IP ROCEED I NGS
Lab.
v.
Scheepsbouwkund
VOLUME 3
Tec'nnische Hocrsclif..c!
Delft
P1975-7
Volume 3
THE SYMPOSIUM WILL BE HELD IN THE NETHERLANDS, THE HAGUE - CONGRESS CENTRE - 27-31 OCTOBER 1975
Statements and opinions expressed in the papers are those of the authors, and do not necessarily represent the views
of the Royal Netherlands Navy.
The papers have been reproduced exactly as they were received from the authors.
SESSION F:
Chairman: R. Wahab
General co-ordinator
Netherlands Maritime Institute
Regional stability of differential equations governing ship
motion.
A.G. Strandhagen and C.B. Mast.
Non-hydrodynamically generated heading instability of
ships. C. Huber.
Automatic control for directionally unstable ships. A.G. Hozos and G.J. Thaler.
SESSION G:
Chairman: M.R. Hauschildt
Technical director machinery systems division. Naval Ship Engineering Center, Philadelphia
Computer control of fast response dynamometers for propeller 3-41 load simulation.
R.V. Thompson, A. Todd, B.C. Richardson.
Use of a simulated model of a turbine operated machinery for 3-72 investigation of regulation and surveillance systems and for the purpose of training.
O. Meiri.
Verification of the dynamic model of a marine boiler.
3-88
R. Whalley.
VOLUME 3
SESSION H:
Chairman: W.W. Rosenberry
Head of automation and control division, Naval Ship Research and Development Center, Annapolis.
Surface ship bridge control system. M.A. Gawitt.
Operational improvement using an integrated conning system. R.H. Sorensen and E.T.St. Germain.
Diginav - automatic navigation for merchant ship officers. L.M. Pearson.
Integrated bridge systems.
J. Dachos, W. Behan and B.V. Tiblin.
Page see Vol.6 3-110 3-126 3-145 3-1 3-16 3-30
SESSION J:
Chairman: J.K. Zuidweg
Reader in control engineering, Royal Netherlands Naval College. Ship model evaluation of automatic control system.
R.W. van Hooff and E.V. Lewis.
A simulator - conception, birth and lifestyle. H.A.R. Beeson.
Simulation as a design aid for ride control systems. C.J. Boyd, W.L. Malone and J.M. Vickery
SESSION K:
Chairman: J. Zalmann
Captain R. Neth. N.
Deputy Head of Research Department, Ministry of Defense (Navy)
Manual steering.
W.K. Wolters. 3-223
Discrete vs. continuous control design for digital
3-232 controllers: A review of experience.
J.R. Ware,
Passing manoeuvre of two large tankers in a channel.
3-242
B.F. Dessing, A. Roos and P.J. Paymans Automatic control for replenishment at sea.
3-258/270 C.G. Lima, G. Astorquixa and G.J. Thaler.
Page
32-163
3-179
REGIONAL STABILITY OF DIFFERENTIAL EQUATIONS
GOVERNING SHIP MOTION
BY
A. G. STRANDHAGEN AND C. B. MAST
University of Notre Dame
Notre Dame, Indiana, 46556
U.S.A.
A ship executing a steady turn corresponds mathematically to a critical point
of the differential equations governing the motion of the ship. For cases in
which this critical, or equilibrium point is locally asymptotically stable, it
is of interest to obtain estimates on the size of the region of stability.
In this
work we present a method which enables such estimates to be obtained and
which also provides a measure of the rapidity with which the ship returns to
the steady turn.
Application of this method to a discussion of the Dieudonne
Spiral Test and of the Pull-Out maneuver is given. For this work we have
selected hulls of the Mariner type, both stable and unstable.
INTRODUCTION
In discussing the motion of a surface ship a system of coordinates x, y, z is
taken with the x axis along the longitudinal axis of the ship, the y axis
a-thwart-ships and the z axis perpendicular to the xy plane (henceforth, called
the horizontal plane).
The origin of this system of coordinates is at the
cen-ter of mass of the ship. We will eliminate rolling of the ship from our
dis-cussion and will thus consider only motion of the ship in the horizontal plane.
The motion in the horizontal plane is specified by three variables u, v and r,
where u is the dimensionless speed in the direction of the positive x axis
(toward the bow of the ship), v is the dimensionless lateral speed along the
positive y axis (toward the port side), and r is the dimensionless angular
speed of rotation about the z axis. The differential equations governing the
motion are those of reference (1), and are of the form
f(u, v, r,6 )
(1)
= g(v,r,8 )
= h(v, r,6),
where 6
is the rudder angle in radians.
For a fixed angle, say 6 =6,, the critical or equilibrium points of system (1)
correspond to values of u, v and r for which CI,
ir and
vanish, and such
crit-ical points correspond to the ship executing a steady turn.
In our work we
will fix our attention on the critical points of the system formed by the second
and third of equations (1).
Those equations do not depend on the variable u,
hence the critical points, which correspond to a constant rate of heading for
the ship, are independent of u.
For the equations of reference (1) there will
be at least one critical point of the system of three equations (1) for ea:h
crit-ical point of the system consisting of the last two of these equations.
We introduce the notation xi = u, x2 = v, x3 = r.
In this notation the
dif-ferential equations of reference (1) can be written (ignoring the equation in xi)
(a)
dX
d7t-= G + EX + G6 +H(X,6 )
where C and G are constant 2 x 1 matrices, E is a constant 2 x 2 matrix,
H is a 2 x 1 matrix whose entries are polynomials of degree three in x2, x3
and 8
having no constant or linear terms, and X is the 2 x 1 matrix whose
entries are xz and x3. These matrices and the polynomials in H are
dis-played in the appendix, and depend on the type of hull considered.
In the
pres-ent work we use hulls of the Mariner type.
If for each fixed 6
there is a unique critical point for system (2), then the
ship is called dynamically stable. It may happen that for some values of 6
there are more than one critical point for system (2).
In such a case the ship
is called dynamically unstable.
For some fixed value 8 =80 of the rudder angle, let (u, 0) denote a critical
point of system (2).
Set
x =y
a
x
+p, A2
2+
3 73= w +60 .
The origin yz = 0, y3 = 0 in the y2y3 plane is now at the critical
point of
system (2) for w = 0 (i.e. 6 =8 o).
In terms of these new variables (2)
becomes
dY
= AY
+ Bw + F(Y, w).
dt
The expressions for the constant matrices A and
B and for the matrix F
are given in the appendix. For a dynamically unstable ship there will be more
than one such transformation (3) depending on the critical
point chosen. The
variable w is our new control variable and it measures
the deviation of the
rudder angle from the fixed angle 6,.
In this report we will be interested in the nature of
the critical point and
hence will treat the case in which w = 0; i.e.,
the rudder held at fixed angle
8 .
For such a situation, equation (4) becomes
0
dY
= AY + F(Y),
at
where, for simplicity, we have denoted F(Y, 0) by F(Y).
Since the components of the vector F are polynomials of degree three in yz
and y3 with no constant or linear terms,
F(0) = 0,
and system (5) is an
al-most linear system. Hence, if the eigenvalues of A have negative real parts,
then the origin is a locally asymptotically
stable equilibrium point.
Indeed,
under these circumstances the origin
is an exponentially asymptotically stable
point, as shown in reference (2).
That is to say, if the system is started at
some point YA sufficiently near the origin, then the distance
11.'1(011of the
system state fiom the origin at any
time
t > 0
satisfies the inequality
Mu l Yo
exp(-Bt),
where M and B are positive constants
and
ilY01! =
Y(0)11. We will obtain
values for M and B and also lower
limits on the size of the region of
sta-bility; i.e., on the set of values
11/.011for which the system is asymptotically
stable.
The procedure we will follow closely parallels that used by Pao (3).
THEORY
If the rudder angle is set at some value, say ow then the corresponding
criti-cal point
(a, 0)
can be found, and the origin of the y2y3 coordinate system
located at this point.
The differential equation for the system is then that given
by (5).
The matrix A is determined by the parameters describing the hull
and by the numbers a and
f3;as is also the vector F.
The points in the
y2, y3 plane correspond to the states of the system, as
dis-cussed in the introduction. A measure of the rapidity with which the system,
when started at some point (y2(0), y3(0)),
returns to the equilibrium state
(the origin) is given by the time rate of change of the distance of the system
state from the origin.
In order to obtain such a measure we need to have a
definition for the distance of a point (y2, y3) from the origin, or, equivalently,
the length
111'11of a vector Y. We adopt the usual Euclidean distance given by
2 2 2
111111
= Yz + Y3
but it will prove useful to consider a more general definition also. We now
examine such a definition.
Let Y and Z be two vectors in the plane and let <Y, Z> = y2.2 + y3z3 denote
the usual inner product in the plane. Let
IYlldenote the norm induced by this
inner product.
Then, if Y is the vector pointing from the origin to the point
(y2, y3),
the quantity
II Yll2
= <Y, Y> = Y2 + Y3
2 2can be interpreted as the square of the distance of the point (y2, y3) from the
origin.
The notion of distance can be generalized as follows: Let V be a real
sym-metric positive definite matrix. A new inner product <Y, Z>v can be defined
in terms of the inner product <Y, Z> by
<Y, Z>v = <Y, VZ>
or in matrix language
<Y, Z>v = YTVZ
where YT means the transpose of the column vector Y.
The inner product
induces a norm
by setting
IYII=<Y,Y>v = <Y, yr, .
Since the matrix V is real symmetric and positive definite,
both of its
eigen-values are positive. Let these eigeneigen-values be denoted by X, and X2,
with
X2 .
Then by transformation of the quadratic form <Y, VY>
to principal
axes it can be shown that the norms IIYII
and j1Y11v
satisfy the inequalities
+ 2<F, y>v,
(10)
X111Y112 11Y112vx211112
It is convenient for our purposes to use this more general notion of distance,
since if we can obtain an estimate for
yiiv , then (10) can be used to give an
estimate for
ilYll .(Obviously, HI% and
are the same if V is the
identity matrix. )
As mentioned above, the time rate of change of distance from the origin gives
a measure of the rapidity of the return of the system to its equilibrium state.
Now
11 112
d
(Y VY) -
T
dYT
VY + YV
TdY
dt
Y v
dt
dt
dt
dY
From (5),
dt
= AY + F(Y),
and therefore
dYT
=
YTAy + FT(Y)
.dt
Inserting this in
(11), we have
ad711Y112v
YT(ATV + VA)Y + FTVY + YTVF,
which can be rewritten
d
-FIyI
yT(ATv vmy
dt
1
Since <Y,AY> = YTVAY = (YT (VA + ATV)Y), this last result may be
2rewritten as
(12)
ci-11Y112, = 2<Y, AY> + 2,<F, Y>
.Suppose that a positive number B can be found such that the right side of
satisfies the inequality
2<Y, AY>v + 2<F, Y>v
-2B
2for all points in some region
flcontaining the origin. Then we have
kllY112v
-
2B11Y112vwhich upon integration becomes
11Y112v 5-
II Yoll2v exp(-2Eit),
where Yo is the state of the system at t = 0. By (10) it follows that
111'(t)11
(X2/k1)
111'011 exP (-13t).
We see from this result that if the state of the system is within a distance
11;311 of the origin at time
t = 0,
then at some later time t
it is within a
distance of(X2/X1)
11II,
0Thus
II
J.
II exp (-Bt).
us the distance of the state of the
system from the origin diminishes exponentially.
The number B, if it exists, is not uniquely determined. Moreover, the
existence, and largest value of B depends on the choice of the matrix V.
In order to simplify the calculations, we will restrict the choices of V. We
note that the first term on the right side of (12) is of the form
YT(ATV + VA)Y
.Now set
Then (12) becomes
ATV + VA
= -R.
d
TIT 111(11,2 =
R Y> + 2<F, y>
Since the eigenvalues of A have negative real parts, and since V is real
symmetric positive definite it follows from a theorem of Lyapunov (4), that
for each given V the matrix R is real symmetric and positive definite, and
is uniquely determined.
The converse is also true.
Remark: If (5) is a linear system; i.e., if F = 0,
then (16), in conjunction
with the inequality (10), shows that the origin is an exponentially
asymptot-ically stable equilibrium point if the eigenvalues of A have negative real
parts.
The quadratic form <Y, Y>
is a Lyapunov function for such a system.
In terms of R, inequality (13) becomes
ii
R Y> + 2,<F, Y>
-2Yli2
1311 .We will restrict the choices of V to those which satisfy (15) when R is
a
diagonal matrix with entries 2/, and 2/2, where Li and /a are positive
numbers.
The inequality (17) involves the variables y2 and y3 and the parameters
which characterize the type of hull.
For the Mariner type hull as described
in the appendix this inequality is quadratic in the variable y3 and can be
written
2 2 2
py3 + Qy3 + S
B(v22y3 + (2v12y2)y3
where v..(i, j=1, 2) are the components of the matrix V, corresponding
to the
the choice of the matrix
(211
0\
R =
0
2/22)' L,12 >0'
and P, Q and S are the functions
P = Lz - 2a[v21'242
v22'34]Y2 - [v21'24 + v22'34,Y2
= z- (2 v
a- 2 + 2'..v21'24 + v22'341
v21 rQ = -[2ac_. V
L4 11 +
[Q1 + 2a c34] v12 + Q2v22JY2
-Ev11'24
v12('34 + '23) + v22'331Y23'
2, 3
4S = -"1y2
Lvlln,
v12'-2JY
r2 Lv11'23v12'33-1Y2'
where Q = 3ac23 +8c24 +6b22
1Q2 =
+13'34
+6b32;
and where (a,8) are the coordinates of the critical point corresponding to
the fixed rudder angle 6 .
CALCULATIONAL PROCEDURE
The inequality (18) was solved for rectangular regions
Iy21
< K2,
y3I <
K3
in the y2y3 plane. For given values of the positive numbers K2 and K3 and
for a fixed value of yz such that
I yzi <
Kz, the smallest value of the ratio
2
Py3 + Qy3 + S
2 2
v2,2Y3
(2v12Y2)Y3
v11Y2
was computed for y3 in the interval
I y3I
< K3. For convenience we refer
to this number as B(y2) The above procedure was carried out for values of
y2 in the interval
Iy21<
K2 and the smallest of the values B(y2) was
de-termined.
This number is the least upper bound for the value of B for which
the inequality (18) is satisfied in the rectangular region
I y2I< K2,
Iy3< K3,
and is the number which we use in the inequality (14).
For a given hull type, rudder angle setting 6 0 and choice of K2 and K3' the
number B as determined by the procedure just described, depends on the
matrix V, which, in turn, depends on the matrix R of equation (15) or,
equi-valently, on the numbers Li and L. The dependence is actually on the ratio
and calculations were carried out for a range of values of this ratio in
each case. Since inequality (18) is satisfied for each B so obtained, it is
sat-isfied for the largest such B. Consequently, the largest such B can be used
in (14) to give a lower bound on the rate of change of distance of the system
state from the origin. That is to say, the distance
IlY(t)lI of the system state
Y from the origin decreases at least as rapidly as given by (14) with the decay
constant taken to be the largest value of B just described.
The calculations were carried out for Mariner hulls of the stable and unstable
types.
Only three values of 60 were considered for each hull; one of these
corresponding to the neutral rudder setting in each case. For each choice of
6
several values for K
and K3 were considered.
It was felt that these
selections would be sufficient for the exposition of the methods of this article
arid their applications. The numerical work, which is obviously extensive,
was done with the aid of a digital computer.
In Tables land II some typical
results are presented.
DISCUSSION AND APPLICATIONS
The results displayed in Tables I and II were obtained for the Mariner
ships whose coefficients are listed in the appendix. The graphs giving the
coordinates of the equilibrium points (states of steady turning) for the stable
and unstable ships are given in Figures 1 and 2. We recall that the variables
ya and y3 are related to the usual variables v and r by the equations:
v = y2 +a and
r = y3 + $,
so that
y2. and yz both vanish at the equilibrium states.
Thus the variables
v and
r
in the geaphs of Figures 1 and 2 can be replaced by a and 0
respectively.
In terms of the variables v and
r,
the rectangular regions
! y21 < K2,
! y3! < K3, correspond to the regions
-K2 + a < v < K2 + a
and
-K3 +0<r<K3 +0,
respectively.
For the stable ship, we considered rudder angles 8 0 = 0.02, 0.10 and 0.20.
The corresponding values of a and 5 (the values of v and
r at the
cor-responding equilibrium) are
The rudder is in a neutral position when the system is in
an equilibrium state
with
r = 0.
For the stable ship we consider here this corresponds to a rudder
angle of approximately 0.02 radians, and will be taken as such in our
discus-sions.
For the equilibrium state corresponding to 6
= 0.02 we found that the
rec-tangular region Iy2! <0.13, Iy31 <0.27, or, equivalently -0.125< v
< 0.135, -0.277< r < 0.263 is a region of
stability.
For this region the value
of B was found to be 0.06. The ratio X2Al of the eigenvalues of V is
51.96.
Thus, if the ship has its rudder set at 6 0 = 0.02, and if some
distur-bance perturbs the motion so that the state of the ship is
inside the rectangle
just mentioned, then the state returns to the equilibrium
state corresponding
to 6
0= 0.02 according to the inequality
3-7
6 =
0
0.02,
a = 0.005,
13 = -0.007
6 0 = 0.10,
a = 0.074,
0 = -0.153
8o = 0.20,
a = 0.104,
0 = -0.227.
11Y(t)11
7.211 Yo 11 exp(-0.06t),
where 11Y011
is the distance of the perturbed state from the origin when t = 0.
Upon inspection of Figure 1 we note that points on the u versus 8
and
versus 6
curves for the interval -0.24
0.28 correspond to points in the
region of stability.
Consequently, if the ship is in a steady turn corresponding
to a rudder setting of 6 = 0.28 and if the rudder is then moved to the neutral
(6
= 0.02) position, the ship pulls out of the steady turn, and the state of
motion approaches the equilibrium state corresponding to the neutral rudder
setting according to the inequality given above. Since
= (,-.)2 + (r-s)2,
we see that
1 r-81
7.211Y011 exp(-0.06t);
that is to say, the rate of turn decreases at least exponentially, with the decay
constant B = 0.06.
The maneuver just described is called the pull-out maneuver, and is discussed
by Burcher (5). He points out that the pull-out characteristic for a stable ship
is essentially exponential. If this is so, then we see from our results that the
exponent for the pull-out maneuver from steady turns corresponding to rudder
settings between -0.24 (-13.7°) and 0.28 (16.3°) cannot be less than 0.06.
By considering smaller regions we can obtain larger values for B. For
in-stance, for the region
ly21 < 0.10,
1y31 < 0.22, we obtain a value B = 0.167.
This region allows pull-out maneuvers from rudder angles as small as
-8.6°
or from rudder angles as large as 10.5°. We have found that for square
re-gions 1y21 <K,
1y31 < K, the value of B is 0.179 for K = 0.1 and it is the
same for smaller values of K.
We notice that the values of B increase (or at least remain the same) as the
size of the region decreases.
Thus if a pull-out maneuver from some steady
turn is contemplated, then a value of B found by our methods gives a
lower
bound on the stability characteristic discussed by Burcher. By choosing the
smallest rectangle with center at the equilibrium point corresponding to
the
neutral rudder position and which also contains the equilibrium point
corre-sponding to the steady turn, the largest such value of B can be
obtained.
Our analysis also lends itself to a discussion of the spiral test.
From our
results in Table I we notice that for the equilibrium state corresponding
to
8
= 0 1
the square 1y21 <0.2,
1y31 <0.2 is a region of stability with
0
'
decay constant B = 0.322.
This region includes the equilibrium point
cor-responding to a rudder angle 8 = 0.5.
Consequently, if the ship is in a steady
turn corresponding to 6 = 0.5 and the rudder
is then shifted to the position
80
= 0 1
'
the state of the system moves to the new equilibrium state
according
to the inequality
11Y(t)11
1.10 exp(-0.322t),
(here
= 0.21). We see from this that a rudder
change from 28.6°
to
5.7°
is allowable in a spiral test. Clearly, smaller steps are also allowed
in going from 28.6° to 5.7°. For smaller changes in rudder angle the
ex-ponent B will be larger; as can be seen from Table I.
The discussion for the unstable Mariner ship parallels that for the stable
Mariner ship in many respects. However, when the rudder angle is
set to
60
= 0
'
there are three equilibrium points; as can be seen from Figure II.
We will only consider the equilibrium point on the upper half of the S shaped
curve.
The coordinates of this point are a = 0.089,
= -0.108.
We found that the rectangular region Iy21 < 0.06,
I y3I < 0.1 is a region of
stability for the above mentioned point.
The exponent B has the value
0.023. This rectangular region includes the equilibrium points with
values
of v and r in the ranges
0.029< v< 0.149, -0.208< r< -0.008.
Some of these equilibrium points are inside the "loop" of the S
shaped
curve.
The region of stability allows for pull-out maneuvers from the
turning state corresponding to a rudder angle of 0.18 (10.2° ), and also
for pull-out from the unstable equilibrium points which
are inside the "loop"
of the S shaped curve.
In contrast to the case of the stable ship, the value of B continues to
in-crease as the size of the region considered dein-creases, even down
to the
value K = .01.
Examination of the behavior of the unstable ship at the points corresponding
to 6
0= 0.1 and 6
0
= 0.2 shows a marked difference from the stable ship.
For comparable size regions the values of the decay constant B are much
larger, whereas the ratio k2/Xi of the eigenvalues of
v are much smaller.
Thus, for instance, if the ship is in a turning state corresponding to 8 = 0.2
and the rudder then is shifted to the angle 0.1, the state of the system moves
to the new equilibrium state with decay constant B
= 1.47.
For the stable
ship the same situation yields the value B
= 0.35 for the decay constant.
We can argue from these observations that the spiral test for the unstable
ship would require considerably less time than for the stable ship.
It must be always borne in mind that
our procedures give only lower
esti-mates on the decay rates and on the size of the
region of stability. It may
happen that the return to equilibrium is much faster than the exponential
rate indicated by our results. Since we are, in essence, requiring
expo-nential asymptotic stability, the region of ordinary asymptotic stability may
be considerably larger.
Table I.
Stable Ship
3-10
Table II.
Unstable Ship
0
K2 K3i1/.02
N1/N2
0.02
0.13
0.27
3.33
51.96
0.06
0.02
0.10
0.22
20.00
146.70
0.17
0.02
0.05
0.12
20.00
146.70
0.17
0.02
0.14
0.14
3.33
51.96
0.04
0.02
0.10
0.10
10.00
95.95
0.18
0.02
0.06
0.06
10.00
95.95
0.18
0.02
0.03
0.03
10.00
95.95
0.18
0.02
0.01
0.01
10.00
95.95
0.18
0.10
0.20
0.20
10.00
27.57
0.32
0.10
0.15
0.15
10.00
27.57
0.32
0.10
0.10
0.10
10.00
27.57
0.35
0.10
0.05
0.05
3.33
10.72
0.46
0.20
0.20
0.20
10.00
20.02
0.49
0.20
0.15
0.15
20.00
38.85
0.50
0.20
0.10
0.10
3.33
6.82
0.64
0.20
0.05
0.05
1.43
2.95
0.97
K2 K3.21/22
N1/N2
0.00
0.06
0.10
3.33
15.32
0.02
0.00
0.05
0.05
1.43
8.85
0.06
0.00
0.03
0.03
3.33
15.30
0.26
0.00
0.01
0.01
3.33
15.30
0.45
0.10
0.06
0.06
1.43
2.52
1.06
0.10
0.03
0.03
1.43
2.52
1.47
0.10
0.01
0.01
1.43
2.52
1.80
0.20
0.09
0.09
3.00
2.01
1.45
0.20
0.06
0.06
3.00
2.01
1.87
0.20
0.03
0.03
1.43
2.25
2.15
0.20
0.01
0.01
3.33
5.03
2.34
ST8.1) a 1 s 0 10 0 0 STB'D PORT -.0 -080 -060-040-020 020 040 060 as o .00 Radians/ -0.25 -0 20 -015 010 -005 -00 -015 ALPHA DELTA -Se -0 20 -030 8ETA us ALPHA 3-11 (Dirsensioniess, I OsHsnmosiess / 030 0.20 010
Figure 1.
Stable Ship
PORT
005 010 015 020 025
ST WO
-025 -aao -0;5 -coo -aos
-ar25 ALARA vs DELTA BETA v DELTA fi1 Dimes...al 0.375 a250 a:25 -a2so 0375 BETA vs ALPHA aos aro 015 020 025 Dimenmonless,
APPENDIX
Setting x2 = v, x3 = r, the differential equations of reference (1) can be put
in the form
dX
= C +EX + Go +H(X,6), (-1 <6 < 1)
dt
where
bc
(c20),
E
=(c21 c22\
-) G
c30
c31 -32,
31and the entries h2 and h3 in H are of the forms
2 3
hk = ck3 x2
3 +ck4 xzx3
2+ bk2 x26 +
2ck 7'2
+ cite
for k = 2,3.
The entries in C. E and G, and the coefficients in H are determined
ex-perimentally for a given hull.
In the following tables we give the values
for two types of Mariner ships.
Stable Ship
3-13
C20 =
-0.002144
C30 =
0.033189
=C21
-0.769435
C31 =
-3.389219
C22 =
-0.334416
C32 =
-2.093985
G23 =
-5.123246
C33 =
18.332967
C24 =
9.579404
C34 = -63.516877
B21 =
0.170701
B31 =
-1.627565
B22 =
0.737781
B32 =
-5.696652
C26 =
-0.055279
C36 =
0.527686
C27 =
-0.001622
C37 =
0.150340
Unstable Ship
020 =
-0.004788
C30 =
0.034744
C21 =
-0.492236
C31 =
-3.767261
C22
=-0.670063
C32 =
-2. 204483
C23 = -10.265325
C33 =
16. 640136
C24 =
19'194019
C34 = -60.351638
B21 =
0.343900
B31 =
-1.402925
B22 =
1.478274
B32 =
-5.452874
C26 =
-0.110762
C36 =
0.509421
C27 =
-0. 003509
C37 =
0.149726
Using the transformations (3)
= Y2 +a' '3 = Y3 + 8, 6 = w +6
0'
the above equations become
dY
=AY +Bw + F(Y,w), (-1-6 0<
1-6o)
(1)2
-
F2
where A =
(a21
\ a31 a32)
a22)
g =
lb /'3
3
where
+ 2aBc
+ 3a2ck3
k4
akl = ckl
2a6Obk2
ck75 02ak2 = ck2
a'2ck4
-k 2 -4-.2.Abk = bkl
-k2.a
Ock6
Ock7
3 2 2
Fk = ck3Y2
ck4Y2Y3
4."3`''ck3
/3ck4
bke
4. ` .,,,,'"ck4Y2Y3 2 2 2 3+ (2abk2 + 26 ock7)y2w +
, ,,k2y2wck7y2w
.ock6
ack7'w
+ c
k6w
3 22 3
+ b
6+ b
a26
+ c 76 a +
6+c a+c
+c
k0
kl
k2
k3a
kl 0
k2
0k
0ck6 0
for each choice of 6 0.
For the first Mariner type ship there is a unique solution of these equations
for each 60 in the interval (-1,1), and, hence, this ship is dynamically
stable.
For the second Mariner type ship there are three solutions for 6
0in the interval (-0.143, 0.053), and, hence, this ship is dynamically unstable.
REFERENCES
3-15
Strom- Tejsen, J.
A Digital Computer Technique For Prediction of Standard Maneuvers of
Surface Ships. David Taylor Model Basin Report 2130, 1966.
Belman, R.
Stability Theory of Differential Equations. McGraw Hill, p. 79, 1953.
Pao, C. V.
On the Stability of Non-Linear Differential Systems. International
Journal of Non-Linear Mechanics, Vol. 8, pp. 219-238, 1973.
Gantmacher, F.R.
The Theory of Matrices.
Vol. 2, Chelsea, p. 187, 1960.
Burcher, R.K.
Model Testing. The Journal of Mechanical Engineering Science,
NON-HYDRODYNAMICALLY GENERATED HEADING INSTABILITY OF SHIPS
BY
CONRAD HUBER
SUMMARY. - Rotary motion of a ship around the roll and pitch axes theoreti-cally can lead to undesired heading rates. This phenomenon, that has affinity with the coning error encountered in gyro applications, is explained. Some possible practical consequences are discussed.
I. Introduction Angular sensors
Coning error Conical platform.
Ship pitch and roll movements
Equal pitch and roll amplitudes and periods
Arbitrary "coning" with equal pitch and roll periods Unequal pitch and roll periods
A selection of idealized ship models Some practical aspects
II. Literature 12. Appendix I. INTRODUCTION
When dealing with the problem of measuring heading rate of change of a ship
by means of a strapdown sensor it appears that under certain conditions of
pitch and roll movements a strap-down rate or rate integrating gyro would indicate that the ship is turning while it actually is following a straight course. This phenomenon is related to the so-called coning error known in gyro technology. Conversely, sets of conditions will exist where the ship actually is undergoing a steady heading rate of change which is not detected by the strap-down gyro. Under these conditions an autopilot, instructed to keep the ship on a straight course and receiving its input from a
strap-down angular sensor, would give no rudder commands. Clearly, if the auto-pilot keeps the rudder set for a straight course but the ship nevertheless does turn, this turn is a type of heading instability of the ship. This type of instability is not caused by the hydrodynamic forces of the water flowing along the ship's hull due to its horizontal speed, but derives from the geometry of the roll and pitch axes. To understand this phenomenon a mini-mum insight into the functioning of angular sensors and the nature of the coning error will be required.
2. ANGULAR SENSORS
We shall be concerned only with single axis angular rate or rate integrating sensors. Single axis sensors have one input axis. They are sensitive to rotations around this input axis only. The rate sensor has an output, say
an electric current or voltage, that is proportional to the angular velocity around this input axis. The angular rate, or angular velocity, can be re-presented as a vector, and only the component parallel to the sensor input
axis will be translated into an output signal.
U U =0
S sensitivityvector; C.).input rate. vector; U = output vottnge
where
U = output signal (e.g. electr. voltage) = sensor sensitivity vector
= rotation rate vector.
Thus, though a single axis rate sensor may experience angular velocities in any direction, it will only show those components at its output that are aligned to the momentary position of its input axis.
3. CONING ERROR
The rate integrating sensor does nothing more than give an output (current or voltage) that is proportional to the time integral of the signal which a rate sensor in its place would have shown. Thus a rate integrating sensor can have travelled over a considerable angle, thus pointing in a different direction than to begin with, and yet show no comparable output since not all angular rates are integrated but only those components happening to have coincided with the input axis. This is the basis of the coning error.
This phenomenon is easiest to understand when demonstrated with rotations around axes orthogonal to the input axis of an integrating sensor. A sensor initially placed for instance with its input axis vertical (position A) is tilted 90 degrees around a horizontal axis to position B:
Figure 2. Tilting the sensor.
3-17
Figure 1. A single axis angular rate sensor's response to an angular rate input.
In equation form this becomes the scalar product of two vectors
No rotation has occurred around its input axis, thus the output remains zero. Next we turn the sensor around the vertical, which still results in zero output.
Figure 3. Turning the sensor.
The next movement, tilting the sensor so that its input axis again becomes vertical, completes the cycle, still without giving rise to an output signal but resulting in a net 900 turn (position D) of the sensor case with respect
to its initial position A.
Figure 4. Reerecting the sensor.
If we had turned the sensor 900 around the input axis from position A directly to position D there would have been an output signal corresponding to this turn. But manoevering the sensor to position D via B and C prevents the generation of an output signal although the resulting net position change is the same.
If the sensor is turned from position A via B and C to D and then directly back to A the result will be zero net rotation but negative output signal.
We need not turn the sensor over full right angles but any angles will de-monstrably result in a corresponding discrepancy between signal output and
net angular displacement (see figures 5 and 6).
Figure 5. Three movements orthogonal to S, one of which is less than 900.
Figure 6. All three angles smaller than 900 and the continuous coning movement.
Likewise the total movement does not have to be broken down to three se-parate angles but finally an infinite number of very small angles will do the job too. Thus we arrive at the actual continuous coning movement
(figure 6c).
4. CONICAL PLATFORMS
The coning movement can be characterized by the path of the arrow point of
-8-as it can be seen in figure 6. The simplest c-8-ase is a circular path belon-ging to a conical movement of the arrow representing the sensitivity vector of the sensor, around the vertical as axis of symmetry.
This type of movement is generated when a circular platform with a central orthonogal rod rolls around on a horizontal plane as indicated in figure 7.
Figure 7. Coning platform.
The rim of the platform will trace a circle on the horizontal plane. If the circumference of this circle is equal to the circumference of the plat-form, the platform will return to its initial position after completing a full coning cycle. We mark this initial position by arrow points on plat-form and horizontal plane. If the trace is shorter than the circumference of the platform the marking on the platform will have advanced beyond the marking on the horizontal plane after the full coning cycle; and, conver-sely, if the trace is longer than the rim, the platform marking will lag
behind.
To further deepen our understanding of the geometry involved we can also imagine the movement to arise from a cone devolving on a second cone like in figure 8.
For the situation with equal circumference of platform and horizontal trace mentioned above the top angles of both cones have to be equal. If the top angles differ we get the cases of advancing or lagging marking.
Figure 8. Double cone model.
SHIP PITCH AND ROLL MOVEMENTS
All that was said up to now was a theoretical introduction to the actual movements of a ship. It is now opportune to establish the analogy between
the coning platform and a ship before investigating the implications of
the model platform.
Let the ship deck be a plane, corresponding to the platform of figure 7 or the flat end of the upper cone in figure 8. Give the ship a central mast representing the central rod of the models in figures 7 and 8. Suppose the ship to have equal rolling and pitching periods, and imagine it to be rolling and pitching with equal amplitudes and with pitch and roll 90° out of phase with each other. The top of the mast then will describe a circle around the vertical through some centre of motion in the ship's body.
Although in practice a ship tumbling around with equal pitch and roll pe-riods might be reasonable, equal pitch and roll amplitudes may seem
far-fetched. Nevertheless, assuming equal amplitudes makes it easier to under-stand the basic implications of the model thereby still retaining relevant information.
The mark on the rim of the platform (figure 7) can be regarded as the bow of the ship. It will be used to indicate the momentary heading of the ship.
As we have seen in paragraph 4 certain coning modes result in a positive or negative heading aberration after each coning cycle. To study the relevant conditions we turn again to the cone as a model for our ship.
EQUAL PITCH AND ROLL AMPLITUDES AND PERIODS
Figure 9 represents a cross section of the double cone model in a plane containing the vertical and the momentary contact line between the two
cones.
The geometric data are:
2a = top angle of the lower (inertial) cone 28 = top angle of the upper cone (= "ship")
y = pitch amplitude = roll amplitude = coning angle.
The momentary motion of the upper cone can be characterized by the momen-tary angular velocity vector M, the direction of which is given by the momentary contact-line between the two cones. This vector M can be split
into two components V and A:
= A
+ V.
(2). See appendix. a = 900
a = F= i(180°-'1)
to
e = 90°.3-21
Figure 9. Double cone model with angular velocity vectors.
horizon
It can be shown* that the vertical angular velocity vector17 represents the coning rate, i.e. the rate at which the top of the mast completes one coning cycle:
V 27/tc (3)
where t is the length of a coning period. A may be called the heading ad-justmeni angular velocity vector.
By varying the top angles of the cones we can find the different modes of coning. Keeping the coning angle y constant we get comparable movements, which even seem identical visually. As we see from figure 9, a and F then become interdependant, since
a 6 y = 180°. (4)
The two principal ranges through which we can move are a < 6 and a > B.
(5)
For our present purpose we can keep within the following boundaries:
Within this frame we select three special cases that exhibit the desired basic characteristics of the cone-model. These are
and you will find them drawn in figure 10. . See appendix
(a)
c- 900
horizontal
A > V
heading rczte
ft c
(b)
Mon bisector
A = V
heading rate zero
Figure 10. Three special coning modes.
It can be shown that the difference between the absolute values of V and A for reasonable angles (y < 69..10°)
is
the approximate heading aberration rate. Only in mode (b) is A = V and the aberration rate zero. This is plau-sible, since only in mode b the "rims" of both cones are equally long. Therefore, after completing a whole coning cycle, the upper cone will have re-established its original "heading". In mode (a), the rim of the upper cone being shorter than the circular trace on the horizontal plane, the"heading" of the upper cone will lag behind after each coning cycle, where-as in mode (c) the cone "heading" will have advanced in the sense of V, its rim being longer than the rim of the lower cone.
From the foregoing we may conclude that a ship on a straight course cannot but tumble in a mode resembling in some way mode (b).
A double cone model, naturally, is deliberately designed with given a, 6, and y. From these the relation between M, A, and V may easily be found. But with a real ship only V and y are readily determined experimentally. To
de-termine M and A we would have to resort to a strap down rate sensor. If aligned to the "mast" with its input axis it would measure the component of M orthogonal to the "ship's deck", designated MA (see figure 9). With V, y, and MA known, M can be found as
M = IV2 sin2y
MA2'
(4)
and thus the triangle A, V, M is determined. (Whether B < 90° or B > 900
can be decided by comparing the sign of MAwith the sign of V). While all the foregoing is true for equal pitch and roll amplitudes and periods it remains to be checked whether a V, y, and MA measurement is possible or
feasible for determining arbitrary coning mode axes.
In a sense case (b) and (c) are complementary, showing the relationship between the gyro coning error and the coning induced heading rate of the
(c)
A.-9 0 °
11 parallel to shipdeck
A < V
ship. In case (b) the ship will not change its heading, but the sensor will give an output since the momentary tumbling rate vector M is not orthogonal to the sensor input axis. In case (c) the sensor will detect nothing, its input axis being orthogonal to 171, whereas the ship does have a heading rate. There is no coning situation in which heading aberration and gyro output match.
7. ARBITRARY "CONING" WITH EQUAL PITCH AND ROLL PERIODS.
From paragraph 6 it follows that the condition for a straight course is equal "rim length" of upper and lower cones. In terms of Fig. 9 this means
r = rb, i.e. equal base radii for both cones, and equal top angles, a = 6. also means that both cone surfaces, devolving the one on the other, must be equal.
It is not difficult to see that the last statement is very general, for if we deform the cones into arbitrarily shaped cap-like bodies (generated by straight lines passing through the top), keeping their surfaces of equal size, one complete "coning" cycle will always restore the original position of the upper "cone".
Many such bodies are imaginable, the most simple of which perhaps are ellip-tical cones. Whereas with the circular cones the top of the "mast" des-cribed a circle, this circle deforms into a warped ellipse when elliptical cones are involved. An elliptical trace of the top of the mast means nothing less than unequal pitch and roll amplitudes.
A very practical case can be illustrated by two saddle-like surfaces as de-picted in Fig. 11. Again, the surfaces must be generated by straight lines starting from the middle of the crest line. Two generating lines of each
inertial "czne
Fig. 11. Horizontal pitch axis and ship-fixed roll axis.
surface form an unbroken straight line, i.e. the y-axis on the inertial "cone" and the x'-axis of the "ship". Of course the area of both surfaces must be equal to give a constant heading.
Suppose the course of the ship to be in the direction of the positive x-axis. The position of the ship as drawn in figure 11 thus is fully pitched at zero roll angle. The contact line between upper and lower "cone" at that moment is the (ship-fixed) x'-axis, which represents the roll axis. A quarter coning period later the upper.cone'will have rolled into a position where
the line marked (y) will contact the (horizontal) y-axis. This then has be-come the pitch axis at zero pitch angle and maximum roll angle.
The reason why this is an interesting and practical "cone" configuration is, that there exists a well-known mechanical device generating a movement com-parable to the above: the cardanic suspension system. Figure 12 shows a mo-del according to this principle.
Figure 12. Model suspension with horizontal pitch axis and ship-fixed roll axis.
However strongly pitched, the model ship will roll around its ship-fixed roll axis, and whatever its banking angle, pitch oscillations will always occur around the horizontal outer gimbal axis only. As we will see in para-graph 9, a long floating cylinder will exhibit the characteristics of such a
model.
A very extreme set of "cones" shall briefly be touched upon to round off the insight to be gained by the "cone" model concept. As shown in figure I3a, both surfaces are shaped like a roof. This model isolates pitch and roll in
Figure 13. Roof-shaped "coning" models.
this sense: At maximum pitch angle there is no pitch rate but the ship rolls over from say left to right maximum excursion, then roll rate stops and pitch rate starts tilting the ship to the opposite pitch amplitude etc. Roll
axis is ship-fixed, pitch axis is horizontal, and the areas of the "cones" involved are equal. So no heading aberration occurs. The mast-top will des-cribe a "rectangular" trace.
If one "cone" has a broken ridge-line like the lower one in figure 13b, there will be a heading aberration because the areas of both "cones" no lon-ger match. Both cones would require a broken ridge-line with the same angle S at the meeting corners of the roof quadrangles to avoid heading aberrations.
UNEQUAL PITCH AND ROLL PERIODS.
Looking down at a double cone model with a coning rate vector V oriented as drawn in figure 9 we see the top of the "mast" describing a circle in clock-wise direction. If the model represents ship-fixed roll and pitch axes as in figure 10c we will see the upper cone changing its "heading" in clockwise direction too. Reversing the coning rate will give a counter-clockwise hea-ding change.
This phenomenon suggests a way to average out coning-induced heading rate: reverse the coning rate vector periodically. Fortunately this can be done very easily by giving roll and pitch different frequencies. For this we must forgo the double cone model as it becomes physically unrealistic although mathematically conceivable. Looking down on a ship rolling with e.g. double the pitch period we will perceive the top of the mast performing 8-like fi-gures. One half of the 8 will be followed clockwise, the other counter-clockwise. Whatever the heading aberration during one half of the total move-ment it will be cancelled by the opposite heading aberration during the
second half, provided there is sufficient symmetry in the roll-axis configu-ration over the positive and negative pitch trajectory.
The figure "8" described by the top of the mast in fact is one of a set of Lissajou figures, well-known to those who have used oscilloscopes. In gyro theory and technology the coning error is an established fact (1). The formulas pertaining to this phenomenon indicate that the error angle is proportional to the area circumscribed by the "top of the mast". Each part of this area is to be counted positive if it is circumscribed in a clock-wise fashion as seen from above, and negative in a counter-clockclock-wise fashion. If the trace of the top of the mast is a symmetric figure"8"(stemming from a roll to pitch period ratio of 2:1) the total area circumscribed is zero, since we may count one half positive and the other negative. Consequently, if the heading aberration is positive due to the first half of the movement it will be negative due to the second half, cancelling out after completion of the whole figure.
It appears that all closed figures such as arise from other pitch to roll frequency ratios different from 1:1 lead to a net zero heading error. The time it takes to average out the aberration, i.e. the time it takes to close the figure can be inferred from the period ratio. Generally speaking, it takes the number of periods that fits into the smallest common multiple. While with simple ratios like 2:1 or 3:1 averaging out is accomplished
rela-tively rapidly, other ratios, especially small ones, can require a conside-rable time as the table of examples on the next page shows.
A SELECTION OF IDEALIZED SHIP MODELS.
We shall now have a look at a few extreme forms of ship bodies to get some insight into the potentially stable and unstable forms.
The form most easily understood will be a cylinder with a length larger than its diameter. Placed in water it will float with its long axis horizontal. If the bow or stern is pushed under the surface a restoring torque will appear around the horizontal athwart-ships axis. Letting go the cylinder will enable it to regain its horizontal position. Because of its rotational sym-metry it will pitch back around the axis of the restoring torque which is horizontal. This it will do regardless of its banking state, so we may con-clude that its pitch axis remains always horizontal. Its roll axis naturally will always be its long axis regardless of its pitching condition. It is thus ship-fixed.
Table I. Examples of coning error cancelling periods (belongs to paragraph 8).
A cylinder therefore will behave like the model drawn in figure 12. It will show no coning-induced heading instability, providedand this seems
ironicalprovided it has no rudder! For a vertical blade will cause the pitch axis no longer to remain horizontal when the cylinder starts rolling to and fro. To remedy this a rudder would have to be used that is indif-ferent to the banking angle. One could think of a hollow cylinder like the guide-ring sometimes used around screws (figure 14).
Figure 14. Body without coning induced heading instability.
In contrast to the cylinder a vertical blade with its long axis horizontal (see figure 15a) would certainly suffer from coning-induced directional
ratio of
roU to pitch
pariocis
smallest
commonmultiple
heading error cancels every...
LiP4Ou
Num
3 :
13
roll period
2 :
12
roll period
2
3 : 2
6
second roll period
. ...
...
5 : 4
20
fourth roll period
;---y:
..,,..,i.
4411: 10
110
tenth roll period
In
aberration. Its roll axis, like that of the cylinder, is structure-fixed, but so is its pitch axis. For it will be readily seen that the banking angle of the principal plane of the blade determines the pitch axis orientation. But if the blade becomes the shape of a semicircular disk (figure 15b) the
Figure 15. Directionally instable (a) and stable (b) blade forms.
roll axis will remain horizontal as long as pitching doesn't alter the sub-merged part of the disk. As compared with figure 14 roll and pitch axes are
exchanged, but directional stability is maintained.
Finally, let us consider a spherical body. It can roll and pitch around any axis and thus may be termed directionally indifferent to coning motion. Ad-ding a rudder and/or fins will tend to fix the axes and thus in theory any
type of coning mode may be achievable. 10. SOME PRACTICAL ASPECTS.
One might ask whether the heading rate introduced by the coning mode could ever become serious enough to be noticed. To calculate an example we shall assume a simple case like that of figure 10c because this coning mode seems realistic when we think of a
ship
comparable to the blade in figure 15a. A coning period of 10 seconds and a coning angle of 3 degrees seem reasonable. From figure 10c we see that the heading rate will be= V - A = V(1 - cosy) 2u radians
With V - and cos 3° = 0,9985
10 secs
we get
= 54.10-3 deg/sec. (6)
In 1000 seconds or 16,7 minutes the ship would make a heading change of 540, and a complete turn of 1800 in less than an hour. With double that coning angle the rate of turn would be approximately 4 times as fast, or one turn in a quarter of an hour.
If the periods of roll and pitch differed by 10% this ship would make course oscillations with a period of about 100 secs and 1 degree amplitude.
Not knowing the cause of the course change the helmsman or autopilot would correct the heading change by rudder commands instead of leaving the ship to its natural course. During the trip it will be difficult to distinguish be-tween coning induced periodic components and genuine errors that have to be
corrected.
(5)
Needless perhaps to say that a ship with a shorter coning period has a lar-ger coning induced heading rate. With y 6° and a coning period of 5 secs. a turn would be effected in about 8 minutes. A small ship will thus probably show the effect more seriously than a larger one.
Buoyancy kinetics may change with a ship's loading condition, affect the roll and pitch axes, and thus change its course stability.
It has been mentioned that one axis should be horizontal if the other is structure-fixed. When the water-surface itself is locally not horizontal with waves longer than the ship's dimensions one must strictly say that the
respective axis must be parallel and stay parallel to the local water surface.
Presumably it will be difficult to isolate the coning effect from other ef-fects in a real ship. For small ships, one way to determine the coning rate may be rocking it in a still basin in the absence of wind. I have
experimen-ted with model ships in a small basin. But it is difficult to make waves without introducing some turbulence in the water, so although these trials
did give an indication of the existence of the coning effect they could not be fully conclusive.
Full scale experiments will have to show how serious the effect is in real ships. Until then we may say that a ship behaving well on flat water but showing a tendency to turn in rough water should be suspected of having an unfavourable combination of roll and pitch axes.
LITERATURE
(I) MAGNUS, K.; Kreisel. Springer, Berlin etc., 1971; chapter 15.5 APPENDIX
One way of explaining the vectors in figure 9 is as follows.
(6)
(c)
Figure 16. Hypothetical double-drive setup to explain figure 9.
Imagine a horizontal platform P1 fixed to the ground, carrying a motor D
1
also carrying a motor with vertical axle AI and a stand S with an axle A2 at an angle y with tte vertical. The axles A1 and A2 are coupled by coni-cal gearwheels with a ratio m:n.
Motor D2 is made to drive axle A1 at speed A'. Then axle A2 will assume a speed A according to
A _n
;
The angular velocity vectors describing the rotation of these axles shall be called W' and
T,
and their orientation is indicated in figure 16b.Then motor DI is started and adjusted so as to give platform P9 a speed
V = - T'.
This will cause axle AI to stand still with respect to the groundand axle A
2 to describe a cone around the vertical with a coning velocity
--equal to V.
For axle AI we can write:
Resultant speed = A' + V = 0.
For axle A2 (representing the "ship's mast" in figure 9) we find: Resultant speed
= T + V = Ti.
This situation is given in figure 16c.
After one complete coning cycle axle AI will still have its original posi-tion, but A, would have to be turned an amount 21-(1 - ;) to regain its ori-ginal direction. Accordingly the heading error rate is given by the following absolute values of the respective vectors:
6 = v(1
= v
= v
V - A.AUTOMATIC CONTROL OF DIRECTIONALLY UNSTABLE SHIPS
BY
A. HOZOS G. J. THALER
HELLENIC NAVY NAVAL POSTGRADUATE SCHOOL
ABSTRACT
Feedback loops are used to stabilize an unstable hull and provide essentially the same dynamics as for an equivalent stable hull. Simulation tests verify calm water behavior and also indicate rudder behavior in presence of waves. Dead zone is used to reduce rudder motions for the stable hull, and is combined with a special compensator design for the unstable hull.
INTRODUCTION
Directionally unstable ships have a smaller turning radius and shorter turn time than comparable stable ships. These advantages are largely counterbalanced
(if guidance is manual) by a need for continual steering. Also, when auto-matic control has been used rudder operation has been excessive. This paper addresses the problem of providing the advantages of an automatic feedback control for stabilization while reducing unnecessary rudder action. This is done in three phases:
Design of a feedback controller using yaw rate and acceleration feedback. Simulation studies of ship and rudder behavior in the presence of waves. Introduction of dead zone to reduce rudder action when waves are present, and modification of the controller design to further reduce rudder action.
THE MODEL
A five foot, Series 60 model (with block coefficient 0.7 and no propellor) as originally developed by the Davidson Laboratory was adopted as a parent model and was modified [1] to provide several stable and unstable hulls. Two of these, A (stable) and C (unstable) were selected as models for this
investi-gation. The nondimensionalized hydrodynamic derivatives for the linearized
equations and the roots of their characteristic equations are given in Table I. The pertinent relationships are the usual ones
(X!-- m')11' + X'Au = 0 (la)
- + Y'vv' + (Y; - + (Y; - m'u'o)r' + Y;6 = 0 (lb)
- m'X)V'+ + (N; -
IZ)fr+
(N; - m'X'Gu'o)r' + N;(5 = 0 (lc)Substitution of numerical values into the second and third of these equations and manipulating provides the characteristic equation. The roots of the characteristic equation for the two models are also given in Table I.
Since the linear equations are valid only for small perturbations, a nonlinear model was also generated using the usual Taylor's series expansion. The
re-sulting equations are
On - xdit
= fi (u,v,r,S)(m - Yv)v + (EXG - Yf)f = f2 (u,v,r,6) (2)
(mXG-Nv)V + (Iz - Ndf = f3 (u,v,r,6)
where:
1 1 3 1
fl (u,v,r,6) = X. + XuAu + -2:XuuAu2 + -i,-XuuuAu + -2- X v2 + (1fXrr+ mXG)r2
1 2 1 1
X66 + -- X v2Au + 1-- X r2Au + X
66u62Au + (X
+15)
vr6 2 vvu 2 rru Vr
+ Xvv6 + Xrdr6 + XvruvrAu + Xvuv6Au + Xr6ur6Au
f (u ,v r 6) = Y + Y Au + Y Au2 + Y v +
-Y
1 v3 + -2Yvrr1 2 v66vr2 + -v62 2' "
0 U UU v 6 vvvY vAu + y vEu2 + Y 6 + 1,Y 63 + 2,Y 6v2 + .-Y 6r2
2 6vv 2 6rr vu
2 vuu6
6 666y6Au + lY
6Au2 + y vr6. du 2 6uu vrS 2 1 f3 (u3 v r 6) = N + N Au + N0 Au2 + N v+ 1N
v3 + 1 -N vr + -N v62 U11
UU v 6 vvv 2 vrr 2 v66NvuvAu + -N2 vAu- + (N - mXu)r + -6N rrrr3 + -1N rv2
vuu G 2 rvv (5) 1
+-Nr66r62 + N
rAu + 1-N r8u2 + N66 + 1,N66 63 + 1-N6 6Au2 ru 2 ruu 6 2 Ouu + Nvr6vr6 + 1-N
6v2 +IN
or2 + Ndu6Au. 2 6vv 2 Orrlie linear model was used in designing the controls, and for all simulations nvolving small perturbations. The nonlinear model was used in those simula-ions involving large perturbatsimula-ions.
iBSIGN FOR STABILIZATION
In order to stabilize the unstable hull, a feedback scheme was chosen as shown in Fig. 1. We wished to reposition the two roots at approximately the same locations as for the stable hull, and thus we did not feed back T (which would have introduced a third root), but we did feed back both r and i to provide better adjustment of root locations.
From Fig. 1 the characteristic equation of the stabilized model is
2 1.3051K1 + 1.965K2 - 2.624 1.965K1 + 0.7373 s
+
s = 1.30511( + 2 - 1 1.30511(2 -0 1the characteristic equation of the stable hull is s2 + 3.4363s + 1.002 = 0
Equating coefficients and solving simultaneously K = -1.0232
K2 = -0.207472
these feedback gains provide the stabilized system with exactly the same roots as those of the stable hull.
SIMULATION TESTS OF THE LINEAR MODEL
From Fig. 1 and from the algebraic relationships it is clear that the stabi-lized model, while possessing a characteristic equation with the same roots as the stable hull, does not have the same system gain or zero. Furthermore, from the block diagrams, a command to the stable hull is directly a rudder
de-flection, but for the stabilized model the rudder action is further modified by the feedback. Thus we would not expect precisely the same behavior from the two systems, even though their roots are the same, and we note in particu-lar that the rudder action must be checked to see that acceptable limits are not exceeded.
To obtain a convenient mathematical form for simulation we rewrite eqn. (lb) and
(lc) as:
AS + Bv + Ci + Dr + Y6 = 0 (le)
ES + Fv + Gi + Hr + N6 = 0 (10
where the coefficients are defined in Table III. By manipulation we obtain the transfer function:
(As + B)N - (Es + F)Y K(s + J)
6(s) (Cs + D)(Es + F) - (As + B)(Gs + H) .2 .4_ I. 4_
where
K = (AN - EY)/(CE - AG) J = (BN - FY)/(AN - EY) I = [(CF + DE) - (AR + BG)]/(CE -AG) M = (OF - BH)/(CE - AG) Numerical values of these constants are given in Table IV.
The transfer function of eqn. (6) may be simulated in the computer as shown in Fig. 2a. Addition of the feedback signals to stabilize the unstable hull results in Fig. 2b. Note that the simulation method permits us to generate the i signal without use of a differentiator. However, Fig. 2b contains an implicit loop (see dotted lines) which the computer cannot solve, so we must add a storage block that permits computation without noticeably affecting the solution. This is done by inserting a remote pole as shown in Fig. 2d. For automatic steering control feedback of the heading is added as shown in Figs. 2c and 2d.
The linear models were subjected to "change of heading" commands and to the zig-zag maneuver (even though the latter is not a small perturbation). Responses of models A and compensated C were essentially the same, except for rudder de-flections. These were somewhat greater for compensated model C, but remained with normal limits as to both deflection angle and ratio [2].
The nonlinear models were also tested using the zig-zag maneuver and a "360°
turn" test. The zig-zag test again showed good agreement between the models except for differences in rudder motion. The time history of the rudder motion for stabilized model C is shown in Fig. 3.
The following assumptions and steps were made and followed in devising the 3600 turn" test:
The simulation time units were taken to be the nondimensionalized time
units.
From t' = 0 to t' = 1 the two ships were moving with constant speed on a straight line. During that interval Y; = = O.
At time t' = 1 a command for a 200 rudder position was given to both ships. The rate of change of rudder deflection was 400 per nondimensionalized time unit.
