Roll stabilisation by rudder

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ROLL STABILISATION BY RUDDER

ByARJMLloyd PhD BSc MRINA

(Admiralty Experiment Works)

SYNOPSIS

The possibility of using rudders to stabilise a frigate is discussed in conjunction with a companion paper byJ B Carley of

the Admiralty Engineering Laboratory. The rudder will be expected to amplify the rolling motions at high and low frequencies and this makes it ineffective in following Seas at high speeds. It is concluded that the rudder stabiliser will not be as effective as a good fin

etabiliser but is probably preferable to a passive damping tank.

A mathematical model describing the motion response to the rudders

is described.

*Copyrii:ht Controller HMSO, London,

1975"

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I. /NTRODUCTION

In recent years it has b,icoMe routine practice to fit ships of the Royal Navy with roll stabiliser fins. Recent developments (References

1,

2 and 3) have led to a deeper understanding of the way in which roll atabiliser fine work and aldo to a realisation of the potential of the rudder as a roll stabilising device.

References 4 and 5 have shown that this concept holds promise in the merchant Ship field and this paper, together with a companion paper by J B Carley of the Admiralty Engineering Laboratory,

considers the case for warships.

In particular, a mathematical model describing the motion responses of the ship to the rudder is developed and the likely performance of the rudder stabiliser in rough weather is assessed.

THE RUDDER AS A ROLL STABILISER

Figure 1 shows the conventional position of roll stabiliser fine on a typical ship. It can be seen that the lift forces developed by the

fins generate away forces and yaw moments

in

addition to the-expected roll moment. The lateral forces and momenta depend on the fin plane depri.noion a and the yaw lever arm

x".

Rudders can be considered, in this context, as ,stabiliser fins with 90 degrees and large x*. It is thus clear that the rudders can be used to generate roll 'moments although the associated sway and yaw motions will be much larger than developed by the fins.

Reference 3 has developed a method for determining the performance of roll stabiliser fins in terms of an apparent lift curve slope

(dCi,)*---dft and a phase difference v1. These can be obtained from forced

tolling trial results -or from simulation.

This paper extends' thie concept to include rudders.

FORCED ROLLING TRIALS

Forced rolling trials using the rudders were conducted in Ship A (the same ship A was used for stabiliser fin trials in Reference 3)-.

The

shipla

frigate of order 100 m in length, was

rtul

at a steed of

19 knots. The stabiliser fins were parked and the rudder was driven by a sinusoidal demand signal generated by a Transfer Function Analyser. The TFA was used to measure the amplitude and phase of roll, yaw and actual rudder motions.

Weather conditions at the time of the trial were very good:. wind speed was 10'-15 knots and the sea Was calm.. Nevertheless > it was found

that at low frequencies-the most 'consistent results were obtained when heading into the Wind.. At high frequencies there was no such effect and for this reason the low frequency measurements were always made in head to wind conditions.

2-214 2-215

The results of the trials are shown in Figures 2 and 3, compared with results obtained in comparable stabiliser fin forced rolling

trials in Figures 4 and

5.

The most severe rolling motions occur close to the natural roll frequency and it follows that any comparison of stabilisation devices must basically be made at this freetincy. Figures 2 and 4 show that, for this ship, the peak roll response to the rudders is about the same as that to the fins. So there is a good case for considering the rudder as a potential roll stabilisation device. Figure 2 also shows that theYaw response at the peak roll frequency is about 0.1 and it is considered that this would not lead to unacceptable yaw motions being generated by the roll induced motions of the rudders if they were used to stabilise the ship.

Figure 3 shows the rudder response, and while this is Much more sluggish than the response of the fins (Figure 5)it is, considered that it would be acceptable.

MATHEMATICAL SIMULATION OF FORCED ROLLING TRIALS

Reference 3 has described the simulation of .stabiliser fin forced rolling trials and took account of a. boundary layer losses, b. interference between fins and bilge keels and c. sway and yaw effects.

For the. special Cade of forced rolling with rudders a. is academic as the flow over the rudders is dominated by propeller wake effects

and b. is non-existent. The rather simple treatment of c. used in Reference 3 is probably not adequate for rudders because the much

greater lateral Motions involved require a more accurate repreden-tation. In particular the treatment adopted here includes non-. linearities and the use Of frequency dependent coefficients.

Details of the mathematical model are given in Appendix I. The results of force rolling the mathematical model are shown in Figure 2, for comparison with the trial.results. The agreement is very good.

EFFECT OF AUTOPILOT

Carley explains, itr his .companion paper, the need for .._ including the effects of the autopilot on the ship dynamics.

Accordingly the mathematical model described in Appendix I was modified to include simulation of an autopilot and the 'rudder servo

dynamics. The characteristics of the autopilot are given

by

ID + T1s)

+ -

1

° + T T

2 3

2-216

Figure 6 shows the predicted effect on the roll response of including

the autopilot and rudder servo in the simulation. Over most of the frequency range there is little change' butat very low frequencies the amplitude response falls to zero and the phase approaches 90 degrees lead instead of 180 degrees. At high frequencies the phase lag is increased (by the rudder servo) ultimately to

360

degrees.

.ROLL STABILISATION

6.1.

Basic Assumptions

The performance of the rudder as

a

roll stabiliser has been predicted using methods in current routine use at AEW for stabiliser fins. These methods are based

on

the theory described in Reference 8 and

predict the response amplitude operators for unstabilised and stabiiiaed roll motions and stabiliser motions'. These are combined with Pierson-Moscowitz wave slope Spectra to calculate roll and

stabiliser motion spectra.. Finally rms values in wave spectra spread withA cosine squared function are calculated.

The lift available from the stabilisers is expressed as an apparent lift curve slope and a phase difference, as explained in Reference

_3.

These are obtained by relating the frequency response results of Figure.6 to simple one degree of freedom frequency responses. Figure 7 shove the results for the rudders of the ship considered

here. The apparent lift curve slope expresses the lift which would Sued to be generated by the rudders if the ship roll motion were described by a simple one 'degree of freedom mathematical model. The phase difference expresses the phase lag between the apparent lift and the rudder motion.

It is assumed that the roll controller will be of a form similar to that used in conventional fin control systems.

K

P +K p+Kb

G 1 2 3

D v

bI + ib2we - b3we2

Typital-values for a. fin system might be _{Kl . 2, ic2}

_{= 5,}

_K3

_=.2.

However the combination of steering and stabilising _functions

_in

the rudder introduces limitations on possible controls. In particular, K1 must be zero.

Otherwise the heel in a turn will always cause the rudder angle and rate of turn to be decreased. 'For simplicity

in the present exercise, K2 _{and K3 have been given the above values.}

Gv is a speed dependent gain set to 1:0 at 20 knots and varying as 1/u7 at other speeds. _{A significant Wave height of}

_7.5

metres has been assumed for all calculations.

6.2. _{Amplification Regions}

Carley shows that the

rudder stabiliser will be expected to reduce the rolling

motions only in an intermediate _{band of}

2-217

frequencies (encompassing the natural roll frequency of

the ship).

At low frequencies (in this case below about 0.24 radians/second)

and at high frequencies (in this case above about

0.76 radians/

second) the rudder stabiliser will be expected to .amplify

the rolling

motions.

The greatest amplification of roll motions would-be

expected

to occur'at'a frequency of about 0,12

radians/second.

The encounter frequency is given by:

U

we 7 w

2 w

cosi,

and

X 21-B-r w2

rearranging we obtain:

X = 2a- - 2211 cosp ±

ff2k2

47g2u cosp

we2 (0.e we4 we3

This equation is plotted in Figure 8 for the

critical values of

encounter frequency as a function of the component ship velocity in

the direction of the wave propagation.

Note that negative values of

the critical frequencies are equally

appropriate; they imply that

the ship is overtaking the waves.

The chart below the figure enables.

the component velocity for a given heading and ship speed to be

determined..

The hatched areas in Figure 8 indicate the zones in which the rudder

stabiliser would be expected-to amplify the rolling motion in wave

lengths up to 800 metres (the Maximum

in which there is any

appreciable wave slope energy).

On headings forward of the beam the amplification zone is confined

to the shorter wave lengths where the encounter frequency exceeds

0.76 radians per second.

The range'of wave lengths in which

this

occurs increases with component velocity-.

On headings abaft the beam the amplification zone.may extend over

almost the entire range of wave lengths.

6.3.

Broaching7to

Also shown in Figure 8 is a

tentative estimate Of the zone in

which

broaching-to is likely to occur, based on

the evidence of Reference 9.

In this region the ship is

liable to be accelerated to wave

speed

.

(so that the encounter frequency becomes

zero) and to lose directional

control.

If for example the ship began to Vali to starboard, very

large port rudder angles might be necessary to regain the desired

heading and in extreme cases. the rudders

might not be powerful

enough to steer the ship and she

would broach-to.

The broach-to

would be accompanied by an

increasing rate Of roll to port.

2-218

A rudder otabiliset Would sense this roll motion and order a starboard

rudder angle (reducing theangle applied to steer the ship) so that

the broach Would become more severe.

6.h.

Roll Spectra

Figures 9-11 show calculated roll spectra in long crested irregular

sees of significant wave height 7.5 metres at a ship speed of

30 knots.

On headings. near to the beam (80 and 100 degrees) the rudder stabiliser

works very well because the predominant scaling motion is at

frequencies between the two critical frequencies.

On headings

forward of the beam (120, 140 and 160 degrees) significant

unstabi-lised roll energy above the higher critical frequency is present and

the rudder amplifies this With consequent reductions in overall

effectiveness.

However, the rolling motions are quite small on these

headings so that the shortcomings of the stabiliser are not very

important.

On headings abaft the beam the unstabilised rolling Motions are much

larger.

At .60 degrees the stabiliser amplifies the peak rolling

motion but reduces the motion over most of-the range of wave lengths,

resulting in an overall reduction. At 40 degrees the atabiliser

amplifies the motion at all wave lengths except where the encounter

frequency is zero, Where it has no effect. 'At 20 degrees the high

frequency motions are reduced but there is a comparable increase in

low frequency-motions over a wide range of wave lengths.

6.5.

Rms Values

The Motion spectra curves have been integrated to obtain ms values

in a spread wave spectrum and the results are shown in Figures 12

and 13

The rudders can provide a-substantial degree of roll stabilisation

at low speeds (15 knots).

At higher speeds the rudders become less effective particularly on

headings abaft the beam and at 30 knots they destabilise the ship

on headings between 0 and about 35 degrees.

The use of the cosine squcared spreading function is probably realistic

but it is known that wave systems are occasionally-much more unidirectional

Figure 14 shows res

s

or

ong cres e

seas and it can

e seen

a

the spread spectrum tends to iron out the peaks and troughs in the

curves.

_{For long crested seas the rudder destablisee the ship for}

headings between 0 and 45 degrees. (at 30 knots);

Figure 15 shows the zones of speed and heading in which the rudder

would be expected to destabilise the ship in long crested seas and

in spread spectra.

6.6.

Yaw Motion.

No accurate calculation of yaw motions have been made but an approximate assessment of the yaw motions likely to result from the roll-induced rudder motions has been made from the rudder motion

spectra (not presented here) and the yaw response curve of Figure

6.

This has shown that the rms yaw motion caused by the rudder

stabiliser is unlikely to exceed about 2 degrees in waves of

7.5

metres significant height. CONCLUSIONS

The results of forced rolling trials using the rudder in a frigate havebeen described and a mathematical model, based on hydrodynamic

principles, has been developed. The mathematical model simulates the trial results quite well.

Standard method of computing fin stabiliser performance have been adapted for use with rudders and have shown that the rudders would be expected to work well in beam seas and adequately on headings forward of the beam.

The rudder stabiliser will not work Well at high speeds on headings abaft the beam and in extreme cases may even amplify the, rolling

Major:.

In considering the merits of the rudder as a roll stabiliser it must be compared with the other devices currently available; namely, active fins and passive damping tanks.

Active fins amplify the rolling motion at high frequencies in the same way as the rudders but they do not have the same problems at

low

frequency. In addition they would not be expected to increase the likelihood of broaching-to.

'Massive damping tanks also amplify the 'motions at high and low frequencies and it is general practice to immobilise them (by filling or emptying them) when the ship is in unfavourable conditions. This may take an hour or more. In these circumstances the rudder stabiliser could be switched off at will and would also have the advantage of .

providing a greater degree of roll atabilisation when conditions are favourable.

ACKNOWLEDGEMENTS

This paper is published by permission of the Ministry of Defence (Procurement Executive) but the responsibility for statements of fact or opinion rests with the author: The author Would like to acknowledge the assistance and co-operation given by his colleagues at the Admiralty Experiment Works, the Admiralty Engineering Laboratory and Director General Ships. In particular the contribution made by Dr J B Carley to this study has been of fundamental importance.

2-220 -Reference 1. Reference 2. Reference 3. Reference 4. Reference 5. Reference

6.

Reference T. Reference 8. Reference

_9.

'Reference 10: References

Design Considerations for Optimum Ship Motion Control by J B Carley and A Duberley. 3rd Ship Control Systems Symposium

1972.

' The Hydrodynamic Performance of Roll Stabiliser Fins

byARJMLloyd.

3rd Ship Control Systems Symposium 1972.

Roll Stabiliser Fins: A Design Procedure.

ByARJMLloyd. NINA

1975..

The Use of the Rudder as a Roll Stabiliser.. By W E Cowley and T H Lambert. 3rd Ship Control Systems Symposium. 1972

Development of an'Autopilot to Control Yaw and Roll. By W E Cowley. The Naval Architect. January

1974.

Programme Scores. Ship Structural Response in Waves. By A I Neff. Ship Structure Committee Report

No SSC 230: 1972.

Calculation of Hydrodynamic Coefficiente for Forced Rolling With Active Fins. By A.11 J M Lloyd. AEW Computer Program No 1130/133. March

1974.

Rolling and its Stabilisation by Active Fins. By J E Conolly. Trans. RINA

1969.

Broaching-to: Exploratory Model Experiments in Following Seas. By K Nicholson. International Symposium on the Dynamics of Marine Vehicles. Universtiy College London.

_1974.

Study on Lateral Motions of a Ship in Waves by Forced Oscillation Tests. By H FUjii and T Takahashi ,Mitsubishi Technical Bulletin No 87 August

1973.

:fr

Appendix I

MATHEMATICAL MODEL FOR FORCED ROLLING WITH RUDDERS

Axes are fixed in theship with the origin at the centre of gravity.

The equations of motion

are:-Iii=1,10+K.L.+Kv+K.r+ Kr

1

v

v

K

_{KrIrl rIrl}

_{KvIrl vIrl}

Kvivi

Ivl

K61r1

IrI + KlvIr Mr + Ks1,61 6161

m2LqYv+

+Y13

p+y6 +

vP. r

+ (Yr - Inu)r + Y66

v1v1

Y61r1 61r1

Yriri r14

Yvl.ri

4. + I

YlvIr

* Y6161 6161

+ N66

I3i=Nr+ N

rPPS V

+Np+N4 + N.L 3.Nv

V

;fivi

N61r1 6111

Nriri dr! +7viri

+Nvivi

M1v1r Hr.+ K6161.

6.161

=

4dt,

Jpdt

°

ILdt, y

f(v + uV)dt

fMt, V . frdt

6=

.6

Sin wt

Frequency Dependent Coefficients

Many of the coefficients in equations (1) to

(3) are frequency

dependent and this dependence has been estimated using data from

the SCORES Computer Program described in Reference

6.

The frequency

dependent parts were obtained by Summation of the sectional added

mass and damping data over the length of the: ship.

A computer

program (Reference 7) was written to accomplish this tedious task.

Frequency dependence of the non linear coefficients was ignored.

This procedure enabled the coefficients K., K., Y' . Y., N. and IL to be

_v

_r

p

r

p

calculated directly.

Other coefficients K,

Y

v

-r

v

p'

r'

r'

-and N

contain a contribution from viscous effects.

This is evident

P

v

from the fact that their calculated

values arezero frequency

in contrast to the results of

,steady state experiments.

Realistic

2-.222

vaLues w,:re obtained by adding the calculated frequency dependent part

to the viscous part determined by experiment or estimation (Bee below).

The coefficient K

representing the roll damping was estimated at the

natural roll frequency from

K'

- 4n mg GM

(8)

puLwo

Tills includes a frequency dependent part, which was calculated at the

natural roll frequency using the programs already mentioned.

This

enables the viscous contribution to be estimated and the total (viscous

and frequency dependent contributions) was then calculated at other

frequencies as for the other coefficients.

The virtual mass and inertia m2 and 13 contain frequency dependent parts

(estimated as described above) plus contributions from the true mass

and inertia Of the ship.

The radius of gyration (in yaw) was assumed

to be 0.225L.

The virtual inertia in roll is given, at the natural roll frequency,

by .

1 .1 mg 0w.

1

(9)

'02

and thin also contains a frequency dependent contribution and a structural

inertia contribution.

The computer programs Were used to determine

the frequency dependent contribution at the natural roll frequency so

that the struccual contribution could then be estimated.

This enabled

the total virtual inertia to be calculated other frequencies.

N0' Y6' N-6 and Y

were assumed independent of frequency.

6 .

Estimation Coefficients

Values of Yv, Yr, Nv and Er were obtained from towing tank and rotating

arm model experiments.

The hydrodynamic side forces were assumed to act at a depth z* below

the centre of gravity.

Kv -z*YV

(10)

K

r

°

zlYr

Some results of model experiments from which z* was derived are presented

in i!eference 10.

These show that z* is frequency dependent and, for

;h.. purpose of this study, z* hole been assumed to be given by

z* 00 + za2(1 + zR ' where zR -g

.6

5111 1,12 4..25

ue

5R2 we2) 2 (12)

Consider a stopped ship listing with an applied roll moment K. If the ship is now driven and the roll angle is 01 .it will be neneesarY

to apply a rudder angle 6i to keep it on course. Ignoring non7 linearities,-equatinns (1) to (3) become:

0 = - mg G141 + Kvv + K661 + K

Yvv +.Y01 + Y661

+Nv+ N661

. 0 1 v

If we now assume that the roll angle under-Wayis the name as the roll angle when the ship is stopped

g mg GM.1

and the equations redline to

61 lv K6 - N6 Kv 0 1 Kv

6v K6 - Y6 K

y v .1 Kv (15) (16) Roman Symbols A b1' b2' b3 etc Notation,

Rudder area (each)

Roll stabiliser control non dimensional compensator coefficients sec, sec2

/radian m

9.81

m/mme2 kg m2 -kg m2

i

/77

non dimensional

Autopilot gain non dimensional

1(1, K' K32 Roll control sensitivities non dimensional

see, eec2

otc Roll moment due to unit roll velocity

etc positive starboard Nm sec, etc

Ship length metres

m Ship mass kg

Virtue/ sway mass kg

yaw moment due to unit roll velocity etc, positive starboard

roll decay coefficient

Number of rudders

meigta of centre of gravity above waterline /radian' Nm sec etc non dimensional non dimensional " The coefficient K6, representing the roll moment

vas estimated in a similar way. However, most

will be generated on the rudder. If it is

is developed on the rudder,

11E1 I).112A dCL

due to the rudder of the lateral force assumed that a proportion

(13)

(i4.)

as follows.

/dC/A*. 1c1C9*

kdO ) kdd

Apparent lift curve slope of fins or rudders'

True lift curve 'slope of rudders.

Speed dependent gain

Metacentric height

Acceleration duo to gravity

Virtual roll inertia

Virtual yaw inertia Gv G14 8 21 q. 2Y6 and K6

=

-In practice The coefficients d6 - pu "Pt 2AR dcl,

I

L. (1-- q)Y6z* + 2 d6 q = 1.

N. and Y. where estimated

1)1

Y

etc

Roll velocity and acceleration

positive starboard

proportion of rudder side force

developed on rudder

Roll lever arm

Yaw velocity and acceleration

positive starboard

Roll energy

Laplace operator

Ship draught

Time

Ship bpeed

Sway velocity and acceleration

positive, starboard

Yaw lever arm

positive forward

Sway force due to unit roll

velocity etc

positive starboard

Sway displacement

positive starboard

Roll lever arm for lateral forces

z82

See equation 12

2-226.

rad/sec

rad/sec2

non dimensional

rad/sec

rad/sec2

deg2/m

sec-1

secs

m/sec

M/sec

.

m/sec2

N sec etc.

non. dimensional

Bee2, sec3

...

:-..-7,,n<144.041F~rlitee&A ;%4XNAVMW

IJ

deg

red

rad

rad

degs

deg

red

1025 kg/10

red

red/sec

red/sec

red/sec

Greek Symbols

a

Fin plane depression

0

Fin angle

6

rudder angle

D

Demanded rudder angle

Wave length

Heading to waves

0 degrees m following seas

.180 degrees . head seas

U'

_{Phase difference.: phase lead of}

apparent lift before true lift

or rudder angle

Roll angle

positive starboard

.Hass den-34y of. sea Water

Yaw angle

positive starboard

Wave frequency

Wave encounter frequency

1.2

IlEk.

FIN

2-228

LCG

FIGURE I.

CONVENTIONAL ROLL STABILISER

FINS.

1.0

2-229

.2

.4

.6

FREQUENCY - RADISEC.

KEY. _.

COMPUTED ROLL

COMPUTED YAW. 5° ROLL TRIAL '

o

10.ROU_ TRIAL.

5 YAW TRIAL.

A

IC YAW TRIAL.

uJ-100

200

EicaLEL.2. FORCED ROLLING WITH RUDDERS.

--

50

75

100

FIGURE 3.

RUDDER RESPONSE.

2-230

8

7

,

.6

1 6

I,

.8

FHEQUENCY- RADISEC.

.

FIGUk 5. FIN RESPONSE.

2-232

(Axe a PREDICTED

FORCED ROLLING WITH

AUTOPILOT AND RUDDER

_{SERVO., !".1}

:-:)3

.6

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