Faculty WbMT
Dept. of Marine Technology Mekelweg 2, 2628 CD Deift
The Netherlands
1. Introduction
The auto-pilot system of ships has been
used for many years, and now only few
ocean sailing ships are not instrumented by the devices. However we have not considered the mission of auto-pilot of ships seriously but have simply deemed it so as to steer the
ship automatically. The author wished to
bring back the problem to the origin and to consider what is the real object of auto-pilot
Takeo KOYAMA,* Member
(From J.S.NA. Japan, Vol. 122, Dec. 1967)
Summary
It has been considered that the mission of the automatic steering system of ships is to maintain ships' course as far as possible.
However, the author would like to point out that this definition is not proper, because, there might be reasonable limitation to the rudder angle in relation to the speed loss, that
is: if an excessive rudder angle was taken to maintain the ship course too. much strictly,
it will induce considerable power loss.
The real purpose of applying the automatic steering to a ship in a seaway is to bring
her from one side to the other end of a route as fast as possible within a limited cost.
Therefore, the author prefers to define the purpose of an automatic steering to be "to
minimize the increase of resistance in a wide sense under the disturbance of wind or waves
in the ocean."
Making the use of this definition, the author obtained the performance index of the automatic steering system of ships at sea, namely:
J=&2+252
where
ó2: mean square of couse error.
52: mean square of rudder angle.
2 : a constant which shows the weight of above two values.
A full scale measurement of yawing motion is also carried out on a cargo ship, "M.S.
Florida-Maru" in Pacific Ocean to obtain the spectrum of angular rate which is affected by
disturbance caused by wind or waves.
Applying these results, some suggestions are obtained to improve the automatic steering
142
of ships at sea, and to obtain the optimum automatic steering system of ships at sea.
Two missions are required for a auto-pilot, one is to keep the ship course automatically
mainly at sea, the other is to change the
ship course automatically from one course to another one. But perfornance indices which
give the basis to evaluate them are com-pletely different.
* University of Tokyo. system of ships, for example,
The rudder angle ratio has to be chosen not too large.
The weather adjustment of conventional system is injurious and cannot keep the rudder angle small, because it necessarily requires rather deep rate control.
First of all, we should pay attention not to make the rudder too much sensitive to each wave and so on.
In case of changing course of ships
auto-natically, it is most important to bring a
;hip to a new course in shortest time. And
e can
use the techniques of automatic:ontrol theory, for example, the traditional nethod of optimum damping or Pontryagin's
Maximum Principle for the shortest time :ontrol problem.'
On the other hand, for the problem
ofautomatic steering for which we are con-cerning, it has been considered so as to keep
the ship
straight on course as much as
possible. But is it the real object ofauto-pilot of ships at sea? May we take anylarge
rudder angle if it can reduce the course
error? The existence of weatheradjustment,
which is built in any existing auto-pilot systems, suggests the incompleteness of this
definition.
It is sure that the object to keep the ship
straight on course has one feature of the mission of auto-pilot, but is not complete. Then what is the real mission of auto-pilot and its complete definition? The most com-plete definition of the mission which may be accepted by any person is;
"to steer the
ship so as to bring her from one port to the port of destination as fast as possible within a limited cost."
This definition is complete but it is too difficult to treat as it contains the routing of
a course, and
the problem of routing isindependent of steering along a ordered
route,
so we remove it
and change the definition;"to steer the ship so as to bring her as fast as possible along the prescribed route within a limited cost." and this definition can be said
in another words; "to steer the ship so as
to minimize the increase of resistance in a wide sense under the disturbances of winds or waves in the ocean."
From this point of view, to keep a ship straight on course is to prevent the
meander-ing motion of the ship and to
keep the extension of the route minimum, in other words, to minimize the increase ofresistance in wide sense.On the other hand, a large rudder angle will cause the increase of resistance and have a bad effect on the propeller performance. So we must find out a meeting point between
the course error and increase of resistance
due to the rudder angle for steering, and
must not take too much
rudder angle to
prevent the course error; our object is notto prevent the course error only but to
prevent the increase of resistance in a wide
sense.
2. The performance index of auto-pilot of
ships at sea
2.1 the increase of resistance in wide sense
We have decided to evaluate the
effective-ness of auto-pilot of
ships at sea with the
increase of resistance in a wide sense, then what kinds of factors should be considered? We should not take into account the in-crease of resistance caused by factors whichhave no relation to steering, for example heaving or pitching, because, however large they may be, we can not reduce them with steering.
Taking the co-ordinate systems as shown in Fig. 1, the equation of motion can be written as follows.
Fig. 1 The Co-ordinate System
(m'+my')ß'= YT' .'+{(m'+mx')
+ Yr'}r'+ Yo'iY+ Y' (1)
Where, prime shows the non-dimensional value. And the equation of motion of x-direction, which is very important to consider
the increase of resistance, can be written
as; (mH-m)ü=
T(1t)(m+n) Ußr
(Rhull+Ro) (2) where T : propeller thrust t : thrust deduction Rh11: resistance of hullR-: resistance of rudder angle
Now, let us consider the factors to cause the increase of resistance in wide sense.
First of all, we should consider the
exten-sion of a route caused by course error, it
may be strange to consider the extension of a route to be increase of resistance, but from the viewpoint of energy, the increase of time is equivalent to the reduction of speed, so we
call it the increase of resistance "in wide
sense,"
The second, (m+m) U18r, the 2-nd term of Eq. (2), is the x-component of the
centrifugal force caused by ship motions,
especially, it is known that the centrifugal force caused by steering becomes considerabl-ly large23, and this force exsists even if a ship moves periodically, for example, when a ship moves sinusoidally.
r=cos wt
¿3=j3 cos(wt+ç5)
where
: amplitude of yaw and sway
phase lead of sway to yaw taking the mean through one period
+ zur) Ucos (3)
We can find the loss as Eq. (2), and we call it "a loss caused by the centrifugal
force."
The third, RhU11, the 3-rd term of Eq. (2), is not so affected by ship motion, Eda and Crane had measured the effect of on Rh11, and found the increase of resistance is small.4
Lastly, the auto-pilot should take the rudder angle so as to minimize these losses, but the loss is caused by helming itself, for example,
it is known that Ra, the 4-th term of Eq. (2),
is large, and as the indirect
effects, thrust deduction or propulsive efficiency will become worse by helming, because the pro-pulsive efficiency will become worse byhelming, because the propeller load is
in-creased or the stream around the propeller
is bent, moreover, when the ship motion
cause.d by helming does not work to prevent the ship motion caused by disturbances, the
loss of centrifugal force is induced by
helming itself.
In conclusion, we will treat the following three factors as the increase of resistance in wide sense to evaluate the effect inveness of auto-pilot.
the extension of a route
the increase of resistance caused by
centrifugal force
The loss caused by helming itself 'caused by waves
caused by winds caused by helming
Other factors, for example the power of steering machine etc. may have some effects
on evaluation, but they will not be
con-siderable.
2-2 the extension of a route
2-2-1 the sinusoidal meandering
When a ship moves as shown in Fig. 2,
the ratio of distance traveled in comparison with straight line is
1 ("
-
7ro
=1+ (4)
Fig. 2 Sinusoidal Meandering
Therefore, when
is small, the rate of
the extension of a route is about and this
value is very small as shown in Fig. 4 so
long as remains small.
different from the tangent of the trajectory
of a ship by the existence of drift angle ß
but the meandering motion becomes con-siderable oniy at low frequency and at low frequency ß will remains small, so we may take the tangent of a trajectory equal to the heading of a ship.
2-2-2 the folded-line meandering
When a ship moves along a wrong course without noticed and adjusted after a while,
the ship trajectory will be a folded line as
shown in Fig. 3.
Fig. 3 Folded-line Meandering
The ratio of the distance traveled in com-parison with straight line will be
i
cos
the increasing rate of a folded line
meander-ing 1/cos-1 is not so large too, as shown
in Fig. 4. 6% 4% 2% o 5 ID ç'
Fig. 4 Increasing-rate of Pass Caused by Meandering
2--3 the increase of resistance caused by centrifugal force
2-3-i the loss due to waves
It has been considered that the phase angle of Eq. (3) is 90 in the ship motion in waves,
and the centrifugal force loss 'in waves is
negligibily small. But when investigated
precisely, we will find it to become con-siderable sometimes.
(a) yawing and swaying of ships in oblique
waves.
It is necessary to estimate the exciting
force of obblique waves to know a ship
motion.
Several methods have been used, for example, (1) Froude-Krilov method, (2) Strip
method using the exact solution of sway
exciting force in beam sea6 or (3) to divide
the exciting force in three terms, namely,
inertia force, damping force and Froude IKrilov force49. The former two methods have some defect in oblique waves, so we used the third method.
The ship motion of a Series 60, Ch=0.6
ship was calculated in case of
r'
0. I
0. 0
o. o s
o
Fig. 5 Ship Motion in Waves (X=60°) 0.8 0.-7 0.9 r'(Yaw) 0.4 0.5 0.6 0.6 '(Sway) 8 -(ç 0.3 0.3 0.4 05 0.9 02 0.4 06 We rad/sec
L=150m U=l8kn
h= i m (wave amplitude)
=each 30° (incident wave angle) using Eda's values4 for derivatives and Tamura's values6 for inertia and damping force, two examples of results are shown in Fig. 5, Fig. 6.
o
0 04
Fig. 7 Increase of Resistance (inertia force) caused by Waves
The losses are considerabily large in beam
or heading waves, and these values are
proportional to the square of wave amplitude, the increase of resistance becomes about 4
tons which is about 10% of total ship
re-sistance for the wave amplitude of 2 m. On the other hand, sometimes it becomes large negative resistance in following sea, it
may be felt funny but, after the Eq. (3),
the increase of resistance will be negative
when the phase angle
between r and
¡3exceeds 90°.
Following above results, it is estimated that the increase of resistance may sometimes become considerable especially in beam or heading sea, but the frequency of each waves are too high for a rudder of ships to reduce the ship motion caused b waves, and if the rudder responds to each waves, the loss due to helming is only added to the loss caused by waves.
2-3-2 the loss due to wind
In general, winds blow fluctuating around the constant velocity. Therefore, it will be unavoidable for a auto-pilot to take a rudder
angle large enough to overcome the side force or moment caused by this constant
component. Therefore the problem is how to steer against the fluctuated component of wind.
Ship motions induced by wind are
cal-culated, estimating the force and its acting
point from Araki and Hamaoka's experi-ments'°, for the ship form of Series 60, Cb=0.6 L=150m U=l8kn ton 0. I 0.01
Fig. 8 Increase of Resistance (inertia force) caused by Winds 0.7 08 7'(Yaw) 06 0.3 0.4 0.8 0.9 5 0 rod/sec 0.3 0.5 0.1 o 0.9 0.01 ,0.l w SQ rad/sec
____iìEEEEEE:\
06 08 1.0 .2 We rad/sec Fig. 6 Ship Motion in Waves (X=120°) (b) the loss caused by centrifugal forceAfter the ship motions are calculated, we can easily obtain the centrifugal force loss due to waves by Eqs, (3), these values are shown in Fig. 7. 0.10 0.05 ton .0 O - LO -2.0
A=1500 m (wind area)
v=10 rn/s (relative wind velocity) the results are as shown in Fig. 8.
Losses
are remarkable only when the
frequency is in very low range, and in this range, O must be very large to make r (wO)
large enough to be the losses remarkable,
therefore, so long as O stays in low level,
the loss due to centrifugal force can be
neglected.
2-3-3 the centrifugal force loss dite to helm ing
It is known that centrifugal force loss due
to helming is very large, and can be
as-certained easily by calculating the loss due to sinusoidal helming as shown in Fig. 9 for
the ship of Series 60, where U=l8kn, ö=
0.1 rad.
ton 10.0
1.0
0. I
Fig. 9 Increase of Resistance (inertia force) caused by Heimming
But the loss due to centrifugal force is induced by the resultant motion of a ship
caused by disturbances and helming. There-fore, the centrifugal force loss is not caused by helming however large it may be, when the rudder moves effectively to reduce the ship motion caused by disturbances.
In general, the auto-pilot works effectively at low frequency range so that the centrifugal
force loss due to helming does not occur,
and at high frequency range rudders fail to move ships so that the loss does not occur too, but at medium range, where the effect of rudders begins to fail, the auto-pilot may
induce considerable loss when its stability is not satisfactory.
The fact that the loss depends upon the stability of auto-pilot makes our effort difficult to get the general expression of perfomance index. But the poor damping motion of ships due to the unsatisfactory stability is most harmful from the viewpoint of the increase of resistance in wide sense, so that we must satisfy the stability of auto-pilot first of all. And after that the centrifugal force loss due to helming can be neglected.
In conclusion, the stability of auto-pilot which is satisfactory enough to minimize the centrifugal force loss is a precondition of our discussion, and after this condition we can
neglect the centrifugal force loss due to
helming.
2-4 the loss caused by helming itself
As stated before in 2-1, it has been given attention only to R5 in Eq. (2), but it is
sure that the helming is harmful for the
performance of propeller.
An experiment was carried out to clarify the all round effects of helming, using the equipments as shown in Fig. 10. A 2.5m Mariner ship model was used at full load condition, the model was fitted to towing
carrige with a single guide for the sake of friction correction, so yawing and swaying of model had been restricted.
Towing corrige
Morter & Thrust-torque dinomometer Fig. 10 Set up of Self-propulsion Test
Torque, thrust and r.p.m. of propeller to maintain a constant speed were measured under the rudder angles of 0°, 100, 20°, and 30°, the results are as shown in Fig. 11'-Fig.
13.
The increasing rate of thrust is shown in Fig. 14 and the increment of thrust obtained is
4T=4ô2 (7)
Single guide Friction correction
w
It is very dangerous to estimate the
in-crement of thrust from only one example, but at least it can be said considerable, and when we take a mean, we get the value of 22 for the loss of sinusoidal helming.
ö=ösinwt X Q g-cm 1,000 LO 0.5 o N rom -loo 600 500
Fig. 14 Increasing-rate of Propeller Thrust Maintain the Ship Speed
2-5 the performance index of auto-pilot
Summarizing the above mentioned, it is
estimated that the loss due to each wave is considerable, but the frequency of wave is
500 too high to reduce it by helming, so that we
_____ should not evaluate the auto-pilot with this
loss however large it may be.
The centrifugal force loss due to wind is remarkable only at very low frequency range,
Oo 1.0
1.1 1.2 1.3 Vm and at this range, course error must be very rn/s
.11
large to make the loss due to the centrifugal
Fig. 12 Increase of Torque caused by Heimming
force remarkable, where the loss due to
)0
/
:.NA
___:T
LU
CQ4 lo 2 1.3 VmFig. 13 Increase of Revolution cansed by Heimming
.0 1.1 .2 .3 Vm
Fig. 11 Increase of Thrust caused by Heimming
00 200 300
400
300
200
Conditions at measuremant
Japan-U.S.A. U.S.A.-JAPAN
Measurements are carried out everyday in our voyage, and course heading and rudder angle recorded continuously.
3-2 the procedure of data processing and results obtained.
The results measured were processed
according to the flow chart of Fig. 15, the power spectra of angular velocity of yawing,
under and without steering, and rudder
angle were obtained.
SMOOTH I NG COMPUTE r0 (t) =901t + I) -8(t) L r0 AUTO-CORRELATION PRINT RYr) Sr,Srd,S COMPUTE rs(t) ro
The following procedure was taken to
145.00 m obtain the angular velocity of yawing
with-Bmld 19.40m out steering, (disturbance only) according to
Dmld d J co Engine 12.20 m 8.72m 17.000 ton 0.68 9.000 BHP/128 at full load cond. r.p.m.
the nomenclature of Fig. 15.
ro(t) = r0(t) + rd(t) rd(t) = ro(t) - rd(t) =ro(t) -
5(r)o(t_r)dr
d 6.30m 8.41m trim 0.83% 0.49% A 11.600ton 16.300tanextension of route caused by course error is larger than the loss due to centrifugal force. Therefore, we should consider the extension
of route caused by very low frequency
disturbances, wind or drifting force of waves, as the increase of resistance in wide sense, and this loss is proportional to
The rudders are used to prevent these
losses at sea, but we must be careful not to take the rudder angle too much, because the helming itself invites the increase of re-sistance in wide sense too, the loss due to
helming is proportional to 2 so long as the stability of auto-pilot is satisfactory.
In conclusion, we get the performance index for auto-pilot of ships at sea
J=2+2
(9)where
02. mean square of course error
mean square of rudder angle
2: weighting constant
(in case of Mariner 2=8)
The optimum auto-pilot at sea is the system which minimizes the value of Eq. (9) in the disturbances along the prescribed route. 3. A full scale measuremant of yawing on
"M.S. Florida-Maru"
3-1 M.S. Florida-Maru
We have obtained the performance index of auto-pilot, but it is necessary to know the charactor of the disterbance acting on ships to design the optimum auto-pilot.
It is very difficult to estimate it with our knowledge, but, fortunately, the chance was
given to measure the yawing of a ship on
M.S. Florida-Maru, a cargo liner sailing in North Pasific.
In this chapter, the results of measurement are given.
Principal Dimension of M.S. Florida-Maru
Fig. 15 Flowchart of Data-processing
Oo heading of ship measured
O rudder angle
ro angular velocity under steering ro anguler velocity due to helming
r anguler velocity due to disturbance
g(r) weighting function of ship
S(w) power spectrum auto-correlation function DATA READ IN CALE L AT ION POWER SPECTRA WINDOW
Some typical results obtained are shown in Fig. 16'Fig. 23, in these figures,
Sr Sprectrum of angular velocity under
steering deg2/ deg 0.! 0.0! 0.001
Fig. 16 Power Spectra
Srd: Spectrum of angulat velocity
with-out steering
S : Spectrum of rudder angle
Fig. 18 Power Spectra
s'
0.01 Date 3.190M Vs l5Kn Wave 8 0l Sea 6 Swell 7 Wind ll0LI5r25 0.1 .0 w rod/S 0.01 Date .I8 0M Vs I6Kn Wave 20CL Sea 5 Swell 5 Wind 80L 0.1 R .0 w S / rad/5 Srd 0.01 Date 3.1 9AP Vs l5kn Wave 80'L Sea 6 Swell 7 Wind ll0[. l5/s R 2 ,W 3 0.1 1.0 w Sx 0.01 rod/S \ 0.0! Date 3.18AP Vs l6kn Wave 20°L Sea 5 Swell 5 Wind 80CL Gtm/s R2,W3 0.1 1.0 w S5x 0.0! cad/SFig. 17 Power Spectra Fig. 19 Power Spectra
ded2/S ded2 S 0.1 0.0! 0.00 deg2/S de92. 5 0. 0.01 0.00! deg2A deg2! 0. I 0.01 0.001
deg2/ deg2 0H 0.01 0.001 deg 2/ deg2 0.1 0.0 I 0.00 I 0.01 Dote 3.220M Vs I4K Wave 120°L Sea 5 Swell S Wir,d 150t 15tm/s S&xO.Ol 1.0 w Srd 0.1 rod/S Sr n 0.01 Date 3.200M Vs I3Kn Wave l7OL Sea 8 Swell 8 Wind l75L Sm/s 0.1 1.0 w
I
rod/S 0.01 Dote 3.20AP Vs l3Kn Wove l70°L Sea 8 Swell 8 Wind 175°L Sm/s R 2, W 2 0.1 1.0 WJA
rod/S 0.01 Date 3.224P Vs l4kn Wave 20 L Sea 5 Swell S Wind 150'L R 2 ,W 2 0.1 .0 w x 0.01 rod/SFig. 21 Power Spectra Fig. 23 Power Spectra
Fig. 20 Power Spectra Fig. 22 Power Spectra
deg2/ deg2. 0.1 0.0 I 0.001 deg2/S deg2 S O.' 0.Ol 0.001
and the conditions are shown, for example: dating, steered by ship speed
direction of swell, left or right
QM: Quater Master AP : Auto-pilot
0° : following
}
Strength of sea or swell from Log Book direction and velocity of wind (relative)
} position of Rudder and Weather Adjustment of Auto-pilot time constant of the ship and rudder angle are considerabily large in the high frequency range in which rudders are not effective at all, these rudder angle are not only useless but also harmful and induce the loss at stated in 2-4.
Weather adjustment device is installed to auto-pilot to prevent the useless helming for each waves, but it does not work properly.
3-3-3 a comparison of manual steering and
auto-pilot
Something was expected to give a hint to design a pilot by comparing the auto-pilot with manual steering, but auto-auto-pilot is far more superior than manual as shown in figures.
Quarter masters are tend to take a larger
rudder angle, but the ship motion is not
reduced in result. The author had tried to steer too, and found that it is very difficult to find the very low frequency disturbance hidden under the high frequency error due to waves.
4. The design of optimum auto-pilot
It is prepared to design a optimum auto-pilot after we obtained the performance index of auto-pilot and disturbances acting on ships at sea.
4-1 the fundamental nature of the proportional
control
The block diagram of this control system is shown in Fig. 24, in this figure O, shows the ordered ship course and we can put it O degree without losing generality in our discussions.
3-3 some comments on results 3-3-1 the spectra of Srd
First of all, the distinguished difference can be found according to the wave direc-tion, heading or following, at high frequency
range (w=O.3 rad/sec); in case of heading sea, spectra are spread in wide band and their peaks are not sharp and high, but in
case of following, spectra are concentrated
in narrow band and their peaks are sharp
and high, these difference is caused by the
fact that the abscissa is chosen to be the
frequency of encounter
/ wU
WeW
g cosx
and a ship is more affected by lower frequency waves.
In low frequency range, there are no effects of waves, so a ship is mainly disturved by winds.
The resemblance is found through spectra that the spectra are flat in very low frequency and begin to fall about at w=0.05 rad/sec. and continue to fall until the effect of waves becomes remarkable.
It shows that the spectra of winds
are white in the frequency range in which winds act to the ship effectively as distur-bance, because the frequency of w=O.O5 rad/ sec is just the inverse of the time constant of the ship.
3-3-2 Sr and S5
The frequency range in which the rudder
acts effectively to reduce the ship motion
(Sr>Sr) is restricted in w<0.05 rad/sec, this result is reasonable when we consider the
Date 1.10 QM(AP) Vs 18 kn Wave 150° L(R) Sea 5 Swell 5 Wind 170° L(R) 20 rn/s
R3
W3
0
+
& K(T3S+ r)
(TSl)(T2S+ )
Fig. 24 Block-diagram of Proportional Control
Oo shows the heading of ship, ra shows
the angular velocity of ship due to
distur-bance and k shows the rudder angle ratio, the proportional constant of rudder angle to
the course error.
O=k(0-0o) 0=O (11)
The transfer function of this system is
i 00 s rd Ya 1+k K(Tas+i)
i
(Tis+1)(Tos+i) sand at very low frequency range (s-O) 00 1
ra kK
o o o 1
ra 0o ra K
therefore, rudder angle ratio is larger the
better; Go/ra is reduced proportionally to k, but O/ra remains constant.
On the other hand, at very high frequency
range (soo)
Ya s
Go i (15)
o k (16)
s
the rudder angle ratio is smaller the better. These relations mentioned above are very important for designing a auto-pilot because ships are affected mainly by waves at high frequency range.
4-2 the rate-control
A rate-control, helming proportionally to angular velocity of a ship, is added to pro-portional control in recent auto-pilot.
In general, rate control improves the stability of control systems, and is necessary when a large rudder angle ratio is required to get quick responce as in case of automatic changing of course, but in case of automatic
steering at sea, a large rudder angle ratio
is
not required so that the
required ratecontrol is not large, or may not be necessary
(12)
(13)
(14)
at all for a very stable ship.
Under the rate control, rudder moves ex-cessively for the high frequency disturbance. For example, the spectra of S0 and S0 are compared under the condition of with and
without rate control in Fig. 25 for the
disturbance Srd shown in Fig. 17. In this figure the hatched part is improved by rate
control, and the meshed
part shows the
increase of rudder anglecaused by rate
control, therefore we should suppress the rate control not to be excessive and filter out when it is not necessary at high frequency. deg2S
LO
0.
0.0
S0 & S with out
rote control
S0 Xc S with rute cout rol
À
S with & with out rute control
Fig. 25 Effect of Rate Control
Let us consider the criterion whether the rate control is necessary or not. The stability of a control system is decided with its phase margin, and should be more than 450, the open loop transfer function of a proportional control system in first order approximation is
kK jw(jwT+1)
0 1
and the phase lag becomes 135 at and
the gain should be less
than unity at this
frequency,kTK<1
S0 with
rote control
therefore, when KT<
rate control is not required at ali.
4-3 the optimum rudder angle ratio
K,' T' values of M.S. Florida-Maru are shown in following table.
JAPANU.S.A. U.S.A.-JAPAN
In case of JAPAN-U.S.A., rate control is not necessary, but in case of U.S.A.-JAPAN, it will not be avoided. Anyway, let us try
to obtain the optimum rudder angle ratio
which minimize the performance index of Eq.
(9).
The mean equare of O or ö can be obtained as follow. Sra dw ra
S dw
2+82 dag2 2.0 1.0 O Oo rThe performance indices are calculated against k for the typical Sra of heading and
following sea as shown in Fig. 26. The
optimum rudder angle ratio is about 0.5 for following sea and 1.4 for heading sea, and
these values are rather small than it has
been considered.
i,
K' 0.93 1.93
T' 0.86 1.62
KT=K'T'
0.80 2.94Fig. 26 Optimum Rudder Angle Ratio
The difference between heading sea and following sea will be found, in case of
follow-ing sea, the ship motion due to each wave is large so that the helming becomes excessive
for large k value. On the other hand, in
case of heading sea, the rudder angle does not become large for larger k value, because the ship motion due to each wave is small.
It may be curious that the rudder angle
ratio should be small when a ship is disturbed largely in following sea, and may be taken more in heading sea, but it is not curious
because the rudder is not effective in both
cases.
As to the problem of stability, rate control is not necessary in case of JAPAN-U.S.A. voyage. On the other hand, in case of U.S.A. -JAPAN voyage it can not be avoided, but we must be careful not to take it excessive
and filter out the rate control at high frequency not to take excessive rudder angle.
4-4 the rudder angle ratio at very low frequency
At very low frequency range, the rudder
angle ratio is larger the better as stated in
Eqs. (13) (14). The value required for practical use is decided by the offset error of course due to wind, and this value will be
degree
for the case of
Series 60, relative wind velocity of 20 rn/s and direction of 120°.The value of k is larger the better, but it is useless to endeaver to make it very large, more than 4 for example.
When it is necessary to make the rudder
angle ratio large only at low frequency range, the integral filter can be applied
GE(S)- TFS+1 a>1 (18) aTES 1 1 s*0
-
iJ. 00(a
If we take the time
constant TF largeenough, this filter does not harm the stability of overall control system.
4-5 the non linear element as a weather
adjustment
It has been used some devices not to take a useless rudder angle to each waves, named "weather adjustment ", for example,
corn-k 4
3
posed of back lash element as shown Fig. 27.
Fig. 27. Back Lash
It seems to be effective at first glance, but
the stability of control system is heavily harmed when a back lash element is
intro-duced into a feed back loop, and large rate control will be required to make it stable,
so
that the effort
to reduce the uselesshelming will result in larger helming by the excessive rate control to improve the stability.
A comparison has been made in Fig. 28
between auto-pilot with rate control and back lash type weather adjustment measured on M.S. Florida-Maru and calculated value of
degS .0 0.1 0.01 Proport lone t Control only Bock Loch Weother Adjust(\ /7
I'
/ i Ss J \ /I I I \Fig. 28 Effect of Back Lash as a Weather Adjust
in proportional control only in the same
distur-bance. It can be seen that the error of course are nearly equal but very large rudder angle
is required and the effect of back lash is
completely lost with the existence of large rate control.
The dead band element is putforward for weather adjustment by Nomoto.'2
A simulation was carried out for dead
band element of band width
of ±2° in the
same disturbance of Fig. 28.The result is shown in Fig. 29, the stability of the loop is satisfactory even under the
condition of without rate control and the rudder angle is suppressed in lower level
too. deg2S LO 0. I 0.01 0.01
Fig. 29 Effect of Dead Band as a Weather Adjust
Dead band element is promissing for a
weather adjustment as mensioned above, but
its short point is large offset error due to
wind is added to dead band, and when the
wind is strong so as to need the
rudderangle of 4° or 5° degree to
maintain the course, the effect will be canceled completely. Any-way the investigation for dead band will be needed.5. Conclusion
We get some concusion from the discussions
above c \ i''' Proportional r:T
\
l
Control only '1 \/\\
Dead Bond ' Weather djust \\\jt
\ Il 0.1 w rod/S 0.0I 0. w rod/SThe effectiveness of auto-pilot of ships
at sea should be evaluated with "the
increase of resistance in wide sense". The performance index shows "the increase of resistance in wide sense" will be written as
J_82+)432
where 2 shows the weight which
contri-butes to the performance index, and in
case of Mariner 2=8, so we should be more
careful not to take large rudder angle
than course error.
We should not forget the stability of
control ioop is the precondition of these
discussions.
It is most important not to take a rudder according to each waves when we design a auto-pilot of ships at sea.
The weather adjustment which is adopted in general is harmful.
It has been used larger rudder
angle ratio than the optimum value, and whenthe rudder angle ratio is small and the
back lash is abandoned, a less rate control is needed, and the rudder angle is reduced considerabily.
The author wishes toexpress his gratitude to Professor Motora for his kind advice and Captain Hon and his crews for their kindness on M.S. Florida-Maru.
Reference
1) H. YAMAMOTO: The Optimal Steering of Ships,
J.S.N.A. Japan, Vol. 121, pp. 20-30, (1967) (in Japanese)
5. MOTORA: On the Automatic Steering and
Yawing of Ships in Rough Seas, J.S.N.A. Japan, Vol. 94, pp. 61-68, (1954) (in Japanese)
K. NoMoTo and T. MOTOYAMA: Loss of
Pro-pulsive Power Caused by Yawing with Particular Reference to Automatic Steering, J.S.N.A. Japan, Vol. 120, pp. 71-80, (1967) (in Japanese) H. EDA and C.L. CRANC: Steering Characteri-stics of Ships in Calm Water and Waves, Trans. S.N.A.M.E. Vol. 75, (1966)
L.J. RYDILL: A Linear Theory for Steered Motion of Ships in Wayes, Trans. R.I.N.A.
Vol. 101, (1959)
K. TAMURA: The Calculation of Hydrodynamic
Forces and Moments Acting on the
Two-Dimensional Body, J.S.N.A. Seibu. Japan, No.
26, (1963)
S. MOTORA and T. KOYAMA: On an Improve-ment of the Maneuvrability of Ships by means
of Automatic Steering, J.S.N.A. Japan, Vol.
116, pp. 38-49, (1964) (in Japanese)
O. GRIM and Y. TAKAISHI: Das Rolimoment in Schräglaufen der Welle, Universität Hamburg Bercht No. 148, (1965)
F. TASAI: Lectured on ship motion in oblique waves at J.T.T.C. (1966)
H. ARAKI and T. HANAOKA: Wind-tunnel
Experiments on Train Ferries, J.S.N.A. Japan, Vol. 84, pp. 61-68, (1952) (in Japanese)
Y. YAMANOUCHI, Y. TAKAISHI, K. St.JGAI and
S. ANDO: On the Analysis of Ship Oscillations
among Waves (Part 4), J.S.N.A. Japan, Vol.
119, pp. 50-59, (1966) (in Japanese)
K. NOMOTO: Stability of Auto-Piloting, J.S.N.A. Japan Vol. 104, pp. 53-71, (1959) (in Japanese)
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