Formulation of the force mathematical model of ship manoeuvring

F O R M U L A T I O N OF T H E F O R C E M A T H E M A T I C A L

M O D E L OF SHIP M A N ΠU V R I N G

M. Pourzanjani

School of Engineering. University of Exeter

Cabnamiuin voor

ScheepshydromechaniQa

Aretâsf

M8tcelw9g^2628CD Delft

1 Introduction. A full description of the original version of the Force mathematical model Is given by McCallum^-^ Since then, hovi/ever. the model has been refined and modified to overcome some of its shortcomings and also to accommodate simulation of new features, T h e basic structure of the model is in many respects similar to the mathematical models developed by Norrbin^^ and Oltman and S h a r m a ' ^ . in that they all adopt a direct approach by considering different effects separately, and use a combination of theoretical and experimental modelling techniques to describe the ship manoeuvres.

2 Systern^of_axes. T h e ship is considered as a rigid body with a moving system of axes fixed within the body as shovvn in figure 1. T h e centre of gravity. C G . is taken as the origin, and alt the arrows indicate the positive direction of motion in six degrees of freedom. A s in most mathematical models of ship manceuvrlng a calm sea and undisturbed water is assumed allowing only three degrees of freedom to be considered, i.e. surge, sway and yaw. Although roll, heave and pitch are ignored, the cross-coupling effects f r o m these motions on surge sway and yaw should be taken into account for a higher degree of accuracy, as discussed by M a t t h e w s ' - .

PITCH

Sl/PSE ROLL

HEAVE

.3 Equations of motioti. The Newtonian laws of rnotion are used to describe the motion of the ship when engaged in m a n œ u v r e s . A s s u m i t i g only three degrees of freedom the equations of nrioti(>n may be written as.

(w • vr - ^^r')m N .Y ( -r nr -t x,j /•" )m - S )

/'(/, -t mu'l) + mu-yii- ^ ur) V \ {})

A s s u m i n g the origin of tbe system of axis to be at the C G the above equations simplify to the following f o r m .

{il - vr)>n — N A" ( r T ur)ïif, = 1"

r]= -- V \ (2)

Terms "mnr and inrr " are theresuFt of having body centred axes T h e mass moment of inertia bf the ship is calciilated frorn the following relationship as suggested by Eda and Crane'

h - (' wa-*t.^ /,".

where <' 0.055 < \ i 0.02!t. (3)

T h e force required to accelerate and change the steady state conditions of â vessel is found to be greater than the product of the mass of the ship times the acceleration

I'ltrt-i niify.v • arr • ^ {A]

T h e extra force > is defined as a product bf the ship's acceleration and a quantity with the same dimensions as the mass, called the added ma^s' T h e térrn added mass is probably misleading since one rtitght imagine that this term implies that a body ts moving in a fluid and dragging witb it certain amount of fluid mass, hence termed added mass. T h i s is not physically correct and a better term fbr the added mass is the coefficient of accderation. T h i s coefficient is dependent on tbe properties of the fluid medium, body geometry and fluid cbnfineitients. M c C a l l u m ^ and N o m o t o ' e s t i m a t e d that the coefficients of acceleration would have \ralues of 10 per çerit of the ship's mass in the longifudinal direction and 100 per cent of the ship s mass in the transverse direction. Added moment of inertia was assumed to be 50 per cent of the actual moment of inertia. Equations 1 may then be written as follows.

imii ~ i'rnt2 N A'

A b k o w i t z ' used the potential flow theory to calculate the coefficients of acceleration for ellipsoids of different dimensions and suggested values of 5 to 10 per cent of mass for longitudinal aod 100 per

cent of mass for transverse directions for marine hull forms In the present work values suggested by M c C a l l u m ^ will be used.

T h e assumption of C G being the origin of the system of axes, reduces the number of terms involved ih the equations o f motion, hence making them easier to handle by digital computers. T h e disadvantage of this assumption is that C G . varies with the operating conditions and also most of the information about ships is related to their longitudinal c e n t r é line and not the C G

4 A n g l e of incidence. A l s o referred tb as the drift angle by some researchers, this is the angle between the direction of the main stream of the flow and the major axis of the body. It is an important parameter in the Force model, since hüll hydrodynamic forces and rudder forces are dependent on its accuracy tt is calculated using the relationship

o ^ t a n " ' ( M (6)

M ó s t digital cbmipùters which support a " t a j i " ' " function return a value between arrd ^ . and to covet the full range of four quadrants a routine must be included in the model to e x a m i n é the vector quantities of "n" and "r" and return an angle between 0 and 2 r T h e angle found in this manner is the incidence at the centre o.f gravity, and unless the ship is drifting with some incidence and zero yaw rate, the local fluid velocities and hence the Ibcal incidence angle along the length would vary, being smaller at the bow and greater at the stern when moving ahead and tiirning

M c C a U u m " assumed an effective drift angle for the rudder which was dejpendent On the manceuvrihg regime. T h i s was later modified by having only a multiplier to the drift angle to get the incidence angle at the rudder position. T h e coiistant multiplier according to M c C a l l u m ^ takes values between 1 2 and 2 0 for different ships

o, - M, n (7) T h i s assumption was o h é of the weaknesses noticed in the Force model T h e concepts o f effective

drift angie or a simple multiplier for "a, " are very poor representations of this variable since it is also dependent on the yaw rate and speed. A more accurate way of calculating "or" is devetoped in this section.

Figure 2 represents a ship engaged In a turning circle manoeuvre. It can be seen that the radius of t u m i n g circle changes along the length, being larger at the stern, which causes a larger drift angle.

T h e geometric incidence angle ä t the rudder "Ory " is calculated f r o m the geometry of figure 2 (flow angle at point B in the figure). The angular velocity is constant along the length but the local fluid velocities and radius bf turn vary If we assume that the rudder positipn is half the length abaft of the C G , and the difference between the incidence at C G and rudder is denoted by then.

O.C - OA + AC - ff + t : I s i n a V L

OC ^ (•-)-|- ( - ) s i n r t r 2

AB^L/S

Fig. 2 A vessel engaged in a taming circle manoeuvre

T h e equations 8 describe the geometric angie of incidence at the rudder position for different s p e ^ s and yaw rates These equations, however tend to overestimate the true angle of incidence at the rudder due to tbe fact that the hull and propeller distort the idealised flow as shown in figure 2. It is assumed that there is a hnear relationship between the geometric and true angle of incidence at the rudder:

a,- - n

where A', = 0.77 (from ref. 11) (9) T h e exact value of A',, for any particiilar ship may be determined f r o m the.open water characteristics of

the rudder and the force curves of the hull fitted with the rudder However, this information is usually not availabte and then value of A', - 0.77 will be used

Full-scale steady-state turning-circle data from four different ships have been used with equations 8 and 9 to calculate o , and results are presented in table 1

Table 1 a c a l c u l a t e d for four differeni ships

Ship Type CH R Ru(i„ ty Ref

V L C C 0 825 246 35:P 30 4 0 3 6 B Carrier 0 804 235 35S 29 40.3 18

Mariner 0.6 424 35P 14 2 4 0 13 Container 0,565 340 35P 12 21.5 2

Values of n.. for the two ships with higher block coefficients ( V L C C and B . Carrier) show that the flbvy angle at the rudder is greater than the rudder angle which indicates that the rudder has an advaise effect on the turn T h i s shows that the hull forces and moments dominate the turning characteristics and indicate the directional instability of these ships. T h i s may be the reason for ships built in more recent years to have a maximum rudder angle of as much as 50^ either side ä s opposed to 'ÄT-i" on most conventipnal ships

5 H ü l l h y d r o d y n a m i c forces. Hull hydrodynamic forces and moments are estimated using the potential and cross-flow lift and drag as discussed by Pourranjani et al

Tf/.tii CVilfid •\ \{C<i'2w\ ("Wl.Hilsin' 'u);

( ' I f , A'tt .4//Miii2oU(is/t ( 'I, ^^('li'Hi s i i i ' J n \ s i i K i Cit Cf„ •• Cl.. Cdf. A ' - .1/^ siit2o sii Cdf -.^ Cdp H CH^ ^ Cd„ Dray I, ]^.p L D Cd, 10

V/, = L ^ f f h cosrt -f- Dragi, s i n a (UV

T h e modulus sign has been used in son>e of the above equations in order to cater for different manoeuvring regimes and covering the four possible quadrants According to the sign convention adopted as shown Iri figurè 1 drag is always negative and lift has a sign opposite to the sway velocity.

T h e yawing moment a r m due to the hull hydrodynamic lateral force is assumed to follow the following relationship derived f r o m experimental results,

I, - 0.72a) - 0.;i5 < ,» 0.35 n.:i5. n ^ TT ~ o.;;5 TT - ü.;i.^ï < a <. TT ^ 0.35 /, ^ J ( 1 .^(i 0.72(2x - ,y)) w + 0.3.5 v a < 2T - 0.35 . \ . k > /, ₁₁₎

Hull forms have a tendency to resist the yawing motion. Ih Its simplest f o r m this resistance may be assumed to be proportional to the yaw rate squared.

.V,, - A , : , v.r.

where K{.t is assumed to be constant and its numerical value for any particular ship may be calculated f r o m the results of the steady-state turning circle trial which is ayailable for most ships. It is obvious that the values of K):; obtained in this manner will represent the steady state of the same trial ade-quately, but errors wijl be introduced during the transient periods or steady state of other r n a n œ u v r e s . The following equations which are derived f r o m theoretical reasoning by considering the moments due to the rotation and cross-flow effects of a flat plate are suggeisted (see appendix B ) ,

A , ; : >ifnr

ti l s /.•• 12 '

/ ' Distance between the pivot point and C G t fi s'ltm (J2)

where the pivot point is the point of zero incidence on the longitudinal centre line

6 Main engine dynamics. Marine engines in use may be grouped into two main categories i e. diesel and steam turbine, although other types of engines are also in service e g nuclear plants-Engine dynamics have an important effect on the manoeuvring behaviour of ships, and their inherent type a n d design characteristics (e g constant torque, constant R P M etc) should be taken into account during the modelling process

Merchant ships normally use a single type of engine (when more than one engine is installed they are of the same type and size). In naval ships a combination of different types of engines is usually used on the same drive (although not necessarily operating at the same time), allowing them to cruise at different speeds for long periods of time: this makes such ships more difficult to model.

Our goal here is not to study the cpmplex dynamic behaviour and response of marine engines but to examine their effects on ship m a n œ u v r i n g . Therefore, as far as we are concerned engines may be looked

upon as means of producing a f o r c é to overcome the ships resistance and making it move at a desired speed. T h e force just mentioned is a variable a n d depends on the operating condition ahd manceuvring regime. T h e main attraction of this type of modelling is that the terms dependent bn the prbpetter characteristics are avoided, herice simplifying the modelHhg and simütätlbn process. Equations for demand and actual throttles are discussed in Èhis section, and thrust equations are developed In the next section after discussing the fluid velocities

A linear differential equation of the first brdèr has been used in mpst models (e.g. McCallurh^. Bûrns^) to represent the engine response to a demand throttle setting.

f h , . - K,{Th,,- Th„) (13) where the demand throttle {Th,i) and the actual throttle take values between -1 and + 1 . representing

full astern to füll ahead conditions. Using equation 13 as a starting point, a more sophisticated model for throttle response of a slow spieed diesel engine is developed, which also takes into account the wind-milting and locking effects o f t h e propeller. T h i s is achieved by testing the s h i p s speed, throttle setting and throttle demand at each iteration Long delays are included where the speed is above a certain limit arKl large throttle change is demanded to simulate the effect of ttie ship s energy in turning the propeller T h e new model is based on the following arguments for a diesel engine:

1- When the difference between the demand throttle and the actual throttle is small (say less than 0.3) then it is expected that the engine can respond quickty and equation 13 applies. T h i s would apply equally at high ahd low s p é e d s as long as it does not Involve reversing the shaft rotation vvKen a delay should be included to stop, reverse and restart the engines

2- When the actual speed is high and a targe throttle demand is ordered (say f r o m full ahead to full astern) the e V ö i t s may be divided in to smaller' time intervals for b é t t e r Understanding of the situation T h e iriltial part of this m a n œ u v r e involves cutting off the fuel supply to the engine. T h i s would ensure that the engine ts not producing any positive thrust f r o m this point onwards. In order to reverse and restart the engine the shaft must be stopped first Various methods are used for this purpose such as allowing the shaft to coast and stop under the influence of the ship s resistance, using äir fbr braking or using shaft brakes T h é points to rernernber here are that under these conditions the engine tim^e constant would be higher than it would be for small throttle changes aod the ship s resistance would also tend l o increase by some amount depending on the mode selected to stop the shaft. After the shaft is stopped it niay be requiired to wait for the speed to drop below a certain limit until the engine can produce an astern propeller torque greater than the opposing propeller torque doe to the ship s motion. A very long delay is Included by choosing a very small time constant so that the shaft would remain at stop position until the speed has dropped sufficiently

3- There is a minimum value for R P M ^ t which the engine can operate which is taken as 20 per cent of the maximum throttle setting

4- Fbr turning m a n œ u v r e s where the actual and demand t h r o t t l é s are the same, as the drift angle and the yaw rate increase the loading of the propeller would also tend to increase. Depending on t h é engine controller installed, the R P M may drop by some amount for high throttle settings (fufl speed turn). A t

slower speeds the engines are expected tp be able to maintarn the R P M if the governor is designed to do so.

T h e following equations are formulated to cope with the first three of t h é abbye arguments. T h e last point may also be incorporated if required.

^ niirn< " I ^ / ^ n j i ( . 1 ' 1

If i ' i . „ r , „ 0.,(i and T b j > 0,() A ' , - 1 '(i.O

if r,„„,,„ n.fi and Th.f -. O.ü

77*,, • 0,<i • I 0(1 Th., • :- A , - 1'I OOOO

if /•„..,.„ - O.fiand Tli,i -0.(1 A , .- I (iO

if / „„,„,. 0.(1 and Tli,i > • 0.0

rh„ •-. - 0 . ( i A , 1 '30 Th,, >r--. 0.(j A , - 1/ 10000

If 0.0 l.„„.„ 0.0 A , 1,30 If f h„ {) and //;,, (( 10 seconds delay

r'h„ A , ( /•/',; fh.,] (14) Equations 14 cover the full possible range bf m a n œ u v r i n g regimes and throttle demands. They havé

been included in the rhpdel and tested in a real-time simulator by experienced sea pilots with extremely good results. This method can be applied in modelling other types of rnarine engines.

7 Fluid velocities. These have been discussed in detail by McCallum'' and only the important points will be mentioned here T h e equilibrium speed is defined as the speed that the ship wpuld eventually reach at that throttle setting and is denoted by T., " Speed departure is defined as the difference between the instantaneous speed and the equilibrium speed for that throttle setting and is d ^ o t e d by

y.=^x\,~u (15) It is assumed that the propetler slipstream velocity may be found from the following relationship .

r.,

Iy, ]\

(16)

TUe equations for the propetler thrust are developed by considering that the propeller must dellvere the thrust needed to maintain the equilibrium speed at that throttle setting and ari additional thrust If the speed differs f r o m the equilibrium speed

A p , A ' u and 7\'5 are used for the vessel.moving ahead. Due to the asymmetry of the bow and stern a different set bf coefficients (A'le. / i i » and A'g) must be used for the ship when rnbylng astern. T h e propefler is assumed to be less effective with the shaft rotating for the astern power:

T {h'u, 4- A',9)r'.,|?^„l •+ 7vV.rjv,,| jr/7„! « < o

/ ; - A , 7' T < 0 (18)

For a t w i n - s a e w ship the thrust is modelled in the same manner with the assumption that each engine delivers only half the required thrust t ö niiaintain the eqUiiibrium speed for that throttie setting.

A s the propeller Is an imperfect propulsive device, it alsb produces a transverse force at the propeller position, which is associated with a turning moment on the hull. T h i s is the reason for even difec-tionally stable single-screw ships to have the position of the z è r b rudder angle slightly o f F s ^ from the longitudinal-centre line in order to navigate in a straight line T h e side-force devefpped by the propeller is assumed to have a linear relationship vinth the longitudinal thrust,

A . , 7' 7^ 0

v;.,.,. A , , / 7 •: 0

y,...,, I,..,.,. (19)

T h e lateral force and yaw mbment due tp propeller are present thrbiighout starting and backing trrals. where a ship is accelerated f r o m rest to a steady state speed ahead or astern as the case may be. Hovvever. the mechanism causing these is quite different for the slow speed case and high speed case. Although we assume a linear relationship with thrust, their effect is more pronounced near zero speed.

A s suggested by Norrbin'-' the value of 0 04 is assumed for A'^i For the astern operation this value is assumed to increase to 0 06 (A^^) T h e value of /^„„,, may found frbm the geometry of the body and is about 0.45 of the length behind the C G

For a single-screw ship with a right-handed propeller t h é force )),•,...p tends to push the stern to P O R T afld bow to S T B D w i i h the engine going astern T h i s is one of the b a s k facts known to all ship handlers and used In advantage to turn the ship around In rivers with very little room to m a n œ u v r e . For a twin-screw ship the side forcé and moment produced by each propeller should be considered separately In a contra-rotating set of propellers (inwards or outwards) with both shafts rotating at the same speed the forces and moments cancel out Any change In R P M In any bf,the shafts results in an effective force and moment.

8.1 Rj|dderjpr«j^^^^ For the purpose bf significant turns, the rudder tends to act as an initiator of the turn and as the yaw angle develops, hull hydrodynamic forces and moments being much larger than t h é rudder forces at large incidence angles, tend tp become the dominating factor in the turning behaviour of the ship. Rudders also affect the directional stability of ships, as they resist the yawing motion in a sirnilar manner as fins and keels resist the rolling motion. Therefore, the directional stability of a bare hull would tend to improve after a rudder is fitted A great deal of research has already been carried

out in determining the characteristics of different t y p é s of rudder and designing more efficient ruddcû'S. e g , M a n d e P ° " .

T h e main problem in describing the rudder forces In mathematical terrns is d u é to the very complex flow conditions over the rudder. T h e dnrectlon of the flow over the hull together with the propeUer rotation and R P M should b è examined each time these forces are calculated. T h e incidence at t h é rudder can be calculated as explained In section 4. and the effective rudder angle may be written as.

Iind, - Hmi„ n.. (20)

Rudder forces are defined by its lift arrd drag characteristics and it is assumed here that the rudder Hft and drag as functions of sine and cosine of the effective rudder angle respectively

i , - \pi';S,.\^miid,)K,A

\f>V;Sr\[\ ':^ys{2Hud,))f<,u-\ (21) where '7 , is the fluid velocity over tbe rudder, and the expressions In square brackets are the lift and

drag coefficients wtibse values depend on the (nanaéûvring regime. For a rudder outside the propeller race, as Is the case for a twin-screw ship with a single rudder at midships, '7',. ' may be calculated simply by taking into account the wake fraction and fibw rectification factor as discussed by Ö l t m a n and Sharma'*' For a rudder in the propeller race the situation is more complex and propeller effects mnst also be taken intb account. '7 / as discussed in section 7 may be used Instead of "7 ,

A better representation of these forces may be obtained by examining lift and drag characteristics of the rudder under study and Including the stall phenomenon in the rnathematical model T h i s information is not always available In vi/hich case the Idealised lift and drag characteristics as mentioned before wilt be lised Rudder fbrces are knoywri to be most effective with ttie ship accelerating ahead f r o m zero speed or when moving ahead with maximum thrust In other m a n œ u v r i n g regimes rudder effectiveness is known to be poor in comparison. T h e following conditions are biiilt intb the mat hé ma tical model fpr a rudder placed behind the propeller in order to take into account different m a n œ u v r i n g regimes

il • 0 A / , 3X1 A</, - 1

il • ₀

r,.

il •.. - 0 and I'r 0 A / , . 0.5 1 (•22)

Rudder forces In surge and sway directions, and tiie yawing moment, may be written as follows.

A / . . s i î t n , - 7*, ros<\, ), -- i . rns.i, - IJ. s i n n ,

y . - I. y (23)

8-2 Rudder movement. T h i s is entirety dependent on the dynamics of the steering aigines. M o s t merchant ships are equipped witti two electro-hydraulic steering engines. T h e normal practice Is to

keep only one of them rurinirig for d e ^ - s è a navli^ation and switch over f r o m one engine to the other at 24 hour intervals In port and dense trafffc areas where a quicker res|>onse is demanded both engines are kept runriirig.

T h e minimum requirements for a merchant ship as set out by the Lloyd's Register classification society is that the rudder should travel f r o m 35 degrees on one side to 30 degrees on the other side in less than 28 seconds, and also that with the ship at her deepest permissible draft and maximum sea speed ahead the steering engine should be capable of deflecting the rudder f r o m - 3 5 " to + 3 5 " . Mandel'** suggests a minimum rudder deflection rate of 2.33 d e g / s for alt merchant ships, which is very close to the values set put by the Lloyd s Registrâ;.

tyiost mathematical rnodels (e.g. M c C a l l u m ' ' . Burns") use a first order linear differential equation (simple tag) to model the dynamics of the steering engines:

nild., K,.,^,^{Hud,i - RudJ (24 )

T h i s is a good a p p r o x i m â t i b n only when vk^e are considering small rudder movements, as Is the case for nrost autp^pilot studies When dealing writh large rudder movements (e g hard over f r p m one side to the other) a more suitable way of mpdellirig the dynamics of the steering engines is to assume a straight line retattonsfilp except for the final part of the rudder movement vvheré a simple lag must be applied tb avoid possible oscillation of the actual rudder angle for a rudder demand

If Hud., liud,, > l\,.,„( 7?W., - K,.,„i If find,. Hnd., •' li'ruj Hud,, A,.,,,/

/ ƒ .Hud,^ Hnd„ < Kru,i Rud^ .- K,.„Al{ud,i Hud,,\ (251

T o be more accurate a short delay and a simple lag should also be included for the initial part bf the rudder m b y é m è n t . Howeyer. the transmission, delay and the transient periods In an electro-hydraulic system are so small that it is q^uite reasonable to Ignore them T h e equations goyerriing the rudder movements have no effect on the steady state behaviour of the model. Hovvever. when ships are navigating in port areas they are continuously changing their m a n œ u v r i n g regime remaining in transierit conditions, which makes it essential to simulate the transierit periods with sufficient accuracy

9 T l l t u s t e r i m i t s . These are devices fitted on some ships at the bbw or bow and astern to provide extra transverse power at these points herice increasing the control o f t h e mariner over his ship. These are mbst useful for turning a ship around in places with little manoeuvring room or berthing operatibris without tugs. The most common type of thrusters is a sirigle propeller, driven by än electric motor, and fitted in a transverse tunnel

Clilstett and Bjorheden^ examihed the effectiveness of lateral thrust units at different ship speeds and jet speeds, tt is assumed that the thruster force is only developed in the lateral direction and varies with ttié forward speed and jet speed according to the foflowing relationships derived f r o m the results presented In figure 6 of the same report.

m = ship speed(r) Jet speed(r,) n

yj=^{TpAf'

y,,^^pAvf

where / / / is the parameter used to evaluate the side force, * V , is t h é jet exit velocity. ''Itt " h t h é distance between the C . G . and the thruster position, and T ' is the thrust developed by the thruster which may be calculated f r o m the open water characteristics of the propulsive device or the full-scale trials of the ship at zero speed.

1 0 S o l u t i o n of the s y s t e m é q u a t i o n s . T h e complete set of system equations to be solved may now be selected from.the previous sections and after rearranging may be vm-itten In an orderly manriêr â s follbws, // ^ ( 7 . V H • V .\ ƒ. • iiuvr) ;f)i -V •-. {)-„ • ) , •• 1;»,.,,. ' T»,f ) f niinr] , 102 r • ( V, A / , A , • A K , ^ A / . ; ) / / I -f r

ih K.ifh.. Th]

(e g wind and current forces): these effects will he discussed in a s e p t a t e paper in the near future. T h e above set of first o r d a differential equatibns may now be solved using a digital computer, and one of the many numerical methods suggested in text books A simple Euler method of integration Is selected, where the value of a variable at time '/ is used to predict its value at time '7 -r bi ' f r o m

y,,.,, 1-, -, V,^7 (28)

T h i s method o f integration is weU-known for its inaccuracies and high computation speed due its siim-pllcity. Because of the large mass and Inertia, arid relatively smaj.l forces involved In ship m a n œ u v r i n g . events occur at a very slow rate justifying the use of such a simple method. Accuracy of this methï>d is increased by taking very smafl values for "bi". T h i s is possible in off-line mode, but in real-time simulation "M" depends on the power pf the prpcessor Care should be taken in real-time s i m u l a t b n that "hi " shall not be greater than the smallest time constant iised in the m o d é l .

11 C o n c l u d i n g r e m a r k s . T h e basic concepts forming the main skeletpri of the Force mathematical 12

model are discussed. It Is endeavoured to keep the model as general arid as mpdular as possible so that each module may be modified or replaced by a module of a higher or tower accuracy as required.

A R u n g e - K u t t a and a Predictor-Cor rector methpd of integration ( Conte and Boor^.) were also exam-ined In off-line simulation Very little difference was noticed in the accuracy when compared with the Euler's method, with the coriiputational time increasing by a factor of five.

T o exanftirie Its yaildity the model must be tested for a series pf différent ships off-line and in r e a l t i m e sintulation. In off-line mode the model predictions must be checked against the avallabfë full-scale trails. In real time simulation expert mariners (Masters and pilots) familiar with the handling of that type of ship must carry out test runs in different m a n œ u v r i n g regimes and comment on the real world likeness of the model These tests have been carried but on this model with good results.

Nomenclature

AH Aspect ratip ( D r a f t / L e n g t h ) ( -, Block coefficient

('d.. Cross-flow drag coefficient Cd,, Potential flow drag cbefficient ( '(/n Drag cbefficierit at zero incidence

Cdlt,u Cross-flow drag cpefflcienl at small incidences ^ '(/2!m Cross-flow drag coefficient at 9 0 " of incidence ("f/,„i Cross-flow drag coefficient

('d, Total drag coefficient ( 7, Cross-flbw lift coefficient CIp Pbteritiat flow lift cbeffident Cl, Total lift coefficient

/.' Draft />, Rudder drag force

/ , Moment of inertia about the z-axis / ] , Virtual moment of inertia abbUt the z axis A , , , Comîpbnent of the rudder drag coefficient A ,, Ç o m p b n è n l pf the rudder lift coefficient A , Engine time constant

A , , . . Model constant A , ,,,; Rudder time constant

i Length between perpendicuiars A RuehJer lift force

/ Moment arm

k. Moment arm fbr the bbw thruster lateral force /,. Mbment arm fbr the rudder lateral force /( Éffective lateral-force momernt-arm in M a s s

ini Virtual mass In the surge direction » ) 2 Virtual mass in the sway direction A Yawing rnoment

Nk, Bow thruster yawing moment

A / , Total hull hydrodynamic yawing moment

A , . ' "'' Yawing moment due to the propeller lateral force X,. Yawing mbment d u é to the rudder lateral force A , . Fluid resistance to yavving mptipn

R Radius of turning circle Rnà^ Actual rudder angle Rudd D e h i ä n d rudder angle Rttd, Effective rudder angle

r ArigUlar velocity in yaw (yaw rate)

{• Angular acceleration Iri yaw

.S,. Rudder area

r

1 imiitt Normalised velocity, i e. I'/l',„.!.<•

!.. Equlhbruim speed

l \ Propeller slipstream velocity il yelpcity component iri surge it Acceleratlbn in surge

r, Bow thruster jet velocity Speed departure

r Velocity component in sway r Acceleration iri sway

A Force In the rbrigitudinal direction Rudder force In surge direction V Lateral Force

y., Bow thruster lateral force

y,. _{Total hull hydrodynamic lateral force} Propeller lateral fbrce

>v

(t Angle of incidesice at C G f l , . Incidence angle at the rudder

' > r,, Geometric angle of Incidence at the rudder Angle by which M , exceeds o

Added mass Fluid density

( • Heading angle

1 • Rate of change bf heading or yaw velocity

References

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5' C o n t é . S D . arid Boor, C . "Elemeritary numerical analysis McGrawHill Internâtibnal Book C o m -pany 1982.

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of S N A M E . Vol 87. 1979. pp.251-283.

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^- M c C a l l u m . I R.. A ship steering mathematical model for all m a n œ u v r i n g regimes' , Symposium on Ship Steering Aulorhatic C o n t r o l , 1980

10- Mandel, P . Some hydrodynamic aspects of appendage design". Transactions of S N A M E . Vol 61. 1953, pf> 464-515

11- Mandel. P Principles bf naval architecture' . Chapter 8. S N A M E . New York. 1976.

12- Mâtthevvs. R . "A six degrees of freedom ship model for computer E m u l a t i o n " . M . S c Thesis. Ü W I S T Cardiff, 1983

13- Morse. R V and Price. D . M a n œ u v r i n g characteristics of the mariner type ship ( U S S Compass Island) in calm seas . Sperry Polaris Management. Sperry Gyroscope Company. Nevv York. Déc. 1961. 14- Nomoto K Problems and requirements of directibnäl stabilitv and control bf surface ships . J Mech Eng Science Vol 14. No 1. 1972

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16- O i l m a n p and Sharma. S D . •'Simulation of combined engine and rudder m a n œ u v r e s Usirig ah improved model of hùlt-propellér-rudder Interactions . I5th symposium on naval hydrodynamics. Hamburg. 1984

17- PooTzanjani. M Zienklewics. H . and Flower. J . Ó ' A hybrid method of estMatiOfi hydrpdyftam-icalty generated forces for use in ship m a n œ u v r i n g simulation' . International Shipbuilding Pi-bgress. rjp 599 Vol 34. November 1987

18- Samsung Shipbuilding L Heavy Industries C o , Ltd . ' S e a trial data bf the vessel Iron P a c i f t c ' . Port K e m b l a . Australia 1986