A moving weight stabiliser for a ship model

MSc PROJECT CONTROL TECHNOLOGY

OCTOBER 1977

A MOVING WEIGHT STABILISER FOR A SHIP MODEL

BY

P

- _e731

gre

'

Contents

Page Frontispiece. Model Instrumentation.

Notation. iv-vi

Introduction. 1

Roll Stabilisers: Introduction and Historical Survey. 1

A Mathematical Model of Ship Rolling. 4

The Model Experiments. 10

Optimal Control Applied to a Moving Weight Stabiliser. 14

Conclusions and Recommendations. 22

References. 23

Table 1. Roll Decay Ordinates at Half Cycle Intervals. 24 Table 2. Data for Frequency Responses (Figure 16). 25

Table 3. Details of Servo Motor. 26

Table 4. Connections to Servo Amplifier Box. 27

Appendix I. The Moving Weight Servo Design. 29

Appendix II. Computer Programs. 32-35

Figure 1. Tank Stabilisers.

Figure 2. Explanation of GM and GZ. Figure 3. Ship Rolling in Beam Seas. Figure 4. Phase Advance Controller.

Figure 5. Moving Weight Servo: Top View.

Figure 6. Moving Weight Servo: Underneath View. Figure 7. Model Tethering Arrangements.

Figure 8. Instrumentation Block Diagram. Figure 9. Forced Roll Response: Zero Speed. Figure 10. Polar Plot of Maximum Weight Response. Figure 11. Effects of Speed on Open Loop Response. Figure 12. Typical Roll Decay.

Figure 13. Decay Coefficients.

Figure 14. Polar Plot of G(jw).H(jw). Figure 15. Rolling at wo.

Figure 16. Frequency Response. Figure 17. Optimal Control Gains. Figure 18. Computed Step Responses.

Figure 19. Comparison of Frequency Responses. Figure 20. Servo Amplifier: First Version. Figure 21. Servo Amplifier: Final Version.

Figure 22. Frequency Response of Moving Weight Servo. Figure 23. Power Measuring Circuit.

Figure 24. Gyro Wiring Diagram.

iv Notation

An attempt has been made to use symbols normal in both ship science and control theory.

A System matrix in general formulation of state variable system.

a11' a12 etc Terms in A matrix.

a Parameter in phase advance controller.

Centre of buoyancy of ship at rest in cairn water. B' Centre of buoyancy of rolling ship.

Beam of ship or parameter in phase advance controller. Input matrix, a vector, in general formulations of state variable system.

b _{b2 etc} Term in b.

Weight servo constant, metres per radian.

Velocity damping coefficient, force per unit velocity.

D2 Non-linear decay coefficient.

a(t) Weight displacement relative to centreline of ship.

D(s) Laplace transform of d(t). PG Gravity force on ship.

Centre of gravity of ship.

GM Metacentric height (see Figure 2).

GZ Roll restoring force lever arm (see Figure 2).

G(s) Transfer function of ship, roll/roll moment.

G(jw) Frequency response of G(s).

G1, G2 etc Optimal control gains.

Acceleration due to gravity.

H(s) Stabiliser feedback transfer function roll moment/roll.

H(jw) Frequency response of H(s).

61 Added mass roll moment of inertia.

Total roll moment of inertia of ship or cost function for optimal control system.

7=T

Matrix of optimal control gains.

K11' K12 etc Terms in K.

Kc Gain of controller.

Gain of controller at

wo

co

Ks Ship roll constant = 1/1.g.GM. General roll moment.

M (t) Total roll moment applied to ship.

(15

M (s) Laplace transform of M (t).

(15

MST(t) Roll moment due to stabiliser.

Mw

(t)Roll moment due to waves.

Mass of moving weight.

Linear roll decay coefficient.

Matrix of values of penalties for state variables in cost function-q11, q12 12 etc R(s) R(jw) u(t) uo(t) x(t) Terms in Q.

Roll reduction factor. Frequency response of R(s).

Penalty for input (for a single input system) in cost function J.

Laplace operator.

Input term in general formulation of state variable system. Optimal control input.

xl(t), x2(t) etc State variables.

ct Wave slope.

ac

Wave slope capacity of stabiliser.

Damping factor of moving weight servo.

0 Roll angle relative to wave.

Tr 3.14159 etc.

Mass of ship.

Roll or heel angle.

(p(t) Roll angle as a function of timer

Laplace transform of 4)(t). Frequency, radians per second.

wo Roll natural frequency of ship.

A MOVING WEIGHT STABILISER FOR A SHIP MODEL

By W B Marshfield INTRODUCTION

1.1. This report describes the design and testing of an active moving weight roll stabiliser for a ship model.

The project was undertaken at the Admiralty Experiment Works (AEW), Gosport, as part of the normal work of the Establishment. Originally AEW were asked to recommend a method of roll stabilisation suitable for stationary or slow moving ships. After surveying the available methods an active moving weight stabiliser, using a solid mass, was recommended. Consequently AEW were asked to design and test a model scale stabiliser.

The design was carried out using classical control techniques as these were most familiar to the author, and time was limited. A stabiliser was then built and tested in a ship model.

The performance of a stabiliser is optimised by including a

controller in the feedback path. In the case of the model stabiliser the feedback signal used was roll angle and the controller consisted of a phase advance circuit. To complete the project it was decided to carry out a theoretical comparison between a controller designed using optimal control techniques and the one used in the model

experiments.

1.2. All the theoretical work and the electronic design of the servo amplifier and controller was the sole responsibility of the author. The mechanical design of the moving weight servo was the work of Mr B S Ireland and Mr N Halewood. The unit was made in the AEW workshops under the supervision of Mr R Lane.

The work of Mr S G Dreier, who supplied the gyros and associated instrumentation, Mr D Sharpe, who assisted in recording the data, and Mr R Jeffries,who ballasted the model and ran the Ship Tank carriage, is also acknowledged.

ROLL STABILISERS: INIRODUCTION AND HISTORICAL SURVEY

2.1. The significant roll motions of a ship occur at a narrow band of frequencies around the roll natural frequency of the ship, wo. This is because the roll damping is small and even small roll moments due to waves encountered at frequencies around wo cause large roll motions.

Roll damping is roughly proportional to the ships forward speed so that a stationary ship will have the largest roll motions for a given

level of excitation.

Roll stabilisers, of any kind, work by applying roll moments to the ship in opposition to the roll moments due to the waves. As the roll moments causing the motion are comparatively small then it

follows that the necessary stabiliser moment is of similar order. In turn this means that the stabiliser can only have a noticeable effect on roll motions at frequencies around

wo.

The most popular roll stabiliser system in use today (in warships at least) is the active fin system. Here low aspect ratio, aerofoil

section, fins provide the necessary roll moments. Unfortunately these do not work well at speeds below about 10 knots because the fin lift is proportional to the square of the ship's speed. Only gyro stabilisers and moving weight stabilisers function at all speeds.

Gyro stabilisers were considered to be unsuitable for naval use

because of the difficulty in designing a unit capable of meeting the required shock specification. Consequently only moving weight

stabilisers are considered here. A moving weight stabiliser is one where the required roll moment is generated by a mass moving back

and forth across the beam of the ship. The mass can either be a solid weight or a mass of liquid. There is a further division into active and passive systems.

An active system is one where the position of the moving mass is controlled by a servo system and therefore requires a power input. A passive system is analogous to a double pendulum where a pendulum

of small mass can dampen the motion of a much larger one to which it is attached, provided that both have approximately the same natural period. A passive roll stabiliser can be described as a

low gain positive feedback system. Its advantage is that it requires no power input.

In general the performance of an active system will be superior to a passive one that can provide an equal roll moment. The way the

capabilities of particular stabiliser installations are compared is to quote their wave slope capacity ac

where:-maximum roll moment of stabiliser 180

a =

g GM where

ac is in degrees.

1 is the mass of the ship, GM is the metacentric height. In practice

ac can range from about to 5 degrees. 2.2. Solid Weight Stabilisers

One of the earliest uses of a closed loop system was the active moving weight stabiliser designed byJ I Thornycroft in about 1890. A prototype was built and fitted to the steam yacht CECILE and the ship trials reported in Reference 1. The mass of the ship was 230 tons and that of the moving weight 8 tons.

Roll reductions of 50 per cent were measured. In spite of this success no commercial units were made.

A passive moving weight system is described in the discussion to Reference 2. This was designed by Professor Cremieu and consisted

of a small heavily laden trolley running on curved rails. The whole device was enclosed in a tank that was filled with glycerine and water in order to provide dampng.

The unit was fitted to a 700 ton steamer which was known to roll badly. The trials took place on a single voyage from Dumbarton to Folkestone. It was not a success. The roll motion of the ship was very irregular and the 'stabiliser' did nothing to reduce it.

It did, however, generate a lot of noise. The ship owner insisted that it be removed when the voyage finished and the experiment was not repeated.

After these early experiments solid moving weight stabilisers have been neglected by shipbuilders. Although some slight interest has been maintained in the technical press.

Recently a joint project between Vosper Thornycroft and Glasgow University was undertaken to consider various types of moving weight stabilisers for fishing vessels. The results of the theoretical study are given in Reference 3 and while these are promising, again, no units have been made.

2.3. Fluid Weight Stabilisers

The only moving weight stabiliser in common use today is the passive tank stabiliser. These are of two types, the free surface tank

pioneered by Watts in 1876 and the U-tube tank pioneered by Frahm, see Reference 2 and Figures la and b.

The correct phasing of the moment applied to the ship by the moving mass of water is achieved by turning the tank to the roll natural frequency of the ship. At the tank natural frequency the roll moment provided by the tank lags the roll motion by 90 degrees. This is the correct phase for the maximum reduction in roll motion. References 4 and 5 are typical design methods for this type of roll stabiliser.

They work well when there is a continuous excitation at the roll natural frequency but their operation under transient conditions is less certain.

At very low frequencies the tank actually increases roll motions and this can sometimes be a problem in quartering seas (following waves at an angle of 45 degrees to the ship). Under these conditions the tanks are often emptied.

Active tank stabilisers are an extension of the U-tube passive tank. For minimal power systems some control can be Obtained by either

controlling the flow of water between the wing tanks or adjusting the air flow between the air spaces above the water in the tanks.

A fully controlled active tank system was designed and fitted to the MONTE-CINCO in about 1970. This is described in Reference

6.

Figure lc is a diagram of the active tank. Details are as

follows:-Ship mass Length Beam GM

Total mass of water in stabiliser tank Weight of water displaced

Peak electrical power consumption Mean electrical power consumption Peak hydraulic power

Wave slope capacity of stabiliser

3,500 tonnes / 96 metres 14 metres 0.85 metre approx 150 tonnes 12.5 tonnes 150 hp 0 to 75 hp 500 hp

2.4

degrees

A special feature of the power system is a flywheel energy storage device which enables energy peaks to be suppled, or if not required to be absorbed while the stabiliser is working.

The system is reported to have worked well on several Atlantic crossings and in spite of some heavy storms the recorded roll angle never exceeded 6 degrees.

One feature of a fluid moving weight stabiliser is that the

effective maximum lever arm is the distance between the wing tanks as compared with half the beam of the ship for a solid weight stabiliser.

2.4.

Conclusion

Having reviewed the current position with regard to stabilisers for slow moving ships an active moving weight system was recommended. A solid moving weight was preferred since it was thought that the

technology involved should be simpler than for an active tank system. AEW were then asked to design and test such a moving weight

stabiliser on a model scale to see if there were any hidden snags in such a scheme.

3. A MATHEMATICAL MODEL OF SHIP ROLLING

3.1. Introduction

A ship treated as a rigid body has 6 degrees of freedom, usually considered as motions of the centre of gravity of the ship relative to the ship's equilibrium position using a right handed set of

orthogonal axes.

These are illustrated in the following diagrams

a. The Symmetric Motions

b. The Lateral Motions

The arrows indicate + ve directions.

PLANE

1

Polk,

SW AY

ST1.0,4444AD

Rolling can be affected by yaw, sway and heave motions but fortunately as far as stabilisers are concerned we are interested in the roll response of a ship due to a roll moment. This can be considered as a single degree of freedom system since rolling itself does not, in general, induce other motions.

This model will be developed from first principles and where waves are introduced only long crested beam waves will be considered. This is not unreasonable because a wave can be considered as a means of providing a roll moment input to a ship and as such the precise derivation of such a moment is unimportant.

3.2. Restoring Force

It is instructive to consider

why a ship

floats upright

and

why it returns to an upright position in calm water after a disturbance. Consider Figure 2. An upright ship in calm water is maintained in equilibrium by the gravity force

FG acting through the centre of gravity, G, being opposed by the buoyancy force FE acting through the centre of buoyancy B. If the ship is disturbed and it rolls an angle (f) the position of the centre of buoyancy changes to B'. The

buoyancy force now acts vertically throu& B' and this line of

action meets the axis of the ship at a point M, called the roll meta-centre, and the distance between G and M (written GM) is called

the metacentric height.

RoLt_

The ship experiences a restoring moment which is equal to FG x GZ where GZ is the horizontal distance between G and B'. For roll anglesup to about 30 degrees GM is fairly constant for most ships and GZ = GM sin 6 (6 is the roll angle).

This is usually approximated to GZ = GM.6 Thus the restoring moment = g GM6

For very large angles of roll GZ varies as shown in Figure 2. So that the restoring moment varies from the value of 1g GM6 near the origin to nothing where the GZ curve crosses the x axis, at 6v, called the angle of vanishing stability.

We are interested in rolling about the origin. Thus if a ship in calm water experiences a constant roll moment M it will roll to an angle

Om such that 1g GM Om = M

If M is suddenly- reduced to zero then the ship will try and regain an upright position, and making the reasonable assumption that the water will provide some damping we get the equation

J(t) + D6(t) + 1g GM 6(t) = 0 (1)

where J is the effective ship roll moment of inertia and D is a linear damping term. J is found to contain an added mass term so that

J = (I + SI) where I is the roll moment of inertia of the ship structure and (SI is the added mass moment of inertia.J is

approximately equal to 1.251 for small angle rolling. We will assume here that D can be considered constant for all ship speeds - a point which will be discussed more fully in Section

4.

Thus dividing (1) throughout by J we get:

"c(t) + ,314)- 6(t) + 6(t) = 0

This is the unforced equation of a simple second order system, so we can write

ci)(t) + 2nw 6 + w2 43(t) = 0

0 0

and define n as the linear roll decay coefficient and wo as the undamped natural frequency.

In practice n varies between about .01 for a stationary ship to about 0.2 for a ship travelling at 30 knots.

If there is a time varying roll moment M (t) being applied to a ship then (1) can be written as:

6

-mcp(t)

.(1)(t) + 2nwo 3(t) + w2(t) = j

Taking Laplace transforms we get

K w 2 q(s) 1 s o

M(s)

J 2 ₂ s + 2nw s + w 2 T s2 + 2nw s w o o where K -s Is -JR

In block diagram form

K w

S 0

2

,

s +

cnwos + wo2

3.3. A Ship in Beam Waves

-A ship in regular beam waves is shown in Figure 3. Here the ship is on the slope of a very long wave of wave slope a. It is displaced an angle 0 (called the relative roll angle) from a so that the

absolute angle of roll cp = a + 8.

The Froude-Kryloff hypothesis maintains that the ship's gravity force now acts normal to the wave slope so that the ship experiences a moment Xig GM0 or more strictly if a and 8 are a(t) and 0(t) we can write

J(t) + Dil(t)* + Xg GM e(t) = o

or J(t) + D3(t) + Xg GM yb(t) = a(t).1.g.GM

Taking Laplace transforms and writing a transfer function we get

w2

K w2

o

cp(s) 1 _$ o

a(s)

s2 +

2nwos + wo2 Ks s2 + 2nwos + wo or in block diagram form

c(s) K w 2 S 0

2,

S + cnwo

+ w2

cp(s)

* Alternatively damping can be considered as being proportional to

_6.

Mw

This effects the transfer function by introducing a high frequency

zero. However this has little practical effect.

Mw(s)

M(s)

a --2 0 o_ 0

where Mw(s) is the roll moment due to waves, in this case Mw(s) = M(s)

3.4. Roll Stabiliser

Introducing a roll stabiliser into the block diagram we get Mw(s)

M (s)

a( s ) where G(s) = K w 2 S 0

2,

s + cnwos + wo2 MST(s) G(s) H(s)

and H(s) is the transfer function of the stabiliser.

If we now consider the input to the ship as a roll moment Mw(s) and do not bother how this arises then it is clear that so long as the

stabiliser itself only introduces roll moments to the ship we can optimise the closed loop system to stabilise the ship.

qb(s) _ G(s) Writing mw ) (s, -1 + G(s) H(s) C1)(S) 1 Or MU

(s)

1

+ G(S)

H(S).G(S)

Now:. M (s) = 1 + G(s) 1.1s).14w(s) 1

the rolling is reduced by the factor 1

+ G(s) H(s) as compared with the unstabilised ship. Call this R(s) roll reduction factor.

If we measure G(jw).H(jw) we can easily obtain the frequency response R(jw).

It now becomes clear why a stabiliser is effective only near the roll natural frequency. This is because G(s) is dominated by Ks

1

(= ) and R(jw) < 1 except around

wo. Now consider H(s).

(1)(s)

lg GM

For a moving weight stabiliser of the type designed for the model experiments H(s) can be written

8

H1 ( s ) H2(s)

where H1 (s) is the controller transfer function' H2(s) is the moving weight servo transfer function and in is the moving mass and C is a

constant with units of metres per radian.

H2(jw) is designed so that it has unity gain at

wo and a phase shift of less than 10 degrees. This implies a weight servo natural

frequency of about 3 x wo if we can assume H2(s) to be second order. While developing the theory of Hi(s) we can assume that H2(s) = 1

and re-introduce it if necessary at a later stage. , 2 Writing G(jw).H(jw) = mFr' . Hi(jw) 1g GM _{(wo2 -} w2) + 2njw0w ' 1 and at wo G(jwo).H(jw0) - m_c 2nj . H1(j) GM

To minimise R(jwo) then G(jaH(jwo) shauld be real. The simple way to do this is to make Hi(jwo)

= Kcoj where Kco is the controller gain at wo.

Thus G(jwo).H(jw ) - mc . co

o 7-24 2n

3.5. Practical Values

It is clear that in theory R(jwo) can be made very small by making in or c or Kc very large but in practice this is not possible.

Obviously there are limits to in and this is usually of the order of

1 to 2 per cent of X. The beam of the ship, b, is limited so that

if c is made very large the weight is continually forced to the stops and the stabiliser would be operating as a bang-bang system and the roll motion of the ship is jerky and unacceptable. (Kco is a non-dimensional constant which can be regarded as integral with c.)

This explains why two stabilisers with the same

ac can have very

different roll responses.

Once in and b are known then it is clearly a matter of judgement as to what value of c is acceptable. A small value of c will mean that R(jwo) is comparatively large (eg i or 1/3) and the stabiliser will not saturate over a wide range of wave slopes. A large value of c means that R(jwo) is small (eg lie to 11/20) and the stabilisation is

very good for small wave slopes but the system soon saturates as the wave slope increases.

o(s)

10

For the best operation over a wide range of wave slopes then an adaptive system is indicated where c is reduced if in a particular sea state the stabiliser is continuously hitting the stops.

3.6.

Controller Design

As stated in the Introduction the classical approach to the control problem was used. In this roll angle is used as the feedback

signal. This is fed to a controller which operates on this signal to produce the desired control signal.

The design criteria for the controller, from the foregoing analysis, was that H1(jwo) = Kcj.

In practice a two stage phase advance circuit was made with each stage providing either 35 degrees, 45 degrees or 55 degrees of phase advance at

wo. This was because the controller was designed before the experiments were carried out and it was felt necessary to provide some flexibility to cope with the weight servo lag and any possible interactions between other motions.

A circuit diagram of the controller is given in Figure 4a and the frequency response of a single stage given in Figure 4b. Note that the DC gain of each stage was maintained at 1 as a matter of

convenience.

An alternative to this type of control would be to use a roll

rate signal for the feedback and scale the gain to give the required R(jw0). Although this was comtemplated for the experiments the rate gyro fitted in the model did not function correctly so this type of control was not used.

4. THE MODEL EXPERIMENTS

4.1. The Purpose of the Experiments

The experiments were designed to validate the theoretical model. First of all forced rolling experiments were carried out. Here the weight was driven back and forth across the model at various

frequencies around wo and the resulting roll amplitude measured. Thus G(jw).H2(jw) was measured ab that appropriate phase advance and gain settings could be chosen for stabilised roll tests. These tests were carried out at zero and two forward speeds.

It soon became Obvious that at zero and low forward speeds that there was a considerable non-linear damping coefficient that had not been introduced into the theoretical study. To investigate this

roll decay tests were carried out and analysed using the following formulae as a model.

Finally stabilised roll tests were carried out in regular beam seas at zero speed (a necessity since the only facility available was a towing tank.

For the forced rolling experiments the moving weight servo was placed initially so that the weight axis passed through the centre of gravity of the model. The servo was then raised and lowered

6.8

cm to see if this would affect the response. In theory raising the weight should produce a slightly enhanced roll moment, while lowering should reduce

it.

In fact there was no measurable difference between the three forced roll responses.

It was not possible to move the servo fore and aft in the model in these experiments.

4.2. Practical Details

The experiments were carried out in No 1 Ship Tank at AEW. This

tank is 145 metres long, 6 metres wide and

2.4

metres deep. The

wavemaker is of the wedge shaped plunger type with a normal working range of 1 to 12 metres wavelength and a maximum waveheight of

0.3

metre.

Details of the model are as

follows:-Model mass

278

kg

Beam of model 572 nmn.

Useful weight travel +

250

mm Boll natural frequency,

wo

0.35 Hz (2.2

radians per second)

GM 31.3 mm

The frontispiece shows the instrumentation in the model. Figures 5 and 6 are photographs of the moving weight servo. Some details of the design of the servo and servo amplifier together with some further instrumentation details are given in Appendix I.

Three values of weight were used in the forced rolling experiments they were

0.364

kg,

0.83)4

kg and

1.304

kg.

For the zero speed experiments the model was lightly tethered across the tank behind the Ship Tank carriage using strings and pulleys as shown in Figure

7.

For the forward speed experiments the model was tethered in a similar manner along the axis of the tank. The model was self-propelled by an electric motor driving two propellers via a gear box. The power for the drive motor was obtained from a controller

on the carriage.

A block diagram of the instrumentation is shown in Figure

8.

For the forced rolling experiments the FRA fed sinusoidal signals of various frequencies between

0.04

and

0.7

Hz to the servo amplifier. The

amplitude was adjusted so that the peak to peak weight travel was about half the beam of the model. The resulting roll moment was detected

by the roll gyro, the output of which was fed to the roll scaling amplifier and fed to the Y input of the FRA while the oscillator signal was fed to the X input. Plots of were made for the various experiments undertaken.

For the roll decay experiments the model was given an initial angle of heel and then released. A recording of the roll decay was

obtained on the UV recorder.

For the stabilised wave tests the FRA signals were fed to the wave-maker controller input. The wave probe signal was fed to the X input of the FRA and the roll signal to the Y input. These and all the other signals were recorded on the UV recorder.

The two stabiliser controllers in Figure 8 have already been described but as stated only the phase advance controller was used.

Note for closed loop operation the scaled roll signal is fed to the controller, the output of which drives the moving weight servo. The

loop is closed by the roll dynamics of the ship.

4.2.1. The Forced Rolling Experiments

The responses obtained with the three different weights is shown in Figure 9. Throughout these tests the FRA output was kept constant at 5 volts amplitude, which gave a weight motion of 110 mm amplitude. The scaling on the amplitude response is arbitrary at this point as the gain of the roll scaling amplifier was set solely to give a reasonable voltage for measurement purposes. A polar plot of the response with the 1.304 kg weight is shown in Figure 10.

To obtain some information on the effects of forward speed forced rolling responses were measured at 0.55 m/sec and 2.0 m/sec

(correspondingto 4.2 and 15.2 knots for a 31/25th scale model). For these responses a single weight was used (1.305 kg) with the weight axis through the CG of the model. The results are shown, together with a zero speed response measured with the model axis along the

tank, in Figure 11.

The zero speed response shows a slight increase in the frequency at which the peak occurs as compared with the response shown in

Figure 10. As some time elapsed4petween the taking of the two responses it is not known whether this effect is due to the model position in the tank or to a slight change in model condition.

4.2.2. Roll Decay Tests

Roll decay records were obtained at 0, 0.55, 1.1 and 2 m/sec.

A typical roll decay for 2 m/sec is shown in Figure 12. The results are tabulated in Table 1 where the peak displacements at cycle intervals is given (ie 21-111, ih2 etc as shown in Figure 12). With hindsight it would have been interesting to conduct these tests at many more speeds in order to give a better plot of the linear and non-linear damping terms.

The anlysis of the roll decay records was carried out as follows. Equation 1 (page 10) was analysed using the Krylov and Bogoliubov method of slowly varying amplitudes and phase and an equation obtained for the amplitude variation in terms of D and D2. A

manual curve fitting operation was then carried out to choose values of n and D2 so that the amplitude equation fitted the curve throne

the peaks of the roll decay records.

Incidently the above analysis indicated that there is no phase

variation due to the presence of D2 in equation 1. The results were as

follows:-The data is given in graphical form in Figure 13.

4.2.3. Discussion of Model Results So Far

It is obvious that it is important to know the value of D2 in order to be able to predict the forced roll response of a slow moving or stationary ship. However it will be noticed that the phase charac-teristic (in Figure

9)

is unaffected by roll amplitude so that the control strategy already outlined still holds good.

Increasing speed reduces the peak of the roll response and reduces the frequency at which it occurs. If the same controller settings are used for all speedsthen there would be some changes in effectiveness with speed, but it would be slight. In practice wo for a ship must vary throughout a voyage as stores and fuel are used up. Therefore a setting for some average condition must surfice unless some sort of systems identification and adaptive technique is used.

For the model stabilised roll experiments it was decided to use the controller settings that gave a 90 degrees phase advance at 0.35 Hz. The 1.304 kg weight was adopted for the experiments and the gain of G(jwo).H(jwo) adjusted to be about

2.

The resulting polar plot of G(jw).H(jw) is shown in Figure

14.

4.2.4.

The Stabilised Model Experiments

With the model placed in regular beam waves the rolling at 0.35 Hz was first of all considered. Stabilised and unstabilised rolling was measured for a variety of wave slopes (up to the maximum

obtain-able). The results are shown in Figure 15.

Speed m/sec n D2 0 0.0115 0.29 0.55

0.048

0.20 1.1 0.093 0.18 2.0 0.115 0.15

Next a frequency response was obtained. This is shown in Figure

16.

The range of frequencies was rather limited, being very much a

function of the wavemakers. It would have been desirable to keep the wave slope constant throughout the frequency range, but this was not possible. The actual readings used to plot Figure 16 are there-fore given in Table

2.

The effects of the non-linear damping is clearly evident in the unstabilised rolling in Figure

15,

where the roll/wave slope decreases quite rapidly with increasing wave slope. It is inter-esting to note that for stabilised rolling the roll/wave slope is much more constant and the model behaves more like a linear system, ie the roll/wave slope is almost independent of wave slope amplitude. The stabiliser worked as expected so that the object of the experiments had been achieved. Clearly more model tests could have been carried out and the effectiveness of the stabiliser at other headings and speeds measured but it was not possible at this time.

4.2.5.

Power Measurements

Figure 8 shows that a circuit was provided to record the power delivered to the weight servo motor. In the event there was a lot of high frequency noise on this signal so that it was difficult to decide what power was actually being delivered to the servo motor to drive the weight. An average bower for the wave experiments was about

1.5

watts.

5.

OPTIMAL CONTROL APPLIED TO A MOVING WEIGHT STABILISER 5.1. Introduction

This part of the report did not form part of the original AEW project because the work of designing stabiliser controllers is not formally

the work of AEW.

However it was decided that the model moving weight stabiliser would provide a useful case study in the use of optimal control techniques because of the comparative simplicity of the system.

A linear fourth order mathematical model of the ship plus moving weight is developed and the optimal system is compared with the phase advance controlled system used in the model experiments.

5.2.

The State Equations

Section 3 established that a ship rolling to small angles can be considered as simple linear second order system for the purposes of stabiliser design. Roll moments are applied to the ship by a moving weight which is driven by a servo that can also be described as a second order linear system. It is assumed that the weight servo loop has already been designed and that reasonable natural frequency and damping characteristics have already been obtained.

The proposed model, using familiar notation

Cu) w2

2 _{+ 2Cws +}

_s2

ww w

Ws)

Where c is a constant with units of metres per radian.

ww is the natural frequency of the moving weight servo and

c is the damping factor of the moving weight servo,d(t) is the weight position relative to amidships.

a(s) is the wave slope input and is the gain between wave slope and

Ks

roll moment.

In the time domain we can write

a(t)...2cww a(t) .4- ww2 d(t) = ww2 c.u(t)

mw

2

and CI;(t) + 2nw0 .4)(t) + w2 A(t) = K3 w₀₂m.g (dt) = d(t)

o GM if we write d(t) = x1(t) and (P(t) = x3(t) à(t) . x2(t)

_$(t)

= )(4(t) a(t) = *2ct) .c.1)(t) = 1c4(t) we can then write the system in state form.

K w

S 0

wo2 + 2nwos + s'

To introduce the phase advance controller into the system we need to introduce two extra state variables.

The block diagram becomes:

a( s ) t ) = 0 _ w 2 w 0

mw

2 1

-2w

0 0 0 0 0 - wo2 0 0 1 - 2nwo x(t) + 0 CW 0 2 u(t) + 0 0 0 w 2 w D(s) 0 2 a(t -0

(s)

S

+b

S + a s2

+w

S + w w2 (s + b)2 where

Kc (s + a)2 is the transfer function of the controller. Defining xi to x4 as before we get:

X1 = x2 X2 = -

w2 x

- x w 1

w2

+ cww2 x5 X3 = 2 m w X4 = -

2_

xi - wo2 x3 - 2nw x4

+ w02a

L GM k5 = - ax5 + k6 + _bx6 X6 = - ax 6 - K

c3

X

- K bx3

Since X3 = x4 X5 = - x5 + (b - a)x6 - x4 Kc - Kc bx3 16 M.8 1 Ks a( s ) K w 2 s 0 + 2nw0s + w, and X6 = In matrix =

-

ax 6 K x4 - K bx3

form this

becomes:-0 1 0 0 0 0

x+

a w 2

_{- 2ww}

0 0 cw 2 0 0 0

mw

2 0 0 1 ₀ 0 o 0

-w02

-2nwo

0 0 woz 0 0 _{- Kcb - Kc} - a (b - a) 0 0 0 _{- Kcb - Kc} 0

-a

X5(S)

_cw 2 - - -- -w

--

-5.3. Values in the Plant Matrices

Let the natural frequency of the weight servo = 1 Hz or 21. radians/second and

= 0.6 w 2 = 39.48

= 7.54

The natural frequency of the model = 2.2 radians/second so wo2 = 4.84. In order to choose a realistic linear damping coefficient for the model the following procedure was adopted.

When deciding the closed loop gain of the model stabiliser it had been decided to reduce the unstabilised rolling at

wo to one third. This meant that the open loop gain at wo would be 2. The resulting value of roll angle/wave slope for the stabilised model was about 2.2 (see Figure 15). For a linear system this would mean that the roll angle/wave slope at wo would be 6.6.

1

Since roll/wave slope at

wo =-ri then n becomes 0.0758, and 2nwo = 0.3335. The value of Kc was decided on the basis that the gain of the controller

at

wo should be equal to 1.

Each stage of phase advance has a transfer function of

V(s)

(0.856 :)s)

ie a = 5 and b = 0.856

V. (s) (

in

The gain of each stage of phase advance at wo = 0.432

Kc = 1

- 5.358 (0.432)2

c is now calculated so that the open loop gain at

wo is 2.

c.K .w 2.m.g

S o

ie 2 = (assuming gain of weight servo at _{wo =} 2n wo2 4n 4 . 0.0758 . 278 . 0.0313 ie c = Ksm.g 1.304 = 2.022 4 c w 2 = 79.83 2 mw Finally --2 = 0.7253 1 GM 2cw -.

The phase advance system becomes: o 1 0 o o o

- 39.48

- 7.54

0 o

79.83

0 o 0 o 1 o o

0.7253

o

- 4.84

- 0.3335

0 0 0 0 -

4.586

- 5.358

- 5

- 4.144

o o

- 4.586

- 5.358

o

- 5

For the application of optimal control the plant matrix becomes

x=

x=

o 1 o o

- 39.48

- 7.54

0 0

O 0 0 1

0.7253

o

- 4.84

- 0.3335

5.4.

The Optimal Control System

For an nth order system 5c(t) = Ax(t) + bu(t)

Then for a cost function J =

Where Q is positive definite.

The input which minimises J, uo(t), is given by

u(t) = - bT K x(t)

where K is positive definite and R = - ATK - KA + b.bTK - Q. r

This is solved with the final value K(tf) = O.

For the moving weight stabiliser which is fourth order.

18 (xT(t) Qx(t) + r u2(t))dt -t

Q=

₁₁ and K = K11 K12 K13 K14 0 q22 o

q33

K12 K13 K22 K23 K23 K33 K24 K34 0 0

°

q44

K14 K24 K34 K44 u(t) x +

79.83

x + 0

4.84

-- - -a 0 0 0

33

To obtain equations program.

equations

where symbols are only given for non zero terms.

the required values of K12'

K22' K23 and K24 the ten K

were solved using a step by step differentiation computer This uses a fourth order Runge-Kutta routine to solve the

going backwards in time from the known final values of K(t) = 0.

uo(t) = - --r- CO b2 01 K12 b2 01 K11 K12 K13 K14 K22 b2 1(12 K22 K23 K24 K13 K14 K23 K24 K33 K34 K34 K44 K23 b2 x(t) K24 b2 . x(t) and A = 0 a12 0 0

b= 0

a21 a22 0 0 b2 0 0 0 a314 0 al41 0 a)4 al-04 0 Let this The compensated = be written at uo(t) = system becomes 0 a12 0 0 G1 G2

x+

G3 G4 . x(t) [G1 G2 G3 G a21 a22 0 0 b2 0 a34

k=

a)411 0 0

a3

_a44 a12 0 0 (a21 + b2 G1 ) (a22 + b2G2) b203 b2G4 0 0 0 a34 al41 0 8'43 a44 [Let j 0

The required values of Ks are the steady values obtained after about 12 seconds of solution.

The ten equations of

it

are:-2 b are:-2

12 2

K11 = _{q11 - 2(a21 K12-+ a41 K14)}

K12 K22 b22

K12 = _{(a21 1(22 + a41 K24 + a12 K11 + a22 K12)}

K12 K23 b22 (

13 3.21 K23 a41 K34 + a43 K14)

K12 K24 b22

"14 (a34 K13 + a44 K14 a21 K24 a41 1(44)

2 b 2 22 2 "22 = r q22 - 2(a12 K12 + a22 K22) K22 K23 b22 (a K

+a

K

+a K)

23 -12 13 22 23 43 24 r7p. K22 K24 b22 ,a34 K__ a44 -44 K-24 a12 K14 a22 K24) 2 b 2 23 2 1(33 = q₃₃ - 2 a,₄₃ K₃₄ K23 K24 b 22 1(34

-r (a34 K33 4- a44 1(34 4- a43 K44)

2 b 2

24 2

"44 = r 2(a34 K_, a44-44 K 44) _{- a44}

The optimal control gains were computed for various values of q33 with and q44 = 0. This meant that (roll angle)2

r = 1 and q11

Q22

was penalised in the cost function. The values of q33 were 2, 4,

6, 8

and 10. The values of K/2, K22, K23 and K24 obtained are shown in Figure 17 plotted against

q33. A system step response was also

computed for each value of q33 considered. For this x3 (roll) was given a value of 1 and xl, x2 and x3 set to zero. The system relax-ation response was then computed using a Runge-Kutta routine.

The rather surprising result is that for some values of q33 + ve feedback of roll angle is required for the optimal control system.

20 r - -r - -22 23

The probable reason for this is that the phase of the feedback signal relative to roll angle should be greater than 90 degrees lead and that for a conventional system feedback of roll acceleration would be considered in addition to the roll velocity term. The only signal of the right phase in the optimal system is + ve feedback of roll.

It is interesting to note (Reference 7) that in early naval stabiliser fin control systems, circa 1941, this technique was used because at that time no measurement of roll acceleration could be obtained.

While academically this is interesting, in fact, removing the roll angle term from the feedback has very little effect on either the step or the frequency response of the system. This supports the view adopted earlier in this report where it was stated that a roll rate signal could provide the necessary feedback in a moving weight stabiliser system.

The feedback of weight position and weight velocity was small which indicates that the initial design of the weight servo was satisfactory.

5.5. Comparison Between the Two Systems

A step response and frequency response of the phase advance system was computed. When the step response was compared with those obtained for the optimal system it was found to be comparable with the one obtained with

q33 = 2. The two step responses together with the corresponding weight motions are shown in Figure 18.

The equation for the optimal system is as follows:-(t) = - 44.71 0.7253 1 - 8.2o4 + 11.74 - 34.19 1 - 4.8o - 0.3335 x(t) +

It is intriguing to notice that the weight moves off in the wrong direction in the case of the phase advance system (Figure 18). It is not known how significant this behaviour would be in a real sea

environment.

The frequency responses of the two systems is shown in Figure 19. These show that the weight motion lags the roll motion by about

90 degrees over the useful range of the stabiliser. In practise this means that the weight is being driven downhill most of the time, a

fact that is confirmed when one looks at the video tape recording of the model wave experiments. This probably explains why the power levels measured in the rolling experiments were small. This in turn means that the power criteria adopted for the servo motor, see A1.2,

Appendix I, was unnecessarily high.

The differences in the + ve and - ve feedback of roll angle (the phase advance controller has a gain at zero frequency) is shown in the low

a(t) -0 0

0

-0 0 4.84

frequency asymptotes of the weight motion phase and the roll/wave slope amplitude.

If roll rate control was used on its own the low frequency asymptote of the weight motion phase would be - 90 degrees and the roll/wave slope would be 0 db.

Since it is difficult, if not impossible, to estimate a cost function to get a particular response from a control system the process adopted here is the one most likely to be used by the practical designer. First of all some system performance criterion is decided. This can be a required step response, as in this case, or a frequency response

or perhaps a maximum power consumption for a particular operation or even a number of maybe conflicting criteria.

Optimal gains are then computed for a range of cost functions and the system chosen that most nearly meets the required performance criteria.

6.

CONCLUSIONS AND RECOMMENDATIONS

This project shows that an active moving weight roll stabiliser provides a useful means of stabilising a slow moving or stationary

ship.

The forced rolling and roll decay model experiments confirmed that a ship rolling in response to a roll moment can be described as a second order system with damping proportional to roll velocity and roll

velocity squared. However it has also been shown that a stabiliser can be designed using an equivalent linear damping (roll velocity) term because the non-linear damping does not affect the phase response of the system.

Optimal control techniques have been shown to be applicable to the design of the control system for a moving weight stabiliser although no clear advantage appears over a system designed using classical techniques.

The simplest, and cheapest form of control would be the - ve feedback of a roll rate signal.

The stabilised experiments were somewhat limited in scope and should be extended to rolling in irregular beam seas and to rolling at other

speeds and headings.

The maximum weight used in the experiments was fairly small

-0.47 per cent of the model mass; in practice this could be increased to about 1 per cent of the ship mass. _{This would increase the wave slope}

capacity from 1.9 degrees to 4.0 degrees for Model used in the experiments.

In the model the moving weight was deliberately placed near the centre of gravity of the model in order to reduce to a minimum possible sway and yaw interactions. Should it be necessary to site a stabiliser at some considerable distance from this position then further theoretical and model work should be undertaken to consider the effects on the forced roll response.

References

Reference 1. J I Thornycroft. Steadying Vessels at Sea. Trans INA Vol

33 1892.

Reference

2.

H Fradiam. Results of Trials of the Anti-Rolling Tanks

at Sea. Trans INA Vol 53 Part I.

1911.

Reference

3.

T R F Nonweiler, P H Tanner, P Wilkinson.

Moving Weight Ship Stabilisers. Third Ship Control Systems Symposium.

1972.

Reference

4.

G J Goodrich. Development and Design of Passive Roll Stabilisers. Trans RINA.

1969.

tm

Reference

5.

J J Van

dA,

Boche J H Vugts. Roll Damping by

Free Surface Tanks. Netherlands Ship Research Centre. Report No

838.

1966.

Reference

6.

G Fournier and J Berne. Description et Resultats D'essais d'um Nouveau System de Stabilisateur Actif Anti-roulis. 1970 Session, Association Technique Maritime et Aeronautique. (Translation: MOD Lunguistic Services GLS(A)

3)40 1976.)

Reference

7.

J Hebditch. Persona/ Communication (AEW

1977).

'Table 1

ROLL DECAY ORDINATES AT HALF CYCLE INTERVALS

24 Model Speed 0 2.0 m/s -Mean Period Seconds -3.08 3-11 . 1 3.06 3,07 3.14 3,16 1 3.06 , 3.107 Peak 0.390 0.379 0.284 0.256 ' 01.169 0.211 1'0.196 HO.203 Roll 0.339 ! 0.328 0.253 0.205 'll G.121 1 0.152 110.132 0.138 1 Angles 0.292 0.286 0.1931 0.167 0.087 0.109 0.088 0.091 Radians 0.2531 0.248 0.159 0.139 0.064 0.080 0.061 0.063 0.222 0.219 0,133 0.114 0.049 0.062 0.045! 0.094 0.197 10.195 0.112 0.094 0.038 1 01.046 0.034 _0.031 0.176 0.175 o.094

o.og6

'0.029 0.036 0.026 0.022 0.159 0_149 0c077 0:06351,0.021 0.0281 0.144 0.143 o,o66 0.053 1 0.0145 0.021 0.131 0.130 0.057 '0.045 0.015 0.120 0.119 0.048 0.039 0.110, 0.102 0.110 0.J02 10404o 0.035 0.033 0.029 . 1 110.094 0.094 0.031 10.023 '1 1 0.087 0.087 0.026 0.0205' 0.081 0.082 0.023 0.017 n ;L0.077 0.077 0.021 1 ' 0.071 0.071 0.017 j ,0.067 _0.0611 H 0.063 0.063 1 0.059' 0.059 - 1 0.057 0.056 , 0.053 10.051 0.053, 0.050' 1 0.048 0.0481 1 0.045 1 0.045 _ 0.55 m/s 1.1 m/s

Accuracy of roll and waveheight + 2

per cent of reading.

Accuracy of phase + 3 degrees.

* Wavelength are calculated according to the formulae

El

2Rd

Wave velocity =

2Tr

tanh

where d is depth of water and A is wavelength.

Table 2

DATA FOR FREQUENCY RESPONSES (FIGURE 16)

Wave Whether Stabilised Roll Amplitude Degrees Roll Wave Slope Frequency Hz Length* Metres Height mm Slope Degrees Amplitude Ratio Amplitude Ratio db Phase Degrees 0.30 14.30 85 1.07 No 3.08 2.85 9.10 - 18 0.30 14.30 85 1.07 Yes 1.72 1.61 4.12 - 32 0.33 12.47 105 1.52 No 7.81 5.14 14.22 - 29 0.33 12.47 106 1.53 Yes 3.44 2.25 7.03 - 67 0.35 11.44 120 1.89 No 12.97 6.86 16.72 - 85 0.35 11.44 119 1.87 Yes 4.38 2.34 7.38 - 90 0.385 9.85 143 2.61 No 9.41 3.50 10.90 - 162 0.385 9.85 140 2.57 Yes 4.48 1.74 4.83 - 123 0.40 9.23 135 2.60 No 7.03 2.70 8.64 - 162 0.40 9.23 136 2.66 Yes 4.38 1.64 4.32 - 130 0.50 6.21 209 6.05 No 3.00 0.496 - 6.10 - 180 0.50 6.21 254 7.36 Yes 3.04 0.414 - 7.66 - 157

-DETAILS OF SERVO MOTOR

Type Vatric Size

18, permanent

magnet, 12 volt Maximum no load current

Maximum no load speed Rated torque

Maximum lOad speed Minimum load speed Maximum load current Maximum input power Maximum output power Minimum stall torque

Armature moment of inertia

Time

constant

(mechanical)

Theoretical acceleration at stall Table 3 26

0.47 amp

7900 rev/min

(827

rad/sec) 325 gm cm

319.1O

metres 5700 rev/min

(597

rad/sec) 4600 rev/min

(482

rad/sec)

3.0 amps

36 watts 19.0 watts 1000 gm cm

981.10-4

Newton metres 56 gm cm2

56.10-7

kg m2

0.038

second

17,500

rad/sec2 Newton

Table 4 CONNECTIONS TO SERVO AMPLIFIER BOX

Plug 1: 12 way Plessey: to moving weight stabiliser.

A Limit switch

- 12V 0 volts

Feedback return (connect to E at servo) Armature Reset signal Tacho Tacho Limit switch

+12V

+15V)

)

- 15V) for reset potentiometer Plug 2: 6 way Plessey: Power input

A

+12V

- 12V

Power 0 volts (straight to OV of battery)

OV (Common) Socket 1:

+15V

-15V

OV common of 15-0-15 supply

6 way Plessey to UV recorder

UV Recorder A FRA A

B

D

F

H

K

Reset Roll Angle Roll Rate Roll Power F B

Socket 2: Cannon D 15 way 1 (via 150 ohms) + 15V 2 (via 150 ohms) - 15V 3 0 volts 4 Roll signal 5 Filter input 6 Filter input 7 Filter output 8 Filter output 9 Roll rate 10 Roll rate 11, 12 No connection 13 TFA input 14 Common

15 To TFA - buffered roll signal

Appendix I

Al.

THE MOVING WEIGHT SERVO DESIGN A1.1. Mechanical Details

Due to the limited time available it was necessary to design the unit with available components.

It was decided to mount the weight on a linear ball bushing, to run on a

25.4

mm shaft. The design is clearly shown in Figures 5 and 6 The drive motor is a vatric size 18 permanent magnet DC servo

motor; details are given in Table

3.

The motor drives a grooved pulley through a

4:1

gear ratio. A steel cable is fixed to the moving weight and both ends lead to the grooved pulley via fixed pulleys at the ends of the weight shaft and spring loaded tensioning pulleys. As the motor drives the weight from its mid position one end of the cable winds onto the grooved pulley and the other winds off. Micro switches switch off the drive current if the weight gets too near the end stops.

A 10-turn potentiometer is connected to the grooved pulley shaft to give the weight position feedback signal. A tacho, providing 3 volts per 100 rpm, is connected directly to the motor shaft.

The weight carriage weighed

0.364

kg and to this could be added four

0.235

kg weights, ie a maximum moving weight of

1.304

kg.

A1.2.

Performance Requirements

In order that the servo transfer function should have a small phase shaft at

wo

(2.2

radians per second) it was required that the natural frequency of the weight servo should be 2 to 3 times wo. It was also thought necessary that the weight servo should be able to drive the maximum weight to the maximum amplitude at a frequency of radians per second. The maximum velocity is this condition is 1.1 metres per second. The motor-to-pulley gear ratio was therefore chosen so that

the motor produced maximum power at this weight velocity. The peak power required for this was

3.47

watts.

ie Power = force x velocity

= 1.304

. Aw2 Sinwt . Aw Cos wt

and in this case

w = 44

rad/sec and A, the amplitude, is

0.25

metre. Peak power occurs when

1

sin wt = cos wt

= T2

Power =

1.304

.

0.252

. .

If the model is assumed to have a 10 degree roll angle at the moment of maximum velocity this adds another 1.73 watts to the peak power.

ie Power to raise weight =m.f4V.sin 10 Where 1 v = 0.25 .

4.4

. -a-= 0.778 metres/second V2 power = 1.304 . 9.81 . 0.778 . sin 10 = 1.73 watts

Total peak power is 5.2 watts, which is well within the rating of the motor.

In the event, for the experiments carried out, this power was not required. The maximum peak power measured was about 2i watts.

A1.3. The Servo Amplifier

Existing servo amplifiers appeared to be somewhat limited in power so it was decided to design one for this servo.

The initial design is shown in Figure 19. The output power transistors are both Darlington input high gain devices. The power for these is

obtained from lead-acid batteries in the model providing + and - 12 volts. One side of the servo motor armature was connected to the battery

centre tap, the other to the servo amplifier output.

In this unsuccessful design it was intended that the amplifier A3 should provide the drive to the power stage and also be the summing anplifierfor the input signals with the feedback coming from the output of the power stage.

When this design was tried it was found to oscillate at about 50 Kc/s and also have limited drive capabilities. At this time the power for the 741 amplifiers was + 10 volts - this being derived from the 12 volt batteries via stabilisers.

The input amplifiers Al and A2 were variable gain buffer amplifiers -the gains of which had been selected as -the result of some preliminary calculations.

The oscillations proved impossible to suppress in spite of the very high frequency at which they occurred.

When a new design was contemplated the National Semi-conductor 301A amplifiers were substituted for the 741C as this is externally compensated and has a higher current capability than the 741C.

It was decided that a separate unity gain output stage would be used, and in order to reduce the chances of oscillations, the gain of the

30

-drive amplifier was reduced to 300. The design is shown in Figure 20. Al is the extra amplifier.

Again there was insufficient drive voltage so that now the supply to the 301A amplifiers was raised to + 15 volts. This necessitated a new power supply which was kept external to the model.

A3 is now used soley as a summing amplifier which presents a gain of 200 to both the reset and the input signals and a gain of 12.12 to the tacho signal. These gains were chosen to provide the sort of response required and still have a stable system.

The original calculations had not taken into account the spring loaded pulleys provided to tension the drive wire. The presence of the

springs meant that the inertial load of the weight was connected to the servo motor via rather soft springs so that the system was much more complicated than the simple second order model which would have been adequate to describe the system had the weight been rigidly coupled to the motor shaft.

The frequency response of the servo is shown in Figure 21. It appears somewhat odd, but it does show that the response is flat up to about 0.7 Hz and that the phase lag at

0.35

Hz (wo) is 5 degrees. It was therefore considered acceptable.

Having obtained a working unit no further time could be spared on producing a better mechanical or electrical design.

A1.4. The Power Measurement

This was carried out by the circuit shown in Figure ,22. The current fed to the motor was detected by the 0.10 resistor in series with the motor armature while the voltage was fed direct to the multiplier circuit. The calibration was arranged so that 1 watt gave a 1 volt

output signal.

It should be noted that the presence of the phase advance controller in the complete closed loop system meant that the gain between the roll gyro and the output of the servo amplifier was +

14 db (5

times)

at 3.5 Hz as compared with the gain at wo. This meant that the output of the servo amplifier was rather noisy and the weight motion was rather jerky at times. It also meant that the power measurement was noisy so that it was necessary to filter the output of the multi-plier before it was fed to the UV recorder. (A standard AEW 6 pole Butterworth 10 Hz filter was used.) Even so it was not easy to

determine the power actually being used to drive the moving weight.

A1.5. Further System Details

In order to complete the project some extra details are now given, albeit with little comment.

Figure 23 shows the wiring diagrams for the gyros. Figure 24 shows the scaling circuits for the UV recorder and the sunning amplifiers intended for the roll-rate-control mode of operation.

All the electronics were housed in one box. This has four plugs and sockets; a list of connections to these is given in Table 4

!2.0. COMPUTER PROGRAMS

it

MAIN PROGRAM

Y(11),1') ,P(11),:.(11),TITL.=.(40)

CO -- "E!\3(2,1)TIT:..E

1 "70:AT(4CA2)

--;EAD(2,2).:;1,1:d2,03, 4,ciR

2 FOAT(5E10.3)

E.AD(2,3)E2,H,CT,N,ISTEP

3 FC.:47(3E12.3,21.10)

51P

$3\SE TIYE = CT*FLOAT(ISTEP)/10.

30 4 I=1,%\

4 _{I,fl J=1,}

5 F,3Ry.A7(4E10.3)

TITLE

6 =-C=1-'-'1,9A,'vDVI.,

_STA6ILISER

_°RC.,C-RA

1VI\3 ',TRIX

E0UATIO':'//40A2) .'.;17E(3,7)

=0.7MAT(1.-1109Xo'TIM;

=11.4, '

SECONDS', 7Xo'RECC'D

E11.4,

10A, 'SYSTEM

0m1.-. 132 =',E11.4

2, 4X,

'01

=',

4X,

='

,11.4,

":

4A,

'C..3t1//:

=,'11.4, 44,

riT's4 3=1

F11.4,4X,

= Ell

.4 pi/ )

30 22 I.1,11'

Y(I)=.

20 9 1=1, TE (3,9) io 4(1 r1 )

Ai

I ,2 )

,A:

1,3) gA( 1,4)

9T-.2,'K, ' 4 '

g 11,

4E16,4

)

3X,

'<1."3.',

c141,

eX,

'<93'

=

!13

11912,12

12 1.".,IT = :TI ; 1 =

IF(

; 114 CY:I) i=1 :L F;.,;1A7(E12.-iolCE11.3) 15 C0'.7IH'..E

:Fr"Sr)

17,18919

7E:3,22) Y 3) oY!6; os(C 7) ,Y )

.2=

=Y(12/-.!

Appendix II '<24', 9,2K, ",',Li31, 32

//t4(9XoE11.4))

s'A

OPTISAL

7

'C2

= 33 )

I-.1 1--K r-,) t a. -...

.

x. ....

4

.. .3

-r

x: Li !--4 4.,-,.. ..., -"K 4 a). 04 I =.1. 1.9 ---_-_, t -e., ... J: C:)

.

.

.

r) >,.... .. .') (A) ss's

.---,

4.4J 4,

.

cr's t. , 4--4-4 CY") 4.;) CI's .

.

e.) VA i /) t -, e., ) r-1 ' P to ,1-() .1 . .. ' -, 11 . 4 ..-, ) (.. ) ''..., '--' 4 t-- )-tri ,,< .6,--, LI 1 r---1 k.) 41. LI / >---1 .. IN/ .-., to

r

Ll r 1 ....1 11

RI 1

.t .::.!

'r.: If) .-. 4

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i N ,

.

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.

.

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: ...

e. 1 ...:1 .. (.) ,f--1 11 (7 ,,) 1::: : 4N 31 II 11 II r1 II 64 li (..) CI, e==. o's rs.J ,..-III 1. r- 1 ,-.--,-, 4r-1

I a,

I ---4.'!

-... ...2.1 .-.

a' 0.% -1-' 11 1:Ji_ (-.. --1'. " 17:iii 1-1-1 rr:'.5: Ill! 11 11) t'l ''..- >--1---- < 11-.----1 ' .4. a_

.

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.

> VII I.--..; , - 4 ri 1 .,.. F-_1 1 -..-14 I--I _I' d) ,'" 1- ::- t---.4 ft. i syi C \J c \I C'sj . -ri ' C 4-4 -t-..--. .1 - 1 4----. -. ' -- I-. Li. .1) c ) t'-' r iv:. ....,- . 11 :. ) '-..-'',:. 0 i ) 1.1 .:* (..) C) < L./ r-=. II (1 if I 11. .1. .r. :1j ,1: -,1 .1. 4 , -,-. I,_ -1,..: 14._ #-. - ) > >-4-:,, uk '-' 4-1 ',..!) (...) 0 .,..' ..' 4,-, ..1.: it 1- .-, (...) 1, f () 4-4 .-4 ( 4 1\1 v's t r_ u s ,)(74 (4) (el I m 74-1, 0. < A- in C\1 (".1N 'J (4 1_11 1.--. 1 ( \ ., 6--r 1 ,-.... Li i

0 ..4

.1% 1/ 1 ..i) !ft U) &t. -4 * *.-,t1 . : (11 .--4 (\4ol 4 I.0 --- r'

0

w t--1 13 .9 (.) t.9 a s., e- 4 41-c.. .9 4-+ i- -4-(..,.) .---. 44 Art C)

li

-= -q: 0 1 a a a a -a a ' a < r a a a V a

-I

----'

). I

2.

RUN GE-KUTTA SUBROUTINE

Y ( 1) Dy (1)

( 1) 9,Qii

IF( :('.::T).1.2.1

2 DO 3

..I=1

3 Ok

=O.

DY ( 1) =1

8 CALL DERYI(N.Y.DY

RETURN

1 DO4 I=1

( I )=.04-DY( I / 1,0 .5

R=(P( I Y-0(

).+`"( I Y( I ) =Y ( I ) +R 4 OC

_{= 3 0* (}

I +R K 1

CALL DER Y (.\1,Y.1DY I

DO 5

I =1 .N

C I )=i-i*DY C I

,R= ( (P ( I

) -3 ( I )

I 5+Y ( I Y C

Y( I )=Y( I

5 I

) = -2.3*Q(

_{+3 .O*P ( /}

CALL DERY ('N 9,Y .DY )

DO 6 I=1N

(

1 ) = -R (

) *(...; 5.1-'1.* Y I

(u( I )+Y( I

) )-Y (

L.N

Y ( I ) =

( 1)+7'

7.)11. )=0(

)+5.C3*(P

CALL Y(P1',.Yy.i.:,`YJ DC 7

I=1 .N

( :)=(-4.0*PI

)+.1-1-4-12Y( I )+,-;

I ,t.)*0.1C;66666,6667

R--=N-'( I

fl )-Y(

7 0( I )= l -°(1)1

r

Note

A similar subroutine iS used with DERY1 .

.;--1) ) ) I) )

) -5.

Y ; = I

II

3. SUBROUTINE FOR OPTIMAL GAINS

-.).7::Y(N9Y9t7)Y)

REA:- K11,<129K13,K149,K229X29K24-9K33.9K341.:<4.4

DIvESIC:\ Y(11)9DY(111

.1.:01ON A(494/9 :1.19029C:39G49(.:Rw:32

<11 =

Y12)

K12 = Y(3Y

<13 = Y(4)

K14 7

Y(5)

K22 =' YC.61,

K23 = Y(7)

K24 = Y(8)

A33 = Y(9)

K34. = Y(101

<44 = Y(11)

Al2 =A(1921

A21 = A(291)

A22 = A:792/

A34 = AA3t4)

A41 = f%491/

A43 = A(493)

A44 =

A(404/

a22 = V62**2)/RR

'QY(2) = -14K12**24*522 '-Q1+ 29,14-Gok2112+A41*K1.4)J

DY(3l

-tK12*K22*b22

,-(A21*K22+A41*K24+Al2*K11+A22*K12))

DY(4)

-tK12*K23*522 .-A21*<23+A41*<34+A43*K141)

OY(5) = =4K12*K24*S22 .+1A21*K241-A41*K44+A34*K13+A44*<141)

";.)6)

-(t<22**2)(*b22 -Q2

24*(Al2*<1.2+A221-22)J

)y 7)- = -K22*K23*522 =0Al2*K13+A22*;<23+A43*K241i

= -4:4<22*<24*522 -Al2*<,14+A22*K24+A34*K23+A44*<24))

OY)) = -:N23**2)*22 - ,03 = 2.*(443*K4))'

OY(1n1= +(<23*<24*522

-+CA43*<44+A34*K33+A44,41-;c3401

-((<24**2)22 -a4 = 2.*A34*<34+A44*K4411

R,ETL:RN =ND

4.

SUBROUT SiMP REsPot4SE

2-3(.)UTIE DE'RY1(i'19YoDY)

.W5)9Dv(5)

COON A(4941

X1= Y(2)

X2= Y(3)

X3= Y(4)

X4= Y15/

OY(2)

= A;1,2)*X2

...)'\%(3;

_{At291)*X1 * AC2921*X2 + A293)*X3}

_AC294)'*X.4

L:Y(4) =

A(394)*X4

12,Y(5)

s A(417j*X1 + 4(493)4X3 + A(414)*X4

'iETUR% = = = DY(8)

DY(11)=

a , FREE S uR.F Ace -Tptt-IK

C.

_{ACTIVE TANK}

<IF me MONTe-- C.ItSTO

FiGu P.E

1

TRN.K S-TABILISERS

W L Waterline of upright ship

W'1, Waterline of heeled ship

Centre of buoyancy of upright ship B' Centre of buoyancy of heeled ship

Heel or Roll angle, in radians Centre of gravity of ship Roll Metacentre of ship

CM Distance between G and M - Metacentric height 727 Restoring force lever arm

Gravity forcP on ship ---;Eg

.2.0noy

force on ship .1Eg

Mass of shi7

i

\1

AVE, SLPE. , RAD vA 1.1s,

o L ANGLE -RELATIVE "Tiz WAVE SL cIP E. , RAD KA44S

R E-t-coA.kN ,FoRc.s --=

. GZ

c. GtA

e

F.,z s,f-iF4LL A-t4G-L-Ss

It)

flitScLUt itaLL G-LE .4PC

PVGURt3

SP csl.LinG

414

B EA M ES

NPUT Ri

a

0 1 0 Cft CL) IT _DIAGRAM RI 37-9K R2 .2e.oK R3 a04C. R4

o62

ci

1 1.

00/4

AMPL F t ER S 3 o Fl alt R41 -ou-r POT

60

4 0 °

ect.

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_:111 ini....11111111111111111111MEI MOW

Eginiormaneas

liON1111111111111M111111111

=6

0111a111111111111111411111111111.

wpm

or

IMNIMUIIIIIIIIIIIIIIMIIIIIIIIIIMIN

==r

2=E

arra

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MN= MIS 11/11/111M NM _{III IMO MIIIIIIIMOIN}

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.

a

_{.4- *5 :7}

sl

Z 4

5

7

10

ao

-so

FR,E__GNENcY RAt AN

_PR

b

Si NG Le ST Q-E.

_FP.UE.Ny

_{R PoN S}

CI

.3 _a

fit r 1,4 tE! 0 E

r

Figure 5

MOVING WEIGHT SERVO TOP VIEW

I/ rgit vo. , .11q1' "II 111 .110 Ite/ t4r,y": , u 1111IIIIII u th1F., malEa, IV

Figure

6

TRN K

WFLL

\AI Pt-TEIre.

VI0I)

L

ETA E.R.1tIC.- Pt RIZ.P.44GE MEN

aac.

SP E

Ek"? RAME.KrrS

FIGuRE.

7

WEL41

//

c-14T MO DE L-rCARIZIIRGE.

S oL FtRTA.N trio

FREQUENcy RESPoNS ANALYsER

ULTRA VvoLE_T RE_c-cA2D1NG

L\J ANOMIETEk

[UV RE0RPEA-21

PHAsE_ ADVANc- coNTRoLLER

tTo

RA i)< ' it4PuT

FoR WAtici EAPERtMEWrS

FoR WAVE E)(PE:Atmew-s

ROLL RATE CoN TgoLi-e"R (tOoT

1.1 SEE)

RoLL

; STAILIsE

p9vJE-R mC_Asues*IG- c_igcutT Sc_AL

(NG-AMPLAF tu-52. _Ral-L

RATE.

GcALING _AMPLIFIER

VJAVE MA kE-Q coniTRoLLER.

VING WEAG44-T Setz.Vc:3 Nnie,t

tfr

PosycleiN ROLL ROLL RATE GYRO

SE.Ry o A MEL (F ICA

WAVE. PRoBE.