Notes on the forward motion stability of ships

Experimental Towing Tank

Stevens Institute of Technology

Hoboken, New Jersey

Lab. V.

Scheepsbouwkuflde

Technische Hogeschoo,,

Deift

NOTES ON TIlE FORWARD-ADTION S2ABILITY

OF SEIPS

NOTES ON THE FORWARDMOTION STABILITY OF SHIPS

by

Marvin Giinprich

Consultant

TECIICAL MEMORANDUM NO. 74

Prepared for the

David Taylor Model Basin Navy Department Washington, D.C.

under Bureau of Ships Contract NObs - 22087

Job Order 17

Experimental Towing Tank Stevens Institute of Technology

Hoboken9 New Jersey

TABLE OF CONTENTS APPENDIX II IIITRODUCT ION 1 DYNAMIC STABILITY Definition 1 3 Definition 2 4 Definition 3 STATIC STABILITY 5

MECHANISM OF STATIC AND DYNAMIC STABILITY

Static Stability 7

Dynaznià Stability ₇

RELATION B)TEN DYNAMIC AND STATIC STABILITY 9

TYPES OF MOTION :

10

MEASURES OF DYNAMIC AND STATIC STABILITY 1].

BEHAVIOR AT STABILITY LIMITS 12

DAMPING MOMENT AND FORCE ₁₄

MOMENT CURVES AS STABILITY INDF(ES ₁₅

BEHAVIOR OF DYNAMICALLY UNSTABLE SHIPS 16

SIGNIFICANCE OF DYNAMIC STABILITY 17

CONCLUDING REMARKS ₁₉

APPENDIX I

MATHATICAL FORMULATION OF DYNAMIC STABILITY 21

Proof of Definition 1 ₂₃

Proof of Definition 2 ₂₄

Proof of Definition 3 ₂₅

RELATION BETWEN DYNAMIC AND STATIC STABILITY 26

TYPES OF MOTION . 28

EASURE OF DYNAMIC STABILITY . 29

MOMENT AND LIFT CURVES AS. STABILITY INDENES . 30

DYNAMICALLY UNSTABLE SHIPS - _{STABILITY IN TURNING}

. 33

DYNAMIC AND DIRECTIONAL STABILITY

35.

LIST OF SYMBOLS

37

REFER ENCES

* The term "forwardmotion" stability is used here in a broad sense to avoId.

the use of such terms as "direôtional" stability, "dynamic" stability, _etc. whióh are later given very specific meanings.

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Th'TRODUCT ION

The problem of the forward-motiôn* stability of ships has been treated

in Reference 1. In an effort to further clarify the fundamental concepts of the subject, this Memorandum is issued as a supplement to Reference l

The discussiOn covers various definitions of dynamic stability, the

definition of static (weathercock) stability, the mechaitisia of dynamic and

statostability, the mehods of testing for dynamic stability, the relation-ship between dynamic and static stability, and numerical scales for evaluating

the two types of stability0 The nature and origin of the damping forces and

moment, which play a large part in determining the dynamic stability, but which sometimes have been neglected in the literature, are diécussed.

It is shown in the earlier sections of the Memorandwn that the only practical measure of the forward-motion stability of a ship is the dynamic

stability.

Later in the Memorandum it is shown that a simple curve of all the

inoiaents acbing on a turning ship with rudder amidships can indicate.whether or riot the ship'is dynamically stable. _{Moment curves of this type are} dis-cussed in Reference 2.

The last section considere the importance of dynamic stability. _It is indicated there that9 although the basic problem of the forward-motion stability of ships has been worked out, ramifications such _{as response to}

various dIsturbances and the steering of ships remain to be discussed. It is planned to treat these in subsequent reports0

Appendix I includes a mathematical discussion of stability _an proofs

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Appendix II is included to add the results of a discussion which was

held after the Memorandum was written0 It concerns en additional distinction that between dynamic stability and directional stability.

DYNAMIC STABILITY

Definition 1

A ship is said to be dynamically stable, in forward motiàn. iny angular velocity and any angle between the heading and he diretion of 'motion of the center of gravity, (yaw angle), resulting from aiinitial

disturbance of the motion of the ship on straight course, decrease in'.

time (without benefit of rudder adjustment), and if the consequent course

deviation is relatively small,

A ship is said to be

dynamically

unstable if the angular velocity and the yaw angle, resulting from an initial disturbance of its motion on straight course, persist or increase in time, the courie deviation

also increasing in time, 'and the ship eventually winding up into a-àir.

010.

In other words, ira dynamical].y Stable ship is travelling along a straightaline course and is disturbed from that course, it will

_assune

a nr straight-line course which makes a relatively small angle with the original course, the angle between the original and final courses depend.

ing on the amount of the initial disturbance and on the degree of dynamic' stability. However, if

a

dynamically unstable ship is travelling along' a straight.-line course and is subjected to an arbitrarily small diiturbaüoe.,

it' will wind up into a circle.

-Thus, there appears one simple method for determining whether or

not a ab3p is dynamically, stable. The ship can be started off on a straight course with the rudder held àiidshipa, and the subsequent motion can be

observed, Since no water is 'completely "still", the ship's motion rill sooner or later be disturbed, If, therefore, the ship continues along it8

original straight.-line course or a course olose'to It, the shIp is'dy

nainically' stable1 while if the ship 'windi up Into 'a circle it 'is dyiamioa11y unstable, It should be noted that the dynamic tability '(or instability) of a ship is a property independent of rudder motion',

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Definition 2

A ship

while a ship

turn against

is dynamically stable if it cannot "turn against" its rudder, is dynamically unstable if, with small rudder angles, it can

its rudder0

Another simple method for determining dynamic stability is thus pro

vided0 This method is easier to use in model size than in full size0 A small rudder angle maybe set on the model and an initial yaw angle, or

angular velOcity, can 'be given in the opposite sense from that in which

the rudder is set0 If the model winds up into a circle turning with, the rudder, or if the path curvature is in the direction which the. rudder angle

is expected to produce, the ship is dynamically stable, while if the ship continues to turn in a circle against the rudder, the ship is dynamically

unstable. Care must be taken to be. sure that small enough rudder angles

are tried, if this method is to be successful0

Definition 3

Closely connected with the first definition (in fact, as is shown

iii Appendix Ion page 25, a limiting case of it),is a third definition

of.dynamic stability. If a ship in a steady turning circle can be brought to a straight course simply by setting the rudder amidships, the ship is

dynamically stable. If,. after setting the rudder amidships, the ship does not assume a straight course, but continues In a circle (normally of djfc. ferent radius)., then the ship is dynamically unstable.

Here, the method for testing is obvious. From experience with this

sort of test, it is known that most ships are dynamically stable, but that

some are dynamically unstable.

Thus far we have given definitions and descriptions of, and methods

of testing for, dynamic stability. Before examining the mechanism and significance of dynamic stability, we shall investigate what is meant by

STATIC STABILITY

First, it should be carefully noted that the stati.o stability of a body in a fluid can be defined only if the body is restrained against

lateral displacement. Vhen this limitation is imposed, the body is robbed

of one degree of freedom. A free body like a ship can translate longi...

tudinally and laterally and can rotate about a vertical axis, so that it has three degrees of freedom in the horizontal planej if lateral motion is prevented, it has only two degrees of freedom. Conditions under

which

static stability can be defined are easily reproduced in tests made in a water tunnel (or wind tunnel), or in tests under the carriage of a towing

tank. In all these testing methods, the center of gravity can be prevented.

from translating laterally, but this is a condition which does not exist

in practice. From this fact alone, the artificiality and inherent im-practicality of the concept of static stability, when dealing with the motion of a ship (or any free body in a fluid) is immediately seen

A ship is said to be statically stable if, when the ship is moving longitudinally and is restrained so that its canter of gravity cannot move laterally, a disturbance of its heading tends to decrease, the ship

tending to return to its "equilibrium" position where the sun of the moments

acting on it is zero.

It is believed on the basis of testa at hand that, in general, all

ships are statically unstable.

The definition of static stability makes no reference to the course deviation resulting from a disturbanoe, since, in order to define static stability, the ship must be restrained so that it cannot undergo a lateral displacement and hence cannot deviate in course. However, whether a ship is statically stable or unstable (which may be determined only in tests which restrict Its lateral motion), it will a'ways, when disturbed, auffa

a course deviation if it moves, as an actual ship moves, with three

de-grees of freedom. The magnitude of the course deviation depends primarily

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-6-on the degree of dynaniá stability and explicitly very little on the

static stability. It is not possible to determine from the motion of a free ship, whether or not it is statically stable0 In marked contrast9

dynamic stability is eaSily determined from the motion of the free ship, by methods already described0

THE MECHAMISM OF STATIC AND DYNAMIC STABILITY

Stato. Stability.

In order for a ship to be statically stable, the center of pressure

of the. hydrodynamio forces must lie aft of the center of gravity, if. this condition is realized, then, when the center of gravity is restrained so that it cannot move laterally, a small yaw angle introduces a moment about

the center of gravity which tends to deorease the yaw angle,. Conversely,

if the center of pressure lies forward of the center of gravity, a small

yaw angle introduces 'a moment which tends to increase the yaw angle, and the ship is statically unstable.

Dynamic Stability

The mechanism of dynamic stability is not quite so easily described0 The underlying principle is that the ship is free to move laterally as well as to rotate0 The lateral motion compensates for, or reduces, an

initial disturbance, For example, suppose the motion of a Statically unstable ship to be disturbed by giving it an initial yaw angle to port. A moment is thereby introduced which tends to increase the yaw angle and

to cause angular velocity. Immediately, due to rotation, a damping

moment is produced which. tends to decrease the angular velocity. But in

addition, and this is most important, a.transverse force to port, is pre .du?ed, by virtue of the yaw angle to port, which moves the ship to port, Giving the ship a transverse velocity in the port direction is in 'effect the same as some starboard yaw angle so that the initial yaw angle to' port

is reduced, Furthermore, this transverse force is augmented by a damping.

force due to rotation, whiàh acts in the same direction, thus giving, the ship an augmented velocity to port. and'further reducing the inital' yaw angle. The over-all result is that, by changing the direction of motion of the ship a small amount, the initial yaw-angle _{angle between heading}

and direction of motion - is reduced to zero if the ship is dynamically

stable0 ' . .

From this explanation of the mechanism of dynamic stability, we can

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-8-see what factors tend to increase it Dynamic stability increases with increasing transverse or "lift" force, damping force, and damping moment coeffioients,while it tends to decrease with increasing yawing moment

coefficients (increasing static instability)0 In addition, the dynamic

stability tends to increase with decreasing mass, since for a given lift the lateral velocity produced will be larger the smaller the mass of the ship and consequently the drift angle will be more greatly reduced0

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RELATION BrWEEN DYNAMIC AND STATIC STABILITY

All ships that have been tested are statically _{unstable; most ships}

.are dynamically stable and only a few are dynamically unstable0 _{From the}

mathematical study of stability given in _{Appendix I on page 27, it appears} that, while it is mathematically possible for _{a ship to be statically stable} and at the same time dynamically unstable, this is very unlikely from a

physical point of view0 Thus, in practice, if a ship istatióal1y stable, it will (very probably) be dynamially stable. _{However, the converse is}

not true. _{If a ship is dynamically stable, this certainly does not imply} that it is statically stable, _{In fact, all ships that have been found to}

be dynamically stable have also proven to be statically _unstable.

The mathematicalcondition for dynamic stability may seem to imply a tendency for increasing static stability to produce increasing _dynamic

stability. But this is certainly not necessarily true. _{If, for example,}

a fin were provided to induce larger lift forces, and if this were placed

at the center. of gravity, the static stability would not be altered, while the dyna.mio stability might be increased considerably. Thus, in general,

the static stability cannot be.assumed to be _{a reliable mea.ure of the}

dynamic stability, for:

it is certain that a statically unstable _ship may very well be dynamically stable,

it Is perfectly possible in principle to alter the dynaniiô stabiltiy without changing the static stability, and vice _versa..

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-TYPES OF MOTION

It is interesting to consider the various types of motion of a ship with its rudder amidships.that are possible both mathematically and

physi-cally0

If a ship is dynamically unstable, it will eventually wind up into

a circle, while if it. is dynamically stable, it will tend to pursue a

straight-line cürse. Both cases are independent of whether the ship is

statically stable or statically unstable. Thus, we have two general types

of motion that i±iclude the motion of all ships.

In the interval after a dynamically stable ship is disturbed from a

straight course, and before it settles down to a fin.l course,, the motion

may, mathematically speaking, depend on the static stability of the ship. To be.preoisé, if a ship were very stable statically (much more than "just" stable statically), the motion could be oscillatory until the ship settl

down to its final steady course. The final course, however, would not be altered by these transient oscillations (since it depends only on the

mi-tia]. conditions and the degree of dynamic stability). In general,

theos-cil].ations would damp out. It is possible, mathematically, for the ampli-tude of the oscillations of a dynamically unstable ship to continue to

increase. But this is practically impossible physically, since it would require some anomalous phenomenon such as the ship absorbing energy from

the surrounding water.

It should be remembered however, that rio ship is known to be

statically sta1le. Consequently, it appears that no ship will tend, to

oscillate when subjected to an initial disturbance, as this would

EAStJRES OF DYNAMIC AND. STATIC STABILITY

Since the nature of the motion of a ship is governed by its dynamic

stability, it is of, interest to establish a numerical scale of dynamic

stability in order to provide a means of comparing ships on a

quantita-tive basis0 Such a scale can be chosen in various ways. A logical index

of dynamic stability, both from mathematical considerations and also be-cause of its clear physical significance, is an exponent which has the dimensions of

1/L

(reciprocal of a ship length), and which we shall call

In Appendix I on page 22 it is shown that a necessary and

suf-ficient condition for dynamic. stability is that p1<Q. The magnitude of

-

Pi.

can serve as a measure of dynamic stability, a higher ma.gnitude

corresponding to greater dynamic stability0 The physical significance of p is that when the motion of a ship. is disturbed, the disturbance will

diminish to approximately

I/e(e ..2.718)

_{times its initial value in a}

distance

_I/pa

ship lengths. An example pf a dynamically stable ship is the,

DD445, where p

=-O.91,

and an example of a dynamically unstable ship is

the AM136, where p = -fOJG _{(These examples are taken from Reference l)}

The simples.t measure of static stability is the yawing moment co-efficient

C,

If the center of pressure of the lift forces due to yaw is forward of the center of gravity, the ship is statically unstable and. the moment coefficient positive; if the center of pressure is aft of the

center of gravity, the ship is statically stable and the momeflt coefficient negative0 The more negative the mpment

coefficient C,., the more

_{stable the}

ship

i8

Statically0 .

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BEHAVIOR AT STABILITY.LIMITS

Having obtained measures and hence scales of dynamic stability and the very much less significant static stability, one can obtain an idea of what will occur near the zeros of stability, namely near _{p = 0}

and

= 0.

When Cm=Q

, the static stability is zero0 The tyre of motion

will not be essentially altered as we go from _Cm<O

_{to Cm>O}

₈₀ long

as the dynamic stability remains the same. 'In other words, if the

dy-namic stability were to remain constant, one would not notice any essen-tial difference in the nature of the motion or the behavior of the shifl as _C changed sign, that is as the ship changed from sts.tic stability to statio instability,

However, as

_Pi

_{approaches zero and changes sign the' effect on the}

motion of the ship will be very marked0 _{This may easily be seen from the}

physical meaning of p Remembering that

_I/ps

is roughly the number 'of

ship lengths required for an initial disturbance to diminish _to

IJe

_of

its initial value, then if

Pt is negative (dynamic stability) and ap-proaches zero,

_VP

_{increases, and the number of ship lengths in which a} die-turbance will diminish to t/ ..TI8 _{of its initial value increases. Thus,}

as the dynamic stability becomes less and less, the distance the ship must traverse in order for a disturbance to damp out will continue to increase, as will the final course deviation resulting from a given disturbance, _The rate of increase of the required number of ship lengths is _{very rapid as}

Pi

approaches zero. When p1=O the number of ship lengths for the

dis-turbance to die out to

l/e

of its initial value is. infinite, _{The disturbance}

does not decrease in time, but remains fairly constant _{and the' ship will.}

eventually end up in a circle, _{A graph.of the number of ship lengths}

re-quired for a disturbance to diminish to

tie

_{if its initial value is roughly}

,_y4 ._I 0 ci g o 5-$. d '-4 . 0 4' 4' r-I '2) o '-4 4' P1

rI

1.4 f4 4' _{0 2.} .0

oj-_

rf

000

If p1 becomes positive (dynamic instability), an initial disturbance

con-tinues to increase, the ship winding up into a circle, and 1/p1 is roughly

the number àf ship lengths required in order for the disturbance to in-crease to times its initial value,

If p is a very small amount negative (the ship being just barely stable dynamically), the methods of testing for dynamic stability

out-lined in the second section may not yield _{very definite answeré, since}

a relatively large number of ship lengths is required for the ship to settle down to a new straight course and this course will make a large angle with the original course. _{The ship may then appear to be dynami.}

cally unstable, particularly as _Pi approaches zeros

Thus in the region of zero dynamic stability, thi behavior _{Of a ship} changes very markedly, while in the region of _{zero static stability there}

is practically no change in the behavior.

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-DAMPING MOMENT AND FORCE

A clearer physical picture of the damping force and moment9 which have sometimes been neglected in the

literatureD

can be gained by con-sidering their origin.. The damping force and moment are defined as the respective differences between the force and moment acting on a ship when on a curved course and when on a straight-line course, the yaw angle (drift angle) at the center of gravity being the same in both cases, When a ship is travelling on a straight-line course with a yaw angle ₍₃ (as can be ar

ranged in model size in a towing tank), every point on the axis of the ship makes the same angle (3 with the direction of motion. When a ship is

travel-ling on a curved course with a yaw angle (3 at the center of gravity, points

on the ship which are for,vard of the center of gravity are erfeotively.at V V

a smaller yaw angle than (3 with the direction of motion, while points on the ship aft of the center of gravity are at a larger yaw angle than (3

with the direction of motion. A point at a distance

r

aft of the center of gravity will have an additional lateral

_{velocityV(L)tl}

because of the ro-tation, where W is the angular velocity of the ship. The lateral Vvelocity amounts, in effect, to an additional yaw angle.. Ct)rj /1)

*

Similarily, for points forward of the center of gravity, the additional yaw

angle is V

The net result of this variation of yaw, angle which occurs in

turn-ing

is that additional forces and 'moments are present which are functions

of the angular velocity & These are called the damping force and moment.

respectively, V

*Strictly speaking, the increase in the yaw angle is tan

(wrI,/v)

_;

MOMIT CURVES AS STABILITY INDCES

It is shown in Appendix I an.. page 32 that if a _{curve is plotted} of the "total-moment-with-zero-rudder" coefficient against the yaw angle

at the center of gravity of a ship in a steady turn, _{as in Reference 2,}

the ship is dynamically stable if the slope of this _{curve is positive at}

the origin, and unstable if the slope at the origin is _{negative. The} "total-moment-with.. zero.rudder' includes, _{as its name implies, both the}

static moment due to yaw and the damping moment due to rotation, _{for the} hypothetical condition of a ship on a curved path with its _{normal drift}

angle but with its rudder amidships.

It is also shown in Appendix I _{on page 32 that the magnitude of}

the slope is not a measure of the dynamic stability0 _{This ourve cannot}

give the magnitude of the dynamic stability, since _{the damping force is} nowhere included.*

is further éhown that a similar curve of "total-liftwjth..zero..rudder"

coefficient (i.eOD liftinstead of moment _{coefficiont)wil]. not yield any} information concerning the dynamic stability _{of the ship.}

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-BEHAVIOR OF DYNAMICALLY UNSTABLE SHIPS

It has been stated earlier that a dynamically unstable ship may be-come dynamically stable when a sufficiently tight turning circle has been reached, the rudder remaining amidships0 It is shown in Appendix I on

page 23 that9 mathematically in the linear approximation, the course

devia-tion of a dynamically unstable ship increases exponentially so that a spiral

course is expected0 On, the other hand, it is observe4 that a dynamically

unstable ship will eventually wind up into a circle,and will not continue

in an ever-tightening spiral0 The explanation for this apparent discrepancy

is quite simple: the angular velocity and yaw angle In the turn continue

to increase beyond the range in which the linear approximation holds, with the result that the drag, lift, moment, damping force, and damping moment

coefficients change. Thus, the hydrodynamlo parameters used for the

cal-culation of stability on straight course no longer apply the parameters

change to such an extent that the motion becomes dynamically stable after a sufficiently tight turn has been reached, In the linear approximation, the hydrodynamic coefficients, are assumed constant and equal to values

which can be determined fOr moderate yaw angles and angular velocities0. That this is a sufficiently good approximation may be seen from the good agreement of computed with experimental determinations of dynamic stability.

As the yaw angles and' angular velocities increase, however, it is just the non-linearities in the hydrodynaraio coefficients that cause the ship to.

SIGNIFICANCE OF DYNAMIC STABILITY

The foregoing discussion has been devoted to dynamic stability and instability, the types of motion resulting when a ship is dynRm1cally stable or dynamically unstable, and various related topics.

It is only natural to inquire into the significance of dynamic

sta.-bility. Why is dynamic stability important? What is its practical effect? Is it desirable for a ship to be dynamically stable? Will a ship which is very stable dynamically respond to disturbances in a much better way? These

are some of the questions whioh must be answered if the significance of

dynamic stability is to be understood.

One important effect of dynamic stability is clear from the des-. criptions of the characteristic niotion with which it is associated. A dynamically stable ship which is traversing a relatively calm sea will

tend to intain a course close to its 8et course without being steered,

while a dynamically unstable ship traversing the same sea will wind up

into a circle if it is not steered. The response of either ship to a

dis-turbance, when being steered, depends on the rules followed by the helma..

nan. This matter will be dealt with at some length in a later report. Certain simple conclusions may be drawn, however, from our present

know-ledge.

If a dynamically stable ship is disturbed slightly, there will be a

course deviation which will not ordinarily be large, and which can be

cor-rected easily by putting the rudder over a small amount for a short time.

If a dynamically unstable ship is disturbed in the same way, a much larg-.

er rudder angle will be required for correction, because the ship may

turn against its rudder. Furthermore, after applying the rudder and aftw

return to the original course, the dynamically unstable ship will go into another circular path, the extent of which will depend on how long the rudder is held over and on the initial

_condit1ona

_{Even if a rule} of steering is followed whereby the ship returns to its original course

with zero yaw angle and zero angular velocity, it will innitediately go

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18-Consequently, the path of a dynamically unstable ship is essentially a succession of curves, and in order to maintain course, successive appli-cations of relatively large rudder angles are required. A dynRinically

stable ship will tend to steer much more easily; it will require smaller

rudder angles

and

much less frequent application of the rudder to stay

on course. ifence, from the point of view of steering and course keep..

ing, a dynamically

stable ship is far superior to a dynamically unstable

ship.

In general, for the same rudder size and rudder

angle, a

dynamically unstable ship will turn in a tighter circle than a dynamically stable ship, and the larger the stability of a dynamically stable ship, the larger its turning circles will be. A small turning diameter is very often required

from a tactical viewpoint. It appears desirable therefore, as a general rule for designing, to keep the dynamic stability small, but on the stable

side

*This may not be absolutely necessary, since the use of a forward rudder might enable a tight turn with a ship which baa high dynamic stability.

CONCLUDING REMARKS

This Memrandum constitutes an attempt to further clarify ground

already covered. Out of the work to date, the degree of dynamic stability

or instability stands out as a very impàrtant consideration from the point of view of the steering and maneuvering of ships.

The degree of dynamic stability or. instability also serves adequately to conpare. the ".nherent" behavior of different designs. Specifically, it

is an index of comparative behaviàr if the same nitia1 disturbance, say .10 _{of yaw or an arbitrary small angular velOcity, is assumed fOr;áll. ships.}

However, it does not necessarily provide an adequate comparison between

ships from a practical point of view, Initial disturbances are produced by exterrjal forces and the same external force acting for the same length of time on-different ships may relult in different initial disturbances, The final course deviations will then no longer be measurable by the

de-grêe of dynamic stability, To be more .pre.oise., if a destroyer and a battleship are given the same initial disturbanoo, thèdistràyer, which

is more stabledynamioally, willhave a smallerfinal _{cov.rse deviation.} However, if both ships receive the same initial impulse, the battleship, which is much larger, will not be given so great an initial disturbance and will therefore undergo a smaller final course deviation.

Consideration of dynanto stability Is necessary baekgro d for taking the next. step.. The zext step, however,

should be clearly the

study ofthe response of ships to various external imülses and initial disturbances, both with rudders amidships and when steered, Work is in progress on this next step.

are:

fl'Lfl

+r(+Cj(3

-C>

n.Q.'±CKfl +C(3

=-C

the coefficients being defined in Reference 1.

APPENDIX I

MATHEMATICAL FORMULATION OF DYNAI!IC STABILITY

From Reference 1 we find that the equations of motion of _{a ship}

(3 = (3e'%5

₊

_(3ae5,

Q =

+

(lateral forces),

(moments),

A necessary and sufficient condition for the ship to be dynamically stable

is that the real parts of _p1 and p be negative so that (3 approaches

zero and

£1 approaches zero.

Inserting the general solution in the equations of motion, they become

I

rrt.C)

+ (rn2p + C)(3

=

0,

(n.p

--

C)c1 +

_CrT

(3 =

where L 1,2,

For these equations to have a non-zero solution, the determinant of

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-21-(1)

(2)

T TM 74 - 22' the coefficients of

flPL

4Cç

or

rLrrt2pt2'+ (rLC+

rrtCK)p + (CCK

- rn.C)

The solution of this equation is:

-(n.C*

2rtrr

must vanish; thus:

+ CQ,

Cm.

4rtnta (ClCKrnC

(3)

where

Pt

s associated with the plus (+) sign, and p is associated with the minus (-) sign.

In order for the real parts of p and

_Pa

to be negative,

(ncz+m..a,cK

>

(cc

-\

arrrL.

By the definition of fl.. and

ma,

fl..>O and

flta)O ,

since fl..

is proportional to the moment of inertia of the ship and flL2 is

propor-tional to the mass of the ship, Also, for all ships,

C1)O

and

C >0

so that the first condition is automatically satisfied, Hence, for a ship to be dynamically stable, it is necessary and sufficient that:

Consequently, increasing C. and C and decreasing

_rn.

and C

tends to increase the dynamic stability. This may be underitood from the

mechanism of dynamic stability as given in the third section of the memorandum, since = rrt1 - C_F

The above is a mathematical formulation of dynamic stability.

Thus, a ship is defined to be dynamically stable if, for that ship,

(CICK -nt. C) >0.

Proof of Definition 1

The general solution of the equations of motion is given in

equa-tion (2) above. If we define

0(s) =e0+f5n.ds,

0 then the course angle is given by:

=O+(=(O3_--L __2)+f(p13i+fli)ePIS

e5.

If the ship is dynamically stable, the real parts of _p1 and

_Pa

are negative so that:

or the final course makes an angle with the initial óourse which is a

constant and is usually small. The quantities

Q

_{and Q}

_a depend on the initial conditions (or on the disturbance given the ahip9 see Ref.= erenee 1..), If the ship is dynamically unstable, however, the real parts of either p1 or

_Pa

are positive, so that

4'

(the final course. angle) continues to increase

in

time, the course being a spiral. Actual-ly, because of non-linearities in the hydrodynamic constants, the ship

TM 74

TM 74 24

-where

C

fraction of ship's length between

C

_K the CG and the rudder stookD

the ship may turn against its rudder even though it is dynamically stable. will eventually wind up into a steady turning circle0

Proof of Definition 2

From equations (1) we see that in a steady turn9 since (3' fl ₌

0,

CLC+CC'k

(5)

CCç

It was shown above that a necessary and sufficient condition for

a ship to be dynamically stable is that:

(CLCK -

0.

Since

C,

,

C

,

and are positive for all ships and

C

is generally positive (most ships are statically unstable), we see that for a dynamically stable ship Q will have the same sign as so

that, the ship cannot turn against its rudder no matter what initial con-ditions are imposed on the ship. On the other hand, for a dynanióally unstable shipD(C2C-

_rn.Cm)<

0 and' the ship may turn against its rud-der (depending on. the initialconditiona).

From relation (5) it is seen that if a ship is very stable

The subsequent motion of a ship in a, steady turn with a certain (3

and

fl

but with the rudder suddenly set amidships is clearly the same as the resulting motion of a ship with zero rudder when given initial

conditions the same as the value of (3 and Q in the steady turn. The

resulting motion will be a straight line course for a dynamically stable ship, while, as has been shown, a dynamically unstable ship will continue

in a circular path.

TM 74 - 25

However, this is equivalent to the requirement that the center of pres-sure of the body forces lie aft of the center of prespres-sure of the rudder forces9 or, roughly, aft of the rudder stock. This requirement will

probably never be fulfilled by any ship..

It should be further noted from (5) that if the numerator remain.s

an, then the larger the value. of C1 C - rn. C

_(which

general-ly implies greater dynamic stability), the larger the turning diameter;

this tendency would be expected intuitively.

TM 74 - 26

RELATION BEFWEEN DYNAJVIIC AND STATIC STABILITY

Perhaps the eas jest way of examining the relationship between

static and dynamic stability is from the requirement for dynamic stabil-,

ity:'

(c2C- rrtC)>O,

or

(CICK

+ C C)>Q.

For a statically unstable ship _C

> 0

so that if

_C

C

_K and rn.

remain constant,, decreasing C (making the ship more stable statically)

will increase the dynamic stability. Hawever, the other constants may be

altered without, changing

_Crr

_{so that the static stability cannot serve} as a measure of' dynamic stability.

For dynamic instability:

(CiCK -

n- C)< 0,

or

r

/

rYLCrn

1

"-1

If _F is the effective moment arm of the damping force aft of the CG,

'

.cK

F

and if is the fraction of the ship's length

by which

the center of

For static stability:

rrt<0'

or

'F >

0

for practically every ship, Therefore, if a ship has zero

static stability (for which

= 0 )

it must be dynamically stable.

However, if a ship is statically stable (for which

-j <0 )

but dynami-cally unstable, then, since

_{> 0}

for practically every ship,

rn1

-

_CF

CF

practically,

j<0.

Hence, for dynamic instability:

rn.1

_°F

CF

by an amount depending upon 'F and

Yj

Both of these requirements are extremely unlikely physically, _{so that}

if a ship is statically stable it will (almost) certainly be dynamically

stable.

TM 74 - 27

TM 74 28

-TYPES OF MOTION

The types of motion described in the text may be readily seen from the exponents in equation (2) of this Appendix. Thua,if

(C,C,(-rn.C)>O,

and if

CC

(n..C.+rrtaCK)2)

or a (rtCR,

-

rrt2CK)

4 n. rn,2 rn.

oscillations will be produced which will diminish in time. It is

seen

that this requires that

Cm< 0

by an amount depending 'on the constants

of the ship, or, that the ship not only be statically table, but quite

a bit statically stable.

If the oscillations are to inoreaae in alitude, it is necessary

that

(nC, #rn.eCK)(O,

arid since

>0

and.rna>O , the ship must have a negative lift coef-ficient or a negative danpiug force coefcoef-ficient (absorb energy), which

is not possible physically.

MEASURE OF DYNAMIC STABILITY

From expressions (2) it is clear that disturbinoes will die out

if

p < 0

and p <

0

It was stated that would be associated

with the (+) sign in equation (3) so that

_{Pu > Pa}

o Hence, the terms in (2) multiplied by

ePa5

will diminish more rapidly than those

multi-plied by

ePiS

.

Approximately then, we oan neglect the

eT'a9

terms

and by ta.king natural logarithms of the expressions (2), it is clear

that /p

is the number of ship lengths required for the disturbance

to damp outto

he

of its initial value.

TM 74

29

-TM 74

30

-where _CM is the restoring moment ooefficient

and

Cç

is the

damping

moment coefficient0

Now,

and

since

MOMENT AND LIFT CURVES AS STABILITY INDF(ES

The criterion for dynamic stability was shown to be:

(ccK-c))O.

We therefore inquire

into

the relation of this expression with that for

the slope of the "total-.monent.owithozero.rudder" coefficient

curve.

The siope of the "totaimoment..with.zeroriidder" coefficient curve

is given by:

MCfl)

C

fl

Since we are considering the "total moment with zero rudder" in a steady

turn9 in equations (1) we may set

or

Therefore,

m.Q +C1

=

-CKQ+Cp3 C,

or, in a steady tu.rn,

(CC

_{+ CC)}

(C+CkC

Then, where

(Cc2+CC)

(c1+C)

-

(C+CKC) -

(4m+CK'

4-Consequently, the slope of the' _{"totalmomont_wjth_zero_dder}

coefficient curve is:

-c

_rn..

CK((Cx4C)

Since for allshipsCK>O,

4>0

_{and it appears that}

, it

TM 74

31

C.C-rttCm

is seen that if the slope of the curve is positive,

(

CK-rnC)> 0,

and the ship is' dynamically stable.

However, sthce(lY

+ C /4)

is certainly not the same for all

ships but will vary, the slope of the curve cannot serve as a quanti-tative measure of dynamic stability, but only as a qualiquanti-tative means of indicating whether or not the ship is dynamically stable,

For the "total.liftwith.-zerorUdder" coefficient curve, the slope is:

(C-C)

_-

Cr+CLCtCCrn.

.

(çm.+C)

The numerator is seen to be:

Cm. +

Since the denominator is positive, it ià clear that if

(CC-YnCm)(O

(condition for dynamic instability) the numerator may either be positive

or negative, so hat the slope of the "totallift.vith-zero-rudder" coef-ficient curve is not even a qualitative indication of dynamic stability.

TM 74

DYNAMICALLY [iNSTABLE SHIPS _{STABILITY IN TURNING}

In Reference 1 it is shown that in a turn, for' a ship having a yaw angle (3 and a space angular velocity _{o the stability}

criter-ion remains in the form:

(CICK

- rrL Cm)) 0.

We may substitute in this:

LCOS

TLICO5(3O_CFO

C(-- Cç0,

[ C1

f

1

From experimental results, it appears

that:

'CD

+CDSIn. (3)

- rn.1Q0sirt(30

S

(tends to be slightly _posiive),

0

(tends to be slightly negative).

For yaw angles which are not very large,

_{rrL, CK}

and

_C

_{are unaltered,}

while

CF]

)II

I

"V Jo

TM 74

33

-T1 74

34

-The transformation is then:

(c +

The value increases fairly rapidly so that going into a tight turn increases the C1 tern. This reduces the magnitude of the dynamic

instability, and may even make the ship dynamically stable (in the turn).

,co

APPENDIX II

DYNAMIC AND DIRECTIONAL STABILITY

In the text of the memorandum a distinction was made between dynamic and static stability and both types of stability were discussed. Another

distinction should also be borne in mind - that between dynamic and direc-.

tional stability0

Direoti'nal stability implies a tendency to maintain a fixed compass heading. _{A ship can be directionally stable only if it has steering tontrol}

(either a helmsman or automatic device) that is always .attempting to keep the

ship pointing in the same compass direction. _{With steering control, the ship} will be returned to the original heading whenever it is disturbed therefrom.

The concept of directional stability is thus directly associated with steer-jug, but is a very different thing from the concept of dynamic stability.

If a ship is very stable dynamically it will tend to approach direc.

tional stability since the course deviations resulting from ordinary

disturb-ances will be very snail. On the other hand, if a ship is unstable dynain..

ioally, steering will always be necessary to maintain any semblance of

direc-tional stability.

Thus, while dynamic stability is a propertyof the ship alone, directional stability depends upon the ship's being steered. The problem

of both steering and directional stability will be discussed in a subse-quent report. .

TM 74 35

-TM 74

LIST OF SYMBOL$

A

total lateral plane area, sq. ft.

B beam, ft.

CD

coefficient of drag force

CF

coefficient of damping force Cpç coefficient of dampin.gmoment

CL

coefficient of lift force

C,

coefficient of drag (C0 ), plus the slope of the lift versus yaw angle curve

(CL/3)

CM

coefficient of hydrodynamic moment in straight-line

towing

C

slope of the moment ve1rsus yaw angle curve

(CM/)

Cp

coefficient of the difiference between the propeller

thrust and the angular velocity dependent part

of the drag

lift coefficient contribution of the rudder to the total lift

moment coefficient contribution of the rudder to the total moment

D hydrodynamic drag force, lbs.

d. angular velocity dependent part of the drag force, lbs.

F

hydrodynariic damping force,. lbs.

H draft, ft.

10

polar moment of inertia of the ship, taken about the CG, slugs ft.2

polar moment of inertia of the displaced water, taken about the CG, slugs ft.2

I

10 + kI

polar moment of inertia of the virtual mass of the

ship, taken about the Cc- of the ship, slugs ft.2;

virtual mass means the mass of the ship increased by the mass of the entrained water.

K hydrodynamic damping moment about the CG, lb. ft. coefficients of accession to inertia

L

hydrodynainic lift force, lbs.

length of loadwaterline, ft.

M hydrodynamic moment in straight-line towing about

T 74 38

-M0

Ma

n-LI ± LL

V

if

mass of the ship, or the displaced water, slugs. virtual longitudinal mass of the ship, slugs virtual transverse coefficient of the the ship coefficient of the ship coefficient of the virtual mass of

mass of the ship, slugs virtual longitudinal mass of

virtual transverse mass of the

polar moment of inertia of the the ship, taken about the CG

propeller thrust, lbs..

distance travelled by the ship in terms of ship

lengths

time,. see.

component of velocity along the ship's longitudinal

axis, ft./séo.

tangential velocity of the ship, ft./sec.

component of velocity perpendicular to the ship's

longitudizal axis, ft./seo.

axes parallel and perpendicular, respectively, to

ship centerline

first derivative with.respect to time, t second derivative with respect to time, t first derivative with respect to distance, a

drift, or yaw, angle of the ship at the CG, radians function of both drift and rudder angle

moment arm in terms of ship lengths

distance by.which the center of pressure of the rud-der force lies aft of the CG

effective moment arm of the damping force aftof the

CG

distance by which the center of pressure of the lift forces lies forward of the CG

ship displacement, lbs. or tons

movement of the ship's path in the transverse direction

in terms of ship lengths

angle between the ship's longitudinal axis and some fixed direction in the horizontal plane, radians rudder angle, deg. or radians

mass density, of water, slugs/ft.3

angle between some fixed direction in. the horizontal

plane and the tangent to the path of the ship,

radians

reciprocal of the radius of the 8hip's turning circle

in ternis of ship lengths

TA 74

-39-TM 74

41

REF RENCES

1. Members of the Staff of the xperimenta1 Towing Tank and Dr0 L.

16

Schiff, Conoultent: "Analysis of Ship Turning

end Steering, with Statement of Theory0" bcperimental Towing Tank Report No0 288, August, 1945. _{(CONFIDENTIAL)}

2 Sutherland, William I: Investigation of Lateral Forces

Acting upon Models of Representative Naval Vessels in Restrained Motion on Straight Course and in Unrestrained Steady Turning." Experimental Towing Tank Report Noe 291,