Maritime University of Szczecin
Akademia Morska w Szczecinie
2010, 20(92) pp. 13–24 2010, 20(92) s. 13–24
Pivot point in ship manoeuvring
Pojęcie chwilowego środka obrotu w manewrowaniu statkiem
Jarosław Artyszuk
Maritime University of Szczecin, Faculty of Navigation, Institute of Marine Traffic Engineering Akademia Morska w Szczecinie, Wydział Nawigacyjny, Zakład Inżynierii Ruchu Morskiego 70-500 Szczecin, ul. Wały Chrobrego 1–2, e-mail: j.artyszuk@am.szczecin.pl
Key words: pivot point, notion of pivot point, ship‟s manoeuvres Abstract
This paper is aimed to clarify the notion of instant pivot point (as being well established in rigid body mechanics) and review its application in practical ship manoeuvring. The extension of the Vasco Costa original results to arbitrary initial ship manoeuvring motions is performed and a transient phase of the pivot point change is studied, especially with regard to ship crash stopping and turning (including inter alia a bow thruster application).
Słowa kluczowe: chwilowy środek obrotu, położenie chwilowego środka obrotu, manewry statku Abstrakt
W artkule podjęto problem położenia chwilowego środka obrotu w różnych etapach manewrowania statkiem. Wskazano na jego praktyczne znaczenie, przedstawiono ścisłe podstawy teoretyczne i wyniki symulacji dla wybranych manewrów. Wzdłużne położenie środka obrotu jest często niepoprawnie lub nie w pełni rozumia-ne i tym samym niewłaściwie wykorzystywarozumia-ne wśród nawigatorów.
Introduction
The phenomenon of instantaneous pivot point existence in ship manoeuvring is well known to navigators, though there is a lot more or less justi-fied disputes among them in recognizing quali-tatively and quantiquali-tatively its location on a ship during various modes of operation. The available literature on ship manoeuvring and handling does not cover all aspects of pivot point in a systematic way. This contribution is aimed to fill in this gap.
The practical role of pivot point arises from that it is easier for a ship‟s operator to control just a single motion, here the angular one around a certain point, the location of which is generally known or perceptible, rather than “tracking” both lateral and angular movements. Moreover, the navigator is familiar and used to apply the pivot point principles while making various kinds of manoeuvres, either at high or at low forward speed. At high speed, like underway, applying the stern rudder will involve the rudder lateral force as
developed on a constant arm. This implies a ship‟s lateral motion (drift) strongly correlated with simul-taneous turning, thus the instansimul-taneous pivot point, as based on time-dependent values of both motions, lies somehow close to the bow and usually changes but in a fairly narrow limit. So from the practical viewpoint the pivot point can be considered steady here – the human rests his first / early shiphandling decisions (e.g. when and how to initiate a sharp turn) on the approximate (rough) estimation of the pivot point location. And then, according to gained experience and skills through his life both in a visual assessment of the ship motion actual state (tendency) and in a response behaviour to a noticed deviation from the planned track, he works out and applies in due time subsequent, small in nature, corrective actions.
On the other hand, when ship berthing is analysed for instance, knowing the instantaneous pivot point under a particular application point of external force (e.g. arising from a tug action), will
ease the operation of keeping a ship‟s heading parallel to a berth.
Though in the present paper a single external force applied at any point is only dealt with, this will also represent any combination of two, or more, forces, that can always be reduced (according to the principles of applied mechanics) to such a single resultant force and a specific arm counted from the body axes origin.
The paper is basically divided into four parts. Firstly, the theoretical background of rigid body mechanics with reference to the ship instantaneous pivot point is thoroughly studied. Secondly, two cases of a ship undergoing an external excitation are discussed in view of the analytical derivation of the pivot point initial position as based on firm hydrodynamic foundations. Thirdly, a steady tur-ning around an arbitrary pivot point is investigated in the aspect of “securing” appropriate external power. Fourthly and finally, an exemplary transient analysis is undertaken to show the pivot point change between a ship resting and entering the steady turning phase.
Pivot point in ship kinematics
For a manoeuvring ship, i.e. moving in a hori-zontal plane, it is common in ship hydrodynamics to assume the reference systems as shown in fi- gure 1. These denote an earth-fixed (geographical) system Ox0y0z0 and a ship-fixed (body axes) moving system Mxyz, where is a ship‟s heading and the origin M is set at the midship section.
xO yO x y M N E z O
Fig. 1. Coordinate systems in a ship planar motion Rys. 1. Układy odniesienia w ruchu płaskim statku
It is well known from rigid body mechanics that instantaneous arbitrary ship planar motions can be uniquely decomposed into a combination of both translational motion of the origin M and angular motion of the whole body around this origin. Adopting here a different body axes origin just only leads to a proportional change in the translational motion while preserving the angular motion (inde-pendent of the origin). Both motions are defined by
the linear velocity vector of the origin (in the xy plane) and the angular velocity vector (perpendi-cular to the xy plane). From the viewpoint of local linear velocities of particular points within a ship body, such an instantaneous state of motion (see Fig. 2) is equivalent to a sole angular motion (with the same angular velocity) around the so-called instantaneous pivot point „PP‟ (in figure 2 located at the starboard quarter). By definition, the latter is a point on the body (formally can be even outside the body physical extents, in this case the body shall be “virtually” extended) in which the linear velocity decays. The radius-vectors of particulars points with regard to the pivot point are normal to the corresponding linear velocities. As the ship linear and angular velocities are time dependent, so the pivot point position instantaneously changes.
PP M
xPP
yPP
Fig. 2. Instantaneous local linear velocities and instantaneous pivot point
Rys. 2. Chwilowe lokalne prędkości liniowe oraz chwilowy środek obrotu
When possessing data on the body axes origin (point M) linear and angular velocity vectors, as for example resulting from the integration of ship motions differential equations, the linear velocity B of arbitrary point B around the ship‟s hull is written by:
B
B r
(1)
where: r – radius-vector of B versus body axes B
origin.
In accordance with the above relationship the pivot point PP is defined by vB = 0 thus:
rPP or rPP (2) In case of ship pure (elementary) manoeuvring motions the vector coordinates are as follows (refer to figure 1 for sign convention):
vx,vy,0
,
0,0,z
PP, PP,0
PP x y r (3) hence: z y PP v x and z x PP v y (4)or non-dimensionally when dividing by ship‟s length L: L v x z y PP ' and L v y z x PP ' (5)
The use of dimensionless values (in general relating to any linear dimension, though ship‟s length is common) is much more advantageous either in the shiphandling practice (tips are universal for any ship size, ship visual positioning against external objects is also easier in relative units), automatic control practice (efficiency of control algorithms), or in the theory of ship manoeuvring. All is arising from similarity laws in ship hydrodynamics. During ship turns with a stern rudder command the longitudinal coordinate of pivot point x'PP is positive and usually of order
0.2L0.5L, while the transverse position y'PP is
close in magnitude to the turning track curvature radius (often in range 1.02.0L) and directed towards the interior of the turning circle.
It is frequently very convenient for a short prediction period (the so-called short-term mano-euvring), as dominating in shiphandling practice, to disregard y'PP and, instead of, to rotate a ship
around the pivot point projection to the ship‟s centre line (that can be called a “working” definition of pivot point for practical purposes, i.e. single dimensional „1D‟) together with its simul-taneous translation in the direction of heading according to the forward speed.
From Equation (5) the following is evident:
' 2 ' 2 ' 1 PP PP z PP PP R L r y x (6)
where: z is the dimensionless yaw velocity, one of the most important kinematic parameter in ship manoeuvring hydrodynamics, defined by:
xy z y x z z v L v v L 2 2 (7)
It shall be reminded that the inverse of dimensionless yaw velocity, or R'pp, is not in
general exactly alike the ship‟s track curvature radius. The latter converges to R'pp only for steady
drift angle.
According to Equations (5) and (7) one can write: z PP x' sin and z PP y' cos (7.1)
which are very fundamental expressions in view of ship manoeuvring hydrodynamics, where the drift angle and dimensionless yaw velocity z are the two major arguments (apart from the resulting / to-tal linear velocity vxy) for evaluating the hull forces.
Also other external excitations, especially the propeller and stern rudder forces and moment can be expressed directly or explicitly (as implied by the internal flow physics) by means of these three factors. As far as the forces are proportional to the square of both the ship‟s length and speed (with regard to moments the ship‟s length cube shall be considered):
F ~ L2v
xy2 M ~ L3vxy2
(which is usually true in pretty wide range), this further leads to the fact that the drift angle, dimensionless yaw velocity and thus the pivot point coordinates (7.1) are independent from the ratio
L/vxy, which is sometimes referred to as tL – ship‟s
specific time(the time of travelling one ship‟s length). As a result, if such parameters are plotted versus t' (dimensionless time) defined as t' = t/tL,
that is equivalent to s' = s/L (dimensionless distance of ship‟s origin as traveled over the track), they assume an identical pattern / chart, being the same for any initial ship speed and ship size (of course, under the conditions of a good geometrical similarity of ship hulls and equal propeller loading as represented by the ratio of ship‟s forward speed and propeller rotational speed, as leading to identical values of propeller advance ratio J or propeller loading coefficient cTh).
Pivot point in ship dynamics under single external force
Ship initially at rest with arbitrary force
For a ship without any initial motion, like sometimes in berthing / unberthing manoeuvres, the differential motion equations referenced to the body axes in case of a single arbitrary external excitation can read (see figure 3):
0 66 0 22 0 11 d dd dd d z z z y y x x M m J t F m m t v F m m t v (8)where: m – ship‟s mass, Jz – inertia moment, m11,
m22, m66 – surge, sway added masses and yaw added inertia, Fx0, Fy0, Mz0 – components of the
external excitation force.
M 0
F
n x y dFo PP 0 Fr
PP r Fig. 3. Symbol definitions for single force application Rys. 3. Znaczenie symboli w zagadnieniu pojedynczej siły
Since considered is here an initial evaluation of ship motions i.e. covering the early period when a ship movement is just started, it is assumed that for null initial velocities, and being still very low in subsequent time instants, the centrifugal terms and Munk moment, and all other hull hydrodynamic forces to the right side of (8) are negligible. The moment Mz0 resulting from the application point of
0
F , allowed in planar problems of rigid body
mechanics to be treated as a scalar magnitude, follows: 0 0 0 0 0 0 0 F F y F x z r F x F y F M (9)
For constant r and F0 F the below is yielded: 0
t m J F y F x t m m F v v t m m F v v z x F y F z z y y y x x x 66 0 0 0 0 22 0 11 0 (10) and finally:
22
0 0 0 0
66 0 x F y F z y z y PP m m x F y F m J F v x (11)
11
0 0 0 0
66 0 x F y F z x z x PP F y F x m m m J F v y (12)
2
0 2 0 2 11 2 0 2 22 2 0 11 22 0 0 , cos y x y x y x F F m m F m m F m m F F rPP F0 (13) It shall be underlined that expressions (11–13) are time independent. If at least one of the following cases is true: added masses, longitudinal m11 and transverse
m22, are equal to each other (real for a symmetric body in both axes x and y);
motion takes place in vacuum or in a medium of low density (e.g. air), as leading to low added masses;
external force F is perpendicular or parallel to 0
the ship,
then the angle
r ,PP F0
is right or very close to right.The conditions specified directly above just constitute the applicability conditions of the method proposed by Vasco Costa [1] for assessing the pivot point position in practical ship berthing with tug assistance. These assumptions are not published, little known, not so obvious and often omitted while solving different manoeuvring problems, thus sometimes leading to significant errors in the achieved results. The origin of the Vasco Costa formulation are the well known basics of rigid body mechanics, by default neglecting the water medium. In such circumstances the pivot point is really lying on the normal n to the force F that is 0
passing through the body axes origin M, at a distance dependent on the force arm dF0 and the
ship radius of gyration, as stated in [1].
In real-world, taking into account typical values for added masses, the maximum deviation of pivot point radius-vector r from the normal n reaches PP
approx. 20. For a positive product Fx0Fy0 the angle
r ,PP F0
is acute, while for a negative value (the case in figure 3) an obtuse angle is a matter of fact.
Introducing radiuses of gyration, but of the so-called hydrodynamic type (i.e. with an inclusion of added masses): 22 66 22 m m m J r z zh and 11 66 11 m m m J r z zh (14)
one can obtain: 2 0 0 4 22 11 0 2 22 2 0 2 0 2 0 4 22 11 2 0 0 2 22 1 1 1 x y zh zh F zh y x y zh zh x F zh PP F F r r d r F F F r r F d r r (15)
The expression (15) is fundamentally very similar to the relationship quoted by Vasco Costa [1], except that on the right side where there is a square root as dependent on the ratio of both components of F and the ratio of the above two 0
hydrodynamic radiuses of gyration.
The hydrodynamic radius of gyration rzh22 can
practically be considered as independent from the nautical area restrictions – horizontal (effect of water width) and vertical one (effect of water depth). Obviously, this is true if the transverse added mass m22 increases in the same degree with the added inertia m66, as the area is more restricted. For typical ship hulls, m11 is relatively low (order of a few percent of ship‟s mass), but m22 is close to the ship‟s mass. Therefore:
2 22 11 22 22 11 m m m m m m m r r zh zh (16)
For the case of null longitudinal force the below is yielded: 0 2 22 F zh PP d r r (17)
It shall be stressed once more that Equations (10) and the final formula (15) are here valid up to a certain time or change of ship‟s heading from the movement startup as contributing to low centrifugal and hydrodynamic forces.
Ship under lateral speed and transverse force
Pure lateral motion (i.e. without forward and angular velocity) is a very frequent case in the navigational practice, especially during berthing manoeuvres of large ships. There are lots of possibilities for developing it that may be more or less governed by human. This can happen due weather conditions (wind, wave, current) or by cautious and purposeful actions of a ship‟s operator (through the use of tugs, lateral thrusters, stern
rudders, propeller “paddle-wheel” effect, etc.). In case of the most interesting transverse external force, i.e. not moving a ship ahead or astern, refer to figure 4, Equations (8, 11 and 12) take the form (ypp = 0):
0 0 66 90 0 0 22 11 d d 5 . 0 d d 0 d d y F z z fyhm y y y y yH y x F x m J t c v LTv F F F m m t v m m t v (18)
m m
t J F x t m m c v LTv F v v v x z y F fyhm y y y y z y y PP 66 0 0 22 90 0 0.5 (19)where: FyH – hull lateral force, – water density,
T – ship‟s draught, cfyhm90 – hull lateral force
coefficient at drift angle 90.
M F0
vy
PP
Fig. 4. Case of a ship with lateral velocity Rys. 4. Przypadek statku z prędkością poprzeczną
Hence:
t F m m v x r F c v LTv t F m m v x r x y y F zh y fyhm y y y y F zh PP 0 22 0 2 22 0 90 0 22 0 2 22 1 0.5 (20)where for getting the last simplified but practically accurate formula (within a reasonably short period t) the usual lateral velocity not higher than 0.5 m/s and available external force in the range of 50 t has been assumed. For such velocity and force the hull related damping can be often neglected.
Equation (20) is sensitive to signs of input arguments in terms of lateral velocity, external force and its application point versus the midship section. The noticed difference with regard to the
former formula (17) lies in a corrective factor embraced by the square bracket. This multiplier is much larger than unity and reflects the elongation of pivot point radius-vector as compared to an initially stationary ship. It shall be mentioned that for a pure translational (without rotation) motion the instantaneous pivot point is in infinity.
Turning a ship with null forward velocity
Typical situations of ship turning at spot are shown in figure 5. It is very often requested to keep here a constant rate of turn. Such manoeuvres happen in turning basins or close to a berth.
To secure the realization process of turning a ship at spot the resultant force of parameters Fx0,
Fy0, Mz0 shall be exerted on a ship that meet the
following steady phase motion equations:
0 66 0 22 0 22 11 d d 0 d d 0 d d 0 z zH z z y yH y x z y m x M M m J t F F m m t v F v m c m m m t v (21)where: cm – transverse added mass empirical
(viscous) reduction factor, MzH – hull yaw moment
(after extracting the Munk moment). This ultimately leads to:
m mzhm PP z z m fyhm PP z y z PP m x c x' L T L M c x' L LT F L x' m c m F , 1 5 . 0 , 1 5 . 0 2 2 2 2 0 2 2 2 0 2 22 0 (22)where cfyhm and cmzhm are hull hydrodynamic
coefficients as usual functions of drift angle and modified dimensionless yaw velocity m (compare
to Equation 7) as defined by:
2 2 2 2 1 z z z xy z m L v L , m 1 , 1 (23)
However, both and m for the motion cases
examined are to be expressed now in terms of the pivot point position:
x'PP z
90sgn ,
2 1 sgn PP z m x' (24)From the last two Equations in (22) the fol-lowing property is concluded:
PP
fyhm PP mzhm y z x' c x' c F L M 00 (25)that means the application point or arm for the transverse component of F has to coincide with 0
the center of the hull hydrodynamic lateral resistance. The latter, described by the right side of (25), can be in general placed outside the hull what suggests an existence of additional hydrodynamic couple of forces. If the relationship: 5 . 0 5 . 0 0 0 y z F L M (26)
is sustained, then the turning with particular value of x'PP can be completed with a single source of
external excitation (e.g. a single conventional tug) positioned between the ship‟s bow and stern at (in ship‟s length units from amidships):
0 0 0 y z F L F M x' (27)
The value of transverse force Fy0, in accordance
with the second Equation in (22), will effect the rate of turn.
Whereas the range for x'F0 given by (26) can not
be satisfied, it is indispensable to employ two
M vx = 0 PP z M = PP z M vx = 0 PP z vx = 0 M z vx = 0 PP M vx = 0 PP z
Fig. 5. Turning at spot Rys. 5. Obrót w miejscu
parallel forces (sometimes of opposite direction) distributed as to produce an additional couple of forces that contributes to the independent moment increase. Thanks to this, the arm of external force can be virtually stretched.
The longitudinal component Fx0 is required to
compensate the centrifugal force arising from turning a ship around a point other than the midship point. For this purpose the ship‟s own propulsive power can be used.
The use of tugs (one or more) for large vessels as to generate the required external force and im-portantly its arm in view of the specific spot turning manoeuvre is subject to some limitations. These are obviously associated with technically possible con-tact points (of discrete nature) around the ship‟s hull for tugs application. Either in pushing mode, where tugs are allowed in specially marked loca-tions coinciding with the hull structural reinforce-ment (generally on bulkheads) or in pulling mode, where tug operation is restrained by the arrange-ment of ship‟s fairleads.
The mentioned application of two independent parallel forces, as providing some flexibility, would certainly solve the problem of discrete attachment points for tugs. The selection process of such forces is defined by (the left side is already known):
2 2 1 1 0 2 1 0 y F y F z y y y F x F x M F F F (28)
Transients of motion under single external transverse force
The transient phase of motion, particularly in view of the instantaneous pivot point alteration, from a motionless ship till its steady turning when subject to external constant transverse force has lots of cognitive advantages, both in theory and prac-tice. Investigations into this intermediate stage, as to supplement and span the analytical results just achieved in previous chapters, can only be realised through numerical simulation (solution) of motion differential equations with full account of hull hydrodynamic forces and moment, and centrifugal terms as well.
These Equations can be written as follows:
0 0 11 22 66 0 11 22 22 11 d dd dd d y F zH y x z z y yH z x y xH z y m x F x M v v m m m J t F F v m m m m t v F v m c m m m t v (29)Introducing the dimensionless hull hydrody-namic forces in four quadrants and added masses for a large tanker, 290 m in length, according to Oltmann & Sharma [2], and keeping the original basic particulars of this ship, the histories as pre-sented in figures 6–8 are obtained. The simulation comprises cases of applying a transverse force of 360 kN (ca. 36 t) ahead of the midship and towards the starboard side every one tenth of the ship‟s length up to the bow (i.e. from +0.0L to +0.5L). Only the first 20 minutes of turning is included. Since the passing time is not meaningful for ship navigators – manoeuvring can be performed either fast or slowly while the navigational situation in terms of angular (heading) and linear (distance) position against fixed objects remains the same. For this reason, the motion velocities are also plot-ted in figure 6 versus the instantaneous heading . The domain of ship‟s heading is maintained for displaying results of the most interesting pivot point location components as well (Fig. 7).
0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 1200 0 4 8 12 16 0 200 400 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 0 60 120 180 240 300 360 0 0.2 0.4 0.6 0.8 1 0 60 120 180 240 300 360 0 4 8 12 16 0 60 120 180 240 300 360 vx[m/s] vy[m/s] z[/min] t[s] t[s] t[s] vx[m/s] vy[m/s] z[/min] [] [] [] xF0[L]= +0.5 +0.3 +0.2 +0.1 +0.5 xF0[L]= +0.1 xF0[L]= +0.5 +0.4 +0.3 +0.2 +0.1 xF0[L]= +0.5 +0.4 +0.3 +0.2 +0.1 +0.5 xF0[L]= +0.1 +0.3 +0.0 +0.3 +0.5 +0.3 +0.2 +0.1 +0.4
Fig. 6. Large tanker motions versus time and heading Rys. 6. Ruchy składowe w funkcji czasu i kursu
0 0.5 1 1.5 2 2.5 0 60 120 180 240 300 360 -1 -0.8 -0.6 -0.4 -0.2 0 0 60 120 180 240 300 360 xPP[L] t[s] [] [] xF0[L]= +0.5 +0.3 +0.2 +0.1 +0.5 xF0[L]= +0.2 +0.3 +0.4 yPP[L] +0.4 +0.1
Fig. 7. Large tanker pivot point change vs. heading Rys. 7. Zmiana położenia środka obrotu względem kursu
It can be concluded that the analytical evaluation of pivot point position, especially with regard to its longitudinal component xPP, under assumptions and
approaches of sub-chapter 3.1 (as aforementioned, intended to be a firm theoretical background for supporting and revising the Vasco Costa formulation), gives practically acceptable results up to 10–20 degrees of the heading change. The trajectories of the tanker body axes origin with superimposed vectors representing the attitude of hull contours is illustrated in figure 8.
An additional analysis of (29), mental or through simulation, leads to the fact that the magnitude of force Fy0 does not affect the general
pattern of time histories of linear and angular velocities (namely their relative changes with the force application point). The only absolute values (y axis scale) are modified. However, when complex parameters of motion, in form of various ratios of basic three velocities (vx, vy, z), are
considered and plotted versus ship‟s heading, these are independent from the external force value at all. We can mention here the following charts: x'pp(),
y'pp(), (), m(), x0(), y0(). Also other combined charts are completely preserved: x0(y) – trajectory, m() – drift-dimensionless yaw
corre-lation. -2 -1.5 -1 -0.5 0 0.5 1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 -0.5 0 0.5 1 1.5 2 x0[L] y0[L] xF0[L]=+0.1 x0[L] y0[L] xF0[L]=+0.2 x0[L] y0[L] xF0[L]=+0.3 x0[L] y0[L] xF0[L]=+0.4 x0[L] y0[L] xF0[L]=+0.1
Fig. 8. Large tanker tracks under single external force Rys. 8. Trajektorie środka statku przy różnych położeniach zewnętrznej siły poprzecznej
Transients of pivot point in ship stopping and moving astern
The propeller when working astern (at whatever direction of a ship movement, ahead and / or astern) produces a side effect called the “wheel effect” or less properly the propeller lateral thrust. The phenomenon is mainly connected with the inflow to the ship‟s stern due to the reverted propeller race. Such twisted slipstream attacks various horizontal sections of ship‟s stern from both sides and genera-tes a resulting lateral force (usually proportional to the developed negative longitudinal thrust) from the pressure distribution around the ship‟s stern hull.
In figure 9, as based on Artyszuk [3], the pivot point longitudinal position for a chemical tanker is shown, as making different crash stop manoeuvres in deep water up to reaching a certain negative (astern) speed. The initial speeds of 14.1 kt, 4 kt,
and 2 kt were assumed and combined with various throttle settings – full, slow, and dead slow astern. Within the range of ship‟s positive speed (up to nearly the ship‟s rest) the pivot point goes in very similar way, independently from the astern throttle, then some “delay lags” appear due to the different stopping efficiency (in terms of time and distance) in view of the negative thrust allocated to particular throttles. When a ship starts a significant sternway the pivot point location changes its sign from the positive (forward) to a negative one (aft). Since the propeller lateral thrust continues to keep its direction (to portside) and hence the accompanied contribution to the negative ship‟s transverse velocity, the lateral projection of the ship‟s centrifugal force („mvxz‟) seems to be primarily
responsible for this behaviour. An in all cases xPP
surprisingly converges the same negative value.
Transients of pivot point in ship standard and accelerated (“kick ahead”) turns
The statements raised at the end of Section 2 are confirmed in the following simulation of ship
turning manoeuvres in deep water–refer to Arty-szuk [4]. The transients of pivot point location (and consequently the steady values) are practically a function of only helm angle. The “standard” (or classical) turn means a ship is initially at steady approach speed (the engine / propeller throttle “coincides” with the forward speed) and only the stern rudder order is applied. So being independent of the ship initial steady speed, where in any case of low or high speed the same propeller loading is experienced (as depicted by the propeller advance ratio J or loading coefficient cTh, which are constant
for any steady speed regime), the pivot point is essentially lower for higher rudder angles.
With regard to the accelerated turns, when additionally the propeller / engine rpm increase is ordered, the pivot point in the range of medium helm angles, irrespective of the rpm acceleration degree, lies very close to that of classical turn at maximum rudder. And the strong “kick ahead” manoeuvres (with maximum rudder) move the pivot point towards midship section, where the rudder mainly “acts through” the yaw velocity.
0 0.2 0.4 0.6 0 1 2 3 4 0 0.2 0.4 0.6 0 4 8 12 16 0 0.2 0.4 0.6 0.8 -40 -20 0 20 40 s'[-] x' PP[-] x' PP[-] s'[-] x' PP[-] [] standard turn (up to 7min) accelerated turn (up to = 110deg) steady turning at DSAH or SFAH DSAH 15 DSAH 35 SFAH 35 stbd port SFAH 15
1) SFAH 35 (or SFAH-SFAH 35) 2) SAH-SFAH 35 3) DSAH-SFAH 35 4) DSAH-SFAH 15 5) SAH-SFAH 15 1) 2) 3) 4) 5)
Fig. 10. Standard (constant throttle) and accelerated turns with stern rudder as affecting the pivot point
Rys. 10. Środek obrotu w cyrkulacjach standardowych i przy-spieszonych („manewrach silnych”)
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 1 2 3 4 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 4 8 12s 16 '[-] x' PP[-] -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 2 4 6 8 x' PP[-] s'[-] x' PP[-] s'[-] SFAH=14.1[kt] 4[kt] 2[kt] DSAS SAS FAS DSAS SAS FAS DSAS SAS FAS
Fig. 9. Pivot point during ship stopping and moving astern Rys. 9. Środek obrotu podczas hamowania aktywnego i ruchu wstecz
Transients of pivot point in ship bow thruster turning
A very interesting case in the aspect of the pivot point is the turning manoeuvre under a bow thruster action, if available. A ship is behaving to some extent in a similar way as observed in the former stopping / aft acceleration manoeuvre, where the
pivot point changes from the initial forward position (established by the propeller lateral thrust at the ship‟s stern) to the aft position (during the phase of astern acceleration, as affected by the hull transverse force and centrifugal force lateral component).
In the following such initial conditions of the turning manoeuvres are considered (deep water in all cases), that are different in view of the pivot point change: ship‟s at rest (vx = 0), forward speed
0.5 m/s (~1 kt) and 1.0 m/s (~2 kt). The latter two will be combined either with the engine throttle (here, the throttle is adjusted at the begin and fixed to keep the initial speed if a straight course run is required, obviously the throttle setting value is relatively low) or without the throttle. The throttle action will of course support the usual forward speed increase – as far as the pivot point is aft, the centrifugal force longitudinal component is directed forwards and slightly accelerates a ship [5].
-200 -100 0 100 200 300 -100 0 100 200 300 [m] [m] speed 0.0m/s -200 -100 0 100 200 300 -200 -100 0 100 200 [m] [m] speed 0.5m/s with throttle -200 -100 0 100 200 300 -100 0 100 200 300 [m] [m] speed 1.0m/s with throttle
Fig. 11. Ship tracks in case of turning with bow thruster – zero and non-zero initial forward speed (throttle initially adjusted & fixed according to speed)
Rys. 11. Trajektorie cyrkulacji na sterze strumieniowym przy zerowej i niezerowych prędkościach początkowych ruchu – silnik główny pracuje
-200 -100 0 100 200 300 -100 0 100 200 300 -200 -100 0 100 200 300 -200 -100 0 100 200 [m] [m] speed 0.5m/s w/out throttle [m] [m] speed 1.0m/s w/out throttle
Fig. 12. Ship tracks in case of turning with bow thruster – non-zero initial forward speed (but without throttle)
Rys. 12. Trajektorie cyrkulacji na sterze strumieniowym przy niezerowych prędkościach początkowych ruchu – silnik główny nie pracuje
speed 0.5 m/s with throttle speed 1.0 m/s with throttle speed 0.5 m/s w/out throttle
Figures 11 and 12 consist of tracks obtained with the mentioned modes of movement. The pivot point is displayed in figure 13.
-0.2 -0.1 0 0.1 0 60 120 180 240 300 360 -0.2 -0.1 0 0.1 0 60 120 180 240 300 360 -0.2 -0.1 0 0.1 0 1 2 3 4 5 -0.2 -0.1 0 0.1 0 2 4 6 8 10 s'[-] x' PP[-] x' PP[-] s'[-] x' PP[-] []
BT turn with throttle
0 m/s 0.5 m/s
1m/s
BT turn w/out throttle
0 m/s 0.5 m/s 1m/s x'PP[-] [] 0 m/s 0.5 m/s 1m/s 0 m/s 0.5 m/s 1m/s
Fig. 13. Pivot point in bow thruster turning versus dimensionless distance and heading
Rys. 13. Środek obrotu przy wykorzystaniu steru strumie-niowego w funkcji bezwymiarowej odległości i kursu
In the case of higher forward velocity, around 1m/s, the pivot point is shifting forward of the midship section (positive values when the heading passed 30 to 60 degs) – the lateral component of the centrifugal force, as proportional to the forward velocity (also refer to Section 5), is able to revert
the sign of lateral velocity and thus the pivot point longitudinal coordinate. However, the maximum forward location is 5% of the ship‟s length, that can practically be deemed to match the centre (midship section) of a ship. It shall be emphasised that also the initial pivot location, i.e. just after the turn initiation, is not so far in the aft direction as could be expected by mariners – 10 to 15% of a ship‟s length counted from the midship. This is easily to be analytically (theoretically) proved using the derivations worked out in Section 3.1 or using the original Vasco Costa approach – the further is the lateral force (for the bow thruster this is pretty far) the closer to the midship is the pivot point. The long lever arm in such a case “produces” a relati-vely higher yaw velocity as compared to practically the same lateral velocity, hence according to Equation 5 the pivot point is near the midship.
Final remarks
The presented study has dealt with most of the instantaneous pivot point aspects. The future research shall concentrate on a direct utilization of the pivot point properties (in terms of rules for its dynamic location) under given external distur-bances in various manoeuvring decision support systems or in manoeuvring automatic controllers (including the demanded track optimisation).
0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 s'[-] 10 vx[m/s] s'[-]
BT turn with throttle
0 m/s 0.5 m/s
1m/s
BT turn w/out throttle
0 m/s 0.5 m/s
1m/s
vx[m/s]
Fig. 14. Longitudinal velocity during turning with bow thruster
Rys. 14. Zmiana prędkości wzdłużnej podczas obrotu statku z wykorzystaniem steru strumieniowego
These disturbances can be classified into active forces (of steering devices) and passive ones (ori-ginated from weather and / or hydrodynamic inter-actions in case of navigational area restrictions).
The majority of modern control systems are one- -dimensional, i.e. they generally control the yaw movement, while the lateral motion is treated as a side effect. This is an unacceptable practice for e.g. classical single-screw ships without a bow thruster, in which the pivot point remains some-where at the bow almost in any regime of manoeu-vring, thus essentially restraining its manoeuvring ability. This should be fully taken into account when planning and executing a passage, especially in very restricted areas.
References
1. VASCO COSTA F.: Berthing Manoeuvres of Large Ships.
The Dock and Harbour Authority, 1968, 48 (569–March), 351–358.
2. OLTMANN P., SHARMA S.D.: Simulation of Combined Engine and Rudder Maneuvers Using an Improved Model
of Hull-Propeller-Rudder Interactions. In 15th Symposium on Naval Hydrodynamics, Hamburg, 1984, Sep 2–7. 3. ARTYSZUK J.: Sensitivity of Ship Berthing Manoeuvre
upon Propeller Lateral Thrust. In International Scientific Conference “The Transport of the 21st Century”, Warsaw, 2004, Sep 20–22.
4. ARTYSZUK J.: The Theory of “Kick Ahead” in Ship Manoeuvring. In International Scientific Conference “The Transport of the 21st Century”, Warsaw, 2004, Sep 20–22. 5. ARTYSZUK J.: Ship Sway / Yaw Motions while Turning
with Bow Lateral Thruster. In 15th International Confe-rence on Hydrodynamics in Ship Design, Safety and Operation HYDRONAV „2003, Ship Design and Research Center, Gdańsk, 2003, Oct 22–23.
Others
6. ARTYSZUK J.: A Reduction of Sway Added Mass in the
Surge Manoeuvring Motion Equation. In Proc. of 16th International Conference on Hydrodynamics in Ship Design / 3rd International Symposium on Ship Manoeuv-ring HYDRONAV‟05 (HYDMAN‟05), Gdańsk / Ostróda, Poland, 2005, Sep 7–10.
7. VASCO COSTA F.: The Berthing Ship. The Effect of Impact on the Design of Fenders and Berthing Structures. London: Foxlow Publications Ltd. 1964.
