THE EFFECT OF A KEEL
O N THE ROLLING CHARACTERISTICS OF A SHIP
by
I r . J . G E R R I T S M A
Publicalion no. 11, Shipbuilding Laboralory. Technological Universily, Delft
I . Summary
The effect of a fairly large keel on the rolling characteristics of a hospital ship, attending the fishing fleet, has been studied by means of the corresponding model performing forced oscillations in still water.
The results of the rolling tests are analysed and the increase of the damping and of the mass moment of inertia due to the keel are determined.
A comparison is made between the rolling of the model without and with the keel in an irregular sea at zero speed of advance.
I I , Introduction
W i t h regard to the use of the' hospital ship i n service conditions the f o l l o w i n g points are of i n -terest:
1. I n most of the cases the ship is stationed i n the neighbourhood of the fishing fleet.
Consequently, the behaviour i n a seaway and particularly the steering qualities at l o w speeds and i n a head sea are important f o r this vessel.
2. A low freeboard over a large portion o f the ship's length was considered as essential f o r the easy handling of patients, who have to be taken on board i n open sea.
This low freeboard implies a relatively large metacentric height G M , to ensure a sufficient range of the transverse stability curve.
I t appeared that a rather s t i f f ship resulted f r o m these considerations.
WITHOUT K E E L
W I T H K E E L
Fig. 1. Side view of the vessel
296
Fig. 2. Botly plan
Furthermore, the steering qualities of the ship are not completely satisfactory under certain cir-cumstances.
The comparatively small draught f o r w a r d , i n connection w i t h the profile of the ship above the waterline, makes i t sometimes d i f f i c u l t to keep the ship on her course at l o w speeds. I n this respect, a staysail appeared to give only a small improvement.
I n general the draught o f the vessel is small (see table 1 ) , as a restricted draught was desired by the owners, because the ship must be able t o enter small tidal harbours.
T o reduce the rolling amplitudes and to improve the steering qualities of the ship, the f i t t i n g of a large keel was proposed by the designer (see Figs. 1 and 2 ) . This keel should increase the damping of the rolling motion, thus reducing the rolling ampli-tudes and the deeper f o r e f o o t of the vessel should improve the steering qualities i n the hove to con-dition at low speeds.
This report is restricted to the influence of the keel on the rolling behaviour of the ship.
The f o l l o w i n g tests have been carried out:
1. Forced rolling at various frequencies i n still water was studied, using the model w i t h o u t and w i t h the keel. T w o conditions were considered: a. Zero speed of advance. I n this case the model
was situated perpendicular to the centre line of the towing tank at about half the length of the tank.
b. A speed corresponding to a Froude number of 0 . 2 6 2 . A t this speed only a negligible w a l l effect could be expected as w i l l be discussed later.
2. R o l l i n g tests i n an irregular beam sea.
n i . Dimensions of the ship and of the ttvo models The main particulars of the ship and of the models are given i n table 1 .
A general view of the ship, w i t h o u t and w i t h the proposed keel is shown i n Fig. 1 and a body plan is given i n Fig. 2 . Both of the models were f i t t e d w i t h bilge keels according t o the prototype (see F i g . 2 ) .
The f i t t i n g of the keel w i l l reduce the distance of the metacenter to the base line, This reduction is compensated p a r t l y by the reduction of the height of the centre of gravity. Consequently a sUghtly smaller G M value f o r the ship w i t h keel is expected, as shown i n table 1 .
O n the models the shell is extended to the bridge-deck, to avoid the shipping of green water over the b u l w a r k on the main deck.
Excluding very extreme conditions the shipping of green water over the sides of the vessel rarely occurred i n practice.
Rudder and propeller were f i t t e d on the models.
I V . Experimental, procedures
The models were equiped w i t h a semi automatic steering apparatus, which corrected any deviation f r o m a preset course by means of rudder action. V e r y slow deviations could be corrected manually.
T A B L E 1 . Main dimensions of ship and models
Length between perpendiculars . . Length on waterline Breadth B''" D r a u g h t f o r w a r d Draught a f t Volume of displacement V Metacentric height G M Inertia o f waterplane
N a t u r a l rolling period f o r small angles at zero speed
ship 3 8 . 4 6 m 3 9 . 4 9 m 8 . 4 0 m 2 . 2 0 m 3 . 0 0 m 4 0 4 m^ 1.05 m 1 0 9 1 m ' 6.0 sec Model scale 1 20 ithout keel 1.923 1.975 0 . 4 2 0 0 . 1 1 0 0 . 1 5 0 0 . 0 5 0 5 0 . 0 5 2 6 m 0 . 0 0 6 8 2 m* ~ 1 . 3 4 sec m m m m m m^ w i t h keel 1 . 9 2 3 1 . 9 7 5 0 . 4 2 0 0 . 1 6 0 0 . 1 6 0 0 . 0 5 1 6 0 . 0 5 0 4 0 . 0 0 6 8 2 m m m m m m m^ 1.42 sec
WHEATSrOIIE DRIOGE
FEED-BiC^ POIHETEFÏ OH RUDEEBSTOCK
Fig. 3. S/eeiing system
A schematic arrangement of the steering control is shown in Fig. 3.
The sensing element of the steering apparatus is a rate gyro, the o u t p u t of which is proportional to . the yaw velocity. This signal was used to
compen-sate deviations f r o m the course.
The system is not very sensitive f o r extremely slow deviations, as the yaw velocity w i l l be smah. Therefore, a manual compensation was provided i n the circuit as shown i n Fig. 3.
The automatic steering of the model ensures a m i n i m u m of mechanical contact w i t h the t o w i n g carriage: except f o r some electrical wires the model is completely free to move.
I n order to force the model to oscillate i n still water, a mechanical exciter, situated i n the model, was used. This exciter consists of two weights, counterrotating i n an athwartship's plane (see the appendix).
As this exciter was not p r i m a r i l y designed f o r rolling tests, some calculating had t o be done to determine the exciting moment, as shown i n the appendix.
I n the case under consideration gravity forces and centrifugal forces on the rotating masses act together to f o r m the exciting moment, whereas i n the more commonly used rolling exciters only-the centrifugal forces are used.
of rolling amplitudes could be investigated. I n all the cases, the height of the centre of gravity re-mained unchanged.
The rolling angles were recorded w i t h a pen-recorder, using a Sperry roU and p i t c h gyroscope w i t h potentiometer pick o f f ' s . The phase of the rotating weights was given by a mark on the record.
V . Experimental results
a. Determination of the transverse stability The transverse stability of the models at zero speed has been measured w i t h the so caUed moment indicator [ 1 ] .
The results are given i n Fig. 4 . I n both cases the stability moment can be expressed w i t h a good ap-proximation by the formulae given i n table 2 .
I n addition, the stability f o r small angles of heel (up to seven degrees) has been measured i n the free r u n n i n g condition. Table 3 gives the results; i n each case the mean value of t w o measurements (one at 95 ~ 3 degrees and one at 99 ~ 7 degrees) is given.
T A B L E 3
Virtual metacentric heigiot as a function of speed
FR = V/VsL G M in m FR = V/VsL
without keel w i t h keel
0 0 . 0 5 2 6 0 . 0 5 0 4
0 . 1 3 1 0 . 0 5 2 5 0 . 0 5 0 1 0 . 2 6 2 0 . 0 5 2 9 0 . 0 5 0 5
For larger angles of heel the influence o f the rudder action, necessary f o r a straight course, is too large to ensure a reliable measurement of the sta-b i h t y moment as a f u n c t i o n of the heeling angle.
The variation o f the stability moment w i t h speed is very small and therefore the values f o r zero speed are taken as vahd over the speed range considered.
Similar tests b y Basihefski [ 2 ] have shown that an increase or a decrease of the stability moment as a f u n c t i o n of the speed of advance is possible. D e -pending on the s h i p f o r m i n the heeling condition, the asymmetric f l o w around the ship creates a hydrodynamic stability moment which can be
T A B L E 2 . Transverse moment of stability at zero speed
Model Y V . kg G M , m FM, m Moment of stability y V GN sin cp, kgm W i t h o u t keel . . W i t h keel 5 0 . 5 2 5 1 . 5 7 0 . 0 5 2 6 0 . 0 5 0 4 0 . 1 3 5 0 0 . 1 3 2 2 ••'•2.59 (p — 0 . 6 6 cp^ 2 . 5 5 — 0 . 6 7 <p^ -cp in radians I n t . S h i p b u i l d i n g P r o g r e s s - V o l . 6, N o . 59 — J u l y 1959
298
O ' 1 0 ' 2 0 ' To 3 0 ' 4 0 '
DEGREES
O 10' 2 0 " «'o 3 0 ' 4 0 '
DEGREES
Fig. 4. Curves of transverse stability
positive or negative w i t h regard to the transverse stability.
I n the case under consideration, this extra stabil-i t y moment appears to be neglstabil-igstabil-ible.
b. Determination o£ amplitude and phase of the rolling motion i n still water
The rolling motions, resulting f r o m the excita-tion, have been measured as a f u n c t i o n of the speed of the model, the frequency of the motion and the
magnitude of the exciting moment. A t each f r e -quency three or f o u r exciting moments were chosen t o cover a wide range of rolling angles.
I n Fig. 5 the rolling amphtudes <po and the phase angle e between the motion and the excitation are shown as a f u n c t i o n of the amplitude of the ex-c i t i n g moment M Q and the ex-cirex-cular frequenex-cy of the motion co.
The I'esults are given f o r the models w i t h o u t and w i t h keel, at zero speed and at a speed correspond i n g to a Froucorresponde number F i l = 0 . 2 6 2 . For a f u r -ther analysis of the data, the amplitudes and phases are plotted on a base of frequency, using Mo as a parameter (see Fig. 6 ) . I t is emphasized that these curves are not to be confused w i t h ordinary reso-nance curves, as the damping of the motion is not constant along these curves.
c. R o l l i n g amplitudes i n an irregular sea
T o get a general impression of the behaviour i n waves, both models have been tested i n an irregular beam sea, at zero speed. As the rolling m o t i o n i n still water showed a strong non linearity (see Fig. 5 ) , a spectral analysis of the records has n o t been made. Therefore, the results are given i n the f o r m of averages, root of variances, averages of Vio or ^/s highest values, as shown i n table 4.
The records of the wave and the rolling angle contained i n each case about 1 0 0 to 1 2 0 cycles corresponding to a total measuring period of about 2 Yz minutes f o r each model.
Due to the repetition of the wave maker program the same wave pattern returns after a preset time which i n this case was 2 J/i minutes.
The repetition of the wave pattern could be verified by a comparison of t w o records w h i c h were shifted 2 Y minutes i n time and by the comparison of the calculated averages of wave height and wave period.
The differences appeared to be so small t h a t the mean of the t w o records is given i n table 4 .
V I . Discussion of the results
The rolling tests i n still water show t h a t the model w i t h keel has only shghtly smaller r o l l i n g
T A B L E 4 . Results of tests in an irregular sea
W a v e model without keel model w i t h keel
height period
rolling
amplitude period
rolling
amplitude period
- c m sec degrees sec degrees sec
5.8 1.45 1 0 . 6 1 . 3 6 9 . 7 1.43
6.7 1.48 1 1 . 4 1 . 3 7 1 0 . 5 1 . 4 4
average of heighest Vs 9 . 6 1 5 . 8 1 4 . 3
« 1
300
Fig. 6. Rolling {implitiidc and phase angle as a junction of frequency
For rolling angles of about 10 to 15 degrees, about the same reduction is f o u n d w i t h the tests i n irregular waves (see table 4 ) .
I t appears f r o m F i g . 5. that the rolling m o t i o n at or near synchronism is essentially non linear.
The main reason f o r this non linearity of the motion must be a n o n linear damping, as the re-storing moment is very nearly linear (see Fig. 4 ) .
Therefore, the f o l l o w i n g analysis of the test data has been made.
The rolling m o t i o n can be described approxi-mately b y the f o l l o w i n g differential equation; A cp + Bl cp + Bi (sign, cp) rp^- -\- Ci cp Cz cp^ =
= Mo sin cot
The solution of forced rolling is taken as: cp — cpQ sin {oit + £)
The slight n o n linearity o f the restoring moment and an analysis of the rolling records j u s t i f y this assumption.
amplitudes than the model w i t h o u t keel, when the same exciting moment is applied (see Fig. 6).
For a comparison, the m a x i m u m rolling ampli-tudes f o r a constant exciting moment are given i n table 5.
T A B L E 5. Maximmn rolling amplitudes for a constant exciting moment
E x c i t i n g moment Mo
Model without keel Model w i t h keel E x c i t i n g moment Mo F R = 0 F i l = 0.262 F R = 0 VR = Q.iei 10 ' k g c m 2 0 k g c m 3 0 k g c m 1 5 . 0 ° • 2 3 . 0 ° 3 1 . 5 ° 1 2 . 0 ° ' 1 9 . 5 ° 2 6 . 0 ° 1 3 . 6 ° 2 0 . 0 ° 2 6 . 0 ° 1 0 . 7 ° 1 7 . 0 ° 2 2 . 4 °
The table shows that the reduction of the rolling amphtudes due to the keel is quite small, consider-i n g the dconsider-imensconsider-ions of the keel.
The total damping moment is now linearized to B (p, where B is determined b y the condition that the total energy dissipated per cycle, is equal i n both cases, thus: '^B
j
cp .q}dt = ABiS
<P . 9 d t + 0 0 + AB2Scp^.^dt 0 or: g B = B l + - — B2 (po CO, 3 71for a pure harmonic motion. Then the d i f f e r e n t i a l equation reduces t o :
Acp + B cp + Cicp + Cscp^ = Mo sin ojt A f t e r substitution of 93 = sin {ojf g) the resulting equation can be solved f o r A and B
as-summg identity at co^ = 0 and cot = 2
This gives:
— Mo sin s + —^ 950^ sin 4 e
B= : cpo CO
and:
^ _ — MQ COSE + C l yo + Ca cpj {cos^t + sin'^e)
9'o
Mo, 9^0, « and co are determined f r o m the r o l l i n g tests i n still water, and C i and Cg f r o m the stability test. The B values are given i n Fig. 7 as a f u n c t i o n of the rolling amplitude 9^0 using the frequency co and the speed of advance as parameters.
I n both cases, w i t h o u t and w i t h keel, the damping coefficient B varies linearly w i t h amphtude, over a f a i r l y large range, and consequently the approx-imation:
8
B = -Bl + - — B2 930 CO
3 n seems justified.
The coefficient B i , w h i c h equals B at zero amph-tude, may be called the wave damping coefficient, and i t is seen f r o m Fig. 7 that this coefficient i n -creases w i t h the speed o f the model.
For zero speed, B i varies very l i t d e w i t h the frequency of the m o t i o n , whereas at a the speed F i l = 0 . 2 6 2 , the value of B i decreases w i t h increas-ing frequency.
The behaviour of wave damping has been studied by Toshio Hishida [ 3 ] , who made a theoretical investigation of the wave damping of an ellipsoid of revolution m o v i n g t h r o u g h the water, parallel to the free surface, w i t h constant speed and per-f o r m i n g a rolling m o t i o n about a longitudinal axis located above the major axis. H i s results regarding
the dependency of wave damping on speed and frequency agree qualitatively very well w i t h the test results discussed i n this publication. Moreover, Hishida calculated the amount of w a l l effect, w h i c h w i l l be present when rolling tests are carried out i n a t o w i n g tank. This wall e f f e c t is a f u n c t i o n of the f o l l o w i n g parameters:
a. The Froude number F i l = V/VgL;
b. The ratio of tank w i d t h and length of the model q/L;
c. The ratio of speed of the model and velocity of
the generated damping waves: y = S
I t was shown by Hishida that high values of y and F i l are beneficial f o r m i n i m u m w a l l effect. Above a Froude number of about 0 . 2 5 the value of q/L has only very little influence, as soon as a certain l i m i t (say q/L = IJS — 2 ) is exceeded. A t low Froude numbers, however, the wave damping varies strongly w i t h q/L and the frequency, and hence the experimental values are useless f o r ana-l y z i n g purposes. D u r i n g the tests at zero speed, wave damping was provided at both ends o f the tank, and moreover, each test was completed before residuary reflections could reach the model.
For the tests w i t h the free r u n n i n g model the f o h o w i n g combination of the above mentioned parameters was used:
F i l = 0 . 2 6 2
y = 0 . 4 6 8 t o 0 . 6 4 5
q/L = 2.1
Therefore, according to Hishida's w o r k , the wah effect is smah i n the case under consideration here. U p to a r o l l i n g amplitude of about 2 0 degrees, the slope of the B curves is only very l i t t l e affected by the speed of advance and the frequency of the m o t i o n . Consequently, Bg can be talten as constant over this range.
The quadratic damping term takes i n t o account the damping moment due to vortex sheet genera-t i o n , f r i c genera-t i o n a l resisgenera-tance and ogenera-ther causes.
The influence of the large keel on the quadratic damping is illustrated clearly by Fig. 7.
The increase o f the mass moment of inertia is shown i n Fig. 8. As a matter of interest, the mass moment of inertia of the model itself has been determined b y f o r c i n g the model to oscillate i n air; the results are also given i n Fig. 8.
The larger damping coefficient of the model w i t h keel, w i l l reduce the r o l l i n g amplitudes. F r o m the B-values as determined, a somewhat larger re-duction w o u l d have been expected than is shown by the direct comparison of the r o l l i n g amplitudes i n both cases.
However, the total mass moment of inertia, is larger i n the case of the model w i t h keel and, hence.
302
0.20e
D> 0.15 m 0.10 0.05 u j = i . 0 rad. secT^ V = 0 m s e c : ' 1 O (p ^ 2 O D E G R E E S . 3 0 d0.20 £ O) 0.15 0.10 J 0.05 = 0.931 F r . = O UJ = 4.5. rad.secT V = D m s e c : 1 0 Q^o 2 0 D E G R E E S 3 O ; 0.20 0.15 J 0.10 0.05 0.828 F r . 0.262 uu -U.ü rad.sec:'' V =1150 m. sec.-'' 1 0 Q)^ ^ 2 0 D E G R E E S 3 0 ao.20 E cn 0.15 0.10 0.05 1^ L / - H . = 0.931 F r . = 0.262 V = G.528 UJ = 4 . 5 rad.sec.-'' V =1.150 m . s e c : ' ' 1 ° % 0.20 i0.15 J ,0.10 0.05 w = 1.035 A WITH K E E L . O WITHOUT K E E L . ;0.20 ^ 2 O D E G R E E S 3 0 F r = O UJ = 5 . 0 r a d . s e c . - ' V = O m sec-'' 0.15 0.10 0.05 F r . 1.138 y = O 5.5 rod. sec-'' .1 1 O ^ 2 O D E G R E E S 3 O 1 O —1— 2 O D E G R E E S 3 O J0.20 ,0.15 0,10 0.05 1 / J . " = 1.035 g UJ = 5.0 rad. sec.-' V =1.150 m . s e c . j0.20 0.15 J 0.10 0.05 UJ 1.138 V q Fr. = 0.262 , 5.5 rad.sec.-1 ; 1.150 m . s e c . 10 D E G R E E S 30 10 (j,^ _ 2 0 D E G R E E S 3 00.15 010 L WITHOUT K E E L 015 010 IN A I R A = 0.085l<gmsec2 0.05 0.15 0 10 F r = 0.262 WITH K E E L D E G R E E S 20 30 0.10 0.05 0.05 0.15 WITHOUT K E E L IN AIR A = 0.095 kgmsec2 0.10 IN A I R A = 0.088 k g m s e c ^ F r = 0 WITH K E E L 10 ( f c — . D E G R E E S 20 30 IN AIR A = 0.088 k g m s e c 2 10 D E G R E E S 20 • 01=4.0 i _ _ 0 U J = 4 . 5 0.05 30 A L U . . 5.0 0 Q U J = 5 . 5
Fig. 8. Mass moment of inertia
10 ( f . • 20
D E G R E E S
30
the beneficial effect of the larger damping is some-what diminished.
The larger inertia results i n a somewhat larger natural period (see table 1 ) . Moreover, the smaUer G M value causes an increase of the natural roUing period.
Finally, attention is directed again to curves o f the amplitude versus frequency i n F i g . 6.
The curves which can be drawn through- the points of m a x i m u m amphtudes, are slightly i n -clined i n the direction of increasing frequency, whereas f r o m elementary theory o f non linear vibrations (assuming constant linear damping) an inclination towards decreasing frequency w o u l d be expected assuming a restoring moment as shown i n Fig. 4 .
However, as has been mentioned before, the damping varies along the a m p l i t u d e - f r é q u e n c y curves, and i t is believed that this may be the reason w h y the points of m a x i m u m amplitudes s h i f t somewhat towards higher frequencies.
F r o m the experimental results discussed i n this article, i t may be concluded that the large keel has only a comparatively small influence on-the rolling behaviour of the vessel. A l t h o u g h i n general scale e f f e c t is present i n rolling tests because of the i n -fluence of viscosity on damping the author is of opinion that i n the case under consideration at least a quahtative comparison between the ship w i t h and w i t h o u t keel is permissible.
Acknowledgement
The author is indebted t o M r . M . Meyer and M r . H . Zunderdorp who designed the experimental technique, and to M r . W . Beukelman w h o carried out the experiments and the calculations.
A P P E N D I X T H E EXCITER
The exciter which has been used to i m p a r t the rolling motion to the model, consists of t w o weights of P k g each.
These weights are counter-rotating w i t h a con-stant velocity i n an athwartships plane (see F i g . 9).
The centre of rotation is i n A, located i n the centre plane of the model at a distance "a" above the centre of g r a v i t y G.
The distance o f A to the centres of g r a v i t y of the weights P is r. W h e n passing through the per-pendicular to the centre plane i n A, the t w o weights are i n coinciding positions, which occurs t w o times d u r i n g a revolution, when:
n n
oJt = ~ (n = 1 , 3 , 5 )
Then, the exciting moment M is:
M = 2 P {a sin (p + r sin ojt cos cp - j - oj" r a . S sin mt)
304
F i g . 9. Exciter S u b s t i t u t i o n o f the a p p r o x i m a t i o n s sm cp = Cp 6 cos cp — \Y
i n t r o d u c e s an e r r o r , t h e m a x i m u m v a l u e o f w h i c h i n this case is o n l y 0.7 % . T h e r o l l i n g m o t i o n m a y be described b y : cp = cpo sin (ojt + 'P),w h e r e 5^ is the phase angle b e t w e e n the m o t i o n a n d t h e passing o f t h e w e i g h t s P t h r o u g h t h e c e n t r e p l a n e .
W h e n t h e h i g h e r h a r m o n i c s : Pa cp(? Pr cpr?
sin 3 {cüt + cp) a n d sin {oH + 2
a n d the t e r m : Pr cpo^ sin (cot + 2 <P) are n e g l e c t e d , t h e f o l l o w i n g expression f o r M is f o u n d : M = 2 Pa {cpo 1 cpo-9 V , 4 sin (ojt + P) +2 Pr 2 \ a OJ sin ojt T h e h i g h e r h a r m o n i c s are n e g h g i b l e i n t h e f r e -q u e n c y r a n g e w h i c h is considered here, a n d , due t o its phase, the t h i r d t e r m t o be n e g l e c t e d has o n l y a v e r y s m a l l i n f l u e n c e o n a m p l i t u d e a n d phase o f M , as c a n be s h o w n b y a n u m e r i c a l c a l c u l a t i o n . I t is o b v i o u s t h a t t h e e x c i t i n g m o m e n t can be w r i t t e n as: M = Mo sin (ojt + v) w h e r e v is t h e phase angle b e t w e e n t h e e x c i t i n g m o m e n t a n d the passing o f t h e w e i g h t s P t h r o u g h
£ uj'r the centre p l a n e , a n d hence:
Mo s i n (cüi + v) = 2 Pai cpo — ^ sin {ojf + ^ ) + / cpQ^ , a cü^\ + 2 P r 1
- ^ H sin cot
\ 4 S J , S o l v i n g t h i s e q u a t i o n f o r v a n d M o : cpo^ , a c o t g V = Mn r/a\ 1-cpa cpo' sin 0
2 Palcpo- cpd + c o t g <3> cp sm V • T h e phase l o g £ o f t h e r o l l i n g m o t i o n w i t h r e g a r d t o t h e e x c i t i n g m o m e n t is d e t e r m i n e d b y : • e = p — V T h e passing o f the w e i g h t s P t h r o u g h t h e c e n t r e p l a n e o f the m o d e l is m a r k e d o f f a u t o m a t i c a l l y o n the r e c o r d o f the r o l l i n g angle w h i c h enables cP t o be d e t e r m i n e d . List of Symbols A = B" = B, B„ B, = C j , Cg = F G
Mass moment of inertia Breadth of Ship or model Damping coefficient
Restoring moment coefficients Centre of boyancy
Centre of gravity . Acceleration of gravity Length of ship or model Metacentre, exciting moment Amplitude of exciting moment False metacentre
Weight of rotating mass Radius of exciter Speed of advance Width of towing tank Froude number
Specific gravity, speed ratio y = Phase angles
Rolling angle
Amplitude of rolling angle Circular frequency Volume of displacement V OJ L M Mo N P r V
A
PR = V/VgLr
S P V cp cpo OJV
References1. Werckmeister: "StabilitatsuntersucKungen m i t dem Model eines Kiisten-Motorfraclitscliiffes. S c l i i f f u n d N>/erft, 1944. 2. Basiliefsky. "Einfluss der Scliiffsgeschwindigkeit auf
Querstabili-ttat und Rollen". Synopsis in S c l i i f f und H a f e n , 1958. 3. Toshio Hoshida: "Studies on tKe wave resistance to t h è rolling
of ships". Society of N a v a l Architects of Japan, V o l . 72 and 87 (English translation).
