Motion stability and control in marine surface vessels
PRINCIPLES OF MOTION STABILITY AND CONTROL OF
MARINE SURFACE VESSELS
Our sincere thanks to Engineer Officer S.K. SARKAR for his support
and invaluable guidance.
1. ANSHUMAN GUHA (5374)
2. HIMANSHU SINGH (5396)
3. PARIJAT SINHA (5429)
4. PIYUSH BUGALIA (5430)
5. ROHIT SHARMA (5444)
6. TARUN AGARWAL (5460)
7. VISHAL ANAND (5467)
8. S.S. MISHRA (5470)
This paper discusses the theory involved in the mathematical modeling of ship motions
and stability, and the recent developments in the control systems being used on modern
day marine surface vessels. The problems of heading control and roll stabilization have
been dealt with and three-term control systems for the same have been discussed in
detail. The newer methods and algorithms for motion control have been discussed
theoretically, but details have been avoided as the mathematical modeling of the same is
beyond our scope. Ship simulation software “GNCtoolbox” developed using MATLAB
6.5 by Prof. Thor I. Fossen has been used to obtain the results for motion stability and the
codes for the same are given in the appendix. Global Positioning System, the modern day
navigational aid, has been discussed in brief at the end.
The most distinguishing feature between shore based vehicles and marine vessels is the
factor of unpredictability. While most shore based transport systems can be simulated
mathematically and analytically because of their predictable behavior, the inherent
randomness in the behavior of oceans makes it difficult to simulate ship’s motion
Generally, for well defined systems, the desired system state can be achieved much more
readily and reliably using automatic control, with the advantage that the automatic
controller will not tire, need relieving, or be erratic in its performance. However, in
systems which are not well defined or understood, the performance of the automatic
controller may be totally inadequate.
The motion control problem for a ship can be classified broadly into two parts:
• The problems of heading control and track-keeping (which is an advanced type of
heading control problem) are the ones which more often than not involve a human
element in the control chain. Most autopilots are not well able to steer a ship
down a heavy quartering sea. This is because the disturbances to the system are
large and can be unpredictable.
• Dynamic positioning and roll control are two problems whose solutions lie in
modification of ship design and development of efficient control systems. The
main reason for this kind of demarcation is that the latter two require multiple
inputs (such as wind speed, wave velocity, roll rate etc coming from different
sensors) to be processed accurately and precisely, as against the heading control
problem which can be and has been, efficiently addressed by human operators
(relying mainly on their senses).
The requirement to be able to produce a desired state or condition of a ship has resulted
in the definition of ship controllability used by the 14th International Towing Tank
Conference (ITTC), 1975:
Controllability is ‘that quantity of a ship which determines the effectiveness of the
controls in producing any desired change at a specific rate in the attitude or position of
the moving ship’.
The above definition accentuates the effectiveness of controls. The fact that
environmental conditions are not mentioned implies the control should be effective
throughout a reasonable range of environmental conditions. Thus a ship which cannot be
adequately controlled in high winds or will not maneuver astern cannot be said to be
adequately controllable, even if its behavior in calm conditions going ahead is
The motion control problem
The definition of controllability as underlined by the ITTC conference and mentioned
earlier in the introduction to this paper gives just a vague idea of controllability of a ship.
Neither the ship operators, nor the ship owners have been able to specify in a meaningful
manner what aspects of controllability are required.
There are two international bodies which do, however, have some interest in the matter.
The International Maritime Organization has a standing committee on ship
maneuverability, which is producing a set of guidelines, and the maneuverability panel
H10 of the US society of naval architects and marine engineers (SNAME) has worked in
The last named panel surveyed a large number of pilots in an attempt to obtain an opinion
on those aspects of a ship’s maneuvers which were most desirable from a controllability
point of view. The most significant factors listed were:
• Slow speed maneuverability (86% of pilots);
• Adequate backing power and straight line stopping ability (79%);
• Short response time following rudder or engine commands (77%);
• Adequate swing control with moderate rudder angles (66%).
Any quantitative measure of controllability must therefore contain elements to assess
each of the above quantities if an adequately controllable ship is to be achieved as far as
the users are concerned.
Looking from the aforementioned perspective, the motion control problems can be
effectively studied under the following subheadings:
• Heading control problem
• Track keeping problem
• Dynamic positioning problem
• Roll control problem
Heading control problem
The two most basic controls required in a ship are those associated with control of the
direction or heading of the ship, and of its speed. For many ships, these are the only
navigational controls available.
Single screw, single rudder ships predominate in the deep sea trades, for obvious
economical reason, and the control arrangements on the bridge consist of a single helm
control, assisted by an autopilot, and a single engine control lever, which may operate in
either bridge control, where the desired shaft speed is set by the position of the lever, or
engine room control, where the lever position is transferred to the engine control room
for action to be taken there.
The basic requirement for heading control is that the ship’s head is maintained to within a
given band of the desired value. The size of the band, or steering error, will depend in
• the dynamic properties of the ship,
• the effectiveness of the steering arrangements,
• the disturbances present (wind and waves), and
• the perceived requirements of the officer of the watch, which will in turn vary
with his assessment of the navigational situation.
Ship steering process
The process of steering a ship is shown diagrammatically in the block diagram. The
difference between the desired course and the actual course is assessed. This may be done
in a number of ways, depending on how the ship is being steered. If under helmsman
control, the helmsman may be given instruction to steer a compass course (i.e. a course
relative to the earth’s north-south lines of longitude), or may be given instruction to head
towards a fixed object, or simply to keep in the middle of the channel. If the ship is in
autopilot control, a desired course will have been set on the autopilot. In either case, the
difference between the desired and the actual course of the ship is defined as the heading
The helmsman or autopilot will act on this error, and will alter the demand to the rudder
control mechanism. This signal is the desired rudder angle. In most large ships, this
action will be in the form of a signal to an amplifier or to a servo valve in a hydraulic
control mechanism. The result of the control action is that the rudder will after a time lag,
assume the value of the desired rudder angle, within the bounds of the error of the control
The rudder will then (usually) act on the slipstream of the propeller, and create a turning
moment (angle of attack, angular velocity) on the ship which will turn the ship in the
appropriate direction. As the ship turns, the error will reduce and eventually the ship will
assume a heading approximately equal to the desired heading at which time the error is
zero. As the link between the rudder angle and the ship’s behavior is complex, the ship
will generally not stay at the desired heading, unless a continuous control action is
applied. The nature of the control action will in turn depend on the ship’s design and
The working of rudders: an overview
The rudder or rudders are usually placed at the stern of the ship, immediately aft of the
propellers. The reasons for fitting the rudders at that position are concerned with the
effectiveness of the controls. The ship will turn under the combined influences of rudder,
inertial and hydrodynamic forces, about a point which will usually be some distance
forward of the mid-point of the ship. In some ships this pivot point, which may be
defined as the point at which there is no sway velocity, is situated some distance forward
of the ship.
A rudder situated at the stern of the ship has three main effects:
• It is able to exert a large lever arm about the pivot point.
• It is able to position the ship such that the hydrodynamic forces assist in the turn.
Some ships are able to turn effectively going astern, when the turning effect is
largely caused by the hydrodynamic forces on the rudder alone, but the turning
ability of a ship is very much enhanced if the hull hydrodynamic forces augment
the rudder forces.
• It can be so positioned that the propeller slipstream augments the flow of water
over the rudder. As rudder forces are heavily dependent on the velocity of flow
across them, the effectiveness of the rudder is enhanced by this positioning.
The effect of disturbances on course-keeping requirements
Disturbances act on a ship as shown in the steering control flow chart. The most common
form of disturbances will be the presence of wind and waves, but other effects which will
affect the course control behavior are:
• The presence of sea bed; ships behave differently in shallow water than they do in
deep water. Turning ability is reduced and the diameter of turn increased.
• The presence of banks and other ships; ships tend to turn away from banks and
there are complex interactions between two ships passing close to each other. In
certain circumstances, a small ship can be uncontrollably drawn under the bow of
a larger ship as a result of interaction forces.
The way in which the steering ability of the ship is affected by the presence of the wind
depends on the shape of both the above water and underwater hull, and on the strength
and direction of the wind. The effect of waves will generally be to reduce the
effectiveness of the control mechanisms and to make steering a course with a given
margin more difficult.
Position control- the track keeping problem
As the aim of most voyages is to get the ship and its cargo from one point on earth’s
surface to another, the track keeping problem is central to the operation of most ships
most of the time. However, for most practical purposes, the track keeping problem can be
reduced to one of course keeping, in that the effects of the disturbances which will tend to
prevent the correct track being steered can be allowed for the accuracy required to ensure
that the ship arrives at the correct destination.
For more exacting applications, however, there is a requirement for a more precise track-
keeping. Examples of this enhanced requirement can be found in survey ships and boats,
in minesweepers, in dredgers and in offshore supply vessels, mobile drilling rigs, etc. All
these vessels need to be able to keep a fixed track rather than a fixed course, within
clearly defined limits. A ship at the end of a voyage will also have to assume a track
keeping role, as it must navigate an approach channel and enter a lock or dock, often with
under 1m (3.3 ft) clearance.
The principle of track keeping is similar to that of course-keeping, in that an error is
defined, and the ship’s controls operated in a manner so as to reduce the error ideally to
zero, but more usually to within a satisfactory limit. The principle is shown in the figure
Desired Track rudder
Track error Track- angle
keeper Steering Ship
+ - Automatic Desired
A reference track is defined, which is fixed relative to the sea bed or in relation to shore
features. The distance from the reference track is determined, either by eye or
mechanically and this distance is used as part of an error signal. Other components of the
error signal might be the yaw rate of the ship and its heading. Decisions on the control of
the ship are made as a function of the track error, as modified by the other inputs, and
signals sent to the rudder control and engine control mechanism.
Roll control problem
Because of the underwater shape of a ship’s hull, which is designed to cause minimum
resistance to motions consistent with adequate cargo or weapon carrying capacity, all
ships will roll to a greater or lesser extent, under the action of wave. This rolling effect is
entirely deleterious to the effective performance of the ship’s role. Among the problems
which occur due to excessive roll action are:
1. Loss of speed for a given power output.
2. Loss of efficiency of ship staff, either directly because of the motion and the
necessity to hang on, or because of actual seasickness.
3. Loss of commercial effectiveness because of the necessity to secure cargo or ship
equipment firmly. An obvious example is the requirement to secure each of
perhaps 100 vehicles for a short sea crossing on a Ro-Ro ferry.
4. Loss of commercial attractiveness of a ship caused by unpleasant motion.
Customers will not wish to pay for a cruise if they are to be sick all the time.
5. Damage to ship equipment or cargo caused solely by rolling.
6. Complete loss of the ship if rolling becomes too severe, or if cargo shifts.
7. Loss of accuracy and effectiveness of warship weapon systems caused by
excessive roll motion.
8. Reduction in operational capability of a warship because it is unable to operate its
helicopters due to excessive roll.
It is not surprising therefore that considerable attention has been paid over the past
century to reducing the extent of roll motion in a ship. Because a ship’s hull is between
five and ten times as long as it is wide, the equivalent problems associated with pitch are
much less severe.
The basic methods of achieving roll reduction are concerned with three main methods of
1. altering the basic hull design so that it does not roll excessively,
2. adding devices to control the roll motion, and
3. operating the ship to reduce its propensity to roll.
The Dynamic Positioning problem
In a number of cases associated with the offshore industry, there is a requirement for a
ship or platform to maintain its position relative to a fixed datum on the sea bed. This is a
special case of track keeping, where the reference track is a single point. Because of the
accuracy requirements for dynamic positioning (DP), a number of specialist devices are
in use for both identifying the positional error and controlling the ship motion.
Position information is obtained from a range of sources. A taut wire may be suspended
from the ship to the sea bed, and its angle from the vertical used to convey information
about the position of the vessel relative to its datum.
Acoustic information may be obtained from reference transponders attached to the sea
bed, and Doppler sonar information can be used to give velocities in both fore and aft and
athwartships direction relative to the sea bed.
The error signal for position is then calculated to produce an optimal vessel response to
get the ship into the correct position. A wider range of effectors is employed in drill ships
and offshore supply vessels, including fixed pitch propellers in nozzles, controllable pitch
propellers, rotatable thrusters and fixed thrusters.
Definition of Motion Stability
The concept of path keeping is strongly related to the concept of stability. A body is said
to be stable in any particular state of equilibrium in rest or motion if, when momentarily
disturbed by an external force or moment, it tends to return, after release from the
disturbing force, to the state of equilibrium existing before the body was disturbed. In the
case of path keeping, the most obvious disturbing force would be a large wave or a gust
of wind. For optimum path keeping, it would be desirable for the ship to resume its
original path after passage of disturbance, with no intervention by the helmsman.
Whether this will happen depends on the kind of motion stability that the ship possesses.
The various kinds of motion stability associated with ships are classified by the attributes
of their initial state of equilibrium hat are retained in the final path of their centers of
gravity. For example, in the figure given below, in all the cases, the ship is initially
assumed to be traveling at constant speed along the same path. In case 1 the final path
after release from a disturbance retains the straight line attribute of the initial state of
equilibrium, but not its direction. This type of stability is termed straight-line stability. In
case 2 the final path after release from a disturbance retains not only the straight-line
attribute of the path, but also its direction. This is termed as directional stability. Case 3 is
similar to case 2 except that the ship does not oscillate after the disturbance, but passes
smoothly to the same final path as case 2. Finally, in case 4 the final path of the ship not
only has same direction as the original path, but also its same traverse position relative to
the surface of the earth. This might be termed positional motion stability.
The foregoing kinds of stability have been defined in a kind of ascending order. A ship
that is directionally stable must perforce also possess straight line stability. It can be
shown that a straight line stability or stability is indicated by the solution to a second
order differential equation, the directional stability or instability is indicated by the
solution to a third order differential equation, and finally positional stability or instability
is indicated by the solution to a fourth order differential equation.
Stability with Controls Fixed and Controls Working
All of the foregoing kinds of stability have meaning with control surfaces (rudders) fixed
at zero, with control surfaces free to swing, or with controls either manually or
automatically operated. The former two cases involve only the last two elements of
control loop of the figure give below, whereas the latter cases involves all of the elements
control loop. In ship and submarine usage the term stability usually implies controls-fixed
stability; however, the term can also have meaning with the controls working. The
following examples will indicate the distinctions:
• A surface ship sailing in the calm sea possesses positional motion stability in the
vertical plane (and therefore directional and straight line stability) with controls
fixed. This is an example of the kind of stability shown by case 4 of the above
figure. In this case, hydrostatic forces and moments introduce a unique kind of
stability which in the absence of these forces could not be introduced either by
very sophisticated automatic controls or by manual control. The fact that the ship
operator and the designer can take for granted, this remarkable kind of stability
does not detract from its intrinsic importance.
• In the horizontal plane in the open sea, a self propelled ship cannot possess either
positional or directional stability with controls fixed because the changes in
buoyancy that stabilize in the vertical plane are non existent in the horizontal
plane. However, a ship must possess both of these kinds of stability with controls
working either under automatic or manual guidance.
• The only kind of motion stability possible with self-propelled ships in the
horizontal plane with controls fixed is straight-line stability. This kind of stability
is desirable but not mandatory. In fact, many ships do not possess it.
With each kinds of control-fixed stability, there is associated a numerical index which
by its sign designates whether the body is stable or unstable in that particular sense
and by its magnitude designates the degree of stability or instability. To show how
these indexes are determined, one must resort to differential equations of motions.
Also ships equations are required for the number of other purposes also, such as:
• To enable the motion of the ship to be studied for primary research purposes;
• As an aid to ship hull design;
• To assist the design of thrust and control surfaces;
• To represent the ship in a range of simulators, for research, design and
LINEAR EQUATIONS OF MOTION
Axis fixed relative to the earth
Figure shown above is self descriptive about the different axes and the motion of the
vessel. The motion of the ship subsequent to time, t=to, is completely defined by the
coordinates xoG, yoG the angle of yaw ψ. In terms of these axes fixed in the earth, the
Newtonian equations of motion of the ship are:
X 0 = m&&0G
Y0 = m&&0G − − − − − − − − − − − − − − − −(1)
N = I zψ&
Where the two dots above the symbols indicate the second derivatives of those values
with respect to time, t, and,
Xo &Yo = total force in xo & yo direction respectively
m = mass of the ship
N = total moment about an axis through centre of gravity of ship and parallel to
Iz = mass moment of inertia of ship about axis just mentioned
Ψ= yaw angle in the horizontal plane measured from the vertical xo zo plane to the x-
axis of the ship.
Axes fixed in the ship
In spite of apparent simplicity of equation (1), the motion of a ship is more conveniently
expressed when referred to the axes x and y fixed with respect to the moving ship as
shown in figure. Now the origin, O, stays at the C.G. of the ship. In particular case shown
in the above figure, both ψ and β are negative.
In order to convert equation (1) from axes fixed in the earth to axes fixed in the moving
ship, the total forces X and Y in the x & y directions, respectively, are expressed in terms
of Xo and Yo:
X = X 0 cosψ + Y0 sinψ
Y = Y0 cosψ − X 0 sinψ − − − − − − − − − −(2)
x0G = u cosψ + v sinψ
y 0G = v cosψ − u cosψ − − − − − − − − − − − (3)
&&0G = u cosψ − v sin ψ − (u sin ψ + v sin ψ )ψ
x & &
&&0G = u sin ψ + v cosψ + (u cosψ − v sin ψ )ψ − − − − − − − (4)
y & &
Substituting equation (4) in equation (1) and inserting the resulting values of Xo and Yo
in equation (2) yields the simple expressions:
X = m(u − vψ )
Y = m ( v + uψ )
X = m(u − vψ ) − − − − − − − − surge
Y = m(v + uψ ) − − − − − − − − sway − − − − − −(5)
N = I zψ& − − − − − − − − − − − yaw
Note the existence of the term muψ in the equation of Y and mvψ .In the equation for
X, whereas terms like these are not present in equation (1). These are the so-called
centrifugal force terms which exist when systems with moving axes are considered, but
do not exist when the axes are fixed in the earth.
Equations (5) have been developed for the case where the origin of the axes, O, is at the
C.G. of the ship. Suppose we chose an origin, O, which is located a distance RG from the
CG of the ship, where RG has components xG, yG and zG along the x,y and z axes which
are parallel to the principal axes of inertia through G. xG will be positive if the CG is
forward of the origin and negative if it is aft. Similarly yG will be positive if G is to
starboard of O and zG will be positive if G is below O. Abkowitz has shown that for the
choice of position for the origin, equations (5) become:
X = m(u − ψv − y Gψ& − xGψ 2 )
Y = m(v + uψ − y Gψ 2 + xGψ&) − − − − − − − − − − − − − − − − − (5a )
N = I zψ& + m[ xG (v + uψ ) − y G (u − ψv)]
& & &
Because equations (5a) describe motions in horizontal plane only, the vertical distance zG
does not appear in the equations.
Assumptions of Linearity and Simple Addable Parts
Expressed functionally X, Y and N are:
X = Fx (u , v, u , v,ψ ,ψ&)
& & &
Y = Fy (u , v, u , v,ψ ,ψ&) − − − − − − − − − − − − − −(6)
& & &
N = Fψ (u, v, u , v,ψ ,ψ&)
& & &
In order to obtain a numerical index of motion stability, the functional expressions shown
in equation (6) must be reduced to useful mathematical form. This can be done by means
of the Taylor expansion of the function of several variables.
The linearized form of the Taylor expansion of function of two variables x & y is a
simple sum of three linear terms as follows:
∂f ( x, y ) ∂f ( x, y )
f ( x, y ) = f ( x1, y1) + ∆x + ∆y …………………….. (7)
Where both x & y must be small enough so that higher order terms of each can be
neglected as well as the product x y.
Motion stability determines whether a very small perturbation from an initial equilibrium
position is going to increase with time or decay with time. Thus, it is consistent with the
physical reality of motion stability to use the linearised Taylor expansions in connection
with equation (6). For example, by analogy with equation (7), the linearized Y-force of
equation (6) can be written as:
∂X ∂Y ∂Y
Y = Fy (u1 , v1 , u1, v1 ,ψ 1,ψ&1 ) + (u − u1 )
& & & + (v − v1 ) + ......... + (ψ& − ψ&1 )
& & − − − − − − − −(8)
∂u ∂v ∂ψ&
Where the subscript 1 refers in all cases to the values of the variables at the initial
equilibrium condition and where all the partial derivatives are evaluated at the
equilibrium condition. Since the initial equilibrium condition for an investigation of
motion stability is straight line motion at constant speed, it follows that
u1 = v1 = ψ 1 = ψ&1 = 0 . Furthermore, since most ships are symmetrical about their xz-
& & & &
plane, they travel in the straight line at zero angle of attack; therefore v1 is also zero. Also
because of symmetry = = 0 since a change in forward velocity or forward
∂u ∂u &
acceleration will produce no transverse force with ship forms that are symmetrical about
the xz plane. Finally, if the ship is in fact in equilibrium in straight line motion, there can
be no Y force acting on it in that condition. Therefore f (u1, v1, u1, v1,ψ 1,ψ&1) is also zero.
& & & &
Only u1 is not zero but is equal to the resultant velocity, V, in the initial equilibrium
condition. With these simplifications, equation (8) reduces to
∂Y ∂Y ∂Y ∂Y
Y= v+ v+
& ψ& − − − − − − − − − − − − − (9a )
∂v ∂v& ∂ψ& ∂ψ&
And similarly the surging force and yawing moment can be written as:
∂X ∂X ∂X ∂X ∂X ∂X
X = u+
& ∆u + v+ v+
& ψ& − − − − − − − −(9b)
& ∂u ∂v ∂v& ∂ψ & ∂ψ&
∂N ∂N ∂N ∂N
N= v+ v+
& ψ& − − − − − − − − − − − − − − − −(9c)
& ∂ψ& ∂ψ&
∂Y ∂Y ∂N ∂N
Where the cross coupled derivatives , , and usually have small nonzero
∂ψ ∂ψ& ∂v
& & ∂v
values because most ships are not symmetrical about the yz-plane even if that plane is at
the midlength of the ship. However, the cross coupled derivatives
∂X ∂X ∂X ∂X ∂Y ∂Y
, , and like and are zero because of symmetry about the xz-plane.
∂v ∂v ∂ψ
& & ∂ψ&
& ∂u ∂u&
Hence, equation (9b) reduces to:
X = u+
& ∆u − − − − − − − − − − − − − −(9d )
∆u = u − u1
For the sake of consistency, before combining equations (9) with equations (5) or (5a),
those equations should also be linearized. If the CG of the ship is in its longitudinal plane
of symmetry, then yG is zero and equations (5a) reduce to:
X = m(u − ψv − xGψ 2 )
Y = m(v + ψu + xGψ&) − − − − − − − − − − − − − − − −(5b)
N = I zψ& + mxG (v + ψu )
Linearization of the expression in the parentheses for Y is performed as follows:
v + ψu + xG = (v1 + ∆v) + (ψ 1 + ∆ψ )(u1 + ∆u ) + xG (ψ&1 + ∆ψ&)
& & & & & & & &
v1 = ψ 1 = ψ&1 = 0
& & &
v + ψu + xGψ& = ∆v + u1 ∆ψ + ∆ψ∆u + xG ∆ψ&
& & & & & & &
The term ∆ψ∆u is second order and must be dropped since similar second order terms
have been neglected in developing equation (9). Since ∆v = v − v1 = v, ∆ψ = ψ − ψ 1 = ψ
& & & & & & & &
and so on, the preceding expression reduces to:
v + ψu + xGψ& = v + ψu1 + xGψ&
& & & & & &
Lenearizing the expressions for X and N of equation (5b) in a similar manner leads to the
following summary result:
X = mu&
Y = m(v + ψu1 + xGψ&) − − − − − − − − − − − − − − − − − − − (5c)
& & &
N = I zψ& + mxG (v + ψu1 )
Notation of Force and Moment Derivatives
In the simplified derivative notation various terms can be written as = Yv , = Nψ&
and so on. Also for motions restricted to the horizontal plane ψ ≡ r , and ,ψ& ≡ r using this
& & &
notation and substituting equations (9) into equations (5c), the equations of motion with
moving axes in the horizontal plane are:
− X u (u − u1 ) + (m − X u )u = 0
− Yv v + (m − Yv& )v − (Yr − mu1 )r − (Yr& − mxG )r = 0 − − − − − − − − − −(10)
− N v v − ( N v& − mxG )v − ( N r − mxG u1 )r + ( I z − N r& )r = 0
Every term of first two equations of (10) has the dimensions of force whereas every term
in the third equation of (10) has the dimensions of a moment. Therefore, to
nondimensionalize equations (10), which are convenient for several reasons, the force
equations are divided through by L2V 2 and the moment equations by L3V 2 .
m′ = ; v′ =
& ; v′ =
ρ V 2
Iz rL rL2
z ; r′ = ; r′ = 2
ρ V V
Yv Yr Nv Nr
Yv′ = ; Yr′ = ;N ′v= ; N r′ =
ρ ρ ρ ρ
L2V L3V L3V L4V
2 2 2 2
Yv& Yr& N v& N r&
Yv&′ = ; Yr&′ = ′
; N v& = ; N r′ =
ρ ρ ρ ρ
L3 L4 L4 L5
2 2 2 2
Nv v N vv
N v v′ =
′ = ;
ρ 3 V ρ 3 2
N v& vL
N v& v
N v& v ′ =
′& 2= ;
ρ 4 V ρ 3 2
G vL =
m′xG v ′ =
′& 2 , etc
ρ L3 L V ρ L3V 2
If the surge equation is neglected and if the previous notation is adopted, equation (10)
becomes in non-dimensional form:
− Y ′vv ′ + (m′ − Y ′v)v ′ − (Y ′r − m′)r ′ − (Y ′r − m′x ′G )r ′ = 0
& & & &
− N v v ′ − ( N v& − m′xG )v ′ − ( N r ′ − m′xG )r ′ + ( I ′ − N r′ )r ′ = 0 − − − − − (11)
′ ′ ′ & ′ z & &
Where the main difference between (10) and (11), aside from the prime notation, is that
u1 is disappeared since u1/V=1 for small perturbations.
Because of the fact that the derivative Yv&′ enters into equation (11) as an addition to the
mass term, it is termed the virtual mass coefficient. It is thus identical to the concept of
“entrained” or “added” mass. Similarly, N r′ is termed the virtual moment of inertia
coefficient. The derivatives Yr&′ and N v& are termed the coupled virtual inertia and the
coupled moment of inertia coefficients, respectively. It should be noted that these
derivatives would be zero if the ship hulls, including their appendages, were symmetrical
about their yz-planes.
Control Forces and Moments
It is important to note that for controls-fixed stability, all of the terms of equation (10) or
(11) must include the effect of ship’s rudder held at zero degrees. On the other hand, if
we want to consider the path of a ship with controls working, the equations of motion
(10) or (11) must include terms on the right-hand side expressing the force and moment
created by rudder deflection as functions of time. These are the control force and
moment. The linearized y-component of the force created by rudder deflection acting at
the CG of the ship is Yδ δ and the linearized component of the moment created by rudder
deflection about the z-axis of the ship is N δ δ where
δ = rudder deflection angle, measured from xz-plane of the ship to plane of rudder;
positive deflection corresponds to a turn to port for rudder(s) located at stern
Yδ , N δ = derivatives of Y and N with respect to rudder-deflection angle δ
For the case of small perturbations, which is the only case where equations (10) and (11)
apply, only small deflections of the rudder are admissible. With this restriction the
derivatives such as Yv′, N v , Yr′, N r′ and so on are evaluated at δ = 0 and are assumed not to
change at other admissible values of δ
With these assumptions the equations of motion including the rudder force and moment
are as follows:
− Y ′vv ′ + (m′ − Y ′v)v ′ − (Y ′r − m′)r ′ − (Y ′r − m′x ′G )r ′ = Yδ′δ
& & & &
− N v v ′ − ( N v& − m′xG )v ′ − ( N r ′ − m′xG )r ′ + ( I ′ − N r′ )r ′ = N δ δ − − − − − (11)
′ ′ ′ & ′ z & & ′
Although not written in functional form, it is clear that the quantities v ′, r ′, v ′, r ′andδ in
equations (10), (11) and (11a) are all functions of time, t.
In the case of the path-keeping ability of ships in rough water, the right hand side of
equation (11a) must, in addition to the rudder force and moment terms, include sea
excitation term which, for regular waves, is harmonic functions of time. However, here
we are considering only the motions of the ship after momentary disturbance has ceased
to act. For this purpose, no terms whatsoever need be considered on the right hand side of
the equations (10) and (11).
The full derivation of the criterion equation for stability is beyond the scope of this paper.
The stability criterion, C, can be expressed simply as
C = Y ' v ( N ' r − m ' x ' G ) − N ' v (Y ' r − m ' ) > 0 .......... .(12 )
According to this equation, stability is improved as (Y ' r − m ' ) increases in positive value,
and as ( N ' r − m ' x ' G ) increases in negative value.
Course-Keeping with Automatic Control
One of the functions of ship control is to maintain a ship’s heading. In performing this
function, a helmsman deflects the rudder in a way which will reduce the error between
the actual and desired heading, designated as ψ on the figure below.
Since the actual heading angle can be determined by means of a compass, the magnitude
of ψ can be readily displayed to the helmsman. A good helmsman will not only deflect
the rudder in response to the heading error,ψ , but he is also sensitive to the angular
velocity of the ship, ψ (≡ r ) , and he will ease of the rudder and apply a little opposite
rudder in order to prevent overshooting the desired heading. It follows that an automatic
pilot (autopilot) should also be responsive to control signals measuring bothψandψ .&
Thus, a rudder under automatic control, might be deflected in accordance with the
following linear expression:
δ = k1ψ + k 2ψ ……………………………… (13)
Where δ ,ψandψ all are functions of time where k1 andk 2 are the constants of
proportionality of the control system.
Both k1 andk 2 should be positive for proper control. Substituting equation (13) in
equations of motion (11a), the following equations are obtained:
Yv′v ′ + (Yv&′ − m′)v ′ + k1Yδ′ψ + (Yψ′ − m′ + k 2Yδ′ )ψ ′ + (Yψ′& − m′xG )ψ& ′ = 0
& & & ′ &
N v v ′ + ( N v& − m′xG )v ′ + k1 N δψ + ( Nψ − m′xG + k 2 N δ )ψ ′ + ( Nψ& − I z )ψ& ′ = 0
′ ′ ′ & ′ ′ ′ ′ & ′& ′ &
Equations (14) are simultaneous differential equations of the first order in v and of
second order inψ . The solutions of these equations for vandψ yields a third order
differential equation which as discussed earlier leads to the concept of directional
stability or instability.
The equations of motion with automatic control, equation (14), differ from the equations
of motion with controls-fixed at δ = 0 , equation (11), in two major respects. Equation
(14) implies a sensitivity to the orientation of the ship,ψ , which is absent in equation
(11). This is, of course, implicit in the concept of directional stability as opposed to
straight-line stability. Secondly, two of the terms which appear in the criterion, C,
equation (12), for controls-fixed, straight-line stability, are altered by the presence of the
controls. The former term (Yr′ − m′) now appears as (Yr′ − m′ + k 2Yδ′ ) and what was
formerly ( N r′ − m′xG ) now appears as ( N r′ − m′xG + k 2 N δ ) . Thus, the second effect of
′ ′ ′
automatic controls is to make the ship behave as if it possessed different values of its
inherent hydrodynamic derivatives. It follows that a ship that is unstable with controls-
fixed can be made to be stable in terms of both straight-line and directional stability by
the use of automatic controls. It is preferable that all ships be stable with controls fixed
and that automatic controls should not be used to overcome the inherent stability of a ship
but rather to provide it with directional in addition to straight-line stability.
It is evident that there are lags in control system. Therefore although ψandψ may be
measured and signaled to the autopilot at time t, it requires finite time, t , for the rudder to
reach the deflection angle prescribed by equation (13). Hence the deflection of rudder, δ ,
at time t is proportional to ψandψ at an earlier time t1 = t − t . In functional form,
equation (13) becomes:
δ (t ) = k1ψ (t − t ) + k 2ψ (t − t ) ………………….. (15)
where t is the time lag of the control system.
Following equation (7) the linearized form of the Taylor expansion of equation (15) is
δ (t ) = k1 [ψ (t ) − tψ (t )] + k 2 [ψ (t ) − tψ&(t )]
& & &
Nondimensionalizing this equation, substituting it in equation (11a), and dropping the
functional notation (t) which is implicit in equation (11a), the following is obtained:
Yv′v ′ + (Yv&′ − m ′)v ′ + k1Yδ′ψ + (Yψ′ − m′ + k 2Yδ′ − k1 t ′Yδ′ )ψ ′ + (Yψ′& − m′xG − k 2 t ′Yδ′ )ψ&′ = 0
& & & ′ &
N v v ′ + ( N v& − m′xG )v ′ + k1 N δψ + ( Nψ& − m′xG + k 2 N δ − k1 t N ′δ )ψ ′ + ( Nψ& − I z − k 2 t ′N δ )ψ&′ = 0
′ ′ ′ & ′ ′ ′ & ′& ′ ′ &
Where t ′ = (t )(V / L)
Again comparing with criterion, C, equation (12), it is noted that the term (Yr′ − m′) now
appears as (Yr′ − m′ + k 2Yδ′ − k1 t ′Yδ′ ) and the term ( N r′ − m′xG ) appears as
( N r′ − m′xG + k 2 N δ − k1 t ′N δ ) .
′ ′ ′
Two important facts emerge from these comparisons:
• The existence of the time lag, t , detracts from the stability of the ship compared
to zero time lag.
• If automatic controls were made sensitive only toψ , and not toψ , (k2=0) and a
time lag existed, the stability of the ships with controls would be less than
without controls. It is conceivable that this decrease in stability could cause a
ship that was stable without controls to become unstable with controls.
A more accurate and realistic, but much more complicate, analysis of the lags in the
control systems can be accomplished by writing the equations which describe the actual
operation of the various mechanisms involved in the system. For example, the equation
describing the build-up of voltage (or amperage) as a function of the quantitiesψandψ , &
the equations describing the actual method of amplification of the signal to produce the
power to activate the rudder motor, the equations describing the electromechanical
response of the electric motor activating the rudder system and the equations of motion of
the rudder system itself can all be written. These equations can then be coupled with the
ship motion equations and the overall response analyzed. The results will give a complete
test of the stability of the overall system, ship and controls. The controls themselves can,
as shown earlier, introduce instability into the system if they are not properly designed.
One solution proposed to remove the time lag from the system is to use some form of
ship-board sensor to determine the presence of disturbances like a large wave. This could
be achieved by some form of laser based device which would be aligned up weather, and
used to sense the presence of a wave or disturbance some seconds before it strikes the
Track Keeping and Position Control
The auto pilot is relatively a simple device, which relies entirely on ship borne sensors to
function. It has, however, a major disadvantage in that its output, the ship’s course, is not
actually required in many cases. The task of most ships is to travel in a predictable
manner along a path or track fixed relative to the earth’s surface, rather on a fixed course
through water. For most purposes, it is acceptable for the ship operator to make the
necessary calculations with respect to the external conditions to provide the desired
course, which is then used as the input to the autopilot.
For certain conditions, however it is necessary for the path over the sea bed to be kept
with greater precision than is feasible with autopilot alone. These applications include:
• Surveying. Clearly, if the water depth is to be recorded at a given point on the sea
bed, it is necessary to know with precision where the ship is at the moment the
depth reading is taken. Also, it is helpful if a known track is steered so that
readings are in reasonable line, so that a uniform coverage is taken.
• Minesweeping and Minehunting. It is vital for minesweeping operations for the
lanes to be known to be clear of mines. Therefore, a swept path must be
maintained relative to the seabed for this operation to be effective.
• Dredging. Similarly, it is necessary to know with degree of precision that the
channel has been dredged adequately. This requirement has assumed rather more
importance in recent years, with large ships regularly operating with very small
• Port approach. In some specialist operations, particularly those where poor
visibility is frequently encountered in conditions where the ship has a very small
clearance for the approach, additional assistance must be given to the operator.
This is a track-keeping operation, as the position of ship relative to the seabed is
• Offshore operations. It is in this area where many of the developments in the
field of track-keeping have arisen in the recent years. As many of the offshore
installations are fixed on the seabed, their support requires ships to keep station
relative to a rig for often prolonged periods while, for example, crane operations
transfer goods from the deck of an offshore supply vessel to the rig. The ability to
maintain a track consisting of a single point is a specialist application of track-
keeping, known as dynamic positioning and is discussed further in the paper.
• Underway replenishment. This is a particular form of track-keeping, where the
requirement is to keep the ship, usually a warship, a predetermined (small)
distance away from another ship, so that stores or fuel may be transferred. The
hydrodynamic forces between the ships and the fact that the store ship may not
be keeping a perfectly steady course exacerbate this problem.
Information requirements - Sensors
In these application, the need is same, to control the position of the ship relative to a
know fixed position relative to the sea bed or, in the case of underway replenishment,
relative to another moving ship. For this it is clearly necessary to know the present
position of the ship. The basic block diagram of a track keeping system is very similar to
that of an autopilot as shown in the figure given below, except that the primary
comparison is between the ship’s desired position and its actual position. Additionally,
many track-keepers will also control the engine performance.
error waves Position
keeper control Ship
Desired position angles angles
Desired Thrusters thruster
Desired speed Speed
error Controls Propeller
All system require as inputs information on the present position of the ship, so that this
may be compared with the desired position to obtain a system error. In some system, a
speed error will also develop. The methods for obtaining this information vary
extensively. One of the most promising methods is the global positioning system, based
on satellites, which can now give a position accurate to a few meters at worst. This may
be sufficiently accurate for many track-keeping tasks and has the advantage of being
contained within the ship.
Close into shore, a number of high precision systems may be used, based on infrared or
radio waves. These all require one or more shore stations to be set up, and so are most
useful for tasks where a ship a ship repeatedly carrying out the same role .Figure given
below shows a high accuracy positioning system used in a dredger owned by Associated
British Ports in South Wales.
The system produces a high definition display, showing the position of the dredger in
relation to the shore, with the dredged channel shown. The display is used only to give
information to the ship’s staff, with the conning of the ship being carried out in a
conventional manner, using helms and engine orders. However, the overall operational
methodology is the same as that shown in the figure shown above with a human in each
of the control loops.
For surveying operations, it may be feasible to set up position lines ashore (leading
mark) and to con the ship down the line thus defined, using normal conning commands.
This is a preferred method of harbor approach for many ships, with the advantage that it
is cheap and reliable. Leading marks cannot be used, however in case of poor visibility.
Radar can be used to fix position in relation to shore features, but can suffer from very
bad distortions close to bridges across a river.
Offshore vessels have particularly stringent requirement for position fixing in relation to
the sea bed or to a fixed structure such as a rig, and a range of systems can be used, based
on sonar, radar, inertial navigation and on a taunt wire attachment to the sea bed. Many
ships employ more than one of these systems.
One factor is common with most position measuring devices is that the data in noisy.
Almost all track-keepers using measured data rely on some form of smoothing or
estimation to obtain the most reliable data. The technique of Kalman filtering is
commonly used for this task.
For most track-keeping application, the ship’s normal complement of propellers and
thrusters is adequate, as the track-keeping may be thought of as simply a development of
the autopilot. For more severe requirements, special effectors are required to ensure the
vessel can maintain its designed station in a range of weather conditions. Figure below
shows the types of thruster commonly fitted aboard many offshore vessels.
Typically controllable pitch propellers are used, often sited inside fixed or azimuthing
nozzles to increase their effective thrust. Both now and stern thruster are used, frequently
in multiple units. Transverse thruster cannot be used effectively in multiple units.
Transverse thruster cannot be used effectively at ship speeds of over about 4 knots, as the
hydrodynamic flow past the nozzles destroys their effectiveness, and so rotatable
thrusters are sometimes fitted. Specialist vessels such as tugs can use thrusters consisting
of a series of vertical vanes, which move in cylindrical manner, such as the Voith-
The methods employed in track-keeping devices are similar to those used in autopilots,
although, as there is greater complexity in the control problem, there is greater variety in
control methods used. In general, some form of strategy is used to control the rudder
which incorporates the position error information as well as error information based on
course and rate of turn information. For example, the control function used to determine
the desired angle rudder angle for the prototype replenishment system developed by the
US Navy involved elements of the following quantities:
• Course error between the two ship;
• Rate of turn;
• Yaw acceleration;
• The distance off the replenishing ship;
• The lateral separation rate;
• The integral of separation distance;
Each of these quantities given an appropriate weighting, and the ruder angle calculated as
the weighted sum of these quantities.
It is useful, with track-keeping device used for assisting in navigating in channel, for
there to be some measure of prediction in the control algorithm, so that the ruder is
applied in the plenty of time before a bend in the channel. In a device used for designing
channel layouts, using simulation techniques, the track-keeping algorithm used the
concept of a ‘look ahead’ distance to evaluate the required ruder angle (figure shown
A point is defined some distance ‘a’ along a reference track, which may for example
be along the centerline of a desired channel. The aim of the track-keeper is to reduce to
zero the angle θ a between the ship’s head and the heading which would bring the ship on
to the designated reference track at the point distance ‘a’ along the track. Later
modification of this system defined the angle θ a as that between the ship’s current track
line and the heading which would bring the ship onto the designated reference track at the
point distance ‘a’ along the track. Control is achieved by a simple proportional and
differential controller acting on the rudder. The system also incorporates a controller to
achieve the desired speed along the ship’s track. The use of the look ahead concept is
analogous to the ship’s pilot anticipating the bend and getting the helm on in good time.
Varying the look ahead distance will represent different pilot behavior. The system is
used to evaluate the feasibility of ships being able to transit particular channel layouts
with the aim of optimizing a port layout for a range of ship types in differing
The minehunting role requires precise positioning of a ship, while a mine is found and
destroyed. For such a ship, with limited thruster capability, it is not always possible to
severe weather conditions for position control to be achieved by the use of thrusters alone
(as shown in the figure below). In severe condition, a different form of control may be
used, is known as Position Control through Maneuvering (PCM).
In light weather condition, the thrusters are sufficient to move the ship in to desired
condition with out change of heading, so that the ship can face into the prevailing
conditions. In heavier weather, the thrusters are insufficient to be able to control the ship
against the weather condition so control is lost. If, however, the thrusters are used to
position the ship so that its main propulsion unit can propel the ship in to its new position,
the system will continue to be operable in very heavy weather. The choice of which
model to use is left to the discretion of the minehunter commanding officer.
Autopilot: How it works?
The main function of an autopilot is to attempt to keep the ship on a constant heading the
autopilot will usually also be able to be used to change that ship’s course. The degree of
success with which the autopilot will be able to control the ship’s course depends on a
number of factors.
The speed of the ship
At zero ship speed, clearly any movement of the rudder will have no effect on the ship’s
course at all. As the speed increases, the amount of rudder to be used for a given ship
response will be less until, at maximum speed, a very small alteration will suffice. For
this reason, the ship’s speed is usually fed into the autopilot.
The environmental conditions
Clearly the presence of wind and waves will affect the ship’s response. In heavy
quartering seas, it is unlikely that the autopilot will be able to control the ship
satisfactorily at all. The presence of a strong beam wind will require a continuous helm
signal and the presence of swells will induce cyclical yaw motions which may be
The conditions of the autopilot
Conventional autopilots will have a number of manual settings with which the user can
obtain what he deems are optimum settings. Adaptive autopilots will attempt to produce
an optimal performance automatically.
Conventional autopilots: the three term control
The essential function of an autopilot can be seen as being the task of changing the course
error signal into a desired helm command. The way in which nearly all conventional
autopilots operate is similar. If we imagine a situation in which a ship is on a course of
028 degrees and the required course is 030 degrees, there will be an error signal equal to
the difference between the desired and actual courses, or 2 degrees. The autopilot will
then calculate a rudder demand of a size and direction such that the ship will come round
to starboard. The size of the rudder setting will depend on the settings of the autopilot and
on the ship’s speed, but could typically in the range of 0.5-5.0 degrees. For larger course
errors, the calculated rudder angle will be correspondingly larger. In other words, there is
an element of proportionality in the calculation of the desired rudder angle.
If the ship is turning towards the desired course, it will be necessary to apply a greater
rudder angle. Similarly, if the ship is turning away from the desired course, it will be
necessary to apply a greater rudder angle. There is thus an element in the calculation
which depends on the rate of turn of the ship. If the inputs do not include a signal for the
rate of turn, it would be calculated internally. With an autopilot containing a
microcomputer, estimating the rate of turn is reasonable simple as an estimate can be
obtained from the difference in heading angles over a short period of time. As this type of
differentiation is likely to be somewhat erratic, a smoothing circuit or filter will be
If the ship were subjected to an asymmetrical force for a long time, such as side wind, it
would be necessary to keep a consistent rudder angle to counter the disturbance. To
achieve this under automatic control would need a constant course error, which would be
unsatisfactory. If, however, the error signal is integrated over a period of time, and a
demanded rudder angle generated dependant on this integral signal, a zero mean course
error will be produced.
This three-part type of automatic control known as proportional, derivative and integral
(PDI) control forms the basis of most commercial and naval conventional autopilots.
Devices with this type of control will enable a ship to be steered effectively in most
The ship’s officer needs to set the controls of a traditional autopilot in such a way as to
provide the optimum performance of the ship for the conditions prevailing and the ship’s
task. As he is able to observe only the rudder angle and the ship’s heading, it can be
difficult to achieve a good or optimal set of control values. Additionally, to continue to
achieve optimum performance, he will need to change the settings from time to time.
This will rarely be achieved in practice, both because of the tedious and difficult nature of
the adjustments and also because it is difficult to know when the optimum performance
has been achieved.
To attempt to solve these problems, a number of adaptive autopilots have been designed
with the aim of producing better course-keeping and course-changing by automatically
adjusting the autopilot parameters. There are several ways in which the necessary
adaptation may be done. Some or all of the parameters of the autopilot can be changed as
simple functions of the external conditions. This process is essentially automating what
the conscientious operator will do.
Model reference technique
A further method whereby the parameters of the autopilot may be adaptively tuned is to
use the model reference technique. In this method, a mathematical model of the ship is
subjected to the same inputs as the actual ship, and to the same disturbances. The ship
model is tuned so as to give an optimal performance, so that, if the actual ship’s autopilot
is giving its best performance in the prevailing circumstances, the performance measures
of the ship will correlate well with those of the model. The difference in output between
the model and the ship are compared and the differences are minimized. It may then be
assumed that the ship’s performance is optimal. Difficulties may be experienced with this
form of autopilot in defining the mathematical model of the ship, with sufficient scope
for it to be able to perform well in an adequately large range of environmental conditions
Achieving optimal autopilot response
An autopilot is designed to achieve a desired course and does this by using the rudder. A
balance has therefore to be struck between the opposing requirements of keeping a tightly
controlled course and using too much rudder activity. If many large rudder commands are
used, there will be an increase in rudder drag, the ship will be set at larger drift angles,
increasing the hull drag, and the rudder movements themselves will increase wear in the
rudder control mechanism. A correct balance has therefore to be struck between the
requirements of the autopilot.
CASE STUDY: Results obtained using ship simulation software
The four kinds of stability for ship motion, straight line stability, directional motion
stability (for critically damped and under-damped cases) and positional-motion stability
have been discussed earlier in the paper. The figure obtained from the references can
be compared with the graphs obtained from ship simulation program using
MATLAB 6.1. The codes of the same can be found in the appendix to this paper.
The first figure shows straight line stability in a controls fixed case. This case deals with
stability of motion when the rudder is fixed at zero angle of attack. The curve shows the
behavior of the ship when a disturbance acts on the ship. It can be seen that the ship,
originally moving in a straight line changes its course but continues moving in a straight
line on a changed course.
The second and third figures show directional motion stability. For the former, a damping
ratio of 1.0 has been considered and for the second case a damping ratio of 0.1 has been
considered. It is evident from the figures that the ship resumes its course in the same
direction but not on the same track. But the behavior of the ship in the two cases is
markedly different. PD control algorithm has been used for these two cases with different
values of zeta (damping ratio).
The fourth figure shows positional-motion stability. In this case, PID control algorithm
has been used. As is evident from the curves, the ship in this case resumes both its
direction as well as the exact track.
Straight line stability
Directional motion stability
Rolling is one of the most undesirable characteristics of ships, setting up stresses in the
structure, causing discomfort to both passengers and crew, generating a risk of cargo
shifting and increasing the cost of operation. Rolling is sometimes a direct cause of speed
reduction, but more often a change in course, which in turn, may result in speed
Fortunately, the forces and moments involved in rolling are comparatively small, and
therefore rolling is much more easily controlled than pitching or heaving. Hence roll
stabilization is being increasingly adopted, both in naval and merchant vessels. Not only
is it of value in its direct effect in reducing rolling, but it gives the master much greater
flexibility in handling of his ship in rough seas. By minimizing the possibility of heavy
rolling, he is able to adjust speed and heading to keep pitching within acceptable limits
and to continue his ship’s mission with less delay. The only type of rolling of practical
interest is resisted rolling among waves. In order to simplify the treatment, however,
rolling is first considered to take place, not in water, but in a liquid with all of the
characteristics of water except viscosity.
In this discussion we have taken:
Tφ& = number of seconds required for a complete free roll from one side to the other and
Tw = is the time in seconds required for successive wave crests to pass a fixed point
Lw = is the wavelength in feet between successive crests.
Unrestricted rolling in still water
A ship can be made to roll in still water only by application and subsequent removal of
some external inclining moment. In the inclined position, a righting moment exists which
is equal and opposite in direction to the moment of external force. If the resistance to the
motion in the liquid is assumed to be zero, the potential energy of the ship is equal to the
work done by the external force in producing the inclination. When the external moment
is removed, the righting moment produces rotation of the ship towards the upright
position and the potential energy in the inclined position is converted into energy of
motion so that the kinetic energy, where the potential energy is zero, is equal to the
potential energy in the inclined position, where the kinetic energy is zero. The ship
therefore continues its rotation to the other side of the vertical with conversion of its
kinetic energy to potential energy. Under the assumed conditions the ship would roll
indefinitely from side to side with constant amplitude.
In a homogeneous medium a body to which a periodic moment is applied tends to rotate
about its center of gravity. Thus an airship or a submerged submarine oscillates in rolling
about an axis through its center of gravity. On the other hand a surface ship which is
partly in air and partly in water does not have any axis of roll, because of different
dynamic effects of the air and water in contact with it. Very little is known quantitatively
regarding the movement of the axis of roll of a ship from its normal position through G,
but it may be stated that, when G is at the waterline or above the axis of roll is below G,
and, when G is very low, as in large sailing yachts, the axis is above G. in general the
axis of rolling is not fixed either in space or in the ship, but describes a curved surface.
For ships of ordinary form at moderate angles of roll, the axis is not far from the center of
gravity; and, where this simplification assists in the solution of the problem, the axis of
roll is assumed to pass through G.
Under these assumptions, the equation of motion of the ship is:
I + M = 0 − − − − − − − (1)
Where I is the mass moment of inertia of the ship about a longitudinal axis through the
center of gravity, M is the righting moment, and φ is the angle of inclination of the ship
from the vertical.
Where k is the radius of gyration of mass of ship about a longitudinal axis through G. For
small angles of inclination
M = ∆GZ = ∆GM sin φ = ∆GMφ
Substituting these values in (1) we have
d 2φ g GM
+ φ = 0 − − − − − − − − − (2)
dt 2 k2
Equation (1) is the equation for simple harmonic motion having the period
Tφ = − − − − − − − −(3)
Therefore the rolling period of the ship is
Tφ = − − − − − −(4)
Unrestricted rolling among waves
The angle of heel of a ship among waves in an unrestricting medium is made up of two
parts, one of which is a result of an inclination impressed upon the ship before the waves
reached it and the other is an inclination produced by wave action.
The inclination due to the wave is also composed of two periodic functions, the period of
one being the wave period Tw, and that of the other the still water rolling period of the
ship Tφ . A ship has a natural period of oscillation and, when subjected to a single impulse,
oscillates in this period until the energy of the original impulse is consumed by the
resistances of motion. Thus, a pendulum moved from its position of equilibrium and then
released swings back and forth in its natural period of oscillation. If, however, the
pendulum is subjected to periodic impulses, it oscillates in the period of the applied
impulses and not in its own natural period. Such motion is known as forced vibration or
oscillation in distinction to free oscillation in the body’s natural period. Rolling in still
water is a free oscillation. Among waves, the impulses produces rolling are periodic and
therefore, as in the case of the pendulum, tend to set up oscillation of the ship in the
period of the wave. Such rolling is called forced rolling. If the waves pass by the ship or
die out, the ship resumes free rolling in its natural period. If waves of constant period act
upon a ship for a sufficiently long time, the ship will ultimately roll in the period of the
waves. If the period of the waves is not constant, the period of rolling will not be constant
because of the ever-present tendency of the ship to revert to rolling in its own natural
Resisted rolling in still water
A ship in still water may be set rolling by the application of small moments synchronized
with the ship’s period of roll.
The sources of passive resistance to rolling are frictional resistance of water on the wetted
surface, resistance due to eddying of water set in motion by the immersed part of the ship,
the generation of water waves by the ship’s rotation and similar resistances due to the
action of air on the above parts of the ship. These sources of rolling have been
characterized as passive because they exist on every ship regardless of any effort on the
part of designer or builder to modify the ship’s natural rolling characteristics. Strictly
speaking, bilge keels are also passive source of resistance to rolling.
These three sources of resistance absorb all the energy damping of a rolling ship not
equipped with active means of reducing rolling and these resistances are functions of
dφ dφ dφ
and . If the resistance of rolling is A , we have the equation of motion in still
dt dt dt
water from Newton’s law and equation (1)
∆k 2 d 2φ dφ
( 2 ) + A( ) + ∆GM = 0
g dt dt
d 2φ Ag dφ g GMφ
+ + = 0.................(5)
dt 2 ∆k 2 dt k2
The solution of equation (5) yields the following results:
Tφ ∆k1 GM
A= − − − − − − − − − (6)
Tφ′ = 1
− − − − − −( 7)
K1 2 2
1 − 2
Tφ′ = * 1
− − − − − −(8)
GM K1 2 2
1 − 2
Where Tφ′ = the period of resisted rolling in still water. In practice K1 is less than unity;
therefore 12 is less than 0.1 and the denominator of the right hand side of equation (8)
is little less than unity. The period of resisted rolling differs but little from the period of
unresisted rolling, and for most practical purposes, equation (3) may be used to obtain the
period of free rolling of a ship. As an example, let K1=0.1; then
K1 2 2
1 − 2 = 0.999 = 0.999
Tφ′ = = 1.001Tφ
Tφ = 15 sec
Tφ′ = 15.015 sec
This shows that the period of rolling is increased slightly by the resistance of the water.
This effect is accentuated by the presence of bilge keels.
Thus the result obtained indicates that the period of resisted rolling at large angles in still
water is slightly greater than the period of unresisted rolling. The general statement can
be made that, regardless of the amplitude of the roll, the period of rolling in still water is
very slightly greater in a resisting medium than it is in a nonresisting medium and that the
period of roll in a resisting medium increases with the amplitude of the roll, but not in
any simple manner.
Resisted rolling among waves
It can be shown that if the rolling were unresisted and the ship’s period synchronized
with the wave period, regardless of the initial inclination and angular velocity of the ship,
the angle of roll would be augmented by the amount Πα m in each swing from side to
side. It can also be shown that when the periods of wave and ship do not synchronize, the
oscillation of the ship goes through repeating cycles whose length depends upon the
relation of period of wave and the period of the ship. In this latter case the existence of
certain initial conditions of inclination and angular velocity results in suppression of the
oscillation of the ship in its own still-water period and in forced rolling in the period of
In view of the fact that in an unresisting medium it is theoretically possible for waves to
set up rolling which capsizes the ship, it is desirable to investigate to what extent this
possibility is modified by the resistance of water. The assumption that the moment of
resistance varies as the angular velocity gives a simple expression for the equation of
motion, which is the same as equation (5) except that (φ − α ) is substituted for φ in order
to take account of the wave slope.
When a ship is among waves the inclination of the ship with respect to the surface of the
water is not φ , as it would be in still water, but (φ − α ) , where α is the wave slope. The
righting moment of a ship among waves is therefore ∆GM (φ − α ) instead of ∆GMα .
Since the ship rolls to angles which are large compared with the wave slope, the moment
dφ d (φ − α )
of resistance is assumed to be A instead of A . The equation of motion
d 2φ Ag dφ g GM (φ − α )
+ + =0
dt 2 ∆k 2 dt k2
Substituting for A its value given in equation (6) and assuming sine waves, so that
α = α M sin ω W t
The equation of motion becomes
d 2φ 4 K 1 dφ 2 2
+ + ω φ φ = ω φ α M sin ω W t
dt Tφ dt
Its solution, when t=0 and φ = φ A is
α M sin(ω W t − x) −2
sin ω φ t 1 − 2 + φ A − − − −(9)
4K 2 Tφ 2 2
1 − 2 + 1
TW ∏2 T 2
2 K 1 TW
x = tan −1
∏ Tφ 2
The second term of equation (9) is the same expression as obtained by the solution of
equation (5), and is the inclination due to resisted rolling in still water, which diminishes
in geometrical expression. The first term is a forced oscillation conforming to the period
of the waves. Therefore after a short time φ will be a maximum when the forced
oscillation is a maximum, which will be when
sin(ω W t − x) = +1or − 1
That is, when
ωW t − x = + 2 n ∏ 0 r + 2n ∏
Where n is zero or any integer.
TW TW 3T T
t= + x, and W + W x
2 2∏ 4 2∏
Effect of synchronism; resisted rolling
Forced rolling occurs only when the period of the ship is not equal to the wave period,
because when Tφ = Tw . The ship rolls in synchronism with the wave, and in an unresisting
medium the angle of roll increases by equal increments to infinity. In a resisting medium,
however, the angle of roll, while great, is finite. To investigate this, let Tφ = Tw . Then
x = tan −1 ∝= Π / 2 and for φ = φ A
t= , or
t = nTφ
Where n is an integer.
When waves and ship are in synchronism, the maximum inclinations therefore occur at
wave crests and hollows: the first value occurs when t = Tφ = Tw and is equal to
∏1 − K 1 + φ
(φ A ) T = + ye sin
2 K1 ∏2
∏α M K12 2
3 ∏1 −
(φ A ) 3T = + ye − 3 K1
The increase in inclination in one wave period is the difference between the last two
terms of the above expressions. Since the value of
2 ∏ t 1 − K 1 + φ
Cannot exceed unity and the value of e decreases in geometrical progression, there
must be some time when the value of the last term is infinitesimally small and the ship
has a maximum inclination of ( Πα M / 2 K 1 ), in the crests and hollows and zero at the
mid-heights. Prior to such time the maximum inclinations are greater, but always finite.
When the energy absorbed by the resistance to rolling per swing is exactly equal to that
supplied by the waves in one-half of a wave length, maximum amplitude of rolling has
been reached and no further increase is possible.
The second term of equation (9), representing the free roll, is damped and in a relatively
short time becomes so small that it can be disregarded. It can be shown that in ships with
very short periods, the inclination of the ship at any time is the same as the wave slope,
therefore they tend to remain with their decks parallel to the water. The maximum
inclination occurs when the wave slope is greatest; i.e., at the mid-height of the wave, and
are equal to α m . The ship is erect in the crests and troughs. The angles of roll are
moderate and the ship does not usually take water on the deck.
For ships of long period, i.e., when → 0, x → 0 , the denominator of the first term of
equation (9) has a large value. Therefore φ F is always small and has its maximum values
at the mid-height of the wave where the wave slope is α M and zero value at crests and
hollows. As increases from 0 to 1, the position of maximum inclination moves up the
wave slope from the mid height toward the and down the other side of the wave toward
the hollow. Similarly, when w increases from 0 to 1, the position of maximum
inclination moves backwards from the mid-height towards the trough on one side and the
preceding crest. Figure below shows the positions on the wave profile at which the
inclination is maximum, for various values of the ratio w of somewhat greater interest
is the next figure, which shows for various values of the ratio the ratio of the
amplitude of the forced oscillation represented by the first term of equation (9) to the
maximum effective wave slope. It can be noted from this figure that only for the values
of w between 0.8 and 1.4 does the forced inclination exceed twice the wave slope,
which seldom exceeds 9 deg. If the ratio of is kept outside of this range, dangerous
rolling is not to be anticipated.
Reduction of roll
Devices which are or have been employed for reducing the rolling of the ships are:
• Bilge keels,
• Antirolling tanks,
• Gyroscopic stabilizers, and
• Stabilizing fins.
Except for bilge keels (which absorb power from the ship when underway) and some
forms of antirolling tanks, all of these require separate expenditure of power. All of them
add weight to the ship and most of them occupy space which could be used for
commercial or military purposes. Generally, but not always, the period of roll damping
mechanism should be the same as that of the ship. There should always be a definite
relation between these two periods. The phase relation is usually such that the device lags
90 deg behind the ship, although this statement also is not without exception; but there
should at any rate be a definite phase relation between the ship and the mechanism. A full
description of the various types of apparatus which have been proposed and developed
for ship stabilization and the mathematical theory underlying their design are beyond the
scope of this paper. So a general discussion is being presented here.
Bilge keels are fins in planes approximately perpendicular to the hull at or near the turn
of the bilge. The two types are shown in the figure.
Ever since their effectiveness in reducing rolling was first demonstrated, about 1870,
bilge keels have been installed on nearly all ocean going vessels, both commercial and
military. The longitudinal extent varies from 25 to 50 percent of the length of the ship
and the depth from approx. 1 to 3 ft. the necessity of keeping the outer edge of the bilge
keel above the base line and within the extreme breadth of the ship limits the depth in
most cases. Very shallow bilge keels, which do not extend beyond the boundary layer of
water around the ship are ineffective. Ships of large mid ship section coefficient usually
cannot be fitted with deep bilge keels without getting the outer edge below the base line
and beyond the extreme breadth.
The resistance of bilge keels to rolling is largely attributed to disturbance of the water in
contact with them and to eddying at their edges. In general, value of rk (mean distance of
projection on hull, in contact with water during roll, from axis of roll) will be maximum
for bilge keels placed squarely at the turn of bilge.
The effectiveness of bilge keels is greatest on ships of low mass moment f inertia. Bilge
keels also have a greater damping effect at large amplitudes of roll than at small ones.
The longitudinal line of bilge keels should conform to the lines of flow over the hull. If
this is done, the increase in hull resistance caused by them is almost entirely frictional.
The earliest application of anti –rolling tanks used compartments in the upper part of the
ship where free water could be carried. The reduction in metacentric height. Due to both
to the added weight high in the ship and free surface of the water, increased the ship’s
period of roll and, if the ship is previously rolling in synchronism with waves, destroyed
the equality of periods. Furthermore, transfer of water to the low side of the ship created a
moment opposing the ship’s righting moment thereby damping the roll. For several
reasons such “water chambers” are no longer fitted. They are potentially dangerous
because any reduction of righting moment, and, if ship and the water should go into
synchronism, the water could cause an increase in rolling.
All other anti rolling tanks are evolved from Frahm’s U-tube type. Horizontal leg of this
type stabilizer is above ships CG because of
1. utilization of space above machinery
2. The moment of force due to the horizontal acceleration of water therein acts in the
same direction as the statical moment of water in vertical legs.
The period of oscillation of liquid in a U-tube of uniform cross-section is equal to
2Π , where l is the half length of the tube. By increasing the sectional area of
vertical legs, virtual length is increased and consequently period of oscillation. This
makes possible the practical design of a tank having a period approximately equal to that
of ship. Recommended period varies from 0.77 Tφ to Tφ , the effort is always made to
have a phase difference of 90 degree between the motion of ship and that of water in the
tanks. The air connections on the top of vertical legs are important features which avoid
formation of vacuum in any of legs.
The experience with Frahm tanks has shown that the average roll with the tanks in
operation is about half of the average roll without the tanks.
Later anti rolling tank installations of Frahm type had no horizontal leg. Vertical tanks at
mid length of ship are connected to sea at bottom and vented to atmosphere from top. The
size of sea connections was determined from model rolling experiments to give a phase
difference of 90 degree.
Activated anti-rolling tanks
In this system tanks are not vented at the top. A high capacity, low pressure air
compressor supplies air at a pressure slightly above atmospheric to the upper parts of
tank, so that water level remains always below the level of outside. Stabilization is
achieved by varying the amount of water in opposite sides of ship, which in turn is
accomplished by varying air pressure above the water. The sensitive element of the
control mechanism is a gyroscope which by its precession actuates electrical signal to
start air compressor and operate the valve which control the air flow in upper part. Phase
angle between ship and water in tanks are kept at 90 degree.
Results have shown that ship rolling 8 degree in still water without tanks in
operation takes 14 oscillation to reduce to 1 degree roll, while with activated anti
rolling tanks for 13 degree roll took 2.5 oscillation to reduce to 1 degree roll. This
stabilizer may be used to reduce heel in turning because it is subjected to control
other than roll of the ship.
The problem of making proper phase difference in ship stabilization can be dealt with the
The stabilizing moment S is given by:
d φ and
S = − Ks ( )
the stabilizing moment of anti- rolling tanks is obtained by a variable quantity of water w,
having constant arm d, the horizontal distance between the vertical legs
S = wd = Ks ( )
w = K T ( ) − − − − − − − A
differentiating twice we have