A theoretical study of the capsize of the ferry HERALD OF FREE ENTERPRISE

A THEORETICAL STUDY OF THE CAPSIZE OF THE FERRY

"HERALD OF FREE ENTERPRISE"

Jianbo Hua

Div. of Naval Architecture, Dept. of Vehicle Engineering Royal Institute ofTechnology, S-100 44 Stockholm, Sweden

Int. Shipbuild. Progr., 43, no. 435 (1996) pp. 209-235

Received: December 1994 Accepted: June 1995

A simplified mathematical model has been established for time-domain simulation of capsize scenarios for RoRo-vessels such as "Herald of Free Enterprise". The capsize can be studied as a consequence of the interaction between heeling and turning motion. The influence of different parameters has been studied. The result shows that quantity of water on G deck before heeling, ingress rate ofwater, ship speed, hull form and KG-value are the main parameters governing the capsize scenario.

1. Background

On Friday 6th March 1987 the RoRo passenger/vehicle ferry "Herald of Free Enterprise" capsized outside the breakwater of Zeebrugge on her route to Dover with heavy loss of human life. The capsize took place in less than one minute. The ferry finally grounded about 800 m starboard from the route with the bow heading nearly by 180 degrees o f f course.

"Herald of Free Enterprise" had a service speed of 22 knots. The rriain particulars can be found in Tabel. 1. Figure 1 shows the general arrangement. The cargo in terms of cars and trailers rolled on and o f f through the openings in the bow {BIH 6.0 ml A.9 m) and stem {BIH 8.5/4.73). Vehicles were carried on D , E, F and G deck.

When departing from the berth, the ferry was trimmed by head and had the bow doors left open. That made it possible for water to ingress to G deck as the ship's speed increased. According to the witness there was water on G deck before heeling started, the heel angle first stayed for five, six seconds at about 20-25 degrees and thereafter the ship continued to capsize. In the meantime, the vessel was turning to starboard and could not be stopped in spite of hard port helm. Water flooded into the super-structure quickly as the vessel went over on her beam.

Technical investigations have been carried out including time-domain simulations of different mathematical models, tank model experiments and full-scale measurements. Even i f much valuable information has been gained a complete picture is still lacking. Among the questions are the following:

1. How large was the quantity of water that ingressed to G deck before start o f heeling and how large was the rate of ingress?

2. What is the low limit of quantity of water on G deck required for capsize?

3. The ferry was turning to starboard while heeling to port and come to rest on a heading very nearly opposite to that of her original course. How large was the maximum turning rate and how large was the effect of the interaction between tuming and heeling on the capsize?

4. How fast was the capsize?

5. How did the ship reach its final position?

Usually, when a ship starts tuming to starboard she heels first to starboard due to the rudder force. Then after a few moments she heels to port. This is because the lateral inertia forces, the mdder force and the lateral hydrodynamic force act at the different levels so that a heeling moment can be induced, which we can call heeling moment of tuming. Experimental measurements and calculation have been shown that the heel angle has a strong influence on the turning radius, see [ 1 ] . The turning radius decreases as the heel angle increases. Sequenntly the heel angle grows due to the increased heeling moment of turning. This effect is apparent f o r ships o f the r o l l -on/roll-off type. Large beam to draft ratio and high vertical position o f mass centre are considered to be the major characteristics o f the hull which make the ship's tuming moment sensitive to this phenomenon.

"Herald o f free Enterprise" had a GZ-curve increasing linearly with G M value o f 1.7 m up to about 30 degrees heel angle and then decreasing down to zero at 57 degrees heel angle. The transverse stability was sufficient i n normal cases and the phenomenon mentioned above was d i f f i c u l t to observe. However, as water had inflooded into G deck the transverse stability would decrease more or less depending on the quantity of water on G deck. When the ship was in upright position G M was reduced due to the free surface effect of the water on G deck. In heeled condition the water on G deck acted as a heehng moment which is a function of heel angle.

The mathematical model i n [1] where the coupling between heel angle and turning motion is taken into account, has shown good agreement with experimental measure-ment i n spite o f its simplicity. I n this paper the turning motion equation has been extended by including the effect of water on G deck. The capsize can then be studied as a consequence o f the interaction between heeling and tuming. The purpose was to find possible circumstance which could have caused the capsize of "Herald o f Free Enterprise", But also to discuss the significance of parameters in the motion equations which are related to the water depth, the hull form and the loading condition.

J. Hua 211 'dD (ID\ iOD QD QD'tn) QD QD I ^ ^ . . ^ d l nm A DECK B DECK C DECK

"Herald of Free Enterprise' Figure 1. General arrangement.

Figure 2. Coordinate system. wreck poj i t i o n \

\ ^

® \

A

J.Hua 213

2. Mathematical Model

The mathematical model comprises four differential equations of motion; surge, sway, roll and yaw motion. Heave and pitch motion due to increasing water quantity are considered here to have less effect on the capsize and w i l l not be coupled into the mathematical model. The degree of freedom of motions is illustrated i n Figure 2. (the mass centre is supposed to be located amidships)

m{ü-vr] = Xh + X{u) + Xr (1)

m{v+u-r]=Yh+Yr (2)

I^.r = Nh + Nr (3)

I^^^Kh + Kr + K,, (4)

The terms with subscript h are the hydrodynamic forces acting on the ship hull. The terms with subscript r are the rudder forces. is the heeling moment due to the water on G deck.

The longitudinal hydrodynamic force is as followed,

Xh=-Mx-u-\-Myv r (5)

where is the added mass in x-direction. My the added mass in }'-direction.

The propeller thrust together with the resistance are expressed approximately as follows,

X{u) =

{-p-Ld-V^-Xm-where XhQ is the resistance coefficient at the initial ship speed Vq. The lateral hydrodynamic force and moment are,

Yh = -Myv - Mx-u-r- My Xa-r + Yh({py) (7)

Nh = -Jzz-f - Myxa-v + Nh({py] + Nhi(f3y,(t>) (8)

where J„ is the added moment of inertia, Xa the coordinate of the centre of lateral [Vo/Vf

added mass, Yho, Nho and Nhi are the lateral lifting forces as functions of / and (p. / = r-UV is the nondimensional tuming rate.

Yho = ^-p-L-d-V' 7 > / ? + YW+ Y'pp-(i\p\ + Y'iir'-P\r'\+Y\y r''\r' (9)

Nm^^-p-L'-d-V' y2

Nhi=i-p-L^-d-V^

(10)

(11)

The hydrodynamic derivatives in (5), (6), (7), (8), (9), (10) and (11) are shown i n Tab.2. They are obtained by using formulas and statistics from model measurements as functions of main hull particulars according to [3] and [4].

The heeling moment consists of hydrodynamic inertia moment, friction damping moment, rudder moment, moment o f the hydrodynamic force and heeling moment due to the water on G deck.

Kh = -Jxx-ip - B44-^+ Yh-Zh (12)

The last term in (12) plus the rudder moment is called heeling moment of tuming. is taken equal to 15% of the critical damping [ 1 ] . The rudder moment is obtained according to [5].

Table 1. Main Hull Particulars.

L 126.1 m B 22.7 m d 5.7 m Cb 0.525 LIB 5.56 Bid 3.98 GM 1.7 m KG 9.73 m Trim -0.8 m

/. Hua

Table 2. Hydrodynamic Derivatives and Other Coefficients

215 MJm 0.035 My/m 0.73 JzAz 0-77 Rr 0.4B Y'p 0.274 N',, 0.09 FV 0.071 /VV -0.0407 F'p/p 0.61 A ^ V -0-133 Y',Y -0.05 A^V;/ 0.019 /i:^^ 0.03/deg K,^ -0.0425/deg

K'^ -0.0076 Z,,o 0.023

r „ 5.00 m

and

The magnitude of the heeling moment of tuming is equal to the lateral hydrodynamic force times Zh, the vertical coordinate, at which the hydrodynamic force acts when the rudder force is not included.

For a ship with a large beam to draught ratio, the effective draught dg (definition see Figure 2) increases significantly as the heel angle grows. In tum, a greater magnitude of heeling moment o f tuming w i l l be induced as a result in increase of Z/,. The model measurements [1] show that Hh can be greater than the effective draught dg. Hh has simply to be considered as a fictions vertical coordinate f r o m the still water surface, on which the lateral force Yh is applied. I n other words, the heeling moment of tuming moment consists of two components, one caused by the lateral hydrodynamic force Yh, the other by the vertical resultant of the hydrodynamic pressure on the submerged hull, which does not act on the mass centre vertically.

By comparing the two ro-ro ships in [1], one w i l l find that the ratio of the beam to draught has a dominant influence on the mean value Hhldg. The mean value of Hhlde for "Herald of Free Enterprise" is estimated to be about 0.81 f r o m interpolation with respect to beam to draught ratio. A n approximate expression for Zh as a function of the heel angle is derived as follows.

Zh = OG cos (p + f ^ - s i n (j) + d\ cos (p •

J d,

(14)

Here OG is the distance f r o m the mass centre to the water line.

For calculation o f the heeling moment of water on G deck, it is assumed that the water surface stays horizontal and flat in spite of the ship's motion in any degree of freedom, a quasi-static state is assumed and the viscosity of water is negligible. The results o f calculations show that this is a reasonable assumption regarding the rate of

heel which is generally less than 3 deg/sec in all time-domain simulations. The water volume can be assumed to have the form of a tetrahedron changing only with trim and heel angle. The magnitude of the heeling moment can then be determined approxi-mately by the following formula,

w B - 0.454. Kv-tan a • 1 - tan^^j - Gg-tm (p (15)

Here is the water volume on G deck as a function of time, a is trim angle and Gg is the distance f r o m the volume centre of the water to G deck.

The heeling moment according to (15) is overestimated to some extent, particularly when the trim is large. A conservative estimate is to assume the water volume to have the f o r m of a hexahedron. The heeling moment is then.

K^ = p-g-V, _w # - 0.529 Kvtan a

tan^0 1 - tan^^ - Gg-tan 0

(16)

The discrepancy between (15) and (16) decreases as the heel angle becomes larger. (16) is applied in the computation.

The equations of motion are solved by a time-domain simulation procedure using numerical method called (1,1) Padé approximant, [6]. The main advantage of this numerical technique is that the procedure is mathematically stable and computation efficient, and produces accurate and convergent solutions.

3. Result of Time-domain Simulation

The initial conditions f o r the time-domain simulation o f the mathematical model consist of f o l l o w i n g parameters; ship speed, d r i f t angle, heel angle, turning rate, amount of water on G deck before heeling starts and the rate of water ingress. Initial conditions with different combinations of these parameters may lead to capsize. The fact that the heeling motion of the vessel made a pause at about 20-25 degrees heel is used as a criterion when evaluating the computation results. Another fact is that the vessel had tumed to starboard during the capsize, but how much is unknown. Each run was stopped when the heel angle had exceeded 50 degrees. Further computation did not seen justified, considering the uncertainties in the computations at very large heel angle.

/. Hua 217

Ingress of water to G deck took place when the bow wave height exceeded the actual free board. The bow wave height would decrease as the ship slowed down during tuming. In the computations, it was assumed that the rate of water ingress into G deck remained constant until the ship speed was down to 90 percent of the speed when the capsize started and more water would not ingress.

Tidal current was present i n a direction f r o m portside during the accident. That could cause the tuming motion to starboard. In the computation an initial drift angle was used to represent the effect of the tidal current. Then, the speed o f current can be expressed as a drift angle times the ship's speed, e.g. a drift angle o f 4.8 degrees to starboard is equivalent to a 1.5 knots current f r o m portside i f the ship's speed is 18 knots. I t is also possible that a sudden heehng motion occurred and initiated the capsize.

So as to visualise the magnitude of the heeling moment, a heel angle is assumed to represent the heeling moments through following transform.

This means that the vessel w i l l heel K' degrees due to the heeling moment K at static equiHbrium. The resultant moment is defined as the sum of the heeling moments due to water on G deck, tuming and restoring moment.

Ifi Run 1 the tuming motion was not taken into consideration. 600 tonnes of water had totally accumulated on G deck at a rate of 30 tonnes/sec during the first 20 seconds. The vessel finally reached a heel angle of about 23 degrees. Obviously, the heeling moment of water was in balance with the restoring moment when the dynamic effect had been damped out. The vessel could carry about 750 tonnes of water on G deck i n static equihbrium without capsize. A rate of ingress of water less than 15 tonnes/sec would give a negligible roll acceleration i f the vessel heels simultaneously.

In Run 2 the heel motion is coupled with the tuming motion. A pause i n a rapid heeling appeared at 22 degrees heel for about 5 seconds and thereafter the vessel tumed over. The heeling moment diagram shows that the heeling moment o f water on G deck increased rapidly i n the first 15 seconds and the heeling moment due to tuming increased during the pause. The two heeling moments together overcame the restoring moment and forced the hull to tum over.

Run 3 and Run 4 have the same initial conditions as Run 2, but f o r ship's speed, 20 knots for Run 3 resp. 16 knots for Run 4. Run 3 shows a quicker capsize than Run 2 and Run 4 was finished without capsize.

Run 5 was made i n order to show a capsize which started with a sudden heel and continued due to the ingress of water at a rate of 15 tonnes/sec to G deck.

The shallow water effect was evaluated in Run 6 by increasing the hydrodynamic derivatives Yp and N'p with 20%. I t corresponds to a water depth about twice the ship

draught. The result shows a greater heeling moment of turning in comparison w i t h Run 2, and a heel pause at a higher heel angle.

In Run 7 the radius of moment of inertia is reduced f r o m 0.24L to 0.22L. The initial drift angle is 4 degrees.

In Run 8 to Run 11 various combinations of / / / / 4 ( o f 1.0, 1.3, 1.5 resp. 1.8), initial amount of water on G deck and ingress rate had been taken into consideration. The heeling moment of turning increases with increasing Hhldg. The heel pause angle depends on the combination initial amount of water, ingress rate and Hhlde for certain ship speed. I n Run 12 the shallow water effect which corresponds to a water depth of about twice the draught was evaluated.

4. Discussion

1) Quantity ofwater on G deck and its ingress rate

The vessel was trimmed by bow all the time. The quantity o f water flooding onto G deck did not cause the loss o f the transverse stability immediately because water did not cover the whole deck area. Rapid heeling would take place only when the heel angle exceed a critical value. In that case the increase of the heeling moment is faster than that of the restoring moment.

A f t e r having passed the out breakwater it took about 120 seconds f o r the vessel to accelerate up to 18 knots and the vessel had proceeded about 570 meter. I t would have needed at least 80 seconds to reach to the wreck position with a speed of 18 knots, Figure 3. When water was found to be running down the stairway thé' quantity of water on G deck could have been 60-100 tonnes. The phone to the information office by M r . Butler can be estimated to be 30 seconds or less. So the maximal water ingress rate was likely not more than about 10 ton/sec at this stage.

2} Turning rate

In no runs did the tuming rate exceed 4.5 deg/sec. The vessel could not have tumed as much as 180 degrees before being heeled on to her beam. I t could happen that the vessel was o f f course before capsize started and tumed a lot after having heeled on to her beam due to the residual tuming momentum.

3} Hull form and the hydrodynamic derivatives

In [7], the hydrodynamic coupling between different degrees of freedom of motion is discussed f o r a trawler i n heeled condition. I t is concluded that degree of coupling exisdng between the hydrodynamic coefficients associated with the symmetric and

J. Hua 219

anti symmetric simultaneous motions is closely related to the non-symmetry of the underwater hull form and not necessary to the angle of heel. In the capsize model the coupling effect on the added masses and damping coefficients has not been taken into consideration and an empirical correction based on the measurement was made for the coupling effect of heel on N'b and N'r.

GZ-curve and Z/,-variation due to heel are two of the most important parameters which govern the behaviour of the ship during capsize. Both parameters are closely related to the hull form.

4) Ship speed

The hydrodynamic forces vary linearly with the square of the ship speed. Therefore a considerable increase in the heeling moment of tuming w i l l follow as the ship speed becomes higher.

5) Shallow water ejfect

The effect of water depth upon the hydrodynamic derivatives o f a ship is dependent upon the block coefficient. A fine f o r m hull (with low block coefficient) is usually less sensitive to water depth. Experimental investigation of a ship model with block coefficient 0.54, see [11] has shown that only Yp and N'p increase considerably as the water becomes shallower (squat effect is included). Computations with correction to the shallow water effect show clearly a considerably increase in the heeling moment of tuming. The trim by bow w i l l increases due to the squat effect, which i n tum w i l l result i n a higher ingress rate of water on G deck.

6) Rudder

The ship had hard port helm but turned to starboard when the heel started. A simulation with 35 degrees rudder to port has shown that the capsize could have been avoided i f the quantity of water on G deck was less than 200 toniies and the ingress rate about 8 tonnes/sec. I f the capsize occurred in spite o f the port helm, the tuming moment of the hydrodynamic force due to the current, and later due to the course instability had to be stronger than the tuming moment f r o m the rudder.

The vessel had two rudders, one forward in locked position and the other aft. I n addition the vessel's profile had large cut-outs for and aft. Unfortunately the effect of the forward rudder and the large cut-outs could not to taken into consideration, when determining the hydrodynamic derivatives. In fact, that is not trivial with regard to the course instability.

7) Trim

Experimental measurements and theoretical calculations are in good agreement conceming the linear variation of some of the hydrodynamic derivatives with trim, [4]. A 0.8 meter trim by the head results i n a change by no more than 10% for the actual hydrodynamic coefficients.

8) Centre ofthe lateral added mass

The computations show that the position of the longitudinal centre of the lateral added mass has some effect on course stability at the initial stage, when a ship turns to starboard, an acceleration componentv to the port side direction w i l l result mainly due to the centrifugal force. According to (8), a position forward of the mass centre can give a tuming moment which w i l l force the ship to tum further to starboard. I t can be looked upon as an unstable mechanism. By strip calculation, it has been found that the centre of lateral added mass is about 4.5 meter forward o f the mass centre. I f the squat effect due to shallow water is accounted for the centre of the added mass comes further forward.

9) Moment of inertia and added moment of inertia

On could expect that a smaller moment of inertia or smaller added moment o f inertia would lead to rapid increase in tuming rate in a unstable condition during turning motion. Computations with various radii of moment of inertia f r o m 20% to 28% o f the ship length do not indicate any significant effect on the capsize.

10} Where the capsize started

The final question is where the capsize started. There are two alternatives to discuss because clear evidence is lacking. One is that the capsize started in the cannel and the vessel floated to the wreck position by the residual speed and the tidal current. The other is that the capsize started about 200/300 m f r o m the wreck position. The foreship after having capsized hit the sea bed and the hull could continue to tum a lot around the foreship by the residual turning momentum (the momentum of transverse motion had transformed into tuming momentum).

The wreck position was about 800 m from the channel. The computation shows that the vessel had moved a maximum of about 200 m to starboard before heeling to her beam ends. We know that the water depth was about 16.2 m in the channel and about 12.2 m at the wreck position. The displacement of the vessel was about 8600 tonnes plus 200 to 400 tonnes water which had flooded into G deck. Another fact is that

J.Hua 221

water gushed into the superstructure after the vessel had capsized. This was possible since the windows on the port side burst due to high water pressure.

It is difficult to explain how the vessel could have turned as much as 80-100 degrees in such water depth by the residual tuming while at the same time it floated about 600 m after having capsized. I t is estimated to be about 5-10 minutes depending on the direction of the residual speed for the vessel to be moved to the wreck position by the current and the residual speed. The floating capability of the vessel seems hardly sufficient taking into account that the water was gushing into the super structure all the time.

I f the capsize started in shallow water it means that the vessel was o f f course about 20 degrees. I t is difficult to judge i f that was possible because there is no evidence to support this. But on the other side, circumstances f o r the capsize would be more available due to the shallow water effect.

5. Conclusion

The computations have shown that the governing parameters of the capsize are the quantity of water on G deck before heeling started, the ingress rate, the ship speed, the hull f o r m and the vertical position of mass centre. The effect of the interaction between heeling and tuming is obviously decisive for the capsize.

The quantity o f water on G deck before heeling started can vary f r o m 150 to 250 tonnes and the ingress rate f r o m 3 to 8 tonnes/sec, depending on the ship speed (between 16 and 20 knots) and the hull form in terms of Hhldg (between 0.8 and 1.8). The tuming rate could be up to about 4 deg/sec according to the computations. Considering that the capsize occurred within about 30-50 seconds, this tuming rate does not seem sufficient to explain the total tum angle. The vessel must also tumed a lot after the capsize.

One should emphasise that this is a theoretical study based on some assumptions. Particularly, i t has to be pointed out that the knowledge about the hydrodynamic behaviour o f a ship with large beam to draught ratio in heeled condition is not sufficient.

Acknowledgement

This work is inspired by the view o f Professor Erik Steneroth on the capsize o f H.O.F.E, who also has given encouragement and support, for what I am grateful. The financial support f r o m the Swedish Shipbuilders Association and the National Swedish Board for Technical Development is gratefully acknowledged.

I n i t i a l c o n d i t i o n ; 0 . 0 5 5 5 . 1 S h i p s p e e d _{S h i p h a s t u r n e d} T u r n i n g r a t e D r i f t a n g l e Water amount I n g r e s s r a t e 1 8 . 0 k n o t s 0 . 0 deg 0 . 0 0 d e g / s e c 0 . 0 deg 0 . 0 t o n n e s 3 0 . 0 t o n n e s / s e c F i n a l c o n d i t i o n s RUN. l A S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e Water amount l B . 0 k n o t s 0 . 0 deg 0 . 0 0 d e g / ' s e c 0 . 0 deg 6 0 0 . 0 t o n n e s

J. Hua I n f 1 1 a l c o n d I t i o n ; S h i p s p e e d 1 8 . 0 k n o t s S h i p h a s t u r n e d 0 . 0 dog T u r n i n g r a t e 0 . 0 0 d e g / s e c • r i f t a n g l e 4 . 8 deg j ^ B 6 . 6 3 2 5 . 0 Water amount I n g r e s s r a t e 2 0 0 . 0 8 . 0 t o n n e s t o n n e s / s e c ( F i n a l c o n d i t i o n ; S h i p s p e e d S h i p h a s t u r n e d 11. 1 1 2 9 . 3 k n o t s deg RUN. 2A T u r n i n g r a t e _{D r i f t a n g l e} 3 . 5 3 _{1 8 . 0} d e g / s e c deg Water amount 4 1 6 . 0 t o n n e s RUN. 2B C u n 1 t : s o c )

2 0 . 0 k n o t s 0 . 0 deg 0 . 0 0 d e g / s e c 4 . 8 deg 2 0 0 . 0 t o n n e s 8 . 0 t o n n e s / s e c 1 5 . E k n o t s G 4 . 4 deg 4 . 0 5 d e g / s e c 1 5 . 4 deg 3 8 4 . 0 t o n n e s RUN. 3B da g de g O •!• of of m a t e r t u r n i ng 1 4Q 1 40 ii o f r e s u 1 t a n t O .75 > \ > ÜJ 30. lli X J G MOMEN -30 + . 5 0 20 lEELI h _.20 .25 10 1 1 1 1 10 A f p 1 1 0 20 40 G0 0 ' • 20 40 S0 ( u n i t : s o c ) I n i t i a l c o n d i t i o n : S h t p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e W a t e r amount 2 2 3 2 . 6 I n g r e s s r a t e RUN. 3A F i n a l c o n d i t i o n ! S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e W a t e r amount

J. Hua I n i t i a l c o n d i t i o n : 2 2 3 . 0 3 2 9 . 9 RUN. 4A S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e Water amount I n g r e s s r a t e 1 6 . 0 k n o t s 0 . 0 deg 0 . 0 0 d e g / s e c 4 . 8 deg 2 0 0 . 0 t o n n e s 8 . 0 t o n n e s / s e c F i n a l c o n d i t i o n : S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e Water amount 1 0 . 7 k n o t s 1 2 9 . 2 deg 2 . 3 4 d e g / s e c 1 4 . 3 deg 4 4 8 . 0 t o n n e s RUN. 4B de g Bs p ₀ + of m a t e r o f t u r n i n g 1 .- 4e 40

_-

I n 1 1 1 a l c o n d 1 1 < o n ; S h i p s p e e d 1 8 . 0 k n o t s S h i p h a s t u r n e d 0 . 0 deg T u r n i n g r a t e 0 . 0 0 d e g / s a c yO^Z-'^ 4 2 8 . 8 D r i f t a n g l e 0 . 0 dog l^ater amount 0 . 0 t o n n e s * I n g r e s s r a t e 1 5 . 0 t o n n e s / s o c F i n a l c o n d 1 1 i o n : S h i p s p e e d 1 6 . 4 k n o t s S h i p h a s t u r n e d 3 5 . 0 dog T u r n i n g r a t e 3 . 7 1 d o g / s o c RUN. 5A D r i f t a n g l e 1 2 . 8 dog N a t e r amount 6 9 0 . 0 t o n n e s RUN. 5B Ol TJ 1 - 40 0 . 7 5 - ^ 30 + .5 - 0 20 . 2 5 - 10 ( 1 1 i 1 ^ |_ 40 / ^ / r / O 30 / £ / ^ / M 20 / _) hi / ^ 1 0 0 o f w a t e r + o f t u r n i n g M o f r e s u l t a n t —-^"-i 1 1 0 20 40 50 0' 20 40 50 C u n i t : s e c )

/. Hua I n i t i a l c o n d i t i o n : , 1 0 2 . 7 2 9 7 . 9 RUN. 6A S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e W a t e r amount I n g r e s s r a t e F i n a l c o n d i t i o n : S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e W a t e r amount 1 8 . 0 k n o t s 0 . 0 deg 0 . 0 0 d e g / s e c 4 . 8 deg 2 0 0 . 0 t o n n e s 8 . 0 t o n n e s / s e c 1 2 . 5 k n o t s S l . B deg 3 . 8 7 d e g / s e c 1 6 . 7 deg 3 9 2 . 0 t o n n e s

I n i t i a l c o n d i t i o n ! 0 5 , 2 3 2 9 . 9 RUN. 7A S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e M a t e r amount I n g r e s s r a t e F i n a l c o n d i t i o n : S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e Water amount 1 8 . 0 k n o t s 0 . 0 deg 0 . 0 0 d e g / s e c 4 . 0 deg 2 0 0 . 0 t o n n e s 8 . 0 t o n n e s / s e c 1 2 . 9 k n o t s 8 7 . 1 deg 3 . 6 1 d e g / s e c 1 6 . 3 deg 4 2 4 . 0 t o n n e s

/. Hua 229 I n i t i a l c o n d i t i o n s 6 . 2 2 8 8 . 9 RUN. 8A S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e M a t e r amount I n g r e s s r a t e F i n a l c o n d i t i o n s S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e M a t e r amount 1 8 . 0 k n o t s 0 . 0 deg 0 . 0 0 d e g / s e c 4 . 8 deg 2 0 0 . 0 t o n n e s 8 . 0 t o n n e s / s e c 1 4 . 2 k n o t s 6 3 . 3 deg 3 . 6 4 d e g / s e c 15.1 deg 4 0 8 . 0 t o n n e s RUN. 8B 0 za 40 60 0 " 20 40 60 C u n i t : s e c )

I n i t i a l c o n d i t i o n : 1-.8 2 8 2 . 7 RUN.9A S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e Water amount I n g r e s s r a t e F i n a l c o n d i t i o n : S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e Water amount I B . 0 k n o t s Q . 0 deg 0 . 0 0 d e g / s e c 4 . 8 deg 2 0 0 . 0 t o n n e s 5 . 0 t o n n e s / s e c 1 4 . 3 k n o t s 6 1 . 2 deg 3 . 6 8 d e g / s e c 1 5 . 1 deg 3 2 5 . 0 t o n n e s RUN. 9B ( u n i t : s e c )

/. Hua 231 I n i t i a l c o n d I t i o n : 7 4 B . 5 2 7 7 . G S h t p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e M a t e r amount I n g r e s s r a t e I B . 0 k n o t s 0 . 0 deg 0 . 0 0 d e g / s e c 4 . 8 deg 1 8 0 . 0 t o n n e s 4 . 0 t o n n e s / s e c F i n a l c o n d i t i o n : R U N . l O A S h i p s p e e d S h i p h a s t u r n e d T u r n i n g r a t e D r i f t a n g l e W a t e r amount 1 4 . G k n o t s 5 7 . 8 deg 3 . 6 7 d e g / s e c 1 4 . 8 deg 2 8 4 . 0 t o n n e s R U N . I O B ( u n i t : s e c )

I n i t i a l c o n d i t i o n s S h i p s p e e d 1 8 . 0 k n o t s S h i p h a s t u r n e d 0 . 0 deg T u r n i n g r a t e 0 . 0 0 d e g / s e c D r i f t a n g l e 4 . 8 deg W a t e r amount 1 5 0 . 0 t o n n e s > ; 7 4 1 . 5 2 7 1 . 9 I n g r e s s r a t e 3 . 0 t o n n e s / s e c F i n a l c o n d i t i o n ; S h i p s p e e d 1 4 . 7 k n o t s S h i p h a s t u r n e d 5 5 . 1 deg T u r n i n g r a t e 3 . 8 0 d e g / s e c R U N . I I A _{D r i f t a n g l e 1 4 . 9 deg} W a t e r amount 2 2 9 . 0 t o n n e s R U N . I I B O l 01 T3 1 - 40 0 . 7 5 - 30 + .5 - O 20 . 2 5 - 10 / ^ 1 u 0 ' ° / ^ U / O 30 / s P O y M 20 / _ j - 1 ^ 1 0 y 1 o of uiater + of t u r n i n g * of r e s u l t a n t ^1 •>! ' I ' I I 1 1 0 20 40 60 0 20 40 60 ( u n i t : s e c )

J. Hua I n 11 i a l cond111 o n : S h i p s p e e d 1 8 . 0 k n o t s S h i p h a s t u r n e d 0 . 0 deg T u r n i n g r a t e 0 . 00 d e g / s e c D r i f t ang1e 4 . 8 deg M a t e r amount 1 8 0 . 0 t o n n e s t ^ ^ 4 5 . 9 2 5 8 . 9 I n g r e s s r a t e 5 . 0 t o n n e s / s e c \

r

Nomemclature

B = breath of ship

d = draught of ship

de •• = maximal draught at heeled condition

GZ{(j)) : = restoring moment arm as function of heel angle

Hh = vertical distance between calm water surface and point upon which lateral

force FHULL acts

= moment of inertia of ship with aspect to x and z-axis

Jxx> Jzz ' = added moment of inertia of ship with aspect to x- and z-axis

K = heel moment

KG = vertical mass centre above the keel

= ship length m -= mass of ship

mx, my -= added mass in x- and y-axis direction respectively

N = yaw moment

OG = distance from the mass centre to the water line

R = tuming radius

r -= tuming rate /

r' = non-dimensional tuming rate (= rUV = LIR)

u -= ship speed in x-axis direction V = ship speed (= (u^ -t- v^)^''^)

= accumulated water volume on G deck

V : = ship speed in y-axis direction

X = force in x-axis direction

Y = force in y-axis direction

Zh = z-corrdinate of the point upon which lateral force I R U L L acts

a = = trim angle

P = drift angle

0

P = water density

Reference

[1] M . Hirano and J. Takashina, " A calculation of Ship Turning Motion Taking Couphng Effect due to Heel into Consideration", Trans, o f the West-Japan Society o f Naval Architects No. 59, 1980.

[2] A . Ogawa and H . Kasai, "On the Mathematical Model of Manoeuvring Motion o f Ship", International Shipbuilding Progress V o l . 25, No. 292, 1978.

J.Hua 235

[3] S. Motora, "On the Measurement o f Added Water and Added Moment o f Inertia for Ship Motion", Journal ofthe Society of Naval Architects of Japan, part 1, Vol.105, 1959, and part 2, V o l . 106, 1960.

[4] S. Inoue, M . Hirano and K. K i j i m a , "Hydrodynamic Derivatives on Ship Manoeuvring", Intemational Shipbuilding Progress Vol. 28, No. 321, 1981. [5] S. Inoue, M . Hirano, K . K i j i m a and J. Takashina, " A Practical Calculation

Method of Ship Manoeuvring Motion", International Shipbuilding Progress V o l . 28, No. 325, 1981.

[6] W .G. Price, E . H . Twizell and Y . Wang, "The Application of Padé Approxi-mant in the Time-Domain, Simulation of the Non-linear Motion of a Semi-Submersible Excited by Wave", PRAD's 87, the Proceeding of the Third Intemational Symposium on Practical Design of Ships and Mobil Units, 22-26 June 1987, Trondheim, Norway.

[7] C.A.L. Conceicao, W . G . Prize and P. Temrel, "The Influence of Heel on the Hydrodynamic Coefficients of Ship L i k e Section and a Trawler Form", International Shipbuilding Progress V o l . 31, No. 355 1984.

[8] R.K. Burcher, "Development in Ship Manoeuvrability", Trans. R I N A V o l . 114, pp. 1-32, 1974.

[9] A . D . Gill and W . G . Prize, "Determination of the Manoeuvring Derivatives o f a Ship Model Using a Horizontal Planar Motion Mechanism in a Circulation / Water Channel", Trans. R I N A Vol. 119, 1977.

[10] A . D . G i l l and W . G . Prize, "Experimental Evaluation o f the Effect o f Water Depth and Speed on the Manoeuvring Derivatives of Ship Models", Trans. R I N A V o l . 120, 1978.