Stability analysis and prediction of performance for a hydrofoil sailing boat, Part 3: Directional stability analysis

178 S T A B I L I T Y A N A L Y S I S A N D P R E D I C T I O N O F P E R F O R M A N C E F O R A H Y D R O F O I L S A I L I N G B O A T Part 3: D I R E C T I O N A L S T A B I L I T Y A N A L Y S I S by Y . Masuyama* Summary

A directional s t a b i l i t y has been analyzed f o r a h y d r o f o i l saihng boat w i t h t w o surface-piercing dihedral f r o n t foils and an inverted ' T ' rear f o ü . F o r the analysis, the interactions o f the m o t i o n s i n sway, y a w , r o l l and surge were accounted. The stabihty discriminant was calculated f o r three rudder c o n d i t i o n s : f i x e d , manual c o n t r o l l e d and windvane c o n t r o l l e d . Even i n the rudder f i x e d c o n d i t i o n , the stability i n directional m o t i o n o f the b o a t was achieved i n relatively w i d e range o f true w i n d directions. A l t h o u g h the range o f stable sailing was n a r r o w e d b y increasing the boat v e l o c i t y , the stability was considerably i m p r o v e d by the manual rudder c o n t r o l . Using the windvane c o n t r o l system, the directional stability was also i m p r o v e d , where the area o f windvane c o u l d be de-creased b y shortening the c h o r d l e n g t h ' o f the r u d d e r and b y l o w e r i n g the gear r a t i o o f the system. Frequency response analysis, w h i c h was also qonducted t o detect the response o f heading angle t o the change i n true w i n d d i r e c t i o n , c o n f i r m e d the v a l i d i t y o f the results o f d i r e c t i o n a l s t a b i l i t y analysis i n c o n j u n c t i o n w i t h resulting i n the quantitative r e l a t i o n between the windvane area and the d i r e c t i o n a l m o t i o n .

1. I n t r o d u c t i o n

For a sailing c r a f t , since heading angle effects larg-ely to the aerodynamic force o n sail, d i r e c t i o n a l stab-i l stab-i t y becomes m u c h stab-i m p o r t a n t compared t o the case o f a powered c r a f t . The d i r e c t i o n a l stability f o r the sailing c r a f t have been studied b y Spens et al. [ 1 ] , Curtiss [ 2 ] . Gerritsma [ 3 ] , Letcher [ 4 ] , Gerritsma and Moeyes [ 5 ] and T i t l o w [ 6 ] . I n these studies, the stabihty was analyzed w i t h considering various m o t i o n s o f the c r a f t such as sway, y a w and r o l l . The m o t i o n i n surge is excluded f r o m their analyses because apparent w i n d w h i c h induces the change i n heading angle is scarcely i n f l u e n c e d b y the m o t i o n i n the o r d i n a r y sailing c r a f t . For the c r a f t w i t h the h y d r o f o i l w h i c l i runs w i t h relat-ively h i g h v e l o c i t y and high acceleration, the m o t i o n i n surge affects considerably to the aerodynamic f o r c e o n sail.

Meanwhile, several researchers [ 7 , 8 , 9 ] have analyz-ed the d i r e c t i o n a l stability o f the sailing boat w i t h windvane c o n t r o l system to steer the boat i n a f i x e d d i r e c t i o n against the apparent w i n d . T h e y have analyzed the r o t a t i n g m o t i o n o f windvane shaft w i t h c o m -b i n a t i o n o f the t w o m o t i o n s : y a w and surge [ 7 ] , y a w and sway [ 8 ] , and y a w and r o l l [ 9 ] .

I n the present paper as a series o f investigations o n a h y d r o f o i l sailing boat [ 1 0 , 1 1 ] , the d i r e c t i o n a l stab-i l stab-i t y analysstab-is was carrstab-ied o u t w stab-i t h consstab-iderstab-ing the m o t i o n i n surge as w e l l as sway, yaw and r o l l . For the analysis, each stability discriminant was calculated f o r rudder f i x e d , manual c o n t r o l l e d and windvane con-t r o l l e d c o n d i con-t i o n s . T h e n con-the scon-table sailing regions were examined by using the parameters such as response

*) Kanazawa I n s t i t u t e o f Technology, Japan.

f a c t o r o f helmsman and span length o f windvane i n t h e latter t w o conditions, respectively. F u r t h e r m o r e , i n the last c o n d i t i o n w i t h the windvane c o n t r o l system, the saihng performance was also examined t h r o u g h the f r e q u e n c y response analysis w i t h the closed l o o p trans-fer f u n c t i o n .

2. Analysis f o r rudder f i x e d and manual c o n t r o l l e d c o n d i t i o n s

2.1. Linearized equations of motion and characteristic equation

F o r the analysis o f directional s t a b i l i t y o f the boat, m o t i o n s i n surge, sway, r o l l and y a w are considered, so t h a t the Unearized equations o f m o t i o n w h i c h derived i n Part 2 [ 11 ] are reduced t o the f o l l o w i n g f o u r equat-ions:

m{ii — rv^) = A X ,

m{v + YUQ - ) = i\Y + mg(i> ,

T h e rudder supposed t o be c o n t r o l l e d t h r o u g h the f o l l o w i n g relationship:

5 = _ q ^ - q i ^ , (2) wliere 6 is rudder angle, C j and are constants

(res-ponse factors) w h i c h depend o n the res(res-ponse o f helms-m a n against y a w helms-m o t i o n o f the boat. Since the angular velocities >// and 0 can be expressed as /• and p, respec-t i v e l y , respec-the Unearized equarespec-tions o f d i r e c respec-t i o n a l m o respec-t i o n are r e w r i t t e n b y using stability derivatives as f o l l o w s :

179

mv + muQ V/ — mwQ0 - mg4) — Y^^u - Y^v - Y^;i} - Y^<t> - - 7 . 0 - 7^ ^ _ Y^ - - 6 = O ,

L<i>'-lJ~^u"-K,v-K.v-K^<l>-K^ii, ( 3 )

5 + q i// + C21//" = O .

Thus, we have the stabihty determinant as.

m X- A ; - A'A - A ; _-'^,>X-'^,>X_^-A^-A_-„X -„X _{-A;.X^ -(X,+} " " ' j ) X -ATj

- ^ i x - ^(mli'u +

rpjX

- ) ' .

- n

- A ' „ - A ' , X - A ' „ - ( / „ + A ' , . ) X - -A'^X - A , - . V , - . V ; . \ - ; V , N\ 0 0 0 c , x + r. 1 (4)

E x p a n d i n g the above e q u a t i o n , the characteristic equation is o b t a i n e d .

fre"^' ^ f y ' ^ f y ' ^ f y ' ^ f y " ' • f y - ' f r , = 0

2.2. Stability derivatives I. Derivatives of liydrofoil

A l t h o u g h most o f the s t a b i l i t y derivatives f o r t h e forces X and Y, and m o m e n t K have aheady been ob-tained i n Part 2, w e have t o calculate the o t h e r derivatives f o r the m o m e n t A ' and f o r the forces and m o -ments w i t h respect t o and /'. These calculations can be done b y t h e same procedure as described i n Part 2, as, ^ri = - U i + X^.l^.cosi,. a = 1,2) = 0 {i= 3) ( 6 ) = -^n-'d;SinAi,.+iV;/^;C0SM. X . = 0 = 0 a = 1,2) a = 3) 0 - = 4 ) , {i= 1 - 4 ) 0-= 1,2) ( / = 3 ) 0 - = 4 ) , (7) (8) where /^; = V ^ f +3^? , Ai,. = tan-i(3^^./xp .

The derivatives f o r t h e m o m e n t N w i t h respect t o

u, V, p, r, V, p and /• are calculated as.

N.= i:^{Y..x^-Xi.y.) ( j = u - i ) , ( 9 )

and w i t h respect t o <p and /2 are also as,

^ = | / ^ ^ ^ . - . - ^ . 3 . ) + 2 ^ ( y , , x , , ~ x . ^ y , )

U = <t>.h) . ( 1 0 )

The derivatives o f r u d d e r angle are derived by t h e

f o U o w i n g relationships w i t h a c c o u n t i n g f o r t h e small-ness o f t h e leeway angle.

1 2 ^^D4

95 '

(11) i 2 ^B ^F4^4 36 ( 1 2 ) = ~Y,z^ > (13) (14) 2. Derivatives of sail

F o r the case o f a sail, we need t o evaluate the derivatives f o r t h e m o m e n t and f o r the forces and m o m e n t s w i t h respect t o \p and r. Where, those w i t h respect t o r were neglected because o f the sail being made by s o f t c l o t h .

T h e derivatives w i t h respect t o ^ are calculated as f o l l o w s :

9 Z , 3C/^ 3X5 3^^

'"^ dU^^ 3 * 3/34 3 *

1 3 * SA _{3 *} ) , ( 1 5 ) as + tl^A ) , 3 * 3 p , 3 * (16)

where b o t h ^C^gl^^A ^^YSI^^A ^ ^ ^ ^ already been obtained i n Part 2. T h e n the apparent w i n d veloc-i t y and d veloc-i r e l t veloc-i o n are expressed as f o l l o w s w veloc-i t h t a k veloc-i n g b o t h 0 and $ t o be zero i n t h e equations ( 5 ) and ( 6 ) i n Part 1 [ 1 0 ] . ^SA = { C ^ + t ^ s r ( c o s 7 ^ c o s * + s i n 7 j , s i n * ) } 2 ^^^^ + { F + £ / ^ j , ( - c o s 7 y , s i n * + s i n T ^ , c o s ^ ^ ) } ^ , = t a n -V + c o s 7 y , s i n * + s i n 7 j . c o s * ) U + f / ^ j , ( c o s 7 j . c o s * + s i n 7 j , s i n « ' ) ( 1 8 ) D i f f e r e n t i a t i n g p a r t i a l l y b o t h sides o f equations ( 1 7 ) and ( 1 8 ) b y we have • = - - — [ { J 7 + f ^ j . ( c o s 7 ^ c o s ^ ' + s i n 7 j , s i n * ) } X ( — c o s 7 „ s i n ' 3 ' + s i n 7 „ c o s * )

180 + {V+ f / g y ( - c o s 7 j , s i n * + s i n 7 j , c o s * ) } X ( - 0 0 8 7 ^ 0 0 5 * ~ 81117^,8111*)] , ( 1 9 ) and 9/?, U„T, — = — ; 7 „ „ ( c o s 7 „ c o s * + 8 i n 7 „ s i n \ [ ' ) } 9 * r r 2 ^ ' ' X ( - c o 8 7 j , c o 8 ^ ' - s i n 7 j , s i n * ) - { F + f / 5 ^ ( - c o s 7 ^ s i n ^ I ' + s i n 7 ^ , 0 0 8 * ) } X ( - c o s 7 y s i n * + s i n 7 j , c o s * ) ] . ( 2 0 )

Consequently, substituting equations ( 1 9 ) and ( 2 0 ) to equations ( 1 5 ) and ( 1 6 ) , we can get the derivatives f o r the forces X and Y. T h e n , the o t h e r derivatives w i t h respect t o * are calculated t a k i n g i n t o aqcount the co-ordinate o f center o f e f f o r t (C.E.) o f sail, {x^,y^,z^), as

( 2 1 )

( 2 2 )

K

The derivatives o f sail w i t h respect t o /• are expressed

as f o U o w s : cos|3^) ,(23) ^ c o s / J , sin/3,) 9 a , 1 rrl A - """NS

N.„ = ~ p U^.

c K rS Yrsh ( 2 4 ) ( 2 5 ) ( 2 6 )

The p a r t i a l l y d i f f e r e n t i a t e d terms i n equations ( 2 3 ) , ( 2 4 ) and ( 2 5 ) can be expressed b y using the equations presented b y E t k i n [ 1 2 ] as 9C, LS C ^^LS = 0 , 9„ i / . . a « . ( 2 7 ) * ( 2 8 ) ( 2 9 )

where h (Ji^ + /7.^^^)/2; h, and li^^^^ are each n o r m -alized distances ( d i v i d e d by the c h o r d length o f sail) f r o m leading edge o f sail t o t u r n i n g center o f y a w , t u r n i n g center o f y a w f o r dC^^/dr = 0 and aerodyna-mic center o f sail, respectively; and C^^^ is the max-i m u m value o f dC^g/dr. F o r the two-dmax-imensmax-ional t h max-i n w i n g , E t k i n gave the f o l l o w i n g values:

/ ' o = 3 / 4 , L = 1 / 4 > C^sr=0 • ( 3 0 )

*) A l t h o u g h E t k i n expressed the equation (27) i n non-dimensional f o r m , the dimensional f o r m is used here. The sign o f equation (27) is opposite to that o f E t k i n because the positive d i r e c t i o n is defined inversely.

Schhchting and T r u c k e n b r o d t [ 1 3 ] have also indicat-ed t h a t these values are weU h o l d f o r the delta w i n g or trapezoidal w i n g w i t h aspect r a t i o o f ~ 3 . Hence, we use these values f o r the calculation o f equations ( 2 7 )

and ( 2 9 ) . The remaining derivatives f o r the m o m e n t A'' are obtained as f o l l o w s :

= Y^s^s -^usys ' ( 3 1 )

= Y.s^s -x^sys ' ( 3 2 )

= Y.s^s - x . s y , • ( 3 3 )

Consequently, summing up the derivatives o f h y d r o -f o i l s and sail we o b t a i n the t o t a l s t a b i l i t y derivatives o f t h e b o a t .

2.3. Results of analysis for rudder fixed condition

T h e directional stabihty analysis was conducted i n all o f the d y n a m i c a l l y stable sailing c o n d i t i o n s w h i c h were c o n f i r m e d i n Part 2. The means o f the R o u t h ' s d i s c r i m i n a n t was apphed t o the c o e f f i c i e n t s o f charac-teristic e q u a t i o n , and at the same t i m e , the equation was solved by Bairstow's m e t h o d as i n Part 2.

UsT^'O "Vs sQii u n s t a l l e d sail s t a l l e d 9 0 6 0 3 0 unstoblo stable not foil-born 1 2 0 V T 1 8 0

Figure 1. Sailing state map f o r w i n d velocity UST = 10 m/s sliowing stable region i n directional m o t i o n f o r rudder f i x e d c o n d i t i o n .

Figure 1 shows a sailing state map i n d i c a t i n g the result o f analysis w i t h rudder f i x e d c o n d i t i o n (C^ = C j = 0 ) f o r the case o f U^j. = 10 m/s, where the abscissa is the t rue w i n d d i r e c t i o n , j j , , and the ordinate the sail angle, e. The heavy sohd line i n the figure d i s t i n -guislies the region i n t o sail unstalled c o n d i t i o n i n l e f t h a n d side and stalled i n r i g h t . The shaded zone i n d i c a t -es the region where the b o a t is unstable i n d i r e c t i o n a l m o t i o n i n spite o f being stable i n b o t h l o n g i t u d i n a l and lateral m o t i o n s . I n the sail unstalled c o n d i t i o n , the directional m o t i o n is stable f o r large e, or l o w boat v e l o c i t y . W i t h decreasing the e and approaching t o c r i t i c a l sailing state d e f i n e d i n Part 2, w h i c h was i n -dicated by dot-dash-line i n the f i g u r e , the b o a t be-comes unstable i n w i d e range o f 7^ except f o r the close-hauled c o n d i t i o n . The unstable region expands w i t h the 7y. When the 7^, exceeds 1 2 5 ° , the sail stalls and the boat v e l o c i t y begins t o decrease a b r u p t l y as indicated i n the polar diagrams i n Part 1 and Part 2.

1 8 1

The boat tends t o slow d o w n u n t i l 7^, = 1 6 0 ° , and i n this region, the d i r e c t i o n a l m o t i o n becomes unstable w i t h regardless o f the e. A t 7^ f r o m 160° to 180° i t becomes stable again.

The variation o f the stable region w i t h w i n d veloc-i t y veloc-is shown veloc-i n Fveloc-igure 2. F o r the saveloc-il unstalled con-d i t i o n , the . s t a b i l i t y l i m i t shifts to the larger e w i t h increasing w i n d v e l o c i t y i n accordance w i t h the similar movement o f the critical sailing state. For the sail

V T = 9 0 ' '

C, =

0 90 6 0 3 0 not f o i l - b o r n or capsize

Figure 2. V a r i a t i o n o f stable region i n directional m o t i o n w i t h w i n d velocity f o r rudder f i x e d c o n d i t i o n .

Stalled c o n d i t i o n , the unstable region shifts t o w a r d larger 7^ side and the stable region i n tlie f o l l o w i n g w i n d c o n d i t i o n narrows w i t h increasing the w i n d velocity.

2.4. Results of analysis for manual controlled con-dition

The d i r e c t i o n a l m o t i o n o f the boat at high v e l o c i t y can be stabilized b y rudder c o n t r o l . Talcing value as zero i n equation ( 2 ) , we may determine the range o f Cj value t o stabilize the directional m o t i o n . Figure 3 shows the v a r i a t i o n o f the stability l i m i t w i t h the value o f f o r the case o f 7^, = 9 0 ° , where the c h o r d length o f the r u d d e r is 0.24 m . I t can be seen t h a t the stability l i m i t i n the d i r e c t i o n a l m o t i o n coincides t o the critical sailing state at C j = 0 . 1 . This means t h a t the stable sailing region is expanded t o the l i m i t o f d y n -amically stable sailing c o n d i t i o n w i t h o u t capsizing. A p p h c a t i o n o f the Cj value, o f 0.1 t o the analysis f o r the other w i n d c o n d i t i o n s succeeded t o stabilize the boat i n the d i r e c t i o n a l m o t i o n .

3. Analysis f o r windvane c o n t r o l l e d c o n d i t i o n 3.1. Windvane system

A basic m o d e l o f windvane c o n t r o l system w i t h gear mechanism is considered. As shown i n Figure 4 , tire windvane rotates around a vertical shaft and the generated t o r q u e is t r a n s m i t t e d t o a rudder t h r o u g h a pair o f gears. A l t h o u g h a linkage mechanism is usually adopted i n the practical systems as described b y Rat-c l i f f e [ 1 4 ] and LetRat-cher [ 1 5 ] , the gear meRat-chanism was selected t o avoid the c o m p l i c a t i o n o f the analysis.

SO AO 3 0 2 0 1 0 s t a b l e without rudder control unstable regions without rudder control c a p s i z e

..critical sailing state

_1 I \ i_

8 1 0 12 U 1 6 % UST

Figure 3. V a r i a t i o n o f stability Umit i n directional m o t i o n w i t h response factor o f helmsman, Ci.

w i n d v a n e

Figure 4 . Arrangement o f windvane system.

O t h e r t h a n t h e windvane-operated r u d d e r m e n t i o n e d above, a p r i m a r y rudder is equipped, b u t the rudder angle is f i x e d t o t h a t o f e q u i l i b r i u m state, 5g. A t the e q u i l i b r i u m sailing state, the angles o f windvane and r u d d e r are coincided t o those o f the apparent w i n d d i r e c t i o n and the boat advancing course, respectively. Mass o f the r o t a t i n g parts is c o u n t e r belanced b y a weight, and a f r i c t i o n a l loss o f the f o r c e is neglected.

3.2. Linearized equations of motion and character-isric equation

The r e l a t i o n between r u d d e r angle and windvane angle is described as

182

where G is the gear r a t i o . D e f i n i t i o n s o f positive sense i n the t o r q u e are shown i n Figure 5. The t o r q u e is t r a n s m i t t e d b y the gears w i t h the mechanical advan-tage o f 1/G as

T„ = ( 1 / G ) ( 3 5 )

T h e e q u a t i o n o f t h e r o t a t i n g m o t i o n f o r t h e w h o l e windvane system w i t h respect to the windvane s h a f t , is described as

( 3 6 )

where I^^^ can be considered as the t o t a l m o m e n t s o f

i n e r t i a f o r the system. E q u a t i o n ( 3 6 ) is hnearized b y using the s t a b i l i t y derivatives as

( 3 7 )

where the q u a n t i t y o f I^^^ is i n c l u d i n g t h a t o f 5 (the added mass), and the 5 is t r a n s f o r m e d t o 77 using the equation ( 3 4 ) . T h e n , the Unearized equations o f direc-t i o n a l m o direc-t i o n o f direc-the boadirec-t w i direc-t h windvane c o n direc-t r o l sys-t e m can be o b sys-t a i n e d by i n sys-t r o d u c i n g sys-the equasys-tions ( 3 4 ) and ( 3 7 ) instead o f f i f t h f o r m u l a i n the e q u a t i o n

( 3 ) . The derivatives w i t h respect t o ri and 77 f o r X, Y, K and A', w h i c h should be supplemented t o the f i r s t f o u r f o r m u l a e i n equation ( 3 ) , are neglected because o f their smallness compared t o those o f 5 and 6 .

Consequently, the stabiUty determinant can be ex-pressed as f o l l o w s :

-,v

A-,,x - A; I/.., - AV)X'-A'^X - (/.,+AV)X--A;X -A, 0

-A'^ - A'^ "•^'rX-A; • l A , + A ' i ) x - - - . v , x i ; „ - . \ - , i x - - . \ ' , x -A; 0 - T . \ - T , - r . x - r „ x - T y - - T i - r , / . x ' - r . x -7;.x- - r x - T

0

UG 1 ( 3 8 )

T h e characteristic e q u a t i o n is t h e n obtained b y ex-panding t h e e q u a t i o n ( 3 8 ) .

+ 4 3 ^ ' + 4 2 ^ ' + 4 i ^ + 4 o = o . ( 3 9 )

3.3. Stability derivatives of windvane system The windvane i n the shape o f f l a t plate w h i c h r o t a t

-es around its leading edge is considered f o r the analys-is. When the angle o f apparent w i n d shifts b y A / i ^ , t h e windvane generates a t o r q u e t h a t can be expressed b y the f o U o w i n g equation.

T=p„ UlA ( 4 0 )

Figure 5. Definitions o f positive sense i n torque and angle. where ly is a distance f r o m C.E. o f windvane t o its leading edge. The derivative o f the Ty w i t h respect t o

u is calculated as f o l l o w s :

3 / i ,

dU

SA - ^ ) _{9 f /} ( 4 1 )

where the f i r s t t e r m i n the parenthesis can be neglect-ed (because C^,^ = 0 at e q u i l i b r i u m state) and using relationship obtained i n Part 2 ( 9 / 3 ^ / 9 t / = —sin/3^/

), the equation ( 4 1 ) can be s i m p l i f i e d as f o l l o w s :

1 Pa^SA^vV 9 C 9f3. sin/3 A ' ( 4 2 )

where 9C^j,/9/3^ is a l i f t - c u r v e slope o f the windvane

and is expressed as f o l l o w s :

9 C ,

LV _ 0.95

2+A ( 4 3 )

I n a similar way other derivatives o f 7"^with respect t o

v,\lj,rt,p and /• are obtained as f o l l o w s :

9 C , ^.V = ^ P . ^ S 4 ^ K 4 ^ C O S / 3 _ , èC^y 9/3^ -^Pn '^SA^V'VT:^ ^ _{9/3, 3 *} Try ^Pa^lA'^vh — T z

9C

LV 97? (44) (45) (46) (47) (48)

-183

vane, d j S ^ / a * has aheady been obtained i n equation ( 2 0 ) , and dC^y/bv is the same as öC^y/d^^ w h i c h is expressed b y equation ( 4 3 ) .

Similarly as the case o f sail w i t h respect to /• i n equation ( 2 9 ) , the derivatives o f Ty w i t h respect t o TJ can be obtained as f o l l o w s ;

Cy bCj^y

and c o m b i n i n g w i t h the equation ( 3 0 ) , we have 1

T_

_{•nV Pa ^SA^V'V} (49)

The derivatives f o r t o r q u e exerting on a rudder can be obtained by using similar equations as ( 4 2 ) , ( 4 4 ) , ( 4 6 ) , ( 4 7 ) , (48) and ( 4 9 ) . I n the equations, instead o f

13^ f o r the f o r m e r equations, the leeway angle, (3, is

used. Since the value o f (3 is very small, the equations can be abbreviated as f o l l o w s : '6R = 0 B "^R 'R 35

T

X ^vR -^^R ' T , R = Pw ^ B \ ^ R 2 '^LR 36 ( 5 0 ) (51) (52) (53) (54) (55)

where b o t h 3C^^ /3(3 and 3C^^ /35 are i d e n t i c a l w i t h a l i f t - c u r v e slope o f the rudder and expressed b y similar f o r m as e q u a t i o n ( 4 3 ) . Considering the added mass e f f e c t o f water, the derivatives o f r „ w i t h respect t o v,

p and /• are given as f o U o w s : T _vR

T.

pR ^vR ^R ' TyR T-VR ^R ( 5 6 ) ( 5 7 ) ( 5 8 ) Consequently, summing up the derivatives f o r b o t h windvane and rudder w i t h accounting f o r the gear r a t i o , the t o t a l derivatives a r o u n d windvane shaft are obtained as f o l l o w s :

T = T

_uV

T. = T^v'^GT^^

Tp=T^v'^GT^R

Tr = T,y^GT^^

• T, = GT,^

Ty + G^T-^

T

4,V ( 5 9 )

3.4. Results of analysis for windvane controlled con-dition

R e f e r r i n g t o commercial models o f the windvane system, the chord length o f windvane, Cp,,was taken as 30 cm and t h a t o f rudder, , was t a k e n as 5, 10 and 15 c m t o investigate the e f f e c t o f o n the d i r e c t i o n a l s t a b i l i t y . T h e submerged length o f the rudder was f i x -ed to the same value as t h a t o f the p r i m a r y rudder. A p p a r e n t l y the area and aspect r a t i o o f the submerged p o r t i o n vary w i t h attitudes o f the boat. I n order t o decrease the m o m e n t o f i n e r t i a a r o u n d the shaft, I^^,^, relatively small value o f Cy (30 c m ) was chosen. T h e estimated values o f I^^^ f o r the possible windvane sys-tems were less t h a n 0.05 kgf.m.s^ i n c l u d i n g the added mass o n the rudder. The a p p l i c a t i o n points o f f l u i d

U5i = 10">*i y^ = 9 0 °

^ = 0 05 k g f m s ' G = 0-5

0 0-5

O 0.2

Figure 6. R o o t locus diagram o f characteristic equation, vary-ing w i t h span length o f windvane , liy (dimensional value).

d y n a m i c f o r c e f o r windvane and r u d d e r were f i x e d o n the quarter-chord line o f t h e m , because the geometry o f t h e i r sections was a x i s y m m e t r i c a l .

The parameters such as span length, hy, and gear r a t i o , G, t o stabilize the boat i n the d i r e c t i o n a l m o t i o n were examined. A n example o f the r o o t locus dia-grams f o r the case o f U^j. = 10 m/s, 7^ = 9 0 ° , e = 22°, ƒ = 0.05 kgf.m.s^, = 5 cm and G = 0.5 is s h o w n i n Figure 6, where the movements o f eight r o o t s w i t h decreasing the hy are represented. F o r large hy, i.e. large windvane area, the real parts o f all r o o t s are negative, i n d i c a t i n g t o be stable i n d i r e c t i o n a l m o t i o n . However, w h e n the hy being smaher t h a n 0.5 m , one o f the roots o n the real axis becomes positive, resulting i n static i n s t a b i l i t y o r divergence.

Figure 7 shows the c o n d i t i o n stabilizing the direct i o n a l m o direct i o n i n directhe r e l a direct i o n bedirectween A and C/^^ w i direct h -i n the range m a -i n t a -i n -i n g the d y n a m -i c a l l y stable con-d i t i o n s con-discussecon-d i n Part 2, f o r various gear ratios f r o m

1 8 4

varve area

Y j . S O * rudder chord: CnaaOSm vane chord: CyiiO-3 m ( vane area s hyx Cy )

ratio vane area Y T = 160* rudder chord: C R I O O S m c 5 90* v Q o e chord: C y» 0 - 3 m (vane area « h y X Cy ) G : gear r o l i o 8 10 1 2 " ) t U 16UsT vana area soil area 2.01-h &6 hy V T = 9 0 * voh 0^ O'^y-^ vans area _{h« Y T = 1 6 0} sail area ' l.Zr = = 9 0 * lo 2 . 0 1-0 0-6 0 0 ' H I — ' 'óil-born ^ CR = 0-10m Cv=0.3 m 8 1 0 12 % U 1 6 U S T 6 10 12 14 16 U 5

vane orea ""'y hy Y T = 1 6 0 '

sail area ' ^ c = 90* "lo 2 . 0 10 1.2 m 0 - 8 0./. 0 0 foil-bom ^ 1 ^ 6 10 12 "Vs 14 16 Uc

(c)

_{s i a b l e}

(c)

Figure 7. Variations o f span length o f windvane f o r stabilizing w i t h w i n d velocity f o r various-chord lengths o f rudder and gear ratios (beam w i n d c o n d i t i o n ) .

Figure 8 . Variations o f span length o f windvane f o r stabilizing w i t h w i n d velocity f o r various chord lengths o f rudder and gear ratios ( f o l l o w i n g w i n d c o n d i t i o n ) .

185

0.2 to 1.0. F o r each gear r a t i o and c h o r d l e n g t h o f rudder, the hy satisfying t h e stable c o n d i t i o n s decreas-ed w i t h increasing the w i n d v e l o c i t y , f / ™ . These results are quite reasonable because t h e higher the w i n d veloci t y , the hvelocigher becomes the steervelocing torque t o t h e r u d der f o r compensating the decrease i n the hy o r w i n d -vane area. As seen i n t h e f i g u r e , the hy can also be de-creased w i t h the smaUer G and because o f decreas-ing the t o r q u e . However, the value o f must have its lower Umit f r o m the strength p o i n t o f view. T h u s the gear r a t i o G should be taken less than 0.5 t o m a k e the windvane reasonable size.

For an example o f sailing w i t h f o U o w i n g w i n d , the results at 7 y = 1 6 0 ° are shown i n Figure 8. I n t h i s case, the boat is stable i n the d i r e c t i o n a l m o t i o n w i t h o u t t h e windvane c o n t r o l u n t i l the w i n d v e l o c i t y reaches t o 10 m/s. A l t h o u g h t h e hy can be reduced w i t h de-creasing b o t h G and as the case o f 7 ^ = 9 0 ° , the increase i n the w i n d v e l o c i t y scarcely reflects t o the shortening o f hy because the apparent w i n d v e l o c i t y does n o t change so m u c h f o r the f o l l o w i n g w i n d con-d i t i o n .

The d i r e c t i o n a l stability o f the boat is t h e n j u d g e d b y using the r o o t locus diagram i n terms o f t h e varia-t i o n o f I^^^ i n Figure 9, where G = 0.2, 5 c m , hy= 0.6 m and w i n d c o n d i t i o n o f 16 m/s and 7 ^ . =

1 6 0 ° . I n this f i g u r e , real parts o f a c o m p l e x pair o f r o o t s become positive w h e n the I^^^ becomes larger t h a n O . ö k g f . m . s ^ , f a l l i n g i n t o the state o f d y n a m i c -ally unstable or divergent osciUation. N a m e l y , such an instabUity i n the d i r e c t i o n a l m o t i o n m a y occur w h e n

0 0 5

0 6

2-0

Figure 9. R o o t locus diagram o f characteristic e q u a t i o n , varying w i t h m o m e n t o f inertia o f windvane system, (dimensional value).

ƒ is large even i f b o t h G and are small. Hence we sUould take tUe value o f I^^^ smaU.

4 . Frequency response analysis

4.1. Closed loop transfer function

A r o l e o f t h e windvane c o n t r o l system is t o keep heading angle o f the boat constant against t h e

direc-t i o n o f apparendirec-t w i n d . The p e r f o r m a n c e o f direc-the c o n direc-t r o l system is checked t h r o u g h the f r e q u e n c y response analysis i n w h i c h the response o f heading angle t o t h e v a r i a t i o n o f t r u e w i n d d i r e c t i o n , 7 , is investigated. The b l o c k diagram f o r the c o n t r o l system is s h o w n i n F i

-gure 10.

3Y

» E oat with sail

Figure 10. B l o c k diagram f o r windvane c o n t r o l system.

We use t h e f o r m u l a e w i t h respect t o Y, N and T i n the equations ( 3 ) and ( 3 7 ) . A f t e r d r o p p i n g the terms o f u and (p, and adding the t e r m o f 7 i n t h e right hand o f the equations, w e get

Y,8=Y J , 5 y I J Y,4'+Y^-Y^^ A ; - * ' - A ^ , ^ - A ^ ^ -N-V • N v - N , 8 - N y , ( 6 0 ) T,8 where i -Y = mu, m - 7 . 0 Y •N-U p o n Laplace t r a n s f o r m i n g b o t h sides o f e q u a t i o n (60) and e l i m i n a t i n g the terms o f v and 6 using the e q u a t i o n ( 3 4 ) , w e consequently o b t a i n t h e closed l o o p transfer f u n c t i o n as.

, ( 6 1 )

where s is t h e Laplace operator and the c o e f f i c i e n t s are expressed as f o l l o w s :

A^^ /^^^^(,-N.Y^^NY.+N-Y-NJ-)IG-T-W.Y.-N.Y.)IG ,

= / „ „ , ( A ' , > ; + A ' , , ! ' , ) / C - T-i-i^.Y^-Nj. + N.Y^~NJ.)IG

-aJG + T^)(i^-)\, -N.Y.) - T-lNyY^ + Y.N^) - T-W-Y^ + l^/V,)

'I„,(N-Y^ *Y-N^)IG .

2 " -T^{NY^.+Nj^)IG-<,TJG-i-T^)(.-NfY^.-NY.+ N^, Y - jV,, r^.) -T,(N,-Y,+Y.N^) + T.(NY^ + Y^N^)-r.(N-Y.^ + Y-N^) - TfiMJ.^ -Y..N,)

= - ( r/ G + 7-,)(A',};,+/v,i;) + r, ( A' , r j + i; A ' , ) + 7;(-/v„}'j-i-);A'j) + ^.i ("V,. r , - 5;^,, )/c + ( r / G + ) (N. Y^ + i;.A-^ )

- r , V, + r , ) - 7; OV, r, + i ' , /v,) ,

\ = + 7 ; ) CV,. - K^'t,) + Ï ; . ^ ^ y, - r , ' V . ) - (W, r, - Y;.N, ) , = LMY+y;N )IG .

186

^1 = " ^ i O V . r - i ; . \ ; ) / c - ( 7 - ^ / ( ; + r j) i, V : r - i - r , , v , )

w i n d and windvane conditions are as f o l l o w s : 7 ^ = 9 0 ° , = 10 c m , G = 0.5 and = 0.05 k g f . m . s ^ . The sail t r i m angles, e, f o r each case are the m i n i m u m values to be stable i n the d i r e c t i o n a l m o t i o n w i t h o u t the windvane c o n t r o l f o r each w i n d v e l o c i t y . The gain starts t o decrease f r o m about 0.1 rad/s o f co f o r the case o f f / ^ j , = 1 0 m / s , whereas i t does t o decrease f r o m about 0.5 rad/s f o r f/^^ = 16 m/s. For the f o r m e r case the gain r e d u c t i o n and phase lag c o u l d be remark-^ 7 = ^ Pfl^s^S4 ^ s r ^ 2 C j , ( - c o s / 3 ^ s i n 7 ^ + sin(3^ c o s 7 ^ ) ^^^y decreased w i t h increasing hy compared t o the

latter case, i n d i c a t i n g the significant i m p r o v i n g e f f e c t o f the response f o r the lower w i n d v e l o c i t y . These results back u p w e l l those indicated i n Figure 7.

As an example o f sailing w i t h f o l l o w i n g w i n d , the result obtained at the c o n d i t i o n o f 11^.^= 16 m/s and

+ 7;,.(- A " ,+ r ) + z;(Av y, + y ,A ; I .

fl„ = - " ' , / c + 7;.)(A;y - y . v ) + 7;,(-A;y,.+ yA', ) + r ( A; i ' , . r^.xj .

The derivatives w i t h respect to 7, i.e. Y , N and T , can be derived i n the same way as expressed by the equations ( 1 6 ) , ( 2 2 ) and (45) w i t h taking * as zero i n the equations (17) and ( 1 8 ) . Namely,

1

9 C ; , 9 ^ .

(cos/3. cos7.^ + sin/i. sin7.^)} ,

N Y X, , 7 5 '

( 6 2 )

( 6 3 )

Ty=\ '"'•'^y^V^SA ^ST (cOS/3^ cdS7

+ sin|3^ s i n 7 y ) . ( 6 4 )

P u t t i n g s = jco i n the equation ( 6 1 ) , we can examine the f r e q u e n c y response f o r the c o n t r o l r a t i o , \p(Jco}/ 7 ( / w ) .

7 ^ = 180° is shown i n Figure 12. Similar tendency was observed as i n the Figure 11(a), and i n this case the gain and phase lag were m u c h a f f e c t e d by the /?,^.Such an increased response o f the directional s t a b i l i t y was caused b y the lower apparent w i n d v e l o c i t y . Thus, the e f f e c t o f windvane area o n the d i r e c t i o n a l stability was c l a r i f i e d t h r o u g h the analysis, p r o v i d i n g an useful aid to design the windvane system.

1 1 . 1 • > . 1 ' ' ' 1 '' " \ \ N \ \ \ N \ YT = 1 6 0 ' c = 9 0 ° c „ = 01 m \ \ \ ^ \ p h a s e \ \ \ \ •v N ^ , N^:15 l O m X Cv = 0-3 m ^ 0-5 m X \ , , , 1 , , , , , , , 1 , , , , , , 1 . , > < ' ' 1 1 1 1 ' ' ^ \ g a i n U s , = 16 "><i Y , = 9 0 ' C r 3 9 ° p h a s e ^yf-\\S / \% \ \y 10m / 0 . 5 m / \ Cf, - 0 1 m p h a s e ^yf-\\S / \% \ \y wilhoot c , -. 0-3 m \ \* \ w i n d v z i r i e / \ y

\ ^

_\ \ \ \ V 1 SA , 1 , , , ,

Figure 11. Variation o f gain and phase angle o f c o n t r o l ratio w i t h span length o f windvane (beam w i n d c o n d i t i o n ) .

4.2. Results of frequency response analysis

Figure 11 (a) and (b) shows the results o f f r e q u e n c y response analysis as the variation o f gain and phase angle o f the c o n t r o l r a t i o w i t h span length o f windvane f o r U^j,= 10 m/s and 16 m/s, respectively. Where the

Figure 12. Variation o f gain and phase angle o f c o n t r o l ratio w i t h span length o f windvane ( f o l l o w i n g w i n d c o n d i t i o n ) .

5. Conclusion

As a succession o f Part 1 and Part 2, the directional stability analysis f o r a h y d r o f o i l sailing boat was car-r i e d out w i t h accounting f o car-r the m o t i o n s i n sway, yaw, r o l l and surge. For w h o l e range o f e q u i l i b r i u m saihng states i n w h i c h the dynamic stabihty was c o n f i r m e d i n Part 2, the d i r e c t i o n a l s t a b i l i t y o f the boat was analyz-ed.

First, stability discriminant was examined f o r the c o n d i t i o n s o f b o t h r u d d e r f i x e d and manual c o n t r o l -l e d . When the sai-l was unsta-l-led, the boat was stab-le i n the directional m o t i o n f o r large sail t r i m angle or l o w boat v e l o c i t y , even t h o u g h the r u d d e r was f i x e d . W i t h decreasing the sail t r i m angle t o the c r i t i c a l sail-ing state, the boat became unstable. On the other h a n d , f o r the sail stalled c o n d i t i o n the boat was unstable i n the o b h q u e l y behind w i n d , b u t stable i n

1 8 7

the foUowing w i n d w i t h regardless o f the sail t r i m angle. TUe unstable regions f o r tUe above conditions, Uowever, c o u l d be eliminated by the manual rudder c o n t r o l .

Secondly, the d i r e c t i o n a l stabUity w i t h windvane c o n t r o l system was analyzed t a k i n g i n t o account the r o t a t i o n o f windvane shaft. T a k i n g chord length as 30 cm, the required span length o f the windvane to satisfy the stable state was examined f o r various values o f cliord length o f the rudder and gear r a t i o . When true w i n d d i r e c t i o n is 9 0 ° , the span length could be short-ened w i t h increasing the w i n d v e l o c i t y , and the l o w e r values o f b o t h c h o r d length o f the rudder and gear ratio were also effective f o r shortening the span length. For the case that the true w i n d d i r e c t i o n is 1 6 0 ° , o n the other hand, although the span length c o u l d be shortened w i t h decreading b o t h chord length and gear ratio as f o r 9 0 ° , the increase i n the w i n d velocity scar-cely reflected to the shortening o f the span length be-cause o f m i n o r change i n the apparent w i n d v e l o c i t y . F r o m the r o o t locus diagram, i t was clarified that the i n s u f f i c i e n c y o f windvane area resulted i n static i n stability or divergence i n the directional m o t i o n . H o w -ever, when the m o m e n t o f inertia i n windvane system

was large, the boat feU i n t o tUe state o f d y n a m i c a l l y unstable or divergent oscillation even i f botU c h o r d length and gear r a t i o were smaU. Namely, the smaller m o m e n t o f inertia is favorable f o r the directional stab-iUty.

F i n a l l y , the results o f the frequency response ana-lysis were indicated t h r o u g h the variations o f gain and phase lag o f the c o n t r o l r a t i o w i t h the span length o f windvane. They backed up weU the results o f direc-t i o n a l Sdirec-tabUidirec-ty analysis and clarified direc-the e f f e c direc-t o f windvane area o n the d i r e c t i o n a l m o t i o n .

T h r o u g h the whole parts o f investigation, the m e t h o d t o predict the performance o f a h y d r o f o U sail-ing boat was discussed w i t h considersail-ing the d y n a m i c stability and the d i r e c t i o n a l stability. A p p l y i n g the m e t h o d t o an example boat w i t h t w o surface-piercing dihedral f r o n t foUs and inverted ' T ' rear f o U , the sail-ing performance at high v e l o c i t y o n foU b o r n m o d e was clarified. I t was also c o n f i r m e d that basic design o f the example .boat was successful t o give a reasonable StabiUty i n d i r e c t i o n a l m o t i o n together w i t h easiness o f manual c o n t r o l . The design and Various analyses described t h r o u g h the investigation are also applicable to the other t y p e o f boat. Therefore, u s e f u l means t o design and construct a h i g h performance h y d r o f o U sailing boat were p r o v i d e d .

A c k n o w l e d g e m e n t

The author w o u l d like t o acknowledge the con-t i n u i n g guidance and encouragemencon-t o f Professor K . N o m o t o o f the World M a r i t i m e University and Osaka University. He also wishes t o thank D r . M . U e k i o f Kanazawa I n s t i t u t e o f T e c h n o l o g y f o r valuable discussion and criticism.

Nomenclature

area o f sail Aj^ area o f rudder

Ay area o f windvane A^^. aspect r a t i o o f windvane

C j , response factors o f helmsman (constants f o r response o f helsman against yaw m o -t i o n )

^LR • G-iv ^^^^ coefficients o f r u d d e r and windvane C^yg y a w i n g m o m e n t c o e f f i c i e n t o f sail

c, Cj^ , Cy chord lengths o f h y d r o f o i l , rudder and windvane

c mean chord length o f sail G gear r a t i o o f windvane system g acceleration due to gravity

, ly m o m e n t s o f inertia f o r r u d d e r and w i n d -vane

/ m o m e n t o f inertia f o r windvane system

^xx'^yy^zz moments o f inertia f o r boat about x, y

and z axes

p r o d u c t o f i n e r t i a f o r boat about z and x axes

/ u n i t o f complex n u m b e r ( = A / ^ ) K, M, N moments o f r o l l , p i t c h and y a w Ip submerged length o f h y d r o f o i l

distance f r o m C.E. o f rudder t o its leading edge

ly distance f r o m C.E. o f windvane to its leading edge

m mass'of boat and crew

p, q, r perturbations i n angular v e l o c i t y f o r r o l l , p i t c h and y a w

s Laplace operator

7^ , Ty torques around s h a f t o f r u d d e r and w i n d -vane

u, V, w perturbations i n v e l o c i t y c o m p o n e n t o f C.G. along x , y and z axes

U^^ apparent w i n d v e l o c i t y U^j true w i n d v e l o c i t y

boat v e l o c i t y

X, Y, Z force components along x, y and z axes (3 - angle o f leeway

apparent w i n d d i r e c t i o n (angle between apparent w i n d v e l o c i t y and centerline o f boat)

T j . t r u e w i n d d i r e c t i o n ( a n g l e b e t w e e n t r u e w i n d v e l o c i t y a n d c e n t e r l i n e o f b o a t ) 6 a n g l e o f r u d d e r e t r i m a n g l e o f s a i l ( a n g l e b e t v / e e n b o o m a n d c e n t e r l i n e o f b o a t ) 7j a n g l e o f w i n d v a n e X r o o t o f c h a r a c t e r i s t i c e q u a t i o n , p^^, d e n s i e t i e s o f a i r a n d w a t e r (p,0,<p p e r t u r b a t i o n s i n a n g l e f o r h e e l , p i t c h a n d y a w ( E u l e r a n g l e s ) CO a n g u l a r f r e q u e n c y f o r t h e d i f f e r e n c e i n t r u e w i n d d i r e c t i o n subscripts i = 1 ; f r o n t s t a r b o a r d h y d r o f o i l = 2 ; f r o n t p o r t h y d r o f o i l = 3 ; r e a r h y d r o f o i l ' = 4 ; p r i m a r y r u d d e r = 5; s a i l R w i n d v a n e - o p e r a t e d r u d d e r V w i n d v a n e c c e n t e r o f s u b m e r g e d p a r t o f h y d r o f o i l t s u r f a c e - p i e r c i n g p o i n t o f h y d r o f o i l 0 v a l u e i n e q u i h b r i u m s t a t e

12. E t k i n , B . , 'Dynamics o f atmospheric f l i g h t ' , John Wiley and Sons, 1972, p . 2 7 3 .

13. Schlichting, H . and T r u c k e n b r o d t , E . , 'Aerodynamics o f the airplane', M c G r a w - H i l l , 1979, p . 186.

14. R a t c U f f e , G., 'Vane self-steerers f o r cruising yachts', 2nd Chesapeake Sailing Yacht S y m p o s i u m , S N A M E , 1975, Annapolis.

15. Letcher Jr., J.S., 'Self-steering f o r sailing c r a f t ' . Inter-national Marine, Camden, 1974.

References

1. Spens, P.G., DeSaix, P. and B r o w n , P.W., 'Some f u r t h e r experimental studies o f the sailing y a c h t ' . Trans. S N A M E , V o l . 7 5 , 1 9 6 7 .

2 . Curtiss Jr., H . C . , 'On the stabiUty and c o n t r o l o f sailing yachts', 2nd A I A A Symposium on tlie A e r o / H y d r o n a u t i c s o f S a ü i n g , A I A A , 1970.

3. Gerritsma, J., 'Course keeping qualities and morions i n waves o f a sailing y a c h t ' , 3 r d A I A A Symposium o n the A e r o / H y d r o n a u t i c s o f Saihng, A I A A , 1 9 7 1 .

4. Letcher Jr., J.S., 'Steering quahties o f f the w i n d ' , 3 r d A I A A Symposium o n the A e r o / H y d r o n a u t i c s o f Sailing, A I A A , 1 9 7 1 .

5. Gerritsma, J. and Moeyes, G., 'The seakeeping performance and steering properties o f sailing yachts', 3 r d S y m p o s i u m on developments o f Interest to Yacht A r c h i t e c t u r e , H I S W A 1973, A m s t e r d a m .

6. T i t l o w , J.D., ' A dynamic model f o r d o w n w i n d s a ü i n g ' , 8 t h A I A A Symposium o n the A e r o / H y d r o n a u t i c s o f Sail-ing, A I A A , 1977.

7. Buchan, A . L . and F l e w i t t , J.R., 'The behaviour and stabil-i t y o f w stabil-i n d opereated steerstabil-ing systems f o r sastabil-ilstabil-ing yachts', Trans. R I N A , V o l . 110, 1968.

8. S m i t h , D . A . , ' A new energy saving automatic steering device f o r sailing vessels'. New England Sailing Yacht Sym-p o s i u m , S N A M E , 1976, New L o n d o n .

9. Letcher Jr., J.S., 'Performance and stability o f wind-refer-enced autopilots f o r sailing vessels'. Marine Technology, V o l . 13, N o . 3 , 1 9 7 6 .

10. Masuyama, Y . , ' S t a b i l i t y analysis and p r e d i c t i o n o f per-formance f o r a h y d r o f o i l sailing boat, Part 1, E q u i l i b r i u m saihng state analysis'. I n t e r n a t i o n a l Shipbuilding Progress, V o l . 3 3 , N o . 384, 1986.

1 1 . Masuyama, Y . , StabiUty analysis and p r e d i c t i o n o f per-formance f o r a h y d r o f o i l sailing boat, Part 2 , D y n a m i c stability analysis', I n t e r n a t i o n a l Shipbuilding Progress, V o l . 3 4 , N o . 3 9 0 , 1 9 8 7 .