\ the Rudder as An Antirolling Device

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1 TECHNSCHE HOGESCHOOL DELFT.AFDELNG DER SCHEEPSBOUW- EN SCHEEPVAARTKUNDE LABORATORUM VOOR SCHEEPSHYDROMECHANCA Fesibility Study of Employing \ the Rudder s An Antirolling Device K. Hung Report no. 518 April 1981 Deift University of Technology Ship Hydromechnics Lbortory Mekelweg CD DELFT The Netherlnds Phone

Contents Summry 1 ntroduction 1 Rolling Motions Excited by the

Responses 5 Roll Velocity Feedbck to the Rudder 7 Roll Angle

Roll Angle to the Rudder 16 Roll Accelertion Feedbck to the

2 Contents Summry 1 ntroduction 1 Rolling Motions Excited by the Rudder 3 Eqution of Forced Rolling Motions nd Open-loop Responses 5 Roll Velocity Feedbck to the Rudder 7 Roll Angle Feedbck to the Rudder 11 Combined Feedbck of Roll Velocity nd Roll Angle to the Rudder 16 Roll Accelertion Feedbck to the Rudder 17 Concluding Remrks 18 Acknowledgement 19 References 21 Ncrnenclture 22

Summry Model tests nd subsequent nlysis of mesurements in n ttempt to employ the rudder s n ntirolling device hve been crried out in the Ship Hydromechnics Lbortory of the Delft University of

The experimentl results showed tht in ddition to steering rudder hs considerble potentil for roll stbilistion t or ner the resonnt roll frequency, nd the fesibility of combining steering nd roll

ntroduction n the light of conventionl conceptions rudders on surfce ships re rrrely ble to control the ship heding only.

3 Summry Model tests nd subsequent nlysis of mesurements in n ttempt to employ the rudder s n ntirolling device hve been crried out in the Ship Hydromechnics Lbortory of the Delft University of Technology, the Netherlnds. The experimentl results showed tht in ddition to steering rudder hs considerble potentil for roll stbilistion t or ner the resonnt roll frequency, nd the fesibility of combining steering nd roll stbilizing by using only rudders exists undoubtedly. Up to now it seems tht roll velocity feedbck gin is the most effective mong vrious types of feedbck. 1. ntroduction n the light of conventionl conceptions rudders on surfce ships re rrrely ble to control the ship heding only. t ws not until 1967 when n Arrericn continership fitted with rudder steeringstbiliser system ws tested on voyge crossing the Atlntic tht nvl rchitects relized tht it my well be possible to use the rudder to reduce undesirble rolling motions of the ship without ffecting its course-keeping cpbilities (1. Subsequently, in 1972 W.E.Cowley nd T.H.Lrnbert (2.) pointed out tht the side force generted by ship's rudder generlly cts beloi the centre of grvity nd thus exerts roll in ddition to yw rrorrents on the ship. When the roll rrorrent cused by the rudder is in phse with the exciting roll morrent considerble rolling would be set up. But if correct phse reltionship cn be estblished between the rudder roll morrent nd the externl excittion, is resonble to suppose tht the rudder my be used s rrens of reducing roll mplitudes. The fesibility of roll stbilistion by rudders ws lso discussed by F.F. Gunsteren (3) with full scle crft experirrents nd the results were published in The uthor held tht rudder stbilistion of smll crft seems to be promising. n 1975, W.E.Cowley nd T.H.Lrnbert pointed out once gin tht the introduction of negtive roll velocity feedbck gin to the rudder enbles the dirrensionless dmping rtio in roll to be incresed C 4). At the sne tirre, study mde by J. B.Crley (5) showed tht. effective stbilistion by rudders is limited to nrrow bnd in the proximity of the nturl roll frequency. n order to coiipletely develop the ide concerning rudder ntirolling devices series of model tests involving it 1

Ljt( -2 -) rolling rrtions excited by the rudder, open-loop responses to exciting rrorrents generted by oscilltors fixed in the model, vrious types of closed-loop responses including vritions in

7 ( Tble 1 ) in the ship model bsin, Delft University of Technology, the Netherlnds.

velocity feedbck is the best one which enbles the rudder to reduce undesirble rolling motions t frequencies higher thn 2.

4 Ljt( -2 -) rolling rrtions excited by the rudder, open-loop responses to exciting rrorrents generted by oscilltors fixed in the model, vrious types of closed-loop responses including vritions in feedbck gins such s roll ngles, roll velocity, roll ccelertion, nd the corrbintion of roll ngle nd roll velocity were conducted by mons of Todd 6 model with C=.7 ( Tble 1 ) in the ship model bsin, Delft University of Technology, the Netherlnds. On the bsis of these tests discussion on the principle of rudder ntirolling devices ws mde in brief nd it ws concluded tht mong the vrious feedbck gins given in this pper, it sens to be tht the velocity feedbck is the best one which enbles the rudder to reduce undesirble rolling motions t frequencies higher thn 2.3 rdsec, with mximum reduction percentge more thn 5% in pek roll ngles s shown in Fig.12 (E). But it cn be seen, from Figs.9 nd 13, tht for the considered cse t frequencies lower thn 2.3 rdsec roll ngle feedbck is better thn feedbck s viewed frn the reduction in pek roll ngles, roll velocity fter introducing roll velocity feedbck gin to the rudder it ppered tht the pek roll ngulr velocity decresed with mximum reduction of 5% s shown in Fig.11. bout Tble 1 Min prticulrs of the model Todd Series 6 model scle 154 tems Ship Model Length, design wterline Length, beeen perpendiculrs Bredth, molded Drft, loded Volwne of displcement, loded Service speed Nturl frequency of roll m Gm 1?.416m 6.968m 1364m3 14 knots.335 rd sec 2.295Gm 2.257?m.3225m.129m 65.?48n3.99msec 2.464rd sec 2

model would be excited in still wter into stedy rolling motion = sin ( w t + S where = roll ngle excited by the rudder = roll mplitude or pek roll ngle = phse ngle

5 )CL -t--) er ) 2. Rolling Motions Excited by the Rudder 1 Mking the rudder ngle be chnged in sinusoidl mnner where 5=S 5mw t o S= rudder ngle mplitude frequency of rudder swing = the model would be excited in still wter into stedy rolling motion = sin ( w t + S where = roll ngle excited by the rudder = roll mplitude or pek roll ngle = phse ngle by which the rolling motion lgs the rudder deflections. The still wter responses to n hrmonic excittion by the rudder nd 1.8 V = 14 V=Q.99msec m 'C.- knots -s_ rd. sec) 3. Fiq.1 _ó =1 6 =2 Still wter roll responses to hrmonic excittion by the rudder The similr results obt:ined from fst continership model tests (2) shown in Fig.3. re S phse ngle t given Vm ( model speed ) nd re shn in Figs.l nd 2 respectively. The fct tht considerble roll moments cn be generted by rudders is verified by the test results of full scle 16,9-ton continership (1) running on voyge of Atlntic crossing, s shown in Fig.4 from which two points re 8 ijdig)

$) 3 V-l4knots Vm=.99msec - 2 4 8 12 16 (degree) =1. 2U Fig.2 Phse ngle ) ) 3 S =2 =1 2 1 \, 5A 9.2 Fig.$ 5 noteworthy: Wnen the rudder reches n ngle of 2 deg to port or to strbord, the swing extremities rech

5 noteworthy: Wnen the rudder reches n ngle of 2 deg to port or to strbord, the swing extremities rech

Therefore, it is unnecessry to mke the mximum rudder ngle greter thn 2 deg.

6 ) 3 V-l4knots Vm=.99msec (degree) =1. 2U Fig.2 Phse ngle ) ) 3 S =2 =1 2 1 \, 5A 9.2 Fig.3.3 W.4 (Hz) Rolling ngles excited by the rudder.5 noteworthy: Wnen the rudder reches n ngle of 2 deg to port or to strbord, the swing extremities rech ngles of ttck to the propeller rce where stll begins nd the side forces fll off drsticlly. Therefore, it is unnecessry to mke the mximum rudder ngle greter thn 2 deg. As fr s full scle ship is concerned, there exists n unsymttricl distribution of side forces to left nd to right becuse of the corrbined effects of wke distribution, chnges in propeller rce, nd so on. 4

iderin (op cijfe i (op cijfer-2 -) lijn) 8O 4 fi 1'-- -6 1 Left swing Right swing 1 2 3 (3 (degree) Fig.

rrost cses by simple liner eqution in one degree of freedom AtBtC=M(t) (1) where A = inertil rrorrent of mss including virtul mss B = dmping coefficient C = restoring

7 iderin (op cijfe i (op cijfer-2 -) lijn) 8O 4 fi 1' Left swing Right swing (3 (degree) Fig.4 Rolling moment excited by the rudder n view of the bove-mentioned sttemnts it my well be imgined tht since rudder cn excite roll it lso hs potentil vlue s n ntirolling device. 3. Eqution of Forced Rolling Motions nd Open-loop Responses By mking simple nd liner ssumptions, the forced rolling motions of ships in wves cn resonbly be described in rrost cses by simple liner eqution in one degree of freedom AtBtC=M(t) (1) where A = inertil rrorrent of mss including virtul mss B = dmping coefficient C = restoring mcment per unit heel ngle M ( t ) = roll exciting morrent which is produced by gyro-oscilltor fitted in the model nd chnged in sinusoidl mnner M ( t ) = Msinuit (2) in which M = mplitude of the exciting morrent w = frequency of the roll exciting mcinent, i.e., the ngulr frequency of the oscilltor. The open-loop responses to the mnent given in Eq. ( 2 ) is s follows = sin( wt + (3) 5

where = phse ngle by which the roll motion lgs the

The nondimensionl dnping rtio is given by * = B 2JAC (4)

8 where = phse ngle by which the roll motion lgs the exciting mnent = pek open-loop response or roll rrplitude. The nondimensionl dnping rtio is given by * = B 2JAC (4) dll w (rd. sec) O M.71kg-rn.47kg-rn Fig.5 Open-loop responses w w 15 V V = 14 5 rn knots =.99rnse ) 1 5 n p A- M=O.O47kg-rn C (rd.sec) U) 4. Fig.6 Phse ngles 6

2 -) S n this cse the open-loop responses nd phse ngles hve been obtined the tests nd re given in Figs.5 nd 6 respectively. 4.

9 2 -) S n this cse the open-loop responses nd phse ngles hve been obtined the tests nd re given in Figs.5 nd 6 respectively. 4. Roll Velocity Feedbck to the Rudder n frn Vrious types of closed-loop responses hve been tested, with the feedbck control system in Fig.7 nd the results given in the subsequent sections. At first the roll velocity feedbck to the rudder will be tken into considertion in this prgrph. f rudder deflections proportionl to the roll velocity, i.e., cs=c2q ( where is coefficient dependent on the roll velocity feedbck gin ) re introduced with correct phse reltionship, negtive roll morrent M() -k2 (5) will be cused by the rudder, nd the right-hnd side of Eq. (1) is rrodified A+B+C=M(t)-k2 (6) The eqution bove my be rewritten in the form where in +c2=m(t) Wn = nturl frequency of rolling rrtions = nondimensionl dmping rtio which is given by B+k 2 2JAC n k2 = moirnt coefficient in kg-m-secde, which is lso proportionl to the roll velocity feedbck gin nd is chosen in ccordnce with the rudder swinq extremities. For such n hrmonic force function s is shown in Eq. (2), the closed-loop resrcnses would be = i" wt + ( in which 1 = pek closed-loop response or roll mplitude = phse ngle by which the rolling motion lgs the exciting mnent. But we only show interest in, the phse ngle difference between the rollng motion nd the rudder swing, insted of The phse reltionship mong M, S nd q is shown in Fig.8, from which we hve 7

.7 Block digrm mesurement with Todd 6 mode1,c=.

mplifier unit for summtion, inversion, differentition cl ibrt ion mplifier d emodul s to r

- L u eceive motor mot or control bttery servo remote chrge rtec n t r ol cont rol

10 .7 Block digrm mesurement with Todd 6 mode1,c=.7o power s u ppl y power suply for bttery chrger trnsmitter HZ genertor for ll gyro' s mplifier unit for summtion, inversion, differentition cl ibrt ion mplifier d emodul s to r low-pss filter 5HZ 5HZ 5HZ L rudder mplifier 5HZ rudder servo remote control rudder 5HZ '4 - L u eceive motor mot or control bttery servo remote chrge rtec n t r ol cont rol regultor gyro Os c ill t or roll oscilltor rollgyro btt ery course- gyro oscilltor period time crrige speed counter u. v. pper recorder printer

b E=EM_EM M Fig.8 Phse reltionship Contrsting Eq. (4) to Eq.

$A V 5 =14 knots V =.99rnsec rn \\\ C''. M =.47kg-rn k2=.$ 24 4 ) 3 2 cd 4J C ) C) ) C 4J C) ) 5 _ Open-loop L-- Closed-loop

11 b E=EM_EM M Fig.8 Phse reltionship Contrsting Eq. (4) to Eq. (9) shows tht the negtive roll velocity feedbck to the rudder enbles the dniping rtio in roll to be incresed. This is ye A V 5 =14 knots V =.99rnsec rn \\\ C''. M =.47kg-rn k2=.24 4 ) 3 2 cd 4J C ) C) ) C 4J C) ) 5 _ Open-loop L-- Closed-loop --Reduction percentge of pek roll respon 1 e (A) (rd.sec) Fig.9 Pek roll responses 9

rified by the open-loop nd closed-loop responses shown in Fig. 9, with the phse ngle difference E2 nd pek rudder ngles given in Fig. 1.

$lrge s shown in Fig. 11. \\ V =14 5 knots V =.99msec m - 3 - M.47kg-m 5 1 \ k2=.24 95 - U 15 9 1.5 2. 2.5 3. 3.5 (rd. sec) w 1 Fig.$ 12, from which it is found tht s k2 increses the pek roll ngle 1 would decrese nd the pek roll ngulr velocity decreses too, with rnging from 95 deg to 16

12, from which it is found tht s k2 increses the pek roll ngle 1 would decrese nd the pek roll ngulr velocity decreses too, with rnging from 95 deg to 16

12 rified by the open-loop nd closed-loop responses shown in Fig. 9, with the phse ngle difference E2 nd pek rudder ngles given in Fig. 1. t is noteworthy tht in generl the roll rte would reduce fter introducing the roll velocity feedbck nd under certin circumstnces this reduction is rther lrge s shown in Fig. 11. \\ V =14 5 knots V =.99msec m M.47kg-m 5 1 \ k2= U (rd. sec) w 1 Fig.1 Phse ngle difference nd oek rudder ngles The effects of k2 on rolling motions re shown in Fig.12, from which it is found tht s k2 increses the pek roll ngle 1 would decrese nd the pek roll ngulr velocity decreses too, with rnging from 95 deg to 16 deg. in mking choice of k2 specil ttention should be pid to the fct tht when the rudder swing reches ngles of ttck to the propeller rce where stll begins the side forces fll off drsticlly. Therefore, there seems to be no need to mke the pek rudder ngle excessive. 1

$5 4C - U S,_ oio ' F K'! 3Q 4J U c 2 - o \ \O 1 2. 2.5 V l6knots S M.47kg-m Vm=l.$ Roll Angle Feedbck to the Rudder f the rudder is controlled by n utomtic pilot

feedbck gin ), negtive roll moment M() =-k will be cused by the rudder, nd the

13 5 4C - U S,_ oio ' F K'! 3Q 4J U c 2 - o \ \O V l6knots S M.47kg-m Vm=l.l3msec k2=.175 Fig.11 Pek roll rte (1) (rd. sec) Roll Angle Feedbck to the Rudder f the rudder is controlled by n utomtic pilot system with roll ngle signl 5=c1 ( c1 is coefficient dependent on the roll ngle feedbck gin ), negtive roll moment M() =-k will be cused by the rudder, nd the eqution of rolling will be motion A + B + ( c+k1 ) = M ( t ) (1) The nondimensionl dmping rtio is given by B - 2JA(c+k1) 11

C) rtj.1j C) 5Q -2-1 V=14 knots M.47kg-m V=O.

4 (A) Open-loop Closed-loop Reduction percentge of

14 C) rtj.1j C) 5Q -2-1 V=14 knots M.47kg-m V=O.99msec C) J G (A) Open-loop Closed-loop Reduction percentge of pek roll responses Pek roll responses i i k C) C) 1 _V=14 knots M=O.O47kg-m C) Fig k2 (B) Effect of roll velocity feedcck gin on rolling motions 12

46rdsec V m 1 5 -- Open-loop Closed-loop Recuction

5 (c) percentge of pek Pek roll responses roll

15 5-2.4J U) U) U) 4 15 ZL o 3. U) 2 1 V =14 knots M 5 =.47kg-m =.99msec =2.46rdsec V m Open-loop Closed-loop Recuction (c) percentge of pek Pek roll responses roll responses U) U) U) :iu1.i. 4, F Fig.12 (D) Effect of roll velocity feedbck gin on rolling motions

1 W (t: 5-4J C) C) C) 4-2 15 1 W C) 3 1 N N N 2 1 5 _.

$..14 U\f 3 3 2 95 VVS =14 knots M =.47kg-rn 2 - V =.99rnsec rn w=2.$

16 1 W (t: 5-4J C) C) C) W C) 3 1 N N N _.1 --Open-loop C losed- Reduction loop.2.3 V =14 5 V =.99rnsec rn Dercentge knots Pek roll responses.4 M =.47kg-rn w=2.7rdsec of pek roll responses k 2.5 (S 5 15 W - U C) () - 5 ) 4 1. i...14 U\f VVS =14 knots M =.47kg-rn 2 - V =.99rnsec rn w=2.7ordsec k 2 bf bln. cij fer Fig.12 Effects of roll velocity feedbck gin on rolling motions 14

-2 nd the closed-loop responses would be wt + (12) where 2 = pek closed-loop response with roll ngle feedbck gin k1 = moment

(4) shows tht roll ngle feedbck gin to the rudder enbles the dmping rtio to be decresed.

corresponding to some frequencies of externl forces, s shown in Fig. 13.

17 -2 nd the closed-loop responses would be wt + (12) where 2 = pek closed-loop response with roll ngle feedbck gin k1 = moment coefficient dependent feedbck gin. on the roll ngle Contrsting Eq.(11) to Eq.(4) shows tht roll ngle feedbck gin to the rudder enbles the dmping rtio to be decresed. When the dmping reduces to such n extent tht the rudder is mde s rollexciting device, the pek roll responses re to be incresed corresponding to some frequencies of externl forces, s shown in Fig. 13. Therefore, roll ngle feedbck to the rudder seems to be unprcticl. 3 2 w cd w ci 2 W 2 1 USL. 4J C-) 1 c j. (rd. sec) w V =14 knots 5 M =.47kg-rn V=.99msec k1= _.Q_ Ooen-loop -- closed-loop 2 Reduction percentge of the pek roll responses Fig.l3 Pek roll responses 15

lerin (opcijfer (op cijfer -2 -) ''erbrokenlijn) 6.

pilot system, negtive roll ment M() =-( k2++ k1 will be cused by the rudder, nd the eqution of rolling motion is A + (B+k2) + (c-f-k1) = M(t) (13) The nondimensionl dmping rtio is given

respectively. to de- Contrsting Eqution (14) to Eqution (4), it is difficult termine whether the dmping rtio increses or decreses fter nd introducing combined feedbck.

18 lerin (opcijfer (op cijfer -2 -) ''erbrokenlijn) 6. Combined Feedbck of Roll Velocity nd Roll Angle to the Rudder f combined signl - c2 + c1 where c1 nd c2 re coefficients previously mentioned is received mo- by the rudder from n utomtic pilot system, negtive roll ment M() =-( k2++ k1 will be cused by the rudder, nd the eqution of rolling motion is A + (B+k2) + (c-f-k1) = M(t) (13) The nondimensionl dmping rtio is given by B + k2 2A(c+k1) (14) nd the closed-loop responses would be = 35ifl t+,i ) (15) 2 3 where = pek closed-loop responses c1 nd k2 = moment coefficients m&itioned in Sections 4 5 respectively. to de- Contrsting Eqution (14) to Eqution (4), it is difficult termine whether the dmping rtio increses or decreses fter nd introducing combined feedbck. The experimentl results given in Fig.14 shows tht the reduction percentge in pek roll responses is less thn in the cse with roll velocity feedbck shown in Fig.9, nd this reduction percentge will be very smll t higher frequencies of externl moments. t cn be concluded tht the combined feedbck is inferior to tht mentioned in Section 4. 1c lnt ci3f 16 (enkel

(opcijfer- )cijfer-2) 3 3 2 2 3 1 1 2. 2.5 3. (rd. sec) w V =14 knots S V m.

Roll Accelertion Feedbck to the Rudder f roll ccelertion signl = -k where k3 is

19 (opcijfer- )cijfer-2) (rd. sec) w V =14 knots S V m.99msec k1=.358 M=O.71k g-rn k2=.24 - Open- loop -t- Closed-loop Reduction percentge of the pek roll responses Fig.14 Pek roll responses 7. Roll Accelertion Feedbck to the Rudder f roll ccelertion signl = -k where k3 is coefficient dependent on the roll ccelertion feedbck gin is given to the rudder by utomtic pilot system, negtive roll moment M() = -k3 ) will be cused by the rudder, nd the eqution of rolling is s follows motion 17

) A+k3) + B q + C q = M(t) (17) en lijn) The nondimensionl dmping rtio is given by B - 2J(A+k3)C nd the closed-loop responses would be (18) where = 4r( wt + ) (19) 4 = pek closed-loop response k3 =

(4), it is evident tht ccelertion 4 nd especilly in the proximity of resonnt frequency the recuction percentge in pek roll responses shown in Fig.9 is much greter thn tht in Fig.15.

Concluding Remrks So fr, the contribution of rudder ntirolling devices to reduce pek roll ngles t or ner the resonnt roll frequency hs been explined.

20 ) A+k3) + B q + C q = M(t) (17) en lijn) The nondimensionl dmping rtio is given by B - 2J(A+k3)C nd the closed-loop responses would be (18) where = 4r( wt + ) (19) 4 = pek closed-loop response k3 = moment coefficient dependent on the roll feedbck gin. By contrsting Eq. (18) to Eq. (4), it is evident tht ccelertion 4 nd especilly in the proximity of resonnt frequency the recuction percentge in pek roll responses shown in Fig.9 is much greter thn tht in Fig.15. This proves tht roll velocity feedbck to the rudder is superior to roll ccelertion feedbck t or ner the resonnt frequency. 8. Concluding Remrks So fr, the contribution of rudder ntirolling devices to reduce pek roll ngles t or ner the resonnt roll frequency hs been explined. n utomtic systems of combining steering nd roll stbilising by using only rudders, roll velocity feed- 4 bck seems to be the best mong ll the feedbck types put forwrd in this pper. As for the effectiveness of rudder ntirolling devices when cripring them with other stbilisers, Fig.16(2) my be good representtion to shcw their reltive merits. n order to mke rudder ritirolling devices cne into use, ships to which these devices re to be pplied must be equipped with lrge rudders, powerful steering mchineries, nd utcmtic control systems. ) 18

) 4 w 2 3 4 2 1 1 2. (rd. sec) 3. V =14 knots S Vrn=O.99rnsec M=O.47kg-rn k3=.

15 Pek roll responses 9. Acknowledgement The uthor is gretly indebted to Prof. r.j.gerritsm nd W.

finished smoothly. The uthor would like to express his grtetude to J.Ooms nd C.V.

21 ) 4 w (rd. sec) 3. V =14 knots S Vrn=O.99rnsec M=O.47kg-rn k3= Onen-loon - Closed-loop EJ-- Reduction percentge of pek roll responses Fig.15 Pek roll responses 9. Acknowledgement The uthor is gretly indebted to Prof. r.j.gerritsm nd W.Beukelmn for their ptient guidnce due to which ll the experiments nd this report could finlly be finished smoothly. The uthor would like to express his grtetude to J.Ooms nd C.V.Jorens for designing nd mking the whole control system in the tested model. The ssistnce nd co-opertion given by R.Brink nd E.C.Post re grtefully cknowledged. (enk 19

$-2 -) ) 6 ci) 4 (\4 ) 2 :.2.3.$ 4 (Hz).

22 erin (opcijfer 1 -) (op cijfer -2 -) ) 6 ci) 4 (\4 ) 2 : (Hz).5 W o Unstbilised Rudder lone Tnk lone + Rudder nd tnk ) Fig.16 Pek roll responses with different stbilising systems {2J ) 2

onderin (op cijfer - r tfl (1.) ciicer - ) References Robert Tggrt, " Anomlous Behvior of Merchnt Ship Steering Systems " Mrine Technology, vol.?, no.2, April 197 W.E.Cowley nd T.H.

23 onderin (op cijfer - r tfl (1.) ciicer - ) References Robert Tggrt, " Anomlous Behvior of Merchnt Ship Steering Systems " Mrine Technology, vol.?, no.2, April 197 W.E.Cowley nd T.H.Lmbert, " The Use of the Rudder s A Roll Stbiliser " 3rd Ship Control System Symposium, Bth, Englnd, September 26-28,1972 F.F.vn Gunsteren, " Anlysis of Roll Stbiliser Performnce ".S.P., vol.21, no.23?, My 1974 W.E. Cowley nd T.H.Lmbert, " Se Tri7s on A Roll Stbiliser Using the Ship's Rudder " 4th Ship Control System Symposium, Oct J.B.Crley, " Fesibility Study of Steering nd Stbilising by Rudder " 4th Ship Control System Symposium, Oct ) (eukeizi 21

A B C Vomenc lture mss of ships including dded mss dmping coefficient restoring moment per unit heel ngle k,k,k 1 2 3 moment coeff-ic-ients w'i-th i-ntroducing feedbck of M(t) M t Vm V5 roll ngle,

$exciting nondimensionl roll dmping rtio unstbilised by the rudder v, v2, \)3 V4 nondimensionl roll dmping rtio fter 34 introducing feedbck of roll velocity, roll ngle, combining both, nd roll$ ccelertion to the rudder respective ly the roll ngle excited by the rudder, or by the gyro-oscilltor with no or vrious types of feedbck mplitude of roll ngle excited by the rudder open-loop response

ccelertion to the rudder respective ly the roll ngle excited by the rudder, or by the gyro-oscilltor with no or vrious types of feedbck mplitude of roll ngle excited by the rudder open-loop response

24 A B C Vomenc lture mss of ships including dded mss dmping coefficient restoring moment per unit heel ngle k,k,k moment coeff-ic-ients w'i-th i-ntroducing feedbck of M(t) M t Vm V5 roll ngle, roll velocity nd roll ccelertion respectively roll exciting moment produced by the gyro-oscilltor fitted in the model mplitude of exciting moments time speed of the model speed of the ship rudder ngle rudder ngle mplitude phse ngle by which the roll motion lgs the rudder deflections phse ngle by which the roll motion lgs the moment phse ngle by which the rudder deflections lg exciting moment exciting nondimensionl roll dmping rtio unstbilised by the rudder v, v2, \)3 V4 nondimensionl roll dmping rtio fter 34 introducing feedbck of roll velocity, roll ngle, combining both, nd roll ccelertion to the rudder respective ly the roll ngle excited by the rudder, or by the gyro-oscilltor with no or vrious types of feedbck mplitude of roll ngle excited by the rudder open-loop response to the moment produced by the pek open-loop response 1, 2, 3, 4 pek closed-loop responses fter gyrooscilltor introducing feedbck of roll velocity, roll ngle, combining both, nd roll ccelertion to the rudder respectively 22

25 w frequency of the roll exciting moment produced by the gyro-osci 1 ltor frequency of the rudder swing 23

TECHNSCHE HOGESCHOOL DELF AFDELNG DEA SCHEEPSBOUW- EN

$Fesibility Study of Employing \ the Rudder s An$ 518 April 1981 Deift University of Technology Ship

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