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Reprinted: 13-09-2001Website: www.shipmotions.nl

Report 158S, March 1972,Netherlands Ship Research Centre TNO,Shipbuilding Department,Schoemakerstraat 97, 2628 VK DelftThe Netherlands.

Prediction of Ship Manoeuvrability

G. van Leeuwen and J.M.J. Journée

Delft University of Technology

Summary

Static sway- and oscillatory yawing tests with a 1:55 model of the 50.000 DWT tanker"British Bombardier" are discussed.The principal purpose of these tests was to determine the coefficients of a non-linearmathematical model to predict a number of standard manoeuvres, which were earlierperformed with the full-scale ship before. Clarke (1965) describes the results of these full-scale manoeuvres.The mathematical model chosen is based on the Abkowitz Taylor-expansion of thehydrodynamic forces and moments; see Abkowitz (1964). However, there is a principaldifference with respect to the variables involved, which enables a more correct description ofsome non-linear phenomena. Comparison of the predicted manoeuvres with thecorresponding full-scale data shows a rather good agreement.For comparison purposes some experiments have been performed as well with a small modelof the same tanker ( 100=α ). However it is found that scale effects, due to the very lowReynolds number, have a considerable influence on the hydrodynamic derivatives. Someinteresting additional figures are given showing the contributions of each term of themathematical model during a turning circle manoeuvre while also the change of the stabilityroots during this manoeuvre is plotted.

1 Introduction

At the Delft Shipbuilding Laboratorymodel tests were performed to determinethe coefficients of a non-linearmathematical model, which describes thestill water manoeuvrability properties of a

ship. Two types of tests are performed.First the static towing tests with a constantdrift and rudder angle and secondoscillation tests to determine the addedmass effect in the swaying motion and thehydrodynamic derivatives of the yawing

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motion. The main particulars of both shipand models are summarised in Table 1.

Table 1 Main Particulars of Ship andModels

The principal purpose of the tests is toobtain information on the possibility topredict the principal manoeuvres of a ship,using an adequate mathematical model andmodel experiments to determine thecoefficients of this model. For comparisonpurposes, Clarke (1965) performed full-scale trials.

Figure 1 Ship Model under Oscillator inTowing Tank

The model experiments have beenperformed at four different initial speedconditions, corresponding with a constantpropeller power each. Originally theresults of these four sets of tests have beenkept separated, because it was assumedthat some of the non-dimensionalcoefficients could change with the Froudenumber, based on the initial speed. In thatcase a set of coefficients which would bedifferent for each initial speed would havebeen found. Clarke (1965) gives the resultsof this tentative analysis.During the analysis of the experimentaldata, it was found that the principaldifferences between some of the non-dimensional coefficients could bedescribed effectively by considering thelocal water velocity near the rudder. In thisway the apparent Froude-effect in thesecoefficients rather could be called a"power-effect" while the differences in theother coefficients were not consideredsignificant, in view of both the availableinformation and the accuracy of themeasurements.It is not to be expected however, that thismethod of describing the phenomenamentioned above will hold for other ships.Especially when the Froude numberbecomes high e.g. 0.30 or higher, it is notvery likely that there will be no realFroude effect in some hydrodynamicderivatives if they are compared with thecorresponding values found at a Froudenumber of 0.10 e.g.Provided a certain mathematical model hasbeen adopted, there are several methods todetermine the coefficients of such a modelby model tests. The main problem is tofind out how far uncoupling of the threemotions in the horizontal plane is allowed.Of course, during an actual manoeuvre themotions are always coupled and even if themathematical model contains terms, whichare meant to describe the cross-couplingeffects one is not sure about the way theseeffects have to be determined. The mostconvenient method might be to performfree running model tests and find the

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coefficients of the mathematical model byanalysing the data concerning position andcourse of a number of representativemanoeuvres. In that case all variablesremain coupled in the natural way. Ingeneral however too less room is availablefor these kinds of manoeuvres, so that oneis pressed to find an acceptable alternative.In this respect the forced horizontaloscillation test provides an alternativesolution. On the other hand, the problem ofuncoupling the motions as mentionedabove is introduced. Another practicalproblem is to choose the rightcombinations of oscillator frequencies andamplitudes. Considering an actualoscillatory motion, due to the harmonicmotion of the rudder, it is found that thecombinations of frequencies andamplitudes involved in these motionscannot easily be simulated by horizontaloscillation tests. This is because the actualrange of amplitudes is of the magnitude ofone half to many ship lengths. In mosttowing tanks sufficient width is notavailable in this respect. These problemsare considered in greater detail in by VanLeeuwen (1969a).The conclusion is that most horizontaloscillation tests involve an unnaturalrelation of amplitudes and frequencies. Inother words the ratios of velocity andacceleration amplitudes are quite differentfrom the actual values. Apparently this isno problem, because most of themathematical models which are now in usedo not contain any cross-coupling termsbetween velocities and accelerations. Thisdoes not imply, however, that such cross-coupling effects could not be introduced, ifthe range of ratios of these variables isextended too far.

Some final remarks on the mathematicalmodel.In the course of the years a lot of studieswere devoted to this subject, starting withDavidson and Schiff (1946), whodescribed a model based on the linearequations of motion. This set of equations,

which originally involves three equations,describing the surging, swaying andyawing motion, has been used by severalauthors, though unfortunately omitting thesurge equation. Abkowitz (1964) hasproposed one of the most extended non-linear mathematical models. In this modelthe hydrodynamic part of the forces isexpanded into a Taylor series of thevariables concerned. This principle is veryuseful, particularly if the constants of themodel are to be determined by the analysisof forced model tests, because allimaginable hydrodynamic effects inprinciple can be described in this way. Animportant question involved in this Taylorexpansion is up to what degree it has to beextended to be sure that the principal non-linear hydrodynamic effects are describedcorrectly. On the other hand it isquestioned to what extend it is necessaryto retain a great number of terms in such amodel for a reasonable accuratedescription of manoeuvres, even if theseparate hydrodynamic effects involvedcan be measured with forced model tests.In other words it is suggested that, on theground that during an actual manoeuvrethe ratios of the variables satisfy just onerelation, it might be possible to describethe joint effect of a number of terms byone term only. In this way a much simplermathematical model would arise, thecoefficients of which had to be consideredfunctions of the coefficients of the originalmodel. Van Leeuwen (1970) describedsome simple non-linear models based onthese grounds.A disadvantage of such simplifiedmathematical models is that its coefficientscannot be determined by uncoupling thethree motions which means that they canonly be derived from free running tests,either full-scale or model tests. Forpractical purposes however, such assimulation studies and automatic piloting,these simplified non-linear models can beapplied successfully.The principle of the mathematical modelused for the present model-tests is, apart

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from some details, the same as has beenused by Abkowitz. The way in whichAbkowitz treats the influence of a changeof the forward speed, however, bringsabout that no insight is gained into thephysical background of this influence.In this paper, the hypothesis is used that ifthe motions are similar, regardingvelocities and accelerations, the principalhydrodynamic forces on the hull areproportional with the square of theinstantaneous forward speed. For theforces, which mainly depend on theeffective angle of attack of the rudder,proportionality with the square of the localwater velocity is assumed.The general concept of this hypothesis isconfirmed by the model experiments. Inthe following chapter this will bediscussed in more detail.

2 Equations of Motion

2.1 Introduction

If we form a mathematical model and westart from the fact that the hydrodynamicforces are functions of the velocities andaccelerations involved in a motion, we canexpand these forces, as has been done byAbkowitz (1964), in a Taylor series ofthese velocities and accelerations. On theground of considerations of magnitude wecan ignore the terms whose order is higherthan e.g. the third. There are someobjections to this procedure, however.Considering a term proportional to thethird power of e.g. the angular velocity theomission of the fourth order terms meansthat the contribution of this term,regardless the forward speed, remainsproportional with the third power of theangular velocity. From model experimentsit is known that this - and similar terms -are reversed proportional with the forwardspeed. Neglecting this speed dependenceconsequently corresponds withunderestimating the non-linear effects

described by these third order terms in thecase of speed reduction. For a speedreduction of 50 per cent, such a non-lineareffect is underestimated by a factor two.Another objection, though of lessimportance, is that if the separatevelocities are considered as lateral,forward and angular velocity, theparticular role played by the forward speedbecomes hardly apparent.In section 2.3, a different basis has beenchosen for the mathematical model, thehypothesis mentioned in chapter 1 beingused.The effects of the fourth degree terms,mentioned above, are involved in the thirddegree Taylor expansion of the forces, ifthey are considered to be functions ofcharacteristic variables, describing thesimilarity of the motions.It is emphasised however, that the conceptof this is not new, because also Davidsonand Schiff (1946), Nomoto (1957) and Edaand Crane (1962) already paid attention tothe importance of these variables. Bothearlier work and the present investigationjustify the adoption of the hypothesisconcerning the forces.

2.2 Components of HydrodynamicForces

The equations of motion describing thebalance of forces and moments during astill water manoeuvre can be written asfollows (see also Figure 2):

( )

( ) XvrUm

NrIYrUvm

x

zz

x

=−⋅

=⋅=+⋅

&&

&

Equation 1-a,b,c

where Y represents the component of thehydrodynamic forces perpendicular to theship and N the corresponding moment,while X represents the component of

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these forces acting in longitudinaldirection.

Figure 2 Co-ordinate System andDefinition of Variables

The sum of the hydrodynamic forces canbe divided into three groups:1. The first group contains the

components, which depend on thecondition of motion of the shipwithout propeller and rudder. Thevariables involved in this case will bediscussed in section 2.3.

2. The second group contains the forces,which act on the rudder. They dependon the effective angle of attack of therudder and as this quantity dependson the ship's condition of motion,these components will depend on thevariables of the first group as well ason the rudder angle itself.

3. The third group contains the forcecomponents, which are caused,among other things, by the change ofcirculation around the ship, due to therudder deflection. In general, these

components are considered to be theresult of the fact that the sum of thehydrodynamic forces is not obtainedby the superposition of the forcesacting on the hull and those on therudder, which may be approximatelytrue for the side forces on sailingyachts.

Concerning the longitudinal force balance,a fourth group has to be considered, whichinvolves the forces due to the resistanceand the change of thrust caused by speedloss during manoeuvring. This groupdetermines the difference between theforward speed of the centre of gravity andthe speed of the water near the rudder.

2.3 Hypothesis Concerning theHydrodynamic Forces Acting onthe Manoeuvring Ship

The hypothesis mentioned in the precedingchapter, concerning the first group forces,is formulated as follows: If two similar motions of a certain ship(or model) are compared, than theseforces will be proportional to the square ofthe forward speeds involved, providedthese speeds do not differ too much.This hypothesis is mainly based on modelexperiments, though the results of full-scale manoeuvres, executed at differentforward speeds, provide an indication for itas well.The explanation for the usefulness of thishypothesis can be derived from theprincipal importance of the inertia forcesand further from the small role played bythe generated waves in a certain speedrange.If comparing the similar motions of a shipand her model, the wave patterns will onlybe similar if the Froude numbers in thiscase are equal, due to the constant value ofthe acceleration of gravity. On the groundof the small influence mentioned of thegenerated waves in a certain speed-range,the hypothesis is to be applied in this casewithin a restricted range of Froude

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numbers, while the forces are proportionalto the factor 22LU .Concerning the forward speed, the upperlimit of the usefulness of the hypothesislogically follows from the increasingimportance of the waves, when the speedis higher.The lower limit, however, cannot bederived only from the decreasing wavegeneration, as this, on the contrary, ratheris a reason to expect its usefulness.Apparently there are other reasons for this,the principal of which probably is theincreasing importance of the frictionalforces, and in general, of the viscouseffects, compared to the inertia forces.The equations of motion of a floatingbody, moving in a horizontal plane, can bewritten in such a form, that the hypothesisis expressed by it. According to thehypothesis the forces, acting on the body,depend linearly on the force unit

225.0 LUρ , so that the forces, divided bythis unit, can only be functions of non-dimensional parameters, which describethe (restricted) similarity of theinstantaneous conditions of motion to becompared.Assuming only the velocities (v, r, U) andthe accelerations (v& , r& , U& ) to play a rolein the equilibrium of forces, the equationsof motion can be written as follows:

( )

⋅⋅⋅⋅⋅

⋅=+⋅

22

2

2*

22

,,,,

21

UUL

UrL

UvL

UrL

Uv

Y

LUrUvm x

&&&

& ρ

⋅⋅⋅⋅⋅

⋅=⋅

22

2

2*

32

,,,,

21

UUL

UrL

UvL

UrL

Uv

N

LUrI zz

&&&

& ρ

( )

⋅⋅⋅⋅⋅

⋅=−⋅

22

2

2*

22

,,,,

21

UUL

UrL

UvL

UrL

Uv

X

LUvrUm x

&&&

& ρ

Equation 2-a,b,c

If the principle of the hypothesis is alsoapplied to the force components of thesecond group, then the local similarity ischaracterised by the actual angle of attackof the rudder, while the forces are thenproportional to the square of the localwater velocity.As the actual angle of attack and themagnitude and the direction of the localwater velocity, these forces may beapproximated as follows:

( )actR YLUY δρ ⋅= 22

21

where RU is the local water velocity and*

3*

21 rpvppact ⋅+⋅+⋅= δδwith:

Uv

v =* and U

rLr

⋅=*

Concerning the components of the thirdgroup, it is assumed that they will mainlydepend on the variables of both the firstand second group. They will partly beproportional to the square of the forwardspeed of the centre of gravity and for therest to the square of the local (rudder)speed.On this basis the mathematical model canbe build up, considering the three groupsto be functions of the non-dimensionalvariables, concerned. The expansion of thethree groups in a third degree Taylor seriesleads to a number of terms a part of whichbeing proportional with the square of theforward speed, while the remaining termswill be proportional to the square of thelocal rudder speed.

Considering e.g. the linear term in *v , thenthe following expression is found:

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{ 2*1

2

21

UvaL ⋅⋅ ρ2*

2 RUva+ ( ) }*232

231 vUaUa R++

1st group 2nd group 3rd group(forces on hull

without rudder andpropeller)

(forces on rudder due toeffective angle of attack)

(interaction rudder-hull;forces due to failing ofsuperposition principle)

The expressions for the other terms willhave similar forms. On this ground themathematical model is to be written in thefollowing form:

( )

( )3

222*

122

232*

132

222*

122

21

21

21

21

21

21

X

XLUXLUvrUm

NLUNLUrI

YLUYLUrUvm

Rx

Rzz

Rx

+

+=−⋅

+=⋅

+=+⋅

ρρ

ρρ

ρρ

&

&

&

Equation 3-a,b,c

In these equations the components markedwith an asterisk contain the termsoriginated from the Taylor expansions ofthe first and second group. The dashedcomponents contain the correspondingterms of the second and third group. Bothasterisk and dash marked components arefunctions of the rudder angle and the fivevariables determine the second ordersimilarity. The longitudinal forcecomponent 3X describes the differencebetween the ship’s straight-line resistanceand the change of thrust due to the speedreduction.On the ground of theoreticalconsiderations and the experience fromearlier investigations a number ofassumptions have been made, whichsimplify the expressions for the variouscomponents. Some of these assumptionshave been investigated particularly, whileothers are not contradicted by themeasurements.

The assumptions concerned aresummarised as follows:1. The forward speed U , as a variable of

Equation 3, can be replaced by itslongitudinal component xU .

Consequently, the variables *v and *rare defined as xUv / and xUrL /⋅

respectively, while the unit *ds isdefined as LdtU x /⋅ .

2. The hydrodynamic lateral forces areindependent of the longitudinalacceleration, and the hydrodynamiclongitudinal forces are independent ofthe sway and yaw accelerations.

3. Non-linear acceleration effects do notoccur in the range of interest.

4. If the ship is on a straight course, theforces due to a certain rudderdeflection are proportional to thesquare or the local rudder speed RU .(This assumption may be consideredthe definition of the quantity RU forthe present investigations).

5. The influence of the rudder rate onadded mass effects is negligible forpractical purposes.

2.5 Set of Equations of Motion

Executing the Taylor expansions of thethree groups of forces and moments andapplying the assumptions indicated above,the equations of motion can be written asfollows:

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( ) ( )

( ) ( )( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )( ){ }

{ } 2'3

2'2'*''2'

2'*

2''

2''*2'

'

2''*

2'2'

2'*2'

2'

2'*

2''3'

2'

'*2''

3'

2'

'*

3'3'

2'

'*3'

3'

2'

'*

''

2'''*'

'

2''*

'2'

2'*'

2'

2'*'

11

11

11

11

111

111

111

111

11

11

11

R

RaaR

vrvr

Rrr

Rvv

Rrrrr

Rvvvv

Rrvvrvv

Rvrrvrr

Rrrrrrr

Rvvvvvv

Rrr

Rvv

Rrr

Rvv

UYYY

UYuYrvu

UYY

ru

UYuYv

uU

YuY

ru

UYYv

u

UYY

vru

UY

uYrv

u

UY

uY

ru

UY

uYv

u

UY

uY

ru

UYumYv

uU

YuY

ru

UYYv

u

UYYm

δδδ

δ

δδ

δδ

δδδδδδ

δδ

δδδδδδδδ

δδδδ

++

++++

++

+

++++

+++

+

+++

++

+

++

++

++

+

+

++

++

++

+

+

+++−+

+++

=

++−

++− && &&&&

Equation 4-a

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )( ){ }

{ } 2'3

2'2'*''2'

2'*

2''

2''*2'

'

2''*

2'2'

2'*2'

2'

2'*

2''3'

2'

'*2''

3'

2'

'*

3'3'

2'

'*3'

3'

2'

'*

''

2''*'

'

2''*

'2'

2'*'

2'

2'*'

11

11

11

11

111

111

111

111

11

11

11

R

RaaR

vrvr

Rrr

Rvv

Rrrrr

Rvvvv

Rrvvrvv

Rvrrvrr

Rrrrrrr

Rvvvvvv

Rrr

Rvv

Rvv

Rrrzz

UNNN

UNuNrvu

UNN

ru

UNuNv

uU

NuN

ru

UNNv

u

UNN

vru

UN

uNrv

u

UN

uN

ru

UN

uNv

u

UN

uN

ru

UNuNv

uU

NuN

vu

UNNr

u

UNNI

δδδ

δ

δδ

δδ

δδδδδδ

δδ

δδδδδδδδ

δδδδ

++

++++

++

+

++++

+++

+

+++

++

+

++

++

++

+

+

++

++

++

+

+

++++

+++

=

++−

++− && &&&&

Equation 4-b

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( )

( ) ( )

( )( )

( ) { }

( ){ } ''*2'*

22'''

2''*

''

2''*''

2'

2''*

2'2'

2'*2'

2'

2'*

'2'

2'*'

1

11

11

1

11

1

uXXuX

UXvu

UXuX

ru

UXuXrv

u

UXmX

ru

UXXv

u

UXX

uu

UXXm

TRR

RR

vv

Rrr

Rvrvr

Rrrrr

Rvvvv

Ruu

+−+

++

+++

+

++++

+++

+

+++

++

=

++−

δδ

δ

δδδδ

δδ

&&&

Equation 4-c

In these equations all variables have beenmade non-dimensional with the initialspeed 0U so that e.g.:

( )( )'*'

'*'

0

'

1

1

1

urr

uvv

UU

u x

+=

+=

−=

Some additional remarks concerning theseequations:a. The side force and moment equations

contain some terms to describe theasymmetrical behaviour of the ship.With zero rudder deflection these

terms are ( ) 2'2'* 1 Raa UYuY ++ and

( ) 2'2'* 1 Raa UNuN ++ while theasymmetrical rudder effectiveness isdescribed by the terms 2δδδY and

2δδδN respectively.b. The longitudinal force component 3X

is divided into two components, thefirst of that describes the balancebetween the original thrust and thestraight-line resistance while thesecond describes the (linearised)increment of the thrust during amanoeuvre.

c. The fourth degree terms, which werementioned in the introduction, are dueto the factors ( )'1/1 u+ , which may be

linearised in the range00.060.0 ' <<− u to '30.284.0 u⋅− .

d. If the change of speed during amanoeuvre is larger than the range inwhich the hypothesis is valid, than thestar and dash coefficients can beconsidered linear functions of thespeed. In that case additionalcoefficients as *

vuY and vuY should beadded. For the present investigation,this appeared not necessary however.

3 Execution of Tests

3.1 Measuring Equipment

The principal property of the measuringequipment is that only harmoniccomponents of the forces are determined.For the present investigation only the firstharmonic components were needed. Thisdetermination is achieved by multiplyingthe forces by tωsin and tωcosrespectively, while these products areintegrated during one or more periods ofthe oscillation. The non-oscillatorycomponents of the forces are determinedby integration of the force during a numberof periods.

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Zunderdorp and Buitenhek (1963) give ina more detailed discussion on the oscillatorand the measuring equipment.The static drift angle adjustment during theoscillation tests is achieved by turning themodel with respect to the connecting lineof the oscillator struts (the “pure yawingline”). This is sketched in Figure 3.

Figure 3 Drift Angle Adjustment ofOscillator

3.2 Determination of Draught andTrim

As the model was restrained from heavingand pitching the draught and trim, asdependent on the forward speed and thenumber of propeller revolutions had to bedetermined.The revolutions adjusted for these testswere estimated from the available full-scale data. A change of the rpm didinfluence neither the mean draught nor thetrim however. In Table 2 the variousdraughts fore and aft for model and shipare given.These draughts have been adjusted for thetests concerned. It is questioned however ifit is correct to restrain the model in verticaldirection during oscillation and other tests.

Table 2 Trim as Function of Speed

A particular investigation might give theanswer, but it is not expected thatundesirable cross-coupling effects woulddisturb the side force measurements, dueto the uncoupling of the swaying andyawing motion. This is also expected withrespect to the rolling motion. As it was notthe purpose of this investigation to find ananswer to this question, it was considereda useful approximation to adjust theconstant draughts, derived from straight-line tests and to restrain the model alsofrom rolling.

3.3 Determination of Resistance andPropulsion Coefficients

The description of the balance betweenlongitudinal resistance and the propellerthrust is partly based on some full-scale ofrpm at 15.5 knots and partly on thepropeller characteristics. From these datathe wake fraction was derived while forthe lower speeds, caused by manoeuvring.This wake fraction was consideredconstant. From the fact that during amanoeuvre the power does not change, theincrement of thrust and the decrement ofrpm could be calculated using thepropeller characteristics. For comparisonpurposes also the full-scale measurementsof these quantities are given in Figure 4and Figure 5.Concerning the other initial speeds, 12.4,9.3 and 6.2 knots, a linear relation betweenthese speeds and the rpm was adopted (seeFigure 6).The increase of thrust and the decrease ofthe rpm during manoeuvres with these

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initial speeds were determined as this wasdone for the 15.5 knots initial speed.

Figure 4 Variation of Thrust with Speedduring Turning Circles at 100 Nominal

RPM at t = 0

Figure 5 Variation of RPM with Speedduring Turning Circles at 100 Nominal

RPM at t = 0

Figure 6 Adopted Initial Speed-RPMRelation

In Figure 7 and Figure 8 the calculatedthrust and torque are plotted for the initialconditions.

Figure 7 Measured and Computed Valuesof Thrust

Figure 8 Measured and Computed Valuesof Torque

It can be shown that on a straight course aparabolic relation between the thrust andthe forward speed exists, provided thethrust curve of the propeller diagram islinearised in the range of interest.Consequently the assumption of aparabolic relation between the longitudinalresistance and the speed is equivalent tothe assumption of a speed-independentthrust deduction fraction. This number wasestimated to be 0.20.On this basis the resistance coefficient

*RX was calculated:

5* 100.54 −⋅−=RX

12

while for the effective thrust incrementcoefficient '

TX was found:5' 100.25 −⋅−=TX

In Table 3 the computed rps, concerningthe various conditions, are summarised.

Table 3 Computed Propeller Rate Values

It is noted however that these data arecorrected for the differences between thefull-scale and model propeller (U 218 - B5.60 respectively).The rps values, adjusted during the modeltests, are not the same as given in Table 3,however, as originally these values werebased on the full-scale relation betweenthrust and speed as given by Clarke(1965). During the analysis of the test datathis relation did not appear to correspondwith the full-scale data of rpm and thepropeller diagram, consequently nor didthe rpm-U relation. This is shown inFigure 9.

Figure 9 Discrepancies between theMeasured Forward Speed and the Speed

Derived from Thrust and TorqueMeasurements (Full-Scale)

From this figure it is assumed that the full-scale speed measurements concerned arenot correct.

As, in addition, the coefficients, whichdescribe the balance between resistanceand thrust during a manoeuvre based onthis relation, resulted in very largedifferences between the manoeuvrescomputed and those executed on full-scale,the linear relation between rpm and speedwas adopted.The coefficients of the force and momentcomponents 2Y , 2N and 2X have beencorrected as far as necessary, which waspossible because their relation with therpm was known from the modelexperiments.

3.4 Determination of Rudder SpeedUR

The quantity RU has been derived fromthe straight-line tests with constant rudderangle. These tests were executed for the 10combinations of speed and rps, while ineach case the rudder angle was varied from36 degrees port to 36 degrees starboardwith steps of 9 degrees. A formaldescription of the water velocity near therudder has been based on the impulsetheory with respect to the propeller.According to this theory the water velocityat a certain distance from the propellerdisk can be written as (see Figure 10):

'''aeR CVU ⋅+= µ

Equation 5

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Figure 10 Change of Water Velocity nearthe Propeller

In both cases the prime denotes making itnon-dimensional with the initial speed 0U .If the curves of the propeller torque andthrust are linearised in the speed range ofinterest the expression for '

aC can bewritten as follows:

212

2

'

'

11Λ

+Λ

+=

+

aaVC

e

a

Equation 6

where the coefficients 1a and 2a describethe linearised torque and thrust.Using the wake fraction ψ (see chapter

3.2), the values of 'aC can be calculated

for each of the 10 speed-rps combinationsgiven in . If these values are applied toEquation 6, the unknown speed '

RU isreplaced by the quantity µ , whichoriginally indicated the distance from thepropeller disk. It must be noted howeverthat in this case the deviations due to theassumptions used culminate at thecomputation of µ , so that this quantityrather has to be considered a calculationquantity than an indication for the distancebetween rudder and propeller. The same isto be applied to the quantity '

RU .

Substitution of the expression for 'RU into

the formal description of the side force andmoment measurements at 0** == rv , foreach of the rudder angles applied, a valueof µ was obtained. In Figure 11 theproducts δµ ⋅ has been plotted versus therudder angle from which the optimal valueof 0.376 was derived.

Using this value of µ , the values of 2'RU

were calculated and plotted in Figure 12versus the relative speed loss 'u .

Figure 11 Experimental Determination ofFactor µ from Equation 5

Figure 12 Linearisation of the Quantity2'

RU

In Figure A 1 (Appendix A), for the 10combinations of speed and rpm themeasured rudder forces and moments areplotted on the basis of 2'

RU while therudder angle is a parameter.The values of aY and aN were too small todistinguish between star and dashcomponents. A mean value, derived fromthe swaying tests and the present tests isobtained.

14

Table 4 Static Sway Test Program

For the computation of the ruddercoefficients the measurements concernedwere corrected with these mean values.

3.5 Determination of RemainingCoefficients

3.5.1 Static Sway Tests

These tests, executed at the 10combinations of speed and rpm, providedthe coefficients of the following variables:Y and N equation: *v , 3*v , 2*δv , δ2*vX equation: 2*v , δ*vwhile also the aY and aN coefficientswere determined.

In Table 4 a scheme of the test programconcerned is given.Using the test results at 0=δ thecoefficients of *v , 3*v and 2*v aredetermined while computing the 2*δv ,

2*vδ and δ*v coefficients; the coefficientsfirst mentioned were considered to beknown quantities.In Figure A 2, Figure A 3 and Figure A 4,the side force and moment measurementsconcerned are plotted, while thelongitudinal force measurements areshown in Figure A 12, Figure A 13 andFigure A 14.As no information was available involvingthe influence of the rudder speed RU , the

dash coefficients concerned could not bedetermined.

3.5.2 Oscillatory Swaying Tests

These tests were mainly executed todetermine the lateral added mass effect.Only two initial speed conditions areconsidered while the influence of a changeof rpm appeared negligible. Consequently

vY & and vN & are set to zero. The range inwhich the non-dimensional acceleration

2/ xUvL &⋅ was changed; it was extended to0.25 though an estimation of the maximumfull-scale value is about 0.15. Neverthelessthe measured force appeared linear withthe acceleration in the whole range.In Figure A 5 the data concerned areplotted where the speeds are consideredparameter.

3.5.3 Oscillatory Yawing Tests withConstant Drift and RudderAngle

In general these tests were also executedfor the ten conditions given in Table 3.The amplitude of the characteristicvariable *r was varied between 0.05 and0.70 corresponding with turning radii ofapproximately 20 and 3 ship lengthsrespectively.

15

The choice of the oscillator frequency hasbeen based on the followingconsiderations:

a. Based on the tank width available andthe properties of the oscillator the non-dimensional frequency gU x /⋅= ωγ ,which is the leading factor determiningthe oscillatory part of the wave pattern,was kept as small as possible.Concerning the influence of thisvariable the reader is referred to VanLeeuwen (1964).

b. The forces involved in the lowestspeed, corresponding with a Froudenumber 0.07, had to be reasonablymeasurable.

In order to judge the frequency range usedat these oscillatory motions, the quantity

*/2 ωπ can be used, indicating the numberof ship lengths sailed during one period.The restricted tank width and oscillatoramplitude involves a disagreementbetween the forced motions of the modeland full-scale manoeuvres e.g. sinus-response tests. These full-scalemanoeuvres involve rather largeamplitudes in the range of practical full-scale frequencies. It is not known,however, how far this discrepancybetween model-scale and full-scalemanoeuvres influences the hydrodynamicderivatives. In the references VanLeeuwen (1969a) and Van Leeuwen(1969b) some more details concerning thismatter are discussed.

In Table 5 a scheme of the completeyawing program is given.

Table 5 Yaw Test Program

16

The measured forces and momentsconcerning these tests are to be dividedinto three components:a. proportional to the angular velocity

(sine component)b. proportional to the angular acceleration

(cosine component)c. the constant component. In Table 6 the various components aresummarised while the variables concernedare mentioned in the sequence ofdetermination.

Table 6 Variables and Components

Due to the criteria given in the precedingchapter, concerning the ranges of oscillatorfrequencies and amplitudes, the maximumvalue of the angular acceleration amplitudeexceeds the corresponding value everoccurring at the full-scale ship, the latterbeing estimated about 1.30.Consequently the coefficients concernedhave been determined in this full-scalerange the more so as outside this range themodel experiments showed a considerablenon-linear effect.In Figure A 6 through Figure A 11 the sideforce and moment measurements areplotted, while in Figure A 14, Figure A 16and Figure A 17 the correspondinglongitudinal force measurements are given.

3.6 Some Experiments with a SmallModel (α = 100)

For comparison purpose the results of arestricted number of tests with a smallmodel are given. These tests have beenexecuted before those with the largermodel. Because of the very low speeds

involved the results are considered notvery trustworthy.The test program consisted of:a. static sway testsb. oscillatory swaying testsc. oscillatory yawing tests (without

rudder angle and drift angle)The influence of the speed reduction andconsequently of the thrust increment wasnot examined in particular. It was assumedthat the hypothesis mentioned in Chapter2.4 would hold. This means that the testsonly had to be executed for the initialspeed conditions.In Table 7 the results of these tests aresummarised and are compared with thoseof the large model. In Figure A 18 throughFigure A 27 the measurements are shown.

Table 7 Coefficients of Small Model(1:100)

As follows from Table 7 for the importantcoefficients, the magnitude of thedifferences between the coefficients of thelarge and the small model is about 10percent while this percentage for the lessimportant coefficients is about 25.

Table 8 Comparison of 190 StarboardCircle for Scale 1:100, 1:55 and 1:1

17

The importance of these differences ispartly shown in Table 8 in which theresults of a turning circle manoeuvre aregiven, computed with the 1:100 modelcoefficients, completed with some of thelarge model.Concerning the range of variables appliedwith this model it is noted that nearly thesame maximum values of rudder angle,drift angle and angular velocity wereadjusted. The difference between thefrequency ranges of the two models isexpressed by the two quantities */2 ωπ ,denoting the number of ship lengthscovered during one period of theoscillation, and γ which quantity governsthe wave pattern during the oscillatorymotion.The first quantity varies from 3.0 at

10.0*0 =r to 1.1 at 50.0*

0 =r , which isnearly the same range as is applied for thelarge model. The maximum value of γ at

50.0*0 =r (0.17) is about two times the

corresponding value of the large modelhowever. Nevertheless this value isconsidered sufficiently low as to avoiddisturbing influences of this parameter.Putting the results of the two models in thelight of the hypothesis, mentioned inchapter 2.4, it appears that the hypothesisholds for both models separately, but not ifcomparing both models. As in both casesthe Froude numbers had the same valuesthe differences between the correspondingcoefficients can be traced to the smallReynolds number of the small model.

3.7 Some Remarks ConcerningComputed Coefficients

It is found that the rpm effect on the linearterms is very small, compared to themagnitude of these terms. Concerning thenon-linear terms, it was not possible todistinguish between the normal scatter ofthe data and this rpm effect, due to therestricted accuracy of the measurementsand the relatively small values of this non-

linear term. This does not apply to the purerudder angle dependant terms of course,the change of which with rpm isconsiderable.Another important result is the usefulnessof the hypothesis, concerning theproportionality of the forces with thesquare of the instantaneous speed and thecharacteristic variables *v and *r . This isclearly shown e.g. in Figure A 2, Figure A3 and Figure A 6. Though it has also beentried to distinguish between the resultsconcerning the four initial speeds by VanLeeuwen (1969c), it follows from thefigures just mentioned that the differences,which could be considered a Froudenumber effect, are not significant however.

Table 9 Lateral Force Coefficients

Table 10 Yaw Moment Coefficients

18

Table 11 Longitudinal Force Coefficients

In Table 9, Table 10 and Table 11 in theleft columns the coefficients of the set ofEquation 3 are given while in the rightcolumns the corresponding coefficients ofEquation 4 are summarised, the latterbeing derived from the preceding ones.This derivation is partly based on thefollowing approximations:

( )

'2'

''

'2'

632.0709.0

300.2837.01

1400.1940.01

uU

uu

uu

R ⋅+=

⋅−=+

⋅+=+

The accuracy of these approximations isshown in Figure 12, Figure 13 and Figure14 in which these quantities are plotted.Concerning the signs of the coefficients inthese tables the rule holds that all terms aretransported to the right hand sides of theequations.

Figure 13 Linearisation of Speed

Dependent Factor ( )2'1 u+

Figure 14 Linearisation of SpeedDependent Factor ( )'1/1 u+

4 Computer Programs

4.1 Least Squares Analysis ofMeasured Data

This IBM 360/65 computer programSBSL#M03 has been developed tocompute the coefficients of a fourth orderpolynomial of four variables:

( ) ∑=

⋅=p

t

nmlkt xxxxCxxxxF

143214321 ,,,

using the least square criterion. Herein is70≤p and 4≤+++ nmlk .

If some of the coefficients are alreadyknown they can be given and thecorresponding terms are subtracted fromthe measured value of the function. If thedistance between a measurement and thecomputed value of the polynomial exceedstwo times the RMS value the dataconcerned are dropped while thecoefficients are computed again. This“data point selection procedure” primarily

19

serves to locate measuring, writing ortyping errors. The factor 2 used in thiscriterion has been found experimentally. Instatistics, usually a factor 3 is applied,though it has been found that if the numberof measurement is relatively small evenlarge errors are not located in that case.This data point selection procedure comesinto operation again, unless:

a. the number of selected dataexceeds ten percent of the totalnumber,

b. the RMS value is already smallerthan a boundary value givenbeforehand.

In the next stage the maximum value ofeach term is computed. If these maximumcontributions of a number of terms is

smaller than the boundary value justmentioned the procedure is repeated,though without the coefficient themaximum contribution of which was thesmallest.Also this “coefficients selection proce-dure” is repeated until the contributions ofall terms remained are sufficient.Further the standard deviation and thecorrelation coefficient of each coefficientare computed. The first quantity is plottedin Figure 15 and Figure 16 concerning thelarge and the small model respectively.

A later PC version of this computerprogram can be found at the Internet:http://www.shipmotions.nl.

Figure 15 Standard Deviations of Coefficients

Figure 16 Standard Deviation of Coefficients (Small Model 1:100)

20

4.2 Solution of DifferentialEquations

In this computer program (IBM 360/65program SBSL#M02) the differentialequations are solved for given timedepending rudder signals, where the

Runge-Kutta procedure is applied. Twocases are considered. In the first the rudderrate is constant or zero, while in the seconda sinusoidal rudder input can be given todetermine the frequency characteristics ofa ship.

Figure 17 Time Histories of Separate Terms of Side Force Equation of Turning Circle D2

Figure 18 Time Histories of Separate Terms of Moment Equation of Turning Circle D2

21

Figure 19 Time Histories of Separate Terms of Longitudinal Force Equation of TurningCircle D2

Figure 20 Time Histories of Stability Roots of Turning Circle D2

The output quantities can also be requiredto obtain the input for the coefficientsprogram M03, to determine thecoefficients of a mathematical model witha reduced number of coefficients. Furtherfor each step the value of each term of theset of equations of motion is computed

which enables to get an insight in theimportance of the various components. Anillustration of this is given in Figure 17,Figure 18 and Figure 19.Another way to observe the process of amanoeuvre is to linearise the equations ofmotion at each step to the set:

22

uaaravauuaaravaruaaravav

∆⋅+∆⋅+∆⋅+∆⋅=∆∆⋅+∆⋅+∆⋅+∆⋅=∆∆⋅+∆⋅+∆⋅+∆⋅=∆

34333231

24232221

14131211

δδδ

&

&&

Using the coefficients of this set ofequations the stability of the system can beobserved. The time constants of the systemcan be found from the roots of the set:

0

343231

242221

141211

=−

−

−

λλ

λ

aaaaaa

aaa

An example of the change of theseconstants, indicating the change ofstability, during a turning circlemanoeuvre, is given in Figure 20.Concerning the time interval between twosteps of the computation, for allmanoeuvres ten steps per ship length,based on the initial speed, was applied.The time intervals following from this aregiven in Table 12.

Table 12 Time Intervals Applied forComputations

5 Comparison of Computed andFull-Scale Manoeuvres

5.1 Turning Circles

The principal data of the turning circles aresummarised in Table 13.

Table 13 Turning Circle Data

The results of the computed turning circlesare shown in Figure B 1 through Figure B4. As follows from these figures, the(final) rates of turn are somewhat smallerthan the full-scale values though,combined with the final speed, the turningdiameters agree very well however.For comparison with the spiral manoeuvreresults some additional turning circleshave been computed. Only the final valuesof the various quantities have been used toobtain complete curves.The computation of the 37 degrees portrudder turning circle has been repeatedomitting certain coefficients, whichsignificance was very little, according tothe model tests. The results are shown inFigure B 2. As appears from this figurethese coefficients have also littleimportance in the mathematical model.

5.2 Zig-Zag Trials

In Table 14, the principal data of the zig-zag trials are summarised. Concerning theinitial conditions, only the course ψ wasconsidered while no other data wereavailable. The values of the rudder rate ofturn were derived from the data andfigures given by Clarke (1965).The computed zig-zag manoeuvres areplotted in Figure B 5 through Figure B 12and compared with the full-scalemeasurements. The overshoot angles aresomewhat smaller, but the meanoscillation periods agree very well. Thesetwo quantities are plotted in Figure 22.Concerning the initial speed conditions ofthese manoeuvres the adopted linearrelation: 0U = 12.5 x rpm has beenapplied, while the rpm values for both full-scale trials and computations are the same.

23

Table 14 Zig-Zag Trial Data

6 Final Remarks

To judge the result of the modelexperiments discussed in this paper, Figure21 and Figure 22 may serve in the firstplace. They provide an overall picture ofthe principal parameters of turning circle,spiral and zig-zag trials:• Figure 21a, *

cr against 0δ : relation

between turning circle diameter */2 cr )and rudder angle.

• Figure 21b, cU against 0δ : final speedreduction of turning circle manoeuvres.

• Figure 22a, pt against 0/1 U : relationbetween oscillation periods of zig-zagtrials and initial speed.

• Figure 22b, 0max /δψ against 0/1 δ :relation between overshoot angle andnominal rudder angle.

From these figures, in which the computedquantities are compared with thosemeasured during the full-scale trials, itappears that it is possible to predict themanoeuvring properties of the shipconcerned by means of oscillation testswith reasonable accuracy. It must be notedhowever that both full-scale data andcomputed data have their uncertainties. Inparticular this may be important if 00 , yxplots are compared, because these plots areobtained very indirectly.Concerning the zig-zag trials it is foundthat a little change of the “executioncourse” has a relatively large effect on themaximum course deviation and theoscillation period. Finally we have to keepin mind that the determination of themanoeuvring properties of a ship viahorizontal oscillation tests is ratherindirect, at least while the motions of themodel during these tests are ratherunrealistic. From the four figures abovementioned, it may then be concluded thatif scale effects play a role in the presentinvestigation their importance is not verylarge and of the same magnitude as theaccuracy of both the full-scale and model-scale measurements.

24

Figure 21 Principal Results of Predicted Circles at 100 Nominal RPM

Figure 22 Principal results of Predicted Zig-Zag Tests

7 Recommendations

The mathematical description of themanoeuvring properties is based on ahypothesis, which describes the relationbetween the forces acting on the ship andthe forward speed. The application of thishypothesis is fully justified by the presentmodel experiments. This does not mean

however that all hydrodynamic effects,which play a part in this mathematicalmodel, are really necessary for a sufficientdescription of the horizontal motions.Therefore, it might be interesting to find asimple mathematical model, whichproperties are not to give and accuratedescription of the hydrodynamicphenomena but rather of the motions.

25

8 List of Symbols

m Mass of the shipzzI Moment of inertia of the ship

ρ Density of water

00 , yx Co-ordinates in space bounded co-ordinate systemY Lateral force, positive to starboardN Moment around 0Z -axis, positive to the rightX Longitudinal force

vY & Added mass in lateral direction

rN & Added moment of inertia

uX & Added mass in longitudinal direction

3,2,12,1,2,1, ,, XNY aa Components of Y , N and X respectively

222 ,, XNY Components proportional to 2RU

'RX Thrust increment coefficient

*RX Longitudinal resistance coefficient

*1Y

≈

= 22

122

1 21

/21

/ LUYLUY xρρ

*1N

≈

= 32

132

1 21

/21

/ LUNLUN xρρ

*1X

≈

= 22

122

1 21

/21

/ LUXLUX xρρ

0U Initial speed

U , U& Instantaneous forward speed and acceleration (vector)

xU , xU& Instantaneous longitudinal speed and acceleration

RU Local velocity near the rudderv , v& Sway (drift) velocity and accelerationr , r& Yaw angular velocity and accelerationδ Rudder angleu Speed reduction: 0UUu x −=

*v ββ tan/sin/ −=≈−== xUvUv*r xr UrLRLUrLdsd //// * ⋅≈≈⋅== ψ*v& ** / dsdv=*r& ** / dsdr=

ψ Heading angleβ Drift angleφ Deviation angle

*φ& vr RLRLRLdsd //// * −=== φs Distance covered by the ship

*s Distance covered by the ship in ship lengthsR Radius of curvature

26

vR Drift radius of curvature ( *// dsdRL v β= )

rR Yaw radius of curvature ( *// dsdRL r ψ= )'m 35.0/ Lm ρ='

zzI 5L0.5 / ρzzI='v 0/Uv='r 0/UrL ⋅='u 0/Uu='v& 2

0/UvL &⋅='r& 2

02 /UrL &⋅=

'u& 20/UuL &⋅=

'RU 0/UU R=

9 References

Abkowitz (1964)M.A. Abkowitz, Lectures on ShipHydrodynamics - Steering andManoeuvrability, HyA Report HY-5,December 1964.

Clarke (1965)D. Clarke, Manoeuvring Trials with the50.000 Ton Dead-weight Tanker “BritishBombardier”, BSRA Report, December1965.

Davidson and Schiff (1946)K.S.M. Davidson and L.J. Schiff, Turningand Course Keeping Qualities, SNAME1946.

Eda and Crane (1962)H. Eda, and C.L. Crane, Research on ShipControllability, D.L. Report, StevensInstitute of Technology, 1962.

Nomoto (1967)K. Nomoto, On the Steering Qualities ofShips, I.S.P. July 1957.

Van Leeuwen (1964)G. van Leeuwen, The Lateral Dampingand Added Mass of a HorizontallyOscillating Shipmodel, T.N.O. Report 65S,December 1964.

Van Leeuwen (1969a)G. van Leeuwen, Enkele problemen bij hetontwerpen van een horizontale oscillator(in Dutch), T.H. Shipbuilding Laboratory,Report 225, January 1969.

Van Leeuwen (1969b)Van Leeuwen, Some Notes on theDiscrepancies between the LateralMotions of Oscillation Tests and Full-Scale Manoeuvres, T.H. ShipbuildingLaboratory Report 232, April 1969.

Van Leeuwen (1969c)G. van Leeuwen, Tentative Results of theHorizontal Oscillation Tests with a 4-Meter Model of the “British Bombardier”,T.H. Shipbuilding Laboratory, Report 239,June 1969.

Van Leeuwen (1970)G. van Leeuwen, A Simplified Non-LinearModel of a Manoeuvring Ship, T.H.Shipbuilding Laboratory, Report 262,March 1970.

Zunderdorp and Buitenhek (1963)H.J. Zunderdorp and M. Buitenhek,Oscillator Techniques at the ShipbuildingLaboratory, T.H. Shipbuilding Laboratory,Report 111, November 1963.

27

10 Appendix A

Figure A 1 Lateral Force and Momentdue to a Rudder Deflection

Figure A 2 Lateral Sway Damping Force

Figure A 3 Lateral Sway DampingMoment

Figure A 4 Rudder Angle – Drift CrossCoupling Effect in Lateral Force and

Moment

28

Figure A 5 Swaying Force and Momentdue to Lateral Mass

Figure A 6 Lateral Yaw Damping Forceand Moment

Figure A 7 Yaw Rate – Drift CrossCoupling Effects in Lateral Force and

Moment

Figure A 8 Rudder Angle – Yaw Rate( 2rδ ) Cross Coupling Effect in Lateral

Force and Moment

29

Figure A 9 Yaw Rate – Rudder Angle( 2δr ) Cross Coupling Effect in Lateral

Force and Moment

Figure A 10 Drift – Yaw Rate – RudderAngle Cross Coupling Effect in Lateral

Force and Moment

Figure A 11 Yawing Force and Momentdue to Moment of Inertia

30

Figure A 12 Longitudinal Force due to aRudder Deflection

Figure A 13 Longitudinal Sway DampingForce

Figure A 14 Longitudinal Yaw DampingForce

31

Figure A 15 Rudder Angle – Drift CrossCoupling Effect in Longitudinal Force

Figure A 16 Rudder angle – Yaw RateCross Coupling Effect in Longitudinal

Force

Figure A 17 Longitudinal Force due toCentrifugal Acceleration

Figure A 18 Swaying Force due toLateral Mass (Small Model 1:100)

Figure A 19 Swaying Moment due toLateral Mass (Small Model 1:100)

Figure A 20 Yawing Force due toMoment of Inertia (Small Model 1:100)

Figure A 21 Yawing Moment due toMoment of Inertia (Small Model 1:100)

32

Figure A 22 Lateral Force due to Rudderdeflection (Small Model 1:100)

Figure A 23 Lateral Moment due toRudder Deflection (Small Model 1:100)

Figure A 24 Lateral sway Damping Force(Small Model 1:100)

Figure A 25 Lateral Sway DampingMoment (Small Model 1:100)

Figure A 26 Lateral Yaw Damping Force(Small Model 1:100)

Figure A 27 Lateral Yaw DampingMoment (Small Model 1:100)

33

11 Appendix B

Figure B 1 Time Histories and X-Y Plotof Turning Circle A at 340 to Starboard

and 100 Nominal RPM

Figure B 2 Time Histories and X-Y Plotof Turning Circle B at 370 to Port and 100

Nominal RPM

34

Figure B 3 Time Histories and X-Y Plotof Turning Circle D2 at 190 to Starboard

and 100 Nominal RPMFigure B 4 Turning Circle Characteristics

at 100 Nominal RPM

35

Figure B 5 Time Histories and X-Y Plotof Zig-Zag Test A (20/20) at 100 Nominal

RPM

Figure B 6 Time Histories and X-Y Plotof Zig-Zag Test C (10/20) at 100 Nominal

RPM

36

Figure B 7 Time Histories and X-Y Plotof Zig-Zag Test D (30/20) at 100 Nominal

RPM

Figure B 8 Time Histories and X-Y Plotof Zig-Zag Test E (10/20) at 85 Nominal

RPM

37

Figure B 9 Time Histories and X-Y Plotof Zig-Zag Test F (20/20) at 85 Nominal

RPM

Figure B 10 Time Histories and X-Y Plotof Zig-Zag Test A (30/20) at 85 Nominal

RPM

38

Figure B 11 Time Histories and X-Y Plotof Zig-Zag Test J (20/20) at 70 Nominal

RPM

Figure B 12 Time Histories and X-Y Plotof Zig-Zag Test L (30/20) at 60 Nominal

RPM

39