but the phasing of the whipping moment after the impact is not correct. Long~crested Js, his 4.0 m, TO 8.0 5, Head O:~deg, Speed O 1m TEST NO 315001 X~1o, 20 40 60 ~-30: ~ 00 SECONDS 16 17 Figure 23 Application of the whipping model. The figure shows a strong relation between the resulting whipping moment and the timing of the impact. l 18 19 20 21 22 23 TIME [s] 24 Figure 24 Example of the measured whipping response in the towing tank compared to the response of the whipping model due to the measured impact force.
1 , , ,1 215 216 -217 218 219 220 221 TIME [s] 1 222 223 224 Figure 25 Example of a slamming event that causes a decrease of the whipping moment. Blot I I I fir, r I , , 382 -383 ~ ~ 384 385 386 387 388 389 390 TIME [s] Figure 26 Example of a slamming event that causes an increase of the whipping moment. Figure 26 gives an example of how a favorable tuning increases the whipping moment with a factor 4. The prediction of the whipping model agrees very well to the actual measured moment. The general good agreement between the calculations using the whipping model with the actual measured vertical bending moment shows that the whipping phenomenon is 'understood', and that the response of the model can be reliably predicted by the whipping model for individual cases. STATISTICS OF WHIPPING LOADS The previous section showed that the amplitude of the whipping moment is very much dependent on the timing of the slam relative to the existing whipping motion. When simulations with the whipping model were carried out, each slamming event could be predicted (using the measured slamming force) if the initial conditions just before the impact were tuned. If this is not done, inevitably some drift will occur, which will change the tuning of the slam with the whipping motion and which will hence have a large effect on the resulting peak . . w ~ppmg moment. We now consider a long record of measured whipping moments and compare the statistics of the measured whipping peaks to those from the whipping model. The initial conditions for the whipping model are not tuned to match the experiments; therefore individual slamming events can be very different between experiment and prediction. l .OE+OO 1 .OE-01 1 .OE-02 1 .0E-03 1 .OE-04 l l l l ' ' ' ' 1 ' ' ' ' O Measured C' ~Iculated _ _ ot+ ~ ~ O 0 500 1000 1500 2000 VBM-whipping [MNm] Figure 27 Extreme values of the whipping part of the Vertical Bending Moment compared to the un-tuned results of the whipping model. Results from a 4.0 meter Sea State The measured peaks of the whipping moment in the 4.0 meter Sea State are plotted in Figure 27 and they are compared to the peaks calculated with the whipping model. The vessel was at zero speed in a following sea condition. This result shows that the distribution of the peaks is similar, so that the differences, which develop due to 'drift' in the whipping model, are inconsequential to the extreme whipping moment. Figure 28 shows a similar plot for a 2.0 meter significant wave height Sea State. The differences between actual measurements and the whipping model at low probabilities of exceedance are important, but the tails of the distributions are again similar. These results mean that a simulation with the whipping model as presented in this paper and using the measured slamming force cannot be used for fatigue assessments. However, the results of the
whipping model can be used for the prediction of extreme values. PREDICTING THE WHIPPING LOADS IN THE ACTUAL SHIP The road is now paved to predict wave loads for the actual ship. This paper demonstrates that the 700- impact load can be constructed from the measured 600- pressures. Applying this load to a structural model of the segmented model as it was in the towing tank. It _ 500 - is possible to reproduce the whipping bending ~ 400 moment at midships. The next step is now to apply ~ the measured impulse to a full 3D structural model of ~ 300 - the ship and do a time domain simulation to > 200 determine the whipping response and peak stresses. inn n 2 1 .OE+OO ~ . cut ' 1.0E-01 x _ 1.0E-02- a - Q 1.0E-03- 0 1 .OE-04 I''' ,lll .~ _ __ _ ' ' ' ' 1 ' ' ' ' O Measured ~ Calculated 0 200 400 600 800 1000 VBM-whipping [MNm] were always significantly lower than those in following wave conditions at zero speed. This again illustrates how important the whipping loads in following waves at very low speeds are for the design of the ship. | )K w eve | | O w hipping | 0 1 2 3 4 5 Significant wave height [m] Figure 29 Vertical bending moment (mean of 1/3 highest peaks) as a function of significant wave height. Vessel at zero speed in following wave conditions. 700 600 500 400 - 300 200 Figure 28 Extreme values of the whipping part of the Vertical Bending Moment compared to the un-tuned results of the whipping model. (2.0 meter Sea State). o IMPORTANCE OF THE WHIPPING LOADS o Although it is not the subject of this paper it is interesting to note the importance of the whipping loads for this vessel. Tests in different wave heights indicate a roughly linear increase of the whipping moment as a function of the wave height as shown in Figure 29. The whipping loads quickly decrease as a function of speed as illustrated in Figure 30. Many tests were carried out; some of these were devoted to extreme wave conditions. These tests were carried out in head and bow quartering waves since it is considered poor seamanship to be in a following sea condition in an extreme sea state. The whipping loads in these conditions (due to bow flare impacts) \ \ |+wave 0 w hipping \B Speed [kts] 6 Figure 30 Vertical bending moment (mean of 1/3 highest peaks) as a function of ship speed. Following waves condition. CONCLUSIONS The objective of this paper was to introduce a method to reliably measure the excitation of a ship hull due to aft body slamming impacts. The measured impact force has been derived from a large array of pressure gauges. Tests with two models with varying stiffness indicated that the pressure due to the
whipping response of the model could be superimposed on the impact pressure. This means that there is hydro-structural interaction, but no important hydro-elastic effects. The impact force has been used in a simple structural schematization of the segmented model that was used in the towing tank. The predictions with this whipping model compared very well to the measured vertical bending moment. It is therefore concluded that the impact force could be applied to a structural model of the ship to determine the whipping moment and peak stresses for the full scale vessel. Further conclusions are: · A vessel having a modern stern shape lying at zero or low speeds in following wave conditions can experience heavy stern slamming. This slamming starts occurring in mild conditions if the wave length is in the order of the ship length. Stern slamming can be considered as an impact that starts very locally and expands first lengthwise and then sideways over the stern. The high pressure ridge passes a single point in the stern in about 0.05 s; due to the size of the stern the total impact duration is about 0.5 s. Instrumenting the aft body of the vessel with a large array of pressure gauges allows an accurate measurement of the impulsive force. The measured impulsive force can be used on a dynamic structural model of the actual ship to predict the whipping loads. ACKNOWLEDGEMENT The authors acknowledge the permission of Northrop Grumman to publish the results in this paper. REFERENCES Bereznitski A. and Kaminski M.L., 2002, "Practical implications of hydroelasticity in ship design", Proceedings Int. Offshore and Polar Engineering Conference, Kyushu. Betts C.V, Bishop R.E.D. and Price W.G., 1977, "A survey of hull damping", Transactions RINA. Vol. 119, pp. 125-142. Bishop R.E.D and Price W.G., 1979, "Hydro- elasticity of ships", Cambridge University Press. Hamalainen R. and Heerd J. van, 1998, "Hydro- dynamic development for a large fast monohull passenger vessel", SNAME Annual meeting. Haugen E.M, Faltinsen O.M. and Aarsnes J.V., 1999, "Application of theoretical and experimental studies of wave impact to wet deck slamming", Proceedings FAST-97 Conference, Sydney. Faltinsen O.M., 1996, "Slamming", Colloquium for Ship and Offshore Hydrodynamics, Hamburg. Faltinsen O.M, 1999, "Water entry of a wedge by hydroelastic orthotropic plate theory". Journal of Ship Research, Vol. 43, No 2, pp. 180-193. Faltinsen O.M., 2000, "Hydroelastic slamming", Journal of Marine Science and Technology, Vol. 5, No. 2, pp. 49-65. Kurimo R., 1998, "Sea trial experience of the first passenger cruiser with podded propulsion", Proceedings of the 7TH Int. SYmposium on Practical Design of Ships and Mobile units (PRADS), The Hague. Kvalsvold J. and Faltinsen O.M, "Slamming loads on wet decks of multihull vessels", Proceedings HYdroelasticitY in Marine Technology Conference, Trondheim, 1994.
