Notes on rolling in longitudinal waves

by

J. E. KERWIN

I - l . Introd lictioii

I f a ship is orientated w i t h its centerHne perpendicular to the crests of a 2 dimensional wave system (see Fig. 1) i t has been f o u n d by G r i m that under certain conditions extremely heavy rolling w i l l result [ 2 ] . I t is assumed that the ship is per-f e c t l y symmetrical and that the waves are regular. Thus i f the ship is in the upright position the forces produced by the wave w i l l only tend to produce

F l S

Fig. 1

pitching, heaving, and surging. A n additional effect of the wave, however, is to change the position of the transverse metacenter. As a result, the initial transverse stability w i l l vary w i t h a period equal to the period o f wave encounter.

This can be seen by comparing the t w o extreme positions of the wave, namely, w i t h a crest at the ends and w i t h a crest amidships. For a normal ship f o r m w i t h flared sections at the ends and wallsided sections amidships i t can be seen that the inertia of the water plane is d i f f e r e n t i n each case. I t is greater than the still water value w i t h the

wave crest at the ends, and less than the still water value w i t h the wave crest amidships. This is sub-stantiated by calculations, and f o r a sinusoidal wave the variation i n the position of the metacenter, KM, is roughly sinusoidal.

I f the ship is now given an initial angle of roll due to some small disturbance, i t w i l l roll at its natural period u n t i l the energy provided by the disturbing force is dissipated i n damping. However, i f thc period of stability f l u c t u a t i o n and the natural rolling period of the ship have the right ratio, i t is possible that thc i n i t i a l rolling angle w i l l n o t only be sustained, but w i l l build up to almost unbeliev-able proportions.

This is shown later i n the solution of the d i f -ferential equation of motion representing this situation. However, a qualitative explanation is as f o l -lows:

Suppose that the stability changes w i t h a f r e -quency twice the natural rolling fre-quency of the ship, and that the phase of the roHing and the stability change is as shown i n Fig. 2.

Starting w i t h the initial angle i/'u> during the swing back to the u p r i g h t position the stability is higher than average, and the stability moment is i n the direction of the motion. A f t e r crossing zero, the stability is now opposing the motion, but its value is now low. Therefore v'l is greater than i/^d. R o l l i n g back to zero again, the stability is now aiding the motion and its value is high. Thus the rolling motion builds up continuously.

Since w o r k must be done against the damping forces, energy must be supplied f r o m somewhere to make the motion possible. This can also be shown w i t h a simple qualitative argument. The stability couple is proportional to G M , and the problem

598

remains unchanged i f G is made to vary rather than M . Assume therefore that the entire weight of the ship is concentrated at G and is moving up and down in the vertical axis o f the ship. So long as the ship IS not rolling, no net energy is required to do this since the energy required i n the up-swing is gained back i n the down-swing. However, once the ship rolls, the weight is no longer moving ver-tically, and the energy required is not necessarily zero, but depends on the phase of the rolling.

E q u i l i b r i u m is established when this energy is equal to that dissipated i n damping. I n the actual case, where M varies, this energy is provided by the wave. This, of course, is not a very rigorous argu-ment, but i t serves to give some physical explanation

for this type of rolling.

Solutions of the equations of motion show that the period of stability f l u c t u a t i o n must be a h a l f -integral multiple of the natural period. That is:

T / r . , = O . J , 1.0, 1.5, 2.0, ( 1 ) Where T is the period of stability f l u c t u a t i o n ,

and T, IS the natural rolling period of the ship.

2. Conditions for resonance at sea

I t is of interest to see the conditions under which such resonances could be encountered. The period

of wave encounter is given by the relation:

X

(2) where i = : wave length

r = wave velocity = ../"^-^ M ) V = ship speed

y is added to c f o r head seas, and subtracted f o r f o l l o w i n g seas. These two relationships are shown graphically in Fig. 3. For rolling to take place, the wave length must be close to the ship length ( i n order f o r the stability f l u c t u a t i o n to be noticable) and the period of encounter must satisfy the rela-tion ( 1 ) . Thus f o r any given ship the speeds neces-sary to produce this type of motions can be quickly determined.

For example, i t can be seen that a cargo ship 400 feet i n length w i t h a rolling period o f 16 seconds w i l l be in the second resonance region i n a f o l -lowing sea w i t h a ship speed o f 12 knots. A fishing vessel w i t h a length of 100 feet and a rolling period of 6 seconds w i l l be i n the f i r s t resonance region when travelling into the seas at a speed of about 6 knots. Obviously there are any number o f pos-sibilities, particulariy i f one includes the case of the ship travelling at an angle to the wave crests.

Once certain s i m p l i f y i n g assumptions are made, it is possible to calculate the change i n stability produced by a wave. The properties of the wave itself are obtained as follows:

The velocity potential f o r 2 dimensional, i r r o -tational, deep water waves of small amplitude at one given instant of time is: '

0 = ^ ff'-^cos kx (4) (see Ref. [ 1 ] )

where

k = ), = wave length r = half wave height, c = \

The pressure given by Bernoulli's equation is:

P = Q^-Q&^ (5) I f the vertical velocities in the wave are

con-sidered small compared to c, the pressure is:

P = Qgr ek-- cos kx — Qgz ( 6 ) The wave profile is obtained f r o m the condition

that the pressure is zero at the free surface, giving:

2TIX

^0 = cos ( 7 ) The effective water density is taken as the sum

of the static and dynamic pressure gradiants. Thus:

yj5 = ^ r = [kre>" COS kx — 1] . ]/4. (8)

where y = Qg

I t is assumed f u r t h e r that the variation of yn w i t h d r a f t can be neglected, and therefore that is constant and is evaluated at the average mean d r a f t . Since the total variation of yn w i t h d r a f t is around 4 % , the use of the average value introduces very little error.

I t is also assumed that the pressure distribution i n the wave is unaffected by the presence of the ship.

I f the section area at a point .v is denoted by A, an element of displacement is:

dV = ( f x ( 9 ) The center of bouyancy B is at the centroid of

A since the variation of ;'/; w i t h d r a f t was neglec-ted. The local metacentric radius BM is given by:

([[_ _ dx _ dV~ Adx ~ t A and the local height of the metacenter KM is:

KM

-hll

KB

4-?>A yiiAdx .. ( 1 2 ) The actual calculation is as follows: A plot of A vs. z (Bonjean curve) is made f o r 2 0 stations along the h u l l . The value of KM is also a f u n c t i o n of z alone, and an alogous set of curves can be made f o r KM. For any position of the wave relative to the h u l l , the value of }'/.; at each station can be calculated. The hull is then "balanced" on the wave in the usual manner, the values of KM are read f r o m the plot, and the integral (12) carried out by Simpsons rule.

Since the end result is to f i n d the change i n GM, which is say 10 % of KM, the calculation must be done quite accurately to produce good results. This coupled w i t h the trial and error process of "balan-c i n g " on the wave results in a rather laborious process.

The results f o r 2 ships are shown i n Fig. 5. The f i r s t is f o r model B, and is calculated f o r a wave w i t h a length to height ratio of 17. The change i n stability, expressed as a percentage of the operating G M of the ship is around 60 % i n this case. The second is f o r a fishing vessel, model A , f o r a wave of length to height ratio of 20. The stability f l u c -tuation i n this case is around 40 % . I n both cases, the f l u c t u a t i o n is roughly sinusoidal.

I t is of interest to note that the average value of G M in a wave is around 10 % higher than the still water value i n both cases. Thus the natural rolling period i n longitudinal waves is not the same as i n still water.

A t h i r d set of calculations were carried out i n order to make a comparison w i t h experimental results. T o c o n f o r m to the conditions of the expe-riment, no t r i m and heave were permitted, w h i c h also simphfied the calculations greatly. The m a x i -m u -m and -m i n i -m u -m values o f KM were calculated for several d i f f e r e n t wave heights, and Fig. 6 shows the result. The comparison of these calculations w i t h experimental results w i l l be discussed later. One of the plots o f KM versus d r a f t is shown i n Fig. 4 f o r the forebody of model A . There i t can be seen immediately that i n the region of the load waterline KM remains more or less constant i n the middlebody, but increases rapidly w i t h w i t h d r a f t at the ends.

I I I - l . Solutions of the equations of Motion A very general f o r m of the equation representing this motion is:

w

v//^

+ R {y>> t) ^ 0 ( 1 3 )

9 I O I I 1 2 13 14 1 5 1 6 1 7 1 8 19 2 0 2 1 2 2 2 3 2 4 K M - C M . M O D E L A C U R V E S O F K M V S . D R A F T FOR F O R E B O D Y - f s T A . I O T O 2 o } F i g . 4 where;

ƒ = total inertia o f ship and water. W = damping term, made up of viscous

and wave m a k i n g effects,

i l = the stability, which depends both on angle and time.

The object is to f i n d the simplest f o r m f o r the above functions w h i c h w i l l give a s u f f i c i e n t l y accurate representation o f the motion.

2. Solufion with no damping

The simplest case is when the damping is zero, the righting arm varies linearly w i t h yi, and the stability f l u c t u a t i o n is sinusoidal. The resulting equation is:

/ ? l + rV (GM-^sln2o>^W = 0 (It \ Jé I

I 7 K M • C M I 6 V \ ^ N V — — M O D E L B^ }^ • L . B . P . A V E R A G E K M IN WAVE C A L C U L A T E D C O S I N E O 2 S T E R N 6 B I O 1 2 P O S I T I O N O F W A V E C R E S T I B 2 0 B O W I 8 K M - C M .

V

\ \ N / —N V \ \ \ / , /

/

— ^ \ ^—- _{7 ^} • '

/ /

A. N K A ^ \ N I O I 2 F i g . 5 M O D E L A Jl •= L . B . P . . A . = 2 0 H - ^ A V E l ^ A G E K M IN WA V E S T I L L W A T E R K M 1 8 2 0 5 0 v' .

/

/ y 3 4 W A V E H E I G H T 5 C M . C H A N G E I N S T A B I L I T Y W I T H W A V E H E I G H T F i g . 6 C A L C U L A T E D F O R S I N E W A V E D E R I V E D F R O M E X P E R I M E N T A L R E S U L T S W I T H F L Y W H E E L S - D E R I V E D F R O M C A L C U L A T E D M O T I O N W I T H F L Y W H E E L S I . S . P . - Vol. 2, No. IG - 1955

6

6 0 2

U N S T A B L E

2. Wo

The stabihty f l u c t u a t i o n is taken as sin 2 w/ so that i n the f i r s t resonance region, the natural frequency O J O is equal to co. Also making use o f the equation f o r the natural frequency:

2 y V • GM

(15)

AGM

and denotmg by b, equation (14) may be re-written:

^ + Wo' (1 — ^ sin 2 w/) y. = 0 (16) This is k n o w n as a Matthieu differential equa-tion, and is f o u n d to have many physical applica-tions. A rigorous solution shows that there are regions of stabihty and instabihty depending on the constants o f the equation. A sketch of these regions is shown i n Fig. 7. (See reference [ J ] f o r a detailed derivation.

Thus i t can be seen immediately f r o m the dia-gram that when W o / o j = 1, 2, 3, 4 , . . . , or equivalently when T/Ts = /a, 1, V„ . . . the solution is unstable. I t can also be seen that as the stability f l u c t u a t i o n increases, the unstable regions become wider.

A l t h o u g h the rigorous solution is quite involved, a simple approximate solution yields perfectly good results. I n all the f o l l o w i n g sections, only the f i r s t resonance region w i l l be considered. Solutions f o r the other regions could be f o u n d i n a similar manner.

A series solution is assumed i n the f o r m :

= sin cDt - f cos wt + sin 3 co/ + . . . , I t is desired to determine the value oi a — and i> f o r which a stable solution is possible. T a k i n g only the f i r s t 2 terms:

dt^

f = A^ sin o)/ + i?, cos cut

= — 0)^ A^ sin cot ~ (jcP- cos mt

2

<iJ2-Substituting i n equation ( 1 6 ) :

— co^ A^ sin cat — co^ i?, cos cot

•\-+ (1 — b sin 2 oot) (A^ sin coi •\-+ B^ cos cot) = 0 The coefficients of sin 2 cot sin mt and sin 2 cot cos OJt can be expressed i n a series w i t h ordinary trigonometric substitutions, g i v i n g the result:

sin 2 cot sin co/ = '/^ cos co/ + . . . . sin 2 CO/ cos co/ = '/^ sin co/ 4- . . . . Equating the coefficients of sin ojf and cos cot:

A, (co,' _ co') _ C A ^ o V ) = 0 A « - co') C A ^ O = 0

(17) In order f o r there to be a n o n - t r i v i a l solution o f these 2 equations, the determinant of the c o e f f i -cients of A , and Bi must be zero:

(cu,' - co') ( - ' A b CO,')

( - >A ^ C0„') (CO,

Expanding (18)

CO'' — 2 CO ' co' _co"

co' (18)

0

2

< ±

A/4 co/ - 4 (1 - V , /;2) co/ _ 2 -= < [ l ± / ' / 2 ] .' (19) This corresponds i n Fig. 7 to 2 lines going through the point cu„yco= = 1 on the vertical axis w i t h slopes of 1. This checks so lortg as b is small, which i n this particular application is always the case.

I f zl G M / G M = 3 0 % , then ^ = 0.15 and periodic solutions can exist when OJ/OJ, = .964} and 1.0397. Thus according to this solution outside the unstable region the roll angle can only be zero, whereas inside the region i t can only be i n f i n i t e . A t the borders of the region, periodic rolling is possible w i t h any arbitrary m a x i m u m angle. A resonance curve f o r the various approximate solu-tions is shown i n Fig. 8.

3. Linear damping

N e x t i n order of complication w i l l be to i n t r o -duce a linear damping term into the equation, which w i l l then become:

d'lp , dip

7J^ + ''1J + COo' (1 — sin 2ajt)y) = 0

(20) The proceedure followed is identical to the pre-ceeding case. Equation (17) then becomes:

A , (co^' — OJ') — B, (moj + ' / ^ b o j , ' ) = 0

A (c"ü' - co') 4- ^ , (moj - ' / ^ b o j , ' ) = 0

NO D A M P I N G -l -l —

[ d V . -mdi^

^ r' ^ J ' / '

1- b s i n 2 a j t ) ^

= 0- L I N E A R DAMPING

y di^

d t

1

1

1

1

1

1

II

1

1

1

1

1 1

1

1

1

1

C O o = 3 . 3 6 0 , T o = I . 8 7 0 Yn = 0 . 0 7 0 9 n = O . 5 5 2

1

1

1

1

C O o = 3 . 3 6 0 , T o = I . 8 7 0 Yn = 0 . 0 7 0 9 n = O . 5 5 2

1

1 /

C O o = 3 . 3 6 0 , T o = I . 8 7 0 Yn = 0 . 0 7 0 9 n = O . 5 5 2

b

= o . i s

A G M = 3 0

1

/

b

= o . i s

G M 3 0

1

1

1 /

1

1

1 /

1

1

1

1

1

1

. 9 6 9 7 . 9 8 9 9 I . O O I . O I I . 0 2 I . 0 3 l . O A X C O ^ T o T 1 1 1 1 1 , 1 1 . 9 4 5 1,926 I . 9 0 7 1.889 1 . 8 7 0 P E R I O D O F R O L L - T Flg. 8 1.851 1.833 S E C O N D S 1.814 1.795

6 0 4

and the solution of the determinant gives f o r the value o f :

C O = |co„ •A m' ± i - co/ ///' + V , + V , b' co/

• (22) A g a i n taking h— 0.15, and using the value of

determined experimentally f o r the ship i n ques-t i o n (see descripques-tion of experimenques-ts) as 0.0709 and c'j„ as 3.360, the critical values are f o u n d to be:

co/oj„ = 0.9632 and 1.03 53

I t can be seen, therefore, that linear damping has practically no effect on the resonance curve. I n the case of rolling caused by an externally applied mo-ment (such as i n a beam sea) i n f i n i t e angles can occur only at the point when cOo/co = 1, and this angle becomes f i n i t e as soon as a linear'damping term is introduced. W i t h this type of rolling, how-ever, i t is not possible to obtain a realistic solution w i t h a linear damping term.

4. 'Non linear damping

The next step is to consider non linear damping. I f i t is assumed that the damping term is: A'^ ^ b u t that the value of N depends on the m a x i m u m roll angle, a realistic resonance curve may be ob-tained.

This damping coefficient was f o u n d experiment-ally to vary linearly w i t h the m a x i m u m roll angle •^>. The damping w i l l then be i n the f o r m A'' = m 4- n ipo- Substituting this f o r i n the determinant i n the preceeding case, equation ( 2 1 ) , the f o l l o w i n g relation is obtained:

z m , w

^ ^» + P •A

oj'Y

Solving this f o r i / ^ gives:

O J Wo = A t resonance then:

+

{by4 - 1) 1 -= 0 ( 2 3 ) (24) where x = to/co,

X = 1, and the maximum angle is

Wo b m Tn

HI

n (25)

A resonance curve given by this expression is also shown i n Fig. 8.

A n important result of this solution is that there is a critical value of b below which no motion is possible. This value is:

, 2 m

h = (26) Further refinements of this solution could be

obtained by taking 4 terms of the series instead of 2, and also by adding a non linear term to the

Fig. 9

r i g h t i n g arm expression. However, f o r the range of angles encountered i n the subsequent experi-ments, the inclusion of these terms w o u l d add only a very small correction.

5. Solution from energy considerations

A n expression f o r the roll angle at resonance can be f o u n d very simply f r o m energy considera-tions, and this approach may give a clearer physi-cal picture of the situation. For clarity, i t is assumed that G varies rather than M , there being no difference i n the result. The ship is assumed t o roll about the average position of G. A t any instant of time, the distance of G f r o m the axis is:

b GM, = b, GM, sin 2 cot

I n the f o l l o w i n g sections only motion at

reso-nance is considered; thus: i d the

script is omitted f o r clarity.

A n d at resonance the m o t i o n is approximately: If = fo sin cot

Reference to Fig. 9 shows that a rolling moment exists whose value is:

M = — GM, y\7 bs'myi ^ ~ GM, y\/ bip = = — GM, b,'if,y\J sin co/ sin 2 o)/ =

= — 2 GM, b,ip,y\J sin' oot cos cot The w o r k done by this moment is:

In],., l7x\u:

M d\p= —2 b, tp,' 7 V Wu GM / sin^ co/ cos^ ootdf =

2 b, y>,' yV 00, GM

j

{sin' oot — sin'' co/) dt =

0

_nb„y\/ ip,' GM

(27) The damping moment is:

rdip

JN

an d t h e energy is:

2 n/m

n 1(11 2jt/(u

JN - j j d f = J N ' f , ' to,' / cos^ oot

= Jnoo,Ny>,' (28) For equihbrium, the t w o energies must be equal,

so that: 2 0) J 2 f r o m w h i c h : 27/ /II (29) w h i c h is the same as the value given by the d i f f e -rential equation,

7. Non steady state sohition

I t was observed while conducting the experi-ments of this type of rolling that the angle built up extremely slowly. I t therefore seems i m p o r t a n t to obtain an approximate solution f o r the transient period.

To s i m p l i f y matters, only motion at resonance w i l l be considered.

The damping can no longer be considered a f u n c t i o n of the m a x i m u m angle (since this is now a f u n c t i o n of t i m e ) and w i l l therefore be considered as:

w ' + « 1 I -yj I (30)

The coefficient w i l l then be evaluated to produce the same steady state angle as before.

Using the method of equating energy, the applied energy is the same as before, and the damping moment is:

> ' ' 3 - I + / ' " I H S ( 3 . )

The energy f o r one half swing is:

0 0 (32) D = D A M P I N G ENERGY > O DC UJ z A P P L I E D ENEP,GY ! — S T E A D Y S T A T E A N G L E 9i

= /|/;/'y)„'cü'ƒ cos'wtdi-{-ii'ff/m'I cos^co/sinco/rt'/j =

(33)

= /

The applied energy is:

71 yig' o) 2 + Vo' w j

and equating the t w o gives the result: Vo '71 , O) T ^ — r

4 «' (34)

Comparing this w i t h ( 2 9 ) , the new damping coefficients are f o u n d to be;

III' = ;//. n' = -r 7tii

4 (35)

A semi graphical method f o r obtaining the n o n steady state region w i l l now be attempted. Fig. 10 shows a plot of applied energy and damping energy f o r a particular case. The steady state angle is where the two curves intersect, and the shaded area between the two curves is the energy available f o r accelerating the motion.

The damping energy f o r one half swing is: „ 71 JllA CO w ' 2 r , , D = „ ^ " 4 - o ƒ « ' CO % 3 and the applied energy is:

A -¬ The difference is:

AE = 71J 00 y),' Ttjoo'b y),' (33) (36) III' (37) The potential energy at any m a x i m u m angle is:

P.E. = y ^ GMy,,=Jo)'yi, . . ( 3 8 ) I t is assumed that the angle increases d u r i n g each

half swing i n such a way that the increase i n dam-ping and potential energy is equal to the excess energy A E. I f the increase i n angle is small, the increase i n energy may be approximated b y :

dD ^ , d(P.E.) , AE = ~r- Aw ^ S '- A y> dy) dy> anc A y) AE dP djP.E.) dy) dy) dP d yi and 7cJoo%p, 2Jn'' oo y), d{P.E.) Fig. 10 d y) (39) (40) (41) (42) I . S . P . - Vol. 2, No. 18 - 1955

6 0 6 3 ' W • I R A D I A N S ) Substituting (41) and (42) i n ( 4 0 ) : 2 A y) •Wo w '2 -"Wo' nm + 2;/yio 4- (ü/y), ••• (^3) This can be evaluated numerically, and a plot o f A ipo vs y>a is shown i n Fig. 11.

From this curve, another curve of vs . t can be constructed simply by drawing aline w i t h a slope of - 1 down f r o m the curve to the \p axis, giving the new angle f o r i/^ swing later. This is repeated u n t i l the whole curve is obtained. I t should be noted that this is a l i m i t i n g process at both ends, that is, the angle approaches the steady state value asymtotically ( w h i c h would be expected) but also that i t leaves zero asymtotically. Therefore, i f the motion is started by a minute disturbance, the build up time is extremely long. O n the other hand, i f the i n i t i a l disturbance is large, the build up time is considerable shorter. W i t h rolling caused by an external moment, on the other hand, the value of the initial angle is not nearly so important.

I V . Experimental results

I t was obviously desirable to check the calcu-lations i n the preceding sections b y means of ex-penments. These are now described chronologically.

and the results of the successful experiments

presented. are

1. Change in stability in a wave

I f the period of encounter of the wave is con-siderable longer than the natural period o f the model, the response to any forces produced by the wave w i l l be practically static. Thus i f the model is given an i n i t i a l heel angle by means of an inclining weight, the angle w i l l be inversely pro-portional to G M , and the variation of the angle w i t h time w i l l give the stability f l u c t u a t i o n directly. The encounter period can be made suf-ficiently long by having the model travel w i t h the waves w i t h a speed slightly slower than the wave speed. A sketch of the set up f o r this experiment appears i n Fig. 12.

The pointer was about 1 meter long, and the assembly was constructed of alumium t u b i n g w i t h s u f f i c i e n t bracing to make a very s t i f f struc-ture. As can be seen f r o m the sketch, the scale is free to heave only, the pointer is free to heave and roll, and neither are permitted to pitch. The scale therefore measures only roll readings were made by photographing the scale.

Since the model must travel at a relatively high speed to keep up w i t h the wave, the change i n stability w i t h speed i n still water was f i r s t

de-I N C L de-I N de-I N G W G T .

Fig. 12

termined. This is shown i n Fig. 13. GM is seen t o be constant up to a speed length ratio of about .8, but increases rapidly as the speed is increased. Since the model must travel at a speed length ratio of 1.2, i t can be seen that the stability is very dependent on speed.

Fot this reason i t was not possible to obtain a good value f o r the change of stability in a wave

w i t h this method. Since the carriage did not travel at a constant speed, the u n k n o w n variation pro-duced by the change i n speed was as large as the variation i n stability produced by the wave, m a k i n g the results useless.

A n attempt was also made to obtain the change i n G M by keeping the model stationary, b u t ad-justing the natural period so that i t was V 2 longer

1.6 1.2

/

/

i

• •

O . : .1 M O 6 . 8 I.O 1.2 1.4 1.6 D E L S P E E D - M E T E R S P E R S E C O N D 1 1 1 2 u 0 . 8 2 0 , 4 . 4 . 8 ' i , 2 S P E E D - L E N G T H R A T I O - _y_ C H A N G E I N G M W I T H S P E E D - S T I L L W A T E R F i g . 13 1.6 I . S . P . - Vol. 2, No. 16 - 1955

6 0 8

Fig. 14

than the encounter period. A t this point the response is also u n i t y (see Fig. 14). However, due to the high G M required i n his case, the motion was so small that the influence of small disturbances w,is so great that accurate readings could not be obtained.

A new approach was therefore decided upon. The idea was to produce rolling i n the f i r s t resonance region w i t h a k n o w n value o f stabihty f l u c t u a t i o n . The same experiment would then be conducted w i t h waves, and the change i n stability would then be f o u n d by comparing the results of the two series of experiments. I n order to improve the accuracy of the measurements, the model was not free to pitch and heave, being held by t w o ball bearings located on the longitudinal axis through the center of g r a v i t y of the model. These bearings were atta-ched directly to the rails w i t h heavy steel angle bars. A n alumium pointer was attached to the model, and a scale was mounted on the side of the tank. The scale consisted of white numbers on a dull black background, and the length of the p o i n -ter was such that one degree was 3 centime-ters on the scale. A small light was attached to the end o f the pointer to facilitate reading and photographing. The accuracy o f this system was 0.2 degrees w i t h visual measurements and 0.02 degrees w i t h photo-graphic measurements.

The k n o w n f l u c t u a t i o n of G M was produced by 2 counter-rotating weights i n the longitudinal vertical plane of the model. These were driven by a motor, through a set of reduction gears.

The period of roll was measured w i t h a relay controlled stop watch. The relay was set o f f when a pomter on the model entered an electrolytic solution, thus completing a circuit. The pointer was adjusted so that contact was made i n the u p r i g h t position. When 10 swings were averaged the accuracy of this set up was ± 0.002 seconds.' A description o f the various experiments w i t h this equipment is as follows:

2. Da III phi g

• The value of the damping coefficient was obtained f r o m an extinction curve. The model was released f r o m a given angle, and the inclination scale and illuminated pointer were photographed The camera shutter was held open f o r 5 complete swings, thus producing a continuous arc on the f i l m . However, at each successive m a x i m u m angle.

the velocity of the light on the pointer was zero, producing a very clear dark spot on the f i l m . The negatives were then projected, and the angles read to ± 0.02 degrees. Some of these extinction curves are shown i n Fig. 1 J.

The value of damping is obtained as follows: The equation f o r free rolling is:

J - j f i + "^^-jj + y V Gy)f y, = 0 and the solution is:

— Wl 'jit p 2 J C O S 2jU T (44) (45) where T is the natural period and the initial angle. The next m a x i m u m after xp, occurs at a time t = Til. Thus:

and the difference i n angle is 4.7 V = Vo — = f r o m w h i c h : — WT = 1 - A ip _{= 1} — WT p * J a WT 4 / = /// (1 - a)

Since r = — the dimensionless damping coef-ficient is:

IF 2

7 - = - - / / / (1 ₍₄₆₎

A plot of a vs. ^! has been made f r o m the results of 11 extinction curves (see Fig. 16). There is some scatter i n the points since this is i n e f f e c t a differentiation. However, since a is small, the value 1—a is determined quite accurately. A straight line seems to f i t the data quite well f o r the range of angles obtained. I t should be noted that the results of the f i r s t t w o swings of each r u n alwavs he below the line. (See the dotted lines on the p l o t ) . This is probably due to the f a c t that the water surface is calm initially, whereas after a few swings, a more or less steady state wave system is produced. O n this basis, these points were disregar-ded when the mean line was drawn. Using the mean values of a, the dimensionless damping coef-ficient is plotted against angle i n Fig. 17.

Fig. 15

6 1 0

)

14 I S 16 17 18

3. Rolling with a known change of stability Rolling resonance curves were determined experimentally, the change i n G M being produced by t w o counter rotating weights. Fig. 19 shows the m a x i m u m angle at resonance f o r 5 d i f f e r e n t values of stabihty f l u c t u a t i o n . The theoretical curve is also shown, and i t can be seen that the theoretical angles are somewhat greater.

A t this point i t was observed that the angular velocity of the flywheels was not constant d u r i n g a revolution. The flywheels were driven by a syn-chronous ,,Magslip" motor via a set of reduction gears, (see Fig. 18).

The field o f the „ M a g s l i p " i n the model is

con-2 O S

2 0 lp Fig. 17

nected electrically to the f i e l d of a similar motor at the side of the tank, which i n t u r n is driven by a large D . C . motor. The motor i n the model, there-fore, follows the motion of the D . C . motor.

When i t was suspected that the speed of the flywheels was not constant, the speed of the D . C . motor was f i r s t checked. A variable capacitance pick-up was located eccentrically on the heavy flywheel. This produced a sinusoidal signal w i t h the same frequency as the flywheel. This signal, after amplification, was applied to the vertical plates of an oscilloscope, and the signal f r o m a sine wave generator was placed on the horizontal plates. When the t w o frequencies were equal, the eliptical pattern was produced on the screen. I f the motor speed changed, the frequency of the t w o signals was no longer the same and this could be observed immediately.

Such observations indicated that the motor speed was quite constant during the course o f one revo-lution o f the excitator, but than a gradual change of speed w i t h time took place.

I t was then attempted to observe the speed of the excitator directly. This was done i n the f o l l o w i n g manner: A black disc on w h i c h there were located a series of transparent radial slits located every 5 degrees, was placed on the excitator. This disc was made by i n k i n g the radial lines on tracing cloth, and then m a k i n g a contact p r i n t on a piece of heavy negative f i l m . The disc then came out black, and the slits, transparent. A small light and a

ger-manium photocel were located on opposite sides of the disc. The photocel thus produced an impulse each time a slit went by. This signal was applied directly to the vertical plates of a 2 beam oscillos-cope. A signal of k n o w n frequency was connected to the other beam to serve as a time base. The result was then filmed.

Runs were made w i t h and w i t h o u t a load on the excitator. The angular velocity is inversely propor-tional to the spacing of the photocel impulses on the f i l m , and a plot of this appears i n Fig. 20. The accuracy is not very great, but i t is sufficient to get some idea of the behaviour of the flywheel. For no load, there seems to be no periodic f l u c t u a t i o n when the results of several revolutions are

super-imposed. W i t h a load, however, the results of 3 successive revolutions show a clearly periodic pat-tern. This consists of a basic sine wave w i t h a period of one revolution, and a rather violent vibration w h i c h occurs shortly after the weights have passed the top position. This vibration was also heard very distinctly. This was due either to motion i n the couplings and back-lash i n the gears, or vibrations i n the „ M a g s l i p " motor. I n any case

it occurred only in the components i n the model, since the speed of the D . C . motor appeared to be constant.

From Fig. 20, the f l u c t u a t i o n i n speed was estimated to be 8.5 % , and the correction i n the rollangle was calculated theoretically. The effect

6 1 2 D I R E C T I O N O F R O T A T I O N + 1.2 -+ • S P E E D F L U C T U A T I O N S - N O L O A D F i s . 20

of the vibrations on the motion was not considered. The corrected roll angles are plotted i n Fig. 19, and i t can be seen that the theoretical and experi-mental results are i n good agreement.

Flg. 21 shows t w o resonance curves, both theore-tical and experimental, and i t can be seen that the agreement is also good.

I t should be remarked at this point that the frequency o f the stability f l u c t u a t i o n is extremely critical. A n error o f one percent i n the frequency at resonance produces a ten percent error in the angle. W i t h a three percent error in the frequency, the motion vanishes entirely. I t therefore takes a considerable amount of e f f o r t to f i n d the resonance region at all, not to mention f i n d i n g the top of i t . This is made even worse by the fact that the motion takes several minutes to build up, so that the effect o f changing the motor speed can not be determined u n t i l several minutes later, d u r i n g which time i t has probably changed to a d i f f e r e n t value.

4. Rolling in waves

The rolling was observed i n a series of waves of d i f f e r e n t height. The natural period o f the model was adjusted so that resonance would occur at a wave length equal to that of the ship. A plot of the m a x i m u m roll angle vs. wave height is also shown i n Fig. 19.

There is a f a i r amount of scatter i n the points, but the explanation is simple. I n order to allow time to adjust the period of the wavemaker exactly and to permit the motion to build up to its steady state value, the wavemaker had to be on f o r at least 5 minutes. B y this time, the quahty o f the waves had deteriorated considerably due to the effects of reflection, and due to the generation o f cross waves in the tank. The waves were, therefore, far f r o m sinusoidal, and the amount of distortion varied f r o m r u n to r u n .

A mean line, f a v o r i n g the higher values of roll angle is drawn through the experimental points, and I t IS seen to lie below the theoretical curve.

By combining the experimental curve w i t h the curve o f roll angle f o r a k n o w n change i n GM, a curve o f change i n G M w i t h wave height can be constructed. This is done using the theoretical curve f o r rolling w i t h a sinusoidal change i n sta-bility, and also using the experimental curve, where the change i n stability is not sinusoidal. From Fig. 6 i t can be seen that f o r low wave heights these two curves lie respectively above and below the calculated curve. Thus the difference between the calculated and experimental change i n stabihty can be explained at least partially by the fact that the waves were not strictly sinusoidal.

There is also an error i n the assumptions made i n the calculation, namely, that the pressure

distribu-6 1 4

uon in the wave is unaffected by the ship. H o w -ever, there is no means of determining the relative magmtude o f these 2 sources of error f r o m the data obtained.

I n any case, experiments w i t h waves show clear-l y that clear-large roclear-lclear-l angclear-les can be obtained w i t h waves of moderate height.

V . Coiichisioiis

I t has been shown i n the previous sections that rolling of very large amplitudes can be caused by regular waves directly f r o m ahead or astern. This rolling, which is caused by a periodic f l u c t u a t i o n in the transverse stabihty, can be calculated quite accurately once the stabihty change is k n o w n . The f l u c t u a t i o n i n stabihty can also be calculated, and although the results could not be checked comple-tely by experiments, the results are definicomple-tely in the r i g h t order of magnitude.

The next step is to attent to predict the motion of a ship under actual sea conditions. I t w i l l be assumed f i r s t that the waves are regular, that is, that the period w i l l remain constant f o r an inde-f i n i t e length oinde-f time. I t has been well established that a ship is very likely to encounter waves of

I t s o w n length w i t h a length to height ratio o f 20,

and w i t h small ships such as fishing vessels, even steeper waves are not uncommon. Therefore, i f the conditions o f resonance are met, roll angles o f 3 0 degrees to each side can easily be produced. For many ships the deck w i l l already be under at this angle, and i f the additional effects o f w i n d and sea are considered, this could easily cause the loss of the vessel.

However, the sea is never completely regular, and f o r this application this is probably a very fortunate fact. While conducting the experiments, three important properties were noted: First, that^the resonance regions were extremely narrow, second, that _the_motion took several hundred swings to build up, and t h i r d , that i f the phase of the motion was disturbed, i t would damp out to zero and then build up again i n the correct phase. I t certainly does not seem possible that i n an actual seaway a ship could encounter 200 waves whose period and phase did not vary more than 2 or 3 per cent. This would indicate that the solution f o r a regular sea

IS not of practical interest.

Is i t possible that under certain irregular sea conditions a state of instabihty could exist, re-sulting i n a similar sort of rohing motion? 'is i t also possible that which the seas approaching par-tially f r o m the side that the combination of the two rolling motions would result i n an unstable condition? I n this case the second resonance region would be of importance, since the period of en-counter would then be the natural rolling period

I f the phase o f the rolling caused by the side ' coniponent of the sea were correct, this would maJ^e an enormous difference i n the build up time l ossibly then a ship could encounter enough waves of the n g h t period to produce dangerous rolling. These possibilities certainly seem great enough •

to warrant f u r t h e r study. \

Acknowledgements

The author would like to express his gratitude to the f o l l o w i n g f o r their assistance in the com-plation o f this project:

The International Educational Exchange Service of the U n i t e d States Government, who made i t possible f o r me to spend a year at the Technical University at D e l f t .

Ir. J . Gerritsma and I r . L . H e r f s t of the" Ship-building Department, who gave advice and assis-tance d u n n g the entire project.

M r . G. de Vries, of the Applied Mechanics De-partment, who assisted w i t h the electronic measu-rements.

M r . L . van der Plas, of the Shipbuilding Depart-ment, who helped w i t h the calculations and w i t h the conduction o f the experiments.

A P P E N D I X Description of model B Model data: Scale 1 : 25.

LWL=mA cm

LBP = 1 1 8 . 5 cm B = 28.0 cm = 19.264 dm" G M = 4.1 cm

(corresponding to operating G M of ship).

Ha = 14.0 cm

Hf = 8.12 cm

H„iea„ = 1 1 . 1 cm

The model was of a normal fishing vessel w i t h a keel 8 m m deep, over the whole length; no bilge keels where f i t t e d . The model was constructed of red wood and f i t t e d w i t h a fore castle, poop and bulwarks along the main deck, corresponding to the f u l l size ship.

Bibliography

1. Couho,,, C.A.: "Waves A Mathematical A c c o u n t of the C o m -mon Types of W a v e Motion," London, 1952.

2. Grin,, Otto: ••Rollschwingimgcn, Stabilitat und Sicherheit im becgang, Forschungshcfte f ü r Schiffstechnik, H e f t I 19J2. 3. Hihlcbraml P. B.: "Advanced Calculus for Engineers," N e w

^ o r k , 1949.

4. Mc lacbl.,,,, N. W.: " O r d i n a r y N o n L i n e a r D i f f e r e n t i a l E q u a t i -ons, O x f o r d , 1950.

5. Mc Lachlan, N. W.: " T h e o r y and Applications of Mathieu functions, London, 1951.