I N Ż Y N I E R I A R U C H U M O R S K I E G O 2 00 5
ZESZYTY NAUKOWE NR 6(78)
AKADEMII MORSKIEJ
W SZCZECINIEJarosław Artyszuk
Towards a Scaled Manoeuvring Mathematical Model
for a Ship of Arbitrary Size
Keywords: ship manoeuvring, mathematical model, similarity, universality, ship
size.
The fundamental fact about manoeuvring of geometrically similar ships, of whatever length (or size) and initial steady approach speed, including scale physi-cal models and the whole series of full-sphysi-cale ships (if they really exist), is that the nondimensional track (in ship length units), the drift angle and the nondimensional yaw velocity are maintained. The only difference is the allocation of time moments to subsequent ship positions and/or heading. The higher is the speed and the shorter is the hull length, the time 'runs' faster. The present paper deals with a me-thodical explanation of these phenomena in view of building the manoeuvring model for a virtual ship of arbitrary size. Problems with satisfying the similarity of propeller parameters, the occurrence of scale effect in the resistance coefficient, and the consequences of variation in the ratios of main dimensions are analysed against the increasing ship length.
Wybrane problemy syntezy uniwersalnego modelu
manewrowania statku o dowolnej wielkości
Słowa kluczowe: manewrowanie statku, model, podobieństwo, uniwersalność.
Przedstawiono wybrane zagadnienia teorii badań modelowych, które są po-mocne w zrozumieniu zachowania się geometrycznie podobnych statków, bądź te-go samete-go statku, lecz o różnych prędkościach początkowych manewru. Przeana-lizowano również problem braku podobieństwa między małymi i dużymi jednost-kami w zakresie parametrów układu napędowego, szczególnie śruby napędowej oraz stosunków wymiarów głównych.
Introduction
It is well known that a geometrically similar ship nondimensionally behaves
in the same way as her reference ship of known hydrodynamic coefficients
there is only a problem of time related data conversion. Although in physical model tests the geometrical similarity of the hull and appendages is of the utmost importance and commonly applied, real ships of increasing size (and length) do not prove such a 'simple' law for different reasons.
In the present study, a few theoretical aspects of making a virtual ship of arbitrary size will be raised, which can be used as the first approximation and/or the best representation of a ship 'requested' for a given shiphandling training scenario, waterway optimisation, or nautical safety and efficiency analysis. The following points will be discussed:
good understanding of ship motions under similar conditions, with a
spe-cial emphasis on manoeuvring, which would prevent making mistakes while interpreting model or full-scale tests, evaluating the quality of such tests, or designing new trials for mathematical model validation and/or identification;
applicability of some theoretical recommendations for a preliminary
de-sign (extrapolation) of the main propulsion;
advantages of various world-fleet printed ship registers and the statistical
processing of the hull dimensions and propulsion related data;
effects of changes in the ratios of main hull dimensions upon hull manoeuvring forces.
The primary focus is set on chemical tankers, but the achieved results are of more universal nature.
1. Basic theoretical background for manoeuvring scale model tests
The scale physical model investigations in relation to ship motions i.e. comprising the whole range of propulsion, manoeuvring and seakeeping prob-lems of the hydrodynamic nature were established very long time ago. Till now-adays they have experienced a lot of progress, especially:
in the instrumentation, thus enhancing the spectrum of physical
phe-nomena which can be quantitatively tested, in other words the applica-tion area for model tests has expanded and the overall accuracy has in-creased;
in the effective procedures for conducting such tests, i.e. mitigating the
so-called scale effect (mostly related to the inequality of Reynold
num-bers for model and ship) – either directly through a special design of the
experiment or indirectly by means of proper analytical methods of tow-ing tank measurement data extrapolation to the full scale.
The best forum of exchanging and analysing the latest views and published results in this field are inter alia the periodic meetings of the International Tow-ing Tank Conference (ITTC) with their widely known proceedTow-ings.
For ship model tests to be adequate, it is necessary to ensure three general similarity laws: geometric (dimensional), kinematic (velocity-related) and dy-namic (force-related) – redears can refer to any textbook on fluid mechanics. The most difficult to fully satisfy is the dynamic similarity, though some minor problems can sometimes exist with regard to both geometric and kinematic simi-larity. Due to some reasons and under some circumstances, even the latter two are not always necessitated to achieve proper results. Reverting to the dynamic similarity, it is frequently impractical to 'model' some force components, for
ex-ample of viscous nature as represented by the mentioned Reynold number the
model velocity must be unfortunately higher than that of a ship and proportional to the model scale. Moreover, it is absolutely impossible to equally model all force components – a compromise has to be then established taking into account the research goals. The best illustration is the classical conflict between the Froude's (gravity, or wave formation related) and Reynold's (viscosity, or fric-tion related) laws of similarity. It shall be finally kept in mind that the ship local kinematic similarity is often out of control for the investigators, it is usually ra-ther assumed as inherent to the geometric similarity.
In the mentioned above areas of interest – propulsion, manoeuvring and seakeeping, it is generally accepted to primarily concentrate on Froude numbers equality for the model and ship and the geometric similarity. The Froude number
FnL in this respect is only once globally defined on the basis of the initial forward
velocity and the length between perpendiculars. The local (specific to particular point on the hull) kinematic and dynamic similarity shall be valid by 'default' in such circumstances. For some shallow water phenomena description, the water
depth related Froude number FnH is sometimes introduced.
There are two distinct goals and methods of model tests:
force measurements (captive models),
records of various kinematic variables (free-running models).
Hereafter our interest is mainly concerned with the latter i.e. free-running models and solely focused on propulsion and manoeuvring.
By considering the common rules of the hydrodynamic force nondimen-sionalising and the scale model behaviour, it frequently happens that both the manoeuvring forces and motion behaviour of scale models can be measured at arbitrary velocity i.e. not necessary corresponding to that the final full-scale simulation is based on. So, the equality of Froude numbers for the ship and the model is not always stipulated, provided that appropriate data extrapolation pro-cedures are applied. Of course, this is generally valid only to some extent, espe-cially for the low speed displacement ships.
Let's assume that the geometric scale of the model is : s m i s i m L L L L const (1) where: Lim and i s
L are corresponding local linear dimensions on the model and
ship, L and m L are the lengths between perpendiculars (serving as the most s
representative references). The scale also applies to all appendages (e.g.
pro-peller and rudder).
Let's next consider the ratio of local velocities (referred to as the velocity scale), of the linear or angular origin, around the hull and appendages as follows:
s m i s i m v v v v v const (2) where: v and mi i s
v are local flow velocities of the model and ship, v and m v are s
briefly the initial forward velocities of the model and ship.
Especially for free running models (remotely controlled models or manned
models), it is sometimes advantageous to use in (2) the so-called time scale t,
which is defined as the ratio of time intervals required to travel specific length units by the model and ship. According to the adopted notations:
v s s m m t v L v L (3)
If the velocities of the model and ship conform to the equality of Froude numbers:
s m s m s nL m nL L L v v F F 2 (4)then the time scale reads:
s m t L L (5)
Since the ratio of model and ship velocities is often much higher than the
model scale , see (4) for instance, the time on the model runs much faster. For
In the case of inequality of Froude numbers for the model and ship, the most general relationship (3) must be used. If the velocity of the model is proportional to its scale, then the time scale equals unity.
The time scale, of course, directly affects the angular (yaw) velocity experi-enced on the model and the frequency of manoeuvring commands (orders) sent to the model. This is because the angular velocity can be treated in terms of local lateral linear velocities as subject to all above kinematic similarity effects, though they are proportional to the distance from the body centre of gravity
(ap-prox. midship point) – i.e. the axis of rotation. The full turn 360 is to be
com-pleted 1/t faster for the model. For example, the following relationships for the
bow movement are valid here:
vyF m
z m0.5Lm ,
vyF s
z s0.5Ls (6)
z s t m z v s m s z m z s yF m yF L L v v 1 (7) where:vyF – bow lateral velocity, z – yaw velocity,
m and s – subscripts denoting the model and the ship.
It can also be easily concluded at least from (6) and (7) that the so-called in-stant nondimensional yaw velocity (as roughly representing the inverse of turn-ing track curvature radius):
2 2 , xy x y xy z z v v v v L (8) where:
vxy – total linear velocity of the body, vx and vy – its surge and sway components,
and the instant drift angle (amidships):
x y v v arctg (9)
are almost (shall be) identical for the model and ship at the 'corresponding' time moments:
1 s m s z m z (10)The expression (10) proves that the nondimensional track of the centre of gravity, i.e. the x-y translations (displacements) divided by the specific lengths between perpendiculars, is exactly the same for the model and ship. Additional-ly, in view of the above statements, the identical nondimensional yaw velocities and drift angles are experienced at the matching positions and orientations of nondimensional contours. Due to a different time scale for the model and ship,
this takes place at different time moments – refer to Fig. 1 (the case of Froude
number equality). - 1 0 1 2 3 4 - 1 0 1 2 3 x/L[-] y/L[-] 50s (m: 10s) 2.4/s (m: 12.1/s) 92s (m: 18s) 2.0/s (m: 9.8/s) 140s (m: 28s) 1.8/s (m: 8.8/s) 194s (m: 39s) 1.7/s (m: 8.3/s) chemical tanker SFAH 35 turning - validated SIMULATION Fig. 1. Demonstration of ship and physical model ('m') manoeuvrability – scale 1:25
In zigzag manoeuvres, the overshoot angle is also unaffected by the model velocity or time scale, though the oscillation period needs to be recalculated ac-cording to the actual time scale.
For reasons of a direct comparison and/or above 'problems' with the time scale conversion of model tests, it is recommended to analyse various manoeu-vring data as a function of the nondimensional (curvilinear by nature) distance travelled by the model or ship, rather than in the time domain. This and the other aforementioned aspects can also be seen in manoeuvres of a ship at various ini-tial speeds, but at the equal propeller thrust loads – the main engine telegraph is then constant and conforms to the steady state approach speed, see [6, 7].
The statements formulated so far with regard to the nondimensional tracks, drift angles, and the travelled distance parameter for the model and ship manoeuvring behaviour analyses have been essentially based on the kinematic similarity law. These would be more obvious by considering the usual ship
manoeuvring differential equations. In [7] a new form of equations was pro-posed, especially suitable for manoeuvring mathematical model validation and identification. It can be briefly written as follows:
, , , 1 , , , , , , , , 1 , , , , , , , , 1 , , , , , * 3 3 * 2 2 * 1 2 1 Th z xy Th z xy z Th z xy Th z xy Th z xy Th z xy xy c f v L c v L f dt d c f v L c v L f dt d c f v L c v L f dt dv (11)where the underlined terms cTh and are the control variables, which indicate the
propeller thrust load coefficient (linked to the propeller rpm) and rudder angle, respectively. The functions f , 1* f , 2* f shall be identical by assumption for the 3*
model and ship.
Taking into account the time derivative of the nondimensional travelled dis-tance: L v dt s d xy (12)
the wanted expressions yield:
, , , , , , * 3 * 2 Th z z Th z c f s d d c f s d d (13)From equations (11) or (13), another very useful differential relationship
can be straightforwardly obtained. This is namely the derivative dz d, which
is essentially the basis for designing new manoeuvring trials of free-running models or full-scale ships from the viewpoint of mathematical model identifica-tion in the maximum possible range of z combinations. As regards the given (required) z trajectories, the inverse problem, i.e. what engine and rudder settings (in terms of cTh and , these as functions of the drift angle ) have
to be ordered, can be solved just by means of the dz d differential equation.
The derivative dz d is also known through a simple numerical
differentia-tion of the input z relationship. Of course, only mutual dependence of cTh
identification process. This approach partially supplements the research con-ducted in [5].
Concerning the propeller performance modelling (also proportionally scaled) in manoeuvring tests, the propeller advance ratio is normally maintained for the model and ship:
s m J J (14) where:
m m m m D n w v J 1 ,
s s s s D n w v J 1 (15) in which:w – wake fraction (assumed equal for model and ship), n – propeller rotational speed (revs/s),
D – propeller diameter.
Therefore, it is required that:
t s m n n 1 (16)
For example, the slow speed diesel engines approximately operate at revo-lutions in the order of 100 rpm, hence the ship model of scale 1:25 at Froude numbers equality has to work at 500 rpm of its propeller.
But more exactly, the propeller thrust is often such (in free running models) as to keep the demanded model forward speed. Due to certain hydrodynamic scale effects, this can imply rather a slight inequality of the advance ratios.
The opposite trend – rpm lowering – shall be seen while attempts are taken to simulate larger ships with the same set of nondimensional hydrodynamic data for the hull, propeller, rudder, etc.
2. Propulsion particulars versus ship deadweight
–case study
2.1. Simple analytical approach
Let the ship propulsion equations be generally written according to:
P ME P P eff nQ DHP T t T R 2 1 (17) where: R – ship resistance,tP– thrust deduction,
DHP – delivered horsepower,
QP– propeller torque.
All the above magnitudes can be simply decomposed as follows:
R c LTv R0.5 2 (18)
2
2 2
2
4 2 , 1 , J D P J k D w v D P J k D n T T T P (19)
2
2 3
2
5 2 , 1 , J D P J k D w v D P J k D n QP Q Q (20) where: – water density, T – ship draught, cR – resistance coefficient, J – advance ratio (see eq. (15)), P/D – pitch ratio,kT and kQ – open water thrust and torque coefficients.
In eqations (17 – 20), the parameters: cR, w, and tP are treated as constant
values, not affected by the ship 'scaling' from one size to another. This is signifi-cantly the weakest link in the reasoning that follows.
The design forward speed (called the sea full ahead or service speed) for a particular kind of ships (e.g. tanker, bulk carrier, container carrier) is more or less fixed as based on economical calculations, in which the dependence on ship size (displacement, deadweight) is practically negligible. The constant forward speed v for any ship size will be assumed hereafter.
With regard to the above equations from (17) to (20), while getting the pro-peller thrust and power demand for the unknown bigger ship (represented by the ship length L), which is of scale :
ref L
L
(21) where Lref is the length of the reference (basic) ship, the easiest though very
rough way is to keep the same proportions for the propellers (their diameters in particular) – the advance ratio J and pitch ratio P/D remain unchanged – which lead to the identical propeller efficiency (p = const.) and thrust load coefficient
(cTh = const.). The latter is crucial for computing rudder forces. The following
re-lationships among propeller diameter, revolutions and main engine power are valid in this approach:
ref D D , 1 ref n n , 2 ref DHP DHP (22)
The problem is that such a drastic reduction in the propeller (and/or engine)
rotational speed is rarely encountered in the real-world – very low rpm engines
are not manufactured at all, partly due to technical disadvantages and market demands.
In [1,2], a few very interesting universal formulas are considered as repre-senting the propeller performance for any pitch ratio – single curve diagrams for thrust and torque coefficients were achieved, which essentially match the stand-ard open water kT and kQ diagrams for the pitch ratio 1.0. This can be
summa-rised as follows:
* * 4 2 J k D P D n TP T (23)
* * 2 5 2 J k D P D n QP Q (24) where: D P J J* , D P k k T T * ,
2 * D P k kQ Q (25)The implication of expressions from (23) to (25) is that one receives some freedom while selecting the propeller rpm-pitch-diameter mutual configuration. It is sufficient to assume here the equality of modified advance ratios J* for both the
reference and extrapolated ship, which also means equal propeller efficiencies for both ships i.e. being independent of the assigned pitch ratios (p = const.). Of
course, the latter consequence is a drawback of this concept, since it is not entirely true. After some rearrangements, the below set of relationships can be produced:
2 3 ref ref n n D D ,
2 1 2 ref ref P D D P D D (26) 2 ref DHP DHP ,
D P D P c c ref ref Th Th (27)If someone chooses in (26) the unchanged propeller pitch ratio for the ref-erence and actual ship, then the expressions (26) and (27) are equivalent to the former approach given by (22), as leading to the very low propeller/engine rpm. The other extreme situation arises if the rpm are kept constant, which necessi-tates a serious reduction of the propeller pitch ratio for a larger ship. It is
be-lieved that a proper compromise between both moderately lower rpm and pitch ratio would neatly represent real ships.
A conclusion can be drawn from eq. (27) that the change of the propeller thrust load coefficient cTh is inversely proportional to the alteration of pitch ratio.
The only question is how this affects the rudder forces under conditions of the identical hull-propeller-rudder arrangement for the reference and actual ship. Theoretical treatment of potential effects is given below.
The rudder lift force, as the most important steering factor, in the very con-venient jet velocity-related form reads (see e.g. [3]):
Th
L PS R LR A v c c F 0.5 2 , (28)
Th PS v w c v 1 1 (29) where: AR– rudder areas,vPS– propeller slipstream velocity, v – ship velocity,
cL– rudder lift coefficient (decreases for lower cTh),
– effective incidence angle.
For the scale = 3, corresponding to the extrapolation of e.g. 100m
sea-going ship to 300m (generally the maximum scale would not be in excess of
this), the increase of propeller thrust load coefficient cTh (due to the lower pitch
ratio, if any) is practically not higher than twice, refer to eqs. (26) and (27) with
n = nref. However, in this case the scale for propeller diameter is 2.06, which
re-sults in the proportionally lower propeller diameter to rudder height ratio D/HR.
The assumed rudder is of the equal scale with the hull, hence the propeller-rudder configuration is completely changed, which makes the reference diagram
of lift coefficient cL no more valid. Nonetheless, the lift coefficient undoubtedly
decreases in such circumstances. In total, the loss of rudder lift coefficient
caused by the increase of cTh and decrease of D/HR is significantly overcome by
the increase of the rudder flow – see eq. (29). A comprehensive simulation re-search concerning a quantitative evaluation of these specific effects with regard to the ship manoeuvring behaviour is planned in the future. It may just appear
that the impact of the cTh increase upon the ship manoeuvring motions is not so
significant as one could expect.
2.2. Statistical data of real ships
In order to investigate the real occurrence of the above problems with a sys-tematic selection (sometimes ambiguous) of the propeller diameter, rpm and pitch ratio for larger ships, a reference has been made to [9]. Since the author's
recent works deal with the mathematical model of a small chemical tanker (6,000 DWT, 97.4m in length), which can also constitute an input ship for fur-ther scaling, instances of the same type of ship have been randomly extracted from the said publication every 5000 tons in deadweight.
The original data of [9] in our interest area consist of all three main
dimen-sions (L, B – breadth, T), nominal main engine power (BHPn) and associated rpm
(RPMn), and ship service speed (vserv). The latter normally ranges between 13
and 16.5 knots for chemical tankers. Because the speed of 14 knots is intended to be the reference point as to enable a mutual comparison of chemical tankers, it is necessary to 'recalculate' the nominal power and rpm according to:
n serv BHP v BHP 3 14kt [kt] 14 , n serv RPM v RPM kt] [ 14 14kt (30)
Figure 2 presents the extended results of processing of the above chemical tanker data. However, these are supplemented with the aforementioned author's tanker, because [9] cover tankers over 10,000 tonnes in deadweight. The subdia-grams of Figure 2, in turn, demonstrate the ship length (between perpendiculars) vs. deadweight and all the following items as function of the ship length: ratios of main dimensions L/B and B/T, engine rpm and brake horsepower as reduced
to 14kt, resistance coefficient cR, ratio of propeller diameter to ship draught D/T
(more useful than pure diameter), propeller pitch ratio P/D, propeller thrust load coefficient cTh.
The nondimensional resistance coefficient cR has been determined in
rela-tion to the LT product by the following:
2 5 . 0 LTv R cR with kt 14 R R and v7.2m/s(14kt) (31) v BHP R14kt 0.85ph 14kt , p 0.5 , h 1.33 (32)
where 15% sea margin, 50% open-water propeller efficiency (p), and 133%
hull efficiency (h, as strictly connected with standard wake fractions and thrust
deductions for chemical tankers) have been uniformly adopted. The propeller di-ameter and pitch ratio have been chosen according to the Wageningen B-series preliminary design method, e.g. [11], with the wake fraction 0.4 in all cases – re-fer to the Bp – chart for B4.40. As an example of the method adequacy (the
propeller selection process is generally more complex) the aforementioned 97.4m chemical tanker can be given – the diameter 4.4m and pitch ratio 0.74 are
obtained instead of real values 4.1m and 0.87, respectively. The magnitude of cTh
is based on the widely available standard kT – J curves of the B-series for the just
0 4 8 1 2 1 6 0 5 0 1 0 0 1 5 0 2 0 0 0 5 0 1 0 0 1 5 0 2 0 0 0 2 0 4 0 6 0 0 2 4 6 8 0 5 0 1 0 0 1 5 0 2 0 0 0 1 2 3 4 0 5 0 1 0 0 1 5 0 2 0 0 0 5 0 1 0 0 1 5 0 2 0 0 0 5 0 1 0 0 1 5 0 2 0 0 0 4 8 1 2 1 6 0 5 0 1 0 0 1 5 0 2 0 0 0 0 .2 0 .4 0 .6 0 .8 0 5 0 1 0 0 1 5 0 2 0 0 0 0 .2 0 .4 0 .6 0 .8 0 5 0 1 0 0 1 5 0 2 0 0 0 1 2 3 4 0 5 0 1 0 0 1 5 0 2 0 0 L[m] L[m] L[m] L[m] L[m] L[m] L[m] L[m] L[m] DWT[kt] L/B[-] B/T[-] RPM BHP[kW] cR[-]x10 3 D/T[-] P/D[-] c Th [-] simul. @14kt @14kt
Fig. 2. Real scaling of chemical tanker particulars as function of ship length
The subchart of BHP in Fig. 2 also includes the 'simulated' values i.e. pro-portional to the actual LT product with the reference power of the first ship in the row – this corresponds to scaling, refer to (22) or (27).
It is evident from Figure 2 that the rpm trend, as expected, is declining. Sta-tistically, from around 150 rpm at 100 m length (6,000DWT) the engine speed goes only to about 100 rpm for a ship 180 m long (approx. 50,000DWT). The most surprising and interesting is a rapid reduction of the resistance coefficient
cR with the increasing ship length – almost twice in the concerned range – which
can be attributed to much lower wave resistance (lower Froude numbers). The propeller diameter practically grows proportionally to ship main dimensions. The pitch ratio is nearly constant for the whole considered series of chemical tankers – at the level of approx. 0.7. The similar trend is visible in the nondi-mensional thrust coefficient kT, which is anyhow not shown in the chart. The
thrust load coefficient cTh follows the change of rpm – but is flatter – which can
be explained by a relatively high increase in the propeller diameter as compared
to the rpm decrease. The cTh becomes thus smaller with the ship length, which is
while selecting the propeller main particulars, the accurate knowledge of the re-sistance coefficient must be possessed with reference to the ship length or Froude number. However, it is further believed that the hull hydrodynamic sway force and yaw moment will not be affected by the ship length – due to somewhat lower local lateral velocities. Nevertheless, it is worthwhile to study this phe-nomenon in the future by model tests in towing tanks.
2.3. The effect of L/B ratio upon hull forces
There are various regression formulas in the literature for estimating the nondimensional hull sway force and yaw moment as functions of the ship hull main characteristics – they primarily rely on ratios of the main dimensions and shape parameters (e.g. the block coefficient cB). The overall accuracy of such
expressions to predict forces in the full scale might be very poor in a lot of cases - just due to different reasons, and this shall be a subject of another well defined research project. As an example, the Kijima formula [10] will been used to ex-amine the effect of L/B for a rough indication of possible effects.
The absolute magnitude of hull sway force FyH and yaw moment MzH (this
excludes the so-called Munk moment by default) can be written in the universal form as follows:
m B mzhm m B fyhm z xy zH yH c T B B L Lc c T B B L c L v LT M F , , , , , , , , 5 . 0 2 2 (33)where the underlined terms denote kinematic variables – drift angle and
modi-fied nondimensional yaw velocity m:
2 2 2 L v L z xy z m , m 1,1 (34)
In order to reliably compare the nondimensional generalised forces, cfyhm
and cmzhm, for different ratios of main dimensions, it is purposeful to preserve the
reference area LT in (33):
const.
LT (35)
Otherwise, the whole formula (33) shall be analysed with regard to all items comprising the hull main dimensions (the ship length L in particular) while in-vestigating the impact of ratios of main dimensions.
Because the product LBT is also constant in view of the assumed equal dis-placement ships, the equation (35) leads to:
const.
B (36)
In this specific case, increasing the L/B ratio must be accompanied by aug-menting the B/T ratio as well. If the L/B is incremented by 1.0, the B/T advances by the value: B L T B x (37)
As examples of initial conditions L/B = 5.87 and B/T = 2.34, the ratios 6.87 and 2.74 are received, respectively. The effect of such new ratios of main dimensions by the formulas [10] is depicted in Figure 3. The following definition is applied (%cmzhm by analogy): ... , , ... , , ... , 4 . 0 , 1 % T B B L c T B B L c T B B L c c fyhm fyhm fyhm fyhm (38)
Generally, both the cfyhm and cmzhm coefficients decrease with the increasing L/B ratio. Though it is clear from Figure 2 that the L/B change with the ship
length is not so distinct for chemical tankers, there are still a lot of other ship types where this ratio significantly changes versus ship length. So, Figure 3 can be more meaningful. - 0 .2 - 0 .1 0 .0 0 .1 0 1 0 2 0 3 0 4 0 0 0 .2 0 .4 0 .6 0 .8 - 0 .3 - 0 .2 - 0 .1 0 .0 0 1 0 2 0 3 0 4 0 0 0 .2 0 .4 0 .6 0 .8 [] [] m m %cfyhm[-] %cmzhm[-]
3. Final remarks
The present study has revealed that extensive statistical studies are the best approach to properly identify the parameters of main propulsion assigned to a given size of ship, and the very crucial forward motion resistance coefficient.
The thrust load coefficient cTh, being of the highest importance while computing
the rudder forces, tends to decrease with the ship length in case of chemical tankers. Other ship types would obviously require similar research to the one
performed above. In general, the potential range of change for the cTh coefficient
does not seem to seriously affect the simulation of ship manoeuvres. Therefore, the major focus shall be set upon mathematical modelling of hull forces with the increasing ship length, with account taken for modified ratios of main hull di-mensions and body lines (at the stern in particular).
There will also be a need for comparative simulation trials to quantitatively assess these two effects – the change of thrust load coefficient and ratios of main dimensions. However, this can be completed if a comprehensive set of special-ised rudder manoeuvres is defined, in which either the rudder or the hull forces dominate. In the very interesting paper of Berlekom and Goddard [8], containing some sensitivity analyses, a few preliminary but limited results were obtained in the field of our concern.
In general, in view of the above analysis it can be stated that the well estab-lished manoeuvring mathematical model of a specific ship (that is often a diffi-cult task) is a very good source for 'scaling' to any ship size. It is only necessary to incorporate in the computer codes some statistical relationships concerning the propulsion particulars, ratios of main dimensions and hence the correspond-ing adequate corrective multipliers of the hull hydrodynamics.
Finally, it shall be mentioned that major differences in manoeuvring of
vari-ous length ships, in terms of relative measures e.g. drift angle, nondimensional
yaw velocity, nondimensional x-y trajectory (ship length related), overshoot an-gle etc., will occur under the second order wave action, which strongly depends upon the ship length [4].
References
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Mathe-matical Model – Thrust Case. 4th Navigation Symposium, Jun 19-20,
Mari-time University, Gdynia 2001.
2. Artyszuk J., Propeller Slip Ratio in the Ship Manoeuvring Motion
Mathe-matical Model – Torque Case. 4th Navigation Symposium, Jun 19-20,
Mari-time University, Gdynia 2001.
3. Artyszuk J., A Novel Method of Ship Manoeuvring Model Identification from
4. Artyszuk J., Drift of a Disabled Ship Due to Irregular Waves. Scientific Bul-letin no. 70 'Marine Traffic Engineering 2003', Maritime University, Szcze-cin 2003.
5. Artyszuk J., Drift-Yaw Correlation in Ship Manoeuvring Model
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7. Artyszuk J., Analysis of steady state turning ability in view of ship
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Recenzenci
dr hab. inż. Tadeusz Szelangiewicz, prof. PS dr inż. Lucjan Gucma
Adres Autora
dr inż. Jarosław Artyszuk Akademia Morska w Szczecinie Instytut Inżynierii Ruchu Morskiego ul. Wały Chrobrego 1/2
70-500 Szczecin
