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 2: D Y N A M I C S T A B I L I T Y A N A L Y S I S by Y u t a k a Masuyama* S u m m a r y
A d y n a m i c stability analysis f o r a h y d r o f o i l sailing b o a t has been carried o u t o n the boat w i t h t w o surface-piercing dihedral f r o n t f o i l s and an inverted ' T ' rear f o i l . The small-disturbance t h e o r y was applied t o the ana-lysis w i t h accounting f o r the i n t e r a c t i o n between the l o n g i t u d i n a l and lateral m o t i o n s . S t a b i h t y derivatives were calculated f o r the h y d r o f o i l s and the sail. As a result o f the analysis, i t was c l a r i f i e d that there was a critical s a ü i n g state f o r the m a x i m u m attainable b o a t v e l o c i t y w h i c h was l o w e r t h a n t h a t o f the conventional predictions f o r the h y d r o f o i l sailing boat. When the v e l o c i t y exceeded the m a x i m u m v e l o c i t y determined as the critical sailing state, the boat f e l l i n t o statie i n s t a b i l i t y or divergence.
1. Introduction
T o p r e d i c t the sailing p e r f o r m a n c e o f the h y d r o f o i l b o a t , i t is essential t o conduct a d y n a m i c s t a b i l i t y analysis, because the boat is balancing w i t h d y n a m i c H f t f o r c e on the h y d r o f o i l s . A l t h o u g h the d y n a m i c s t a b i h t y analysis has already been c o n d u c t e d b y K a p -lan et al. [ 1 ] f o r a powered h y d r o f o i l c r a f t , there is l i t t l e i n f o r m a t i o n about the analysis available f o r the case o f sailing boat. The m o t i o n o f the powered c r a f t can be analyzed t h r o u g h classifying i n t o long-i t u d long-i n a l and lateral dlong-irectlong-ions because the c r a f t long-is sym-m e t r i c a l w i t h respect t o the centerline w i t h advancing s t r a i g h t f o r w a r d at e q u i l i b r i u m state. However, f o r the case o f the sailing boat the d y n a m i c s t a b i l i t y analysis m u s t be conducted w i t h accounting f o r the i n t e r a c t i o n o f the m o t i o n s i n a l o n g i t u d i n a l and lateral d i r e c t i o n .
I n t h e present paper as a succession o f the Part 1 [ 2 ] w h i c h presented an estimation m e t h o d o f the e q u i l i b r i u m state and the h i g h v e l o c i t y sailing charac-teristics w i t h f o i l - b o r n m o d e , the d y n a m i c s t a b i l i t y f o r a h y d r o f o i l sailing boat is examined. F o r the ex-a m i n ex-a t i o n , the stex-ability derivex-atives f o r sex-ail were ob-tained b y partially d i f f e r e n t i a t i n g the aerodynamic forces acting o n i t , and those f o r the h y d r o f o i l s were calculated using the procedure proposed b y K a p l a n et al. [ 1 ] . The results o f analysis were t h e n c o n f i r m e d b y the numerical s i m u l a t i o n conducted using nonhnear equations o f m o t i o n .
2 . Linearized equations of motion and characteris-tic equation
The m o t i o n o f h y d r o f o i l sailing boat r e f e r r i n g t o the b o d y axes is described b y the Ruler's e q u a t i o n as f o l l o w s :
m{Ü + QW ^ RV) = X , miV + RU-PW)=Y ,
*) Kanazawa Institute of Technology, Ishikawa, Japan.
m{W + PV-QU) = Z ,
I j - I ^ i P - Q R ) ~ PQil,, -Iyy)=N . (1)
The Euler angle rates are also given b y * = - f + ( Q s i n * + i ? c o s * ) t a n G ,
0 = ö c o s ^ - i ï s i n ï ' , (2) * = ( ö s i n * - l - i ? c o s # ) see© .
F r o m the results o f e q u i l i b r i u m saihng state analysis, i t was indicated that the angles o f p i t c h , heel and lee-w a y lee-were relatively small, especially i n p i t c h . Thus the p e r t u r b e d equations o f m o t i o n can be reduced f r o m the Euler's equation as f o l l o w s :
m ( M + Wp — /• V Q ) = A Z — mg 6 cos0g , miv + r UQ - p WQ) = A Y + mg 4> C O S 0 Q , m ( w + p - <7 M Q ) = A Z - 0 sin^^ ,
( 3 )
I n the equation the symbols w i t h zero s u f f i x indicate the values i n e q u i l i b r i u m state, and the symbols w i t h -o u t s u f f i x f -o r vel-ocities (u, v and w), angular vel-ocities
ip,q and r) and angles (0 and Ö) indicate the
deviat-ions f r o m their e q u i l i b r i u m values. The AX ~ AZ and
AK ~ A A ^ are p e r t u r b e d forces and m o m e n t s ,
respect-ively.
Since the sailing boat runs usuaUy accommodating w i t h sideslip and heel, causing t h a t aerodynamic f o r c e o n sail directs d i f f e r e n t l y t o the advancing course, the deviations i n any directions a f f e c t simultaneously o n b o t h m o t i o n s i n l o n g i t u d i n a l and lateral directions. F r o m this aspect, i t is r e q u i r e d t o carry o u t the stab-i h t y analysstab-is usstab-ing a l l o f the f o r m u l a e stab-i n e q u a t stab-i o n ( 3 ) .
The m o t i o n around z axis, however, w o u l d n o t a f f e c t f o r t h e dynamic stability analysis, although i t should be treated i n the directional stability analysis. Thus the last f o r m u l a i n equation ( 3 ) and the terms of r and /• are eliminated. As consequence, the linearized equat-ions o f m o t i o n can be expressed b y t h e f o l l o w i n g equation (4).
m{i(+qwQ) = AX-mge , miv W Q ) = AY + mg<l, ,
miw+pvQ -quQ)= AZ-mg<l>,pQ , ( 4 )
Iyyq = AM .
The angular velocities are also reduced t o 0 = p and e = ^ . The velocity o f t h e C.G. i n the true vertical direction is t h e n expressed as f o l l o w s :
h = w - U Q e + V Q 0 . ( 5 )
The perturbed force along x axis is . expressed i n terms o f derivatives as
AX = X^^u+X.v+X.v+X^w+ X.^ w + 0 + Z 0
+ Z . 0 + Z , 9 + Z , ( 6 ) A y , A Z , AK and AM are also expressed by similar
f o r m u l a e as equation ( 6 ) . Subsrituting these expres-sions i n t o equations ( 4 ) and ( 5 ) , we can rewrite the linearized equations o f m o t i o n as f o h o w s :
mii+mw^é +mge -X^,u-X^v-X.v-X^^w ^X-^w -^^1>~^p<P-Xp$-X,e-Xj^X.ë-X,h = 0 ,
mv-mWf^4,-mg<l>-Y^^u~Y^v-Y.v-Y^^w-Y.^w
- Y,1> -
Yp<P - Y.'^'- Y^e-Y^e-
Y.ë- Y,h = 0 ,mw + m 0 - m w g e +mg4>Q ^-Z^u-Zy-Z.v-Z^^ w -z.w-z^<p-zJ-z.:p-z^e-zJ-z.e-z,h=o, 4 v ''P'-K^y-Xv-K,v^K^^ w -K. w^K^rf -K^4>-K.4-K^e-Kj-K.e-K,Ji = Q , lyyd -M^^u-M^v-M.,v-M^^ w -7W,. v i ' - M ^ 0 -M^i-M.,p-Mg6 -M^Ö ~M.é-MjJi = O , h-w+u^e -1^00 = O . ( 7 ) Since t h e equations Of m o t i o n are linear, their
solu-tions can be expressed as f o l l o w s :
u =u^e^', v = e^' , vv = vv^e^' •
0 = 0 J e ^ ^ e =e^e^', h = h^e^' .
Substituting these expressions i n t o equation ( 7 ) and cancelling t h r o u g h o u t b y e'^^ w e obtain the six
simul-taneous equations: (mx - X , , ) - ( Z , X + ) - ( Z . X+XJ vvj - ( Z . x 2 + Z p A + Z ^ ) 0 j + { - Z . x 2 + ( m W o - Z ^ ) X + img-X^)}9^-X,Ji^ = 0 , -Y^^u^+{(m^,)X-Yyv^-(Y^X + YJw^ -{Y.X^+imwQ+Yp)X + {mg+Y^) }<p^ -iY.X^+Y^X + Y^)B,-Y,^h,=0 , -Z,u, ~ ( Z ; , X + Z ^ ) ) . j + { ( m - 4 J X ~ Z , J v V j + {-Z.X^ + {mvQ-Z^)X + img^Q~Z^)}^^ -{Z.X^+imu^ +Z^)X+Z^ - Z , / z j = 0 , -K^y,-iK,X+K^)v,^{K.X+KJw^ + {(I^-K.)X^ -K^X-K^}<p^-(K^X^ +K^X -M^u^~(M.X+M^)v^- (M^. X+MJ - (M. X^ X )0^+{(I^^-M^)X^-M^X -Me }Öi -M,^h, = 0 , - v v ^ - v ^ ^ ^ + u ^ e ^ + X h ^ = 0 . ( 9 ) N o w we have the stability determinant as,
- A - . X = + ( „ n . . „ - A r )X Hmg-xJ ->„ ( » i - r ; ) X - r - 5 ; , -z.x-z. - Z ^ X ^ ( „ „ . „ - Z )X - Z , x ' - ( m » „ + Z )X - 4 + ( » . f « „ - Z , ) -z. + ( » . f « „ - Z , ) -A-.x2-A-_,X-A-„ -'''/, - M . x 2 - / l / ^ x 4 / , " A / , ; ) X ' - A / , X -"'/, 0 0 - 1 « 0 X = 0 (10)
The characteristic equation is then obtained b y ex-panding the above equation.
fs^^+fj^^ + fe^^ + f.x'+f^X^ + f^x'+f^X^
+ / i ^ + / o = 0 ( 1 1 ) The necessary and s u f f i c i e n t c o n d i t i o n t o make t h e
solutions stable is determined b y either applying t h e R o u t h ' s discriminant m e t h o d t o t h e c o e f f i c i e n t s o f equation ( 1 1 ) , o r solving equation ( 1 1 ) t o f i n d direct-l y the r o o t s w i t h o u t positive readirect-l parts.
3 . Stability derivatives
3.1. Derivatives of hydrofoil
The results o f e q u i l i b r i u m sailing state analysis i n -dicated that t h e leeway angle was relatively smaU. Hence we m a y f o l l o w t h e calculation procedure o f K a p l a n et al. [ 1 ] f o r t h e stability derivatives o f each
h y d r o f o i l . Formulae f o r the derivatives are described i n the A p p e n d i x using the h y d r o d y n a m i c forces. The derivatives o f the f o r c e acting i n n o r m a l d i r e c t i o n t o the h y d r o f o i l panel, F.., are resolved i n t o y and z direc-t i o n s as
Y.. = F..smV. ,
" " ( i = \ - 4 ) ( 1 2 ) Z.. = ^;..cosr,. ,
where P is a dihedral angle o f the f o i l and ƒ = u, v, w, p,
q, <p, e, h, V, w, p and q. Consequently, the derivatives
o f f o i l system f o r the forces X , 7 and Z are obtained as. (13) The derivatives f o r the moments K and M w i t h respect t o u, v, w, p, q, v, w, p and q are calculated as
K , = 2 ( Z , y , - Y..Z.)
/ = l ƒ!•'/ n I'
(14) M , = S ( A ^ , z . - Z , . , V
and considering the difference o f a p p l i c a t i o n p o i n t o f p e r t u r b e d forces w h i c h are caused b y the change o f b o t h submerged area and d e p t h o f submergence, the moments w i t h respect t o 0 , Ö and h are also calculated as
(15)
3.2. Derivatives of sail
Since we may take b o t h * and e t o be zero i n the equation f o r the v e l o c i t y components o f the apparent w i n d f o r the sail w h i c h is described i n Part 1, the e q u a t i o n can be s i m p h f i e d as f o l l o w s :
= U + Ugj.cosjj. ,
V^=V + f/5j,sinT7,cos$ , ( 1 6 ) W^=W - L f y j . s i n T j . s i n * .
A m o n g these components we m a y neglect the value o f , because o f its relative smallness and its i r -relevancy w i t h the sail performance. T h e n the apparent w i n d v e l o c i t y , U^^ , and apparent w i n d d i r e c t i o n , jS^ , are expressed respectively as f o l l o w s :
= (U+Ugj,cosyj.r+(V+Ugj.smyj.cos'S>y , = s i n - l ( K / C / , . ) I tan - 1 V+ [ / ^ j . s i n 7 j , c o s * C/+C/^j.cos7^ (18)
The aerodynamic forces, and Y^, are calculated b y the f o l l o w i n g relationships:
where
Cxs ^-Cj^s'^^^h +^isSin/^.4 ' CYS =-^DS'^^'I^A - ^LS^^^^A • The derivative of X^ w i t h respect t o u is
J X , _ 5 X , _ ÖX, öU,^_^bX,
U p o n partially d i f f e r e n t i a t i n g b o t h sides o f equations (17) and (18) b y U, we have (22) (23) (19) (20) (21) dU . UA UsA cosfl^ sin (3^ UIA UsA
S u b s t i t u t i o n o f equations ( 2 2 ) and ( 2 3 ) i n t o equation (21) yields 3 C xs sin/3. F u r t h e r m o r e , f r o m equation ( 2 0 ) we have 3 C y „ 3 C , . 3 ^ 5 (24) + QjC0S/3^ 3
a
^ ^ s i n , 3 ^ - C ^ , c o s ^ ^ - ^ W ^ (25) 3|3_4 3^^ + CisSin/3^ •F i n a l l y substituting equations ( 2 0 ) and ( 2 5 ) i n t o equation ( 2 4 ) , we o b t a i n \s-\PaUsA^s'^-'^<^DS^''''-h 3C, + ^ + Cis]cos(3^sin'5.4 3C 3/3 (26)
Similariy, the derivatives o f Y^ w i t h respect t o u is given as
1
3C LS 9/3.
'DS
(27)
The derivatives o f and 7^ w i t h respect to v are obtained i n a similar w a y as f o l l o w s : 'SA ^ 5 { -3C. 'DS 9/3. cos^fi^ bC 'LS 9/3. cos/3^ sinj3_4 + 2 q 5 s i n 2 / 3 ^ } , (28) 3C, 'Ö5 9/3.
+ c
C0s2/3, •2q,5sin2;3^ } (29)The (3Cp5/3)3^) and (3C^_5,/3|34 ) i n the above equat-ions correspond t o the slope o f the lines representing the relationships o f and Q versus attack angle o f the sail i n d i c a t e d i n Figure 6 o f Part 1.
The derivatives o f X^ and w i t h respect to p and q are obtained as f o l l o w s : YpS ^ S " ^ 5 X YqS ^ ( S ^ 5 (30) ' Y^s = Y , s h ( 3 1 ) where (negative value) is coordinate f o r C E . o f the
sail. I n the same manner the derivatives o f m o m e n t s and My are given as f o l l o w s :
Ks = - Y u s h • K s - \ s h ' (32) ^ s = - \ s h >
"^vS^S ' (33)
\ s ' l • % s ~ ^ v s ^ i ' (34)
-Yus^l . (35)
The derivatives o f are assumed t o zero because o f negligible i n f l u e n c e o f the sail f o r c e o n the m o t i o n i n the d h e c t i o n . Consequently, summing up the deri-vatives 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 stab-i l stab-i t y derstab-ivatstab-ives o f the boat, where the derstab-ivatstab-ives o f aerodynamic f o r c ë on the h u l l were n o t taken i n t o ac-count because they are considerably smaller than those f o r the sail.
4. Results o f stabihty analysis
The d y n a m i c stabihty analysis was p e r f o r m e d i n all o f the possible e q u i l i b r i u m sailing state as a f u n c t i o n o f sail t r i m angle, e. Table 1 shows an example o f the calculated derivatives f o r the h y d r o f o i l and the sail i n the dimensional value f o r th6 case o f f/^^ = l O m / s , T j , = 9 0 ° and e = 2 5 ° . The submerged l e n g t h o f f r o n t starboard and p o r t f o i l s are 0.30 and 0.79 m ,
respec-Tablc 1
Example or stability derivatives of hytjrofoils a n d s a i l f o r t h e c a s e o f ü s r " 1 0 m / s , 7 r = 9 0 ° a n d e . 2 5 ° (dimensional value) S t . P o r t R e a r R u d d e r S a i l T o t a l D i m e n s i o n ( i - l ) ( i = 21 ( i = 31 [ i = 4t ( i - 5 1 D i m e n s i o n u -1 . 5 9 - 4 . 8 9 - 4 . 2 3 -1 . 2 0 - 9 . 4 8 - 2 1 . 3 9 k q £ - s V 0 . 2 0 - 4 . 0 9 0 . 0 0 . 7 4 2 9 . 0 8 2 5 . 9 3 m VJ 0 . 2 3 4 . 8 8 - 0 . 5 7 0 . 0 -- 4 . 5 4 p 0 . 1 6 - 4 . 9 6 0 . 0 -1 .31 5 5 . 2 6 4 9 . 1 5 k g f • s q - 3 . 2 1 - 1 2 . 4 8 - 9 , 2 5 - 2 . 1 1 1 8 . 0 2 - 9 . 0 3 r a d i- -1 0 3 . <5 1 3 1 , 4 0 . 0 0 . 0 2 8 . 0 k g f e 4 2 . 1 4 8 . 6 0 . 0 -81 . 7 9 . 0 r a d h - 5 2 . 0 - 6 0 . 0 0 . 0 - 3 5 . 2 - 1 4 7 . 2 k g f / m V 0 . 0 0 . 0 0 . 0 0 . 0 -- 0 . 0 k g f • s 2 w 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 m p 0 . 0 0 . 0 0 . 0 0 . 0 -- 0 . 0 k g f - s ^ a 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 r a d u - 6 . 6 3 6 . 3 0 . 0 6 . 4 5 . 7 4 1 . 8 k g f - 5 V - 2 6 . 7 - 1 4 9 . 7 0 . 0 - 5 6 . 7 -4 1.9 - 2 7 5 . 0 m w - 3 1 . 8 1 7 6 . 4 0 . 0 0 . 0 1 4 6 . 6 p - 2 2 . 1 - 1 8 1 . 6 0 . 0 9 9 . 9 - 7 9 . 7 - 1 8 3 . 5 k g f - 5 q 1 3 . 1 - 8 1 .2 0 . 0 1 1 . 2 - 1 0 . 9 - 6 7 . S r a d - 5 0 5 . - 1 1 7 2 . 0 . 0 . - 1 6 7 7 . k g f Ö 1 9 6 . - 4 1 2 . 0 . 5 9 9 . 383 . r a d h - 2 4 1 . 5 0 3 . 0 . 2 5 8 . 5 2 5 . k q f / m V - 0 . 4 6 - 1 . 2 5 0 . 0 - 0 . 8 8 - 2 . 5 9 k g £ - s 5 w - 0 . 5 6 1 . 4 9 0 . 0 0 . 0 -- 0 . 9 3 m p - 0 . 3 9 -1 .51 0 . 0 1 . 5 5 - 0 . 3 5 k g f • q 0 . 4 5 -1 . 2 0 0 . 0 0 . 0 - 0 . 7 5 r a d i; - 7 . 9 - 4 3 . 2 -1 . 4 0 . 0 - - - 5 2 . 5 k g f - 5 V - 3 1 . 8 1 7 8 . 4 0 . 0 0 . 0 1 4 6 . 6 w - 3 7 . 9 - 2 1 2 . 7 - 5 1 5 . 9 0 . 0 - 7 6 6 . 5 P - 2 6 . 4 2 1 6 . 5 0 . 0 0 . 0 -- 1 9 0 . 1 k g f - s q 1 6 . 9 7 . -1 1 9 9 . 0 . -- -1 0 8 6 . r a d i' - 6 0 1 . 1 3 9 7 . 0 . 0 . 7 9 6 . k g f Q 2 3 3 . 4 9 0 . - 6 7 . 0 . 6 5 6 . r a d h - 2 6 8 . - 6 0 5 . - 2 9 . 0 . -- - 9 2 2 . k q f / m v - 0 . 56 1 . 4 9 0 . 0 0 , 0 -- 0 . 9 3 k g f - 3 2 w - 0 . 6 6 - 1 . 7 7 - 4 . 1 6 0 , 0 - - - 6 . 5 9 m P - 0 . 4 6 1 . 6 0 0 . 0 0 . 0 1 . 3 4 k g f - s ? Ó 0 . 5 4 1 . 4 3 - 9 . 6 4 0 . 0 __ - 7 . 6 7 r a d u - 5 . 5 4 4 . 0 0 , 0 - 1 1 . 2 1 0 . 9 3 8 . 2 k g £ - s V - 2 2 . 1 - 1 8 1 . 6 0 . 0 9 9 . 9 - 7 9 . 7 - 1 8 3 . 5 k g £ - s w - 2 6 . 4 2 1 6 , 5 0 . 0 0 . 0 -- 1 9 0 . 1 p - 1 8 . 3 - 2 2 0 , 3 0 - 0 - 1 7 6 . 3 - 1 5 1 . 4 - 5 6 6 . 3 k g f • s - m q 1 0 . 9 - 9 8 . 5 0 . 0 - 1 9 . 8 - 2 0 . 7 - 1 2 8 .1 r a d - 5 1 8 . - 2 0 2 2 . 0 . 0 , -- - 2 5 4 0 . k g f - m 2 0 3 . - 7 2 1 . 0 . 0 . -- - 5 1 8 . r a d h - 2 5 0 . 8 9 0 . 0 . 0 . 6 4 0 . k g f V - 0 . 3 9 - 1 . 5 1 0 . 0 1 , 5 5 -- - 0 . 3 5 k g f • s 2 w - 0 . 4 6 1 . 8 0 0 . 0 0 . 0 1 . 3 4 k g f • s 2 p - 0 . 3 2 -1 . 8 3 0 . 0 - 2 , 7 4 -- -4 . 8 9 k g f . s?m 0 . 3 7 -1 . 4 6 0 . 0 0 , 0 -1 . 0 9 r a d U 3 . 4 2 6 , 5 - 1 1 . 2 - 2 . 1 1 Ü . 0 3 4 . 6 k g f . 5 V 2 6 . 1 - 1 5 1 . 7 0 . 0 1 . 3 - 5 5 . 3 - 1 7 9 . 6 k g f . 5 w 31 . 181 . - 1 1 9 8 . 0 . - 9 8 6 . • P 22 . - 1 8 4 . 0 . - 2 , - 1 0 5 . - 2 6 9 . kg f - s • m •"1 - 1 9 . - 1 0 0 . - 2 8 0 0 . - 4 . -3-5 . - 2 9 5 7 . r a d 3 0 0 . - 9 3 6 . 0 . 0 . -- - 6 3 6 , k g f • m e - 1 1 3 . - 3 2 5 . - 1 5 5 . 0 , - 5 9 3 . r a d h 1 3 9 . 401 . - 6 7 . 0 , -- 473 . k q f V 0 . 4 5 -1 . 2 0 0 . 0 0 , 0 - 0 . 7 5 k g f -w 0 . 5 4 1 . 4 3 - 9 . 5 4 0 , 0 -- - 7 . 6 7 k g f -p 0 . 3 7 - 1 . 4 6 0 . 0 0 . 0 -- -1 . 0 9 k g f . s ? m q - 0 . 4 4 - 1 . 1 6 - 2 2 . 3 6 0 . 0 - - - 2 3 . 9 6 r a d
l i v e l y . The derivatives o f the f r o n t p o r t and rear f o i l are d o m i n a n t among those o f h y d r o f o i l s , because these f o i l s have larger submerged area then the others. I t also can be seen that the derivatives o f the sail oc-c u p y the large p o r t i o n t o those f o r the f o r oc-c e X and the m o m e n t K. U5T= 10 V T = 9 0 ° e X 5 0 ° -0 2 5 ° G 2 0 ° Imaginar y axi : - 2 0 - 1 5 , > i - i ; 1 (1 ^ 1 1 1 ' 1 „. - 1 0 b R e a l a;ii - - 5
Figure L Root locus diagram for the case ofU^j.= 10 m/s and f j = 90° (dimensional value).
The e q u a t i o n ( 1 1 ) is solved using Bairstow's m e t h o d . The examples o f the r o o t locus diagrams f o r the cases o f U^j,= 10, 14 and 18 m/s at e = 9 0 ° are s h o w n i n Figures 1, 2 and 3 respectively. The simultaneous movements o f the eight roots are represented w i t h decreasing the e, and the corresponding variations o f sailing state parameters have already been i l l u s t r a t e d i n Figures-8, 9 and 10 o f Part 1. F o r large e w h i c h
VT = 9 0 ° X 50 0 3 5°
- 3 0 - 2 5 - 2 0 -15
Figure 2. Root locus diagram for the case of f / j j . = 14 m/s and 7jn = 90° (dimensional value). Usi = 1 6 " H V , = 9 0 ° X 5 0 ° 0 i O ° O 3 8 ° ^ 4 J J U s r
stable sailing limit (including sail derivatives)
do.
(excluding sail derivatives) highest velocity by the equilibrium equations
Figure 3. Root locus diagram for the case of U^j, = 18 m/s and
yj. = 90° (dimensional value).
makes boat velocity l o w , the real parts o f all roots are negative and the movements o f t h e m are relatively small. W i t h decreasing the e the movements o f the r o o t s become large gradually, and t h e n one o f t h e m o n the real axis becomes positive at 2 0 ° , 3 0 ° and 3 8 ° f o r
U^j, = 10, 14 and 18 m/s, respectively. This means
t h a t , regardless o f w i n d v e l o c i t y , the boat falls i n t o a static i n s t a b i l i t y or divergence at small e w h i c h yields h i g h v e l o c i t y . As m e n t i o n e d i n Part 1, i t should be n o t e d that the boat becomes unstable w h e n the sub-merged length o f starboard ( w i n d w a r d ) f o i l , / ^ j , be-comes smaller than about 0.1 m , a l t h o u g h the equi-h b r i u m equations can be solved u n t i l /^^ becomes t o zero, i.e. taking o f f f r o m water surface.
Figure 4 is polar diagram w h i c h compares the boat v e l o c i t y at the Umit f o r stable sailing determined b y the present analysis (sohd curve) w i t h the highest v e l o c i t y derived b y the e q u i l i b r i u m equations (dot-dash-curve) f o r Ugj= 10, 14 and 18 m/s. Each s h i f t o f the solid curve f r o m the dot-dash-one is shown b y the shaded zone where the boat is actually unstable a l t h o u g h the e q u i h b r i u m equations give some solu-t i o n s . I solu-t appears solu-thasolu-t solu-the convensolu-tional performance p r e d i c t i o n t h r o u g h the e q u i l i b r i u m equations results i n the over-estimation by 5 t o 10% f o r the m a x i m u m attainable v e l o c i t y . D o t t e d curve i n the f i g u r e shows the stabihty l i m i t derived b y the calculation excluding saü derivatives f o r each w i n d v e l o c i t y . C o m p a r i n g the curve w i t h t h a t f o r i n c l u d i n g sail derivatives (solid cui-ve), the latter is s h i f t e d t o higher v e l o c i t y side t h a n the f o r m e r b y about 5% f o r the cases o f (7^^ = 14 and
150
Figure 4. Polar diagram comparing the boat velocity at the limit for stable sailing with the highest velocity derived by the equili-brium equations.
18 m/s although b o t h curves almost coincide f o r
Ugj, = 10 m/s. A c c o r d i n g to the result o f analysis
pre-sented i n Figure 4, the sail has the roles n o t o n l y as the thrusting device b u t as a stabilizer at high w i n d v e l o c i t y .
5. Numerical simulation using non-linear equations of motion
I n order t o c o n f i r m t h e v a h d i t y o f b o t h Hnearization o f equations o f m o t i o n and calculation o f the deriva-tives described i n the previous sections, the m o t i o n o f the boat was simulated n u m e r i c a l l y using non-linear equations at the same c o n d i t i o n i n w h i c h the d y n a m i c stability analysis was c o n d u c t e d . The s i m u l a t i o n was carried out under the five-degree-of-freedom ( w i t h * = 0 ) w i t h f i x i n g the rudder angle t o the e q u i l i b r i u m state value b y a p p l y i n g the Runge-Kutta-GiU m e t h o d . T h e forces and m o m e n t s f o r e q u a t i o n ( I ) are calculat-ed b y the f o U o w i n g proccalculat-edure. First, the length and d e p t h o f submerged p a r t o f h y d r o f o i l s at any m o m e n t are determined f r o m the c o n t e m p o r a r y a t t i t u d e o f the boat. T h e n , v e l o c i t y components f o r the center o f submerged part o f each f o i l and C E . o f the sail are calculated as
U. =U+Qz.-Ry.
W. =W + Py.-Qx; .
Since attack angle o f each part can be obtained f r o m these velocity components, we have the forces and m o m e n t s using the f o r m u l a e described i n Part 1. Where, the added mass e f f e c t o f the f o i l was also i n -cluded f o r the m o t i o n n o r m a l to the f o i l panel. The appUcabihty o f the calculation m e t h o d f o r the h y d r o -f o i l system was c o n -f i r m e d t h r o u g h t o w i n g tank test [ 3 ] i n c l u d i n g regular wave conditions.
The simulation results f o r e = 3 0 ° , 2 5 ° and 2 0 ° under the w i n d c o n d i t i o n o f Ugj,= 10 m/s and 7^, = 9 0 ° are shown i n Figures 5, 6 and 7 respectively. F o r r o h disturbance o f = - 0 . 5 rad/s at t = 5s, the t i m e dependent f l u c t u a t i o n s o f # , 0 and f o f the boat were illustrated i n the figures. I n accordance w i t h the re-sults o f dynamic s t a b i l i t y analysis, these parameters representing the m o t i o n s o f the boat f l u c t u a t e d w i t h -o u t divergence f -o r e = 3 0 ° and 2 5 ° , whereas f -o r e = 2 0 ° they varied a b r u p t l y , f o l l o w e d by the. immediate capsize o f the boat.
I n Figure 5, f o r e = 3 0 ° , the f l u c t u a t i o n s damped r a p i d l y and the boat exhibits h i g h stabihty. Since the nearest complex eigen-values t o the imaginary axis indicated i n Figure 1 are - 0 . 8 8 7 + 2 . 6 7 2 j , the period and the t i m e to h a l f a m p h t u d e o f the f l u c t u a t i o n are 2.35s and 0.78s respectively. These values agree well w i t h the simulation results.
"'/s ^ deg 1 0 Us, = 1 0 " t / s V , = 9 0 E = 3 0 initial r o l l d i s t u r b a n c e p = - 0 - 5 r a d / g ^ ^ { a t 5 s e c ) 35 4G
Figure 5. Numerical simulation result showing the fluctuations of roll, pitch and velocity for roll disturbance (e = 30°).
/s . deg UsT = 1 0 m/s V T = 9 0 ° E = 2 5 ' initial r o l l d i s t u r b a n c e p = - 0 - 5 ^^'^/ssc ( d l 5 s e c )
W\AMAAyVvWv.wvvV\A/Wv'
\ / V W W W W V \ A A / \ / \ / v W \ / \ A A A ' - 1Ö l i i'B 25 30 35 46Figure 6. Numerical simulation result showing the fluctuations of roll, pitch and velocity for roll disturbance (e = 25°).
Figure 6, f o r e = 2 5 ° , indicates that the i n i t i a l f l u c -t u a -t i o n damps modera-tely, b u -t -the f l u c -t u a -t i o n con-tinues w i t h constant a m p l i t u d e . Since i t is apparent
"Vs ^ deg "r
UsT=10 "i/s V , = 9 D ' • E = 2 0 "
i n i t i a l roll d i s t u r b a n c e p = - 0 5 r a d / g g j . ( a t 5 s e c )
28 25 35
Figure 7. Numerical simulation result showing the fluctuations of roll, pitch and velocity for roll disturbance (e = 20°).
m / s d e g 40-20¬ 0¬ -20-U s T = 1 0 m / s Y , = 9 0 ° e = 2 0 ' w i t h o u t i n i t i a l d i s t u r b a n c e 0 10 12 16 8
Figure 8. Numerical simulation result showing the fluctuations of roll, pitch, velocity and leeway angle without initial dis-turbance (e = 20°).
f r o m the simulations w i t h various values o f r u d d e r angle t h a t the amphtude o f the f l u c t u a t i o n is sensitiv-ely a f f e c t e d whereas the period o f the fluctuation is u n a f f e c t e d b y the angle, the c o n t i n u e d fluctuation observed i n Figure 6 can be considered as so-cahed self-excited v i b r a t i o n due to the deviation o f the value o f r u d d e r angle f r o m its e q u i l i b r i u m one. The occur-rence o f the self-excited-vibration-hke fluctuation is a signal o f approaching the critical sailing state. T h e nearest complex eigenvalues t o the imaginary axis f o r this case are - 0 . 4 9 6 ± 2 . 8 9 0 j , giving the p e r i o d o f fluctuation o f 2.17s, although the period derived b y the s i m u l a t i o n is 1.4s. This discrepancy seems t o be caused b y the enlarged non4inear e f f e c t due t o de-crease i n the f o ü submerged length w h i c h is used f o r the calculation o f h y d r o d y n a m i c f o r c e .
T h e s i m u l a t i o n was also conducted at the c o n d i t i o n w i t h o u t i n i t i a l disturbance t o c l a r i f y the i n s t a b i h t y o f the boat f o r e = 2 0 ° as shown i n Figure 8. A t this value o f e, the boat tends t o heel t o leeward side and capsizes w i t h o u t disturbance as same as the case o f Figure 7. The $ and 0 change m o n o t o n e o u s l y i n ' F i -gure 8, w h i c h apparently corresponds t o the static i n s t a b i l i t y determined b y the dynamic stabihty analy-sis. The eigenvalue on the real axis is 0.767 i n this case, giving the t i m e t o double amphtude o f 0.90 s, w h i c h agrees w e l l w i t h the s i m u l a t i o n result i n Figure 8.
non-linear equations resulted i n the c o n f h m a t i o n o f the v a h d i t y o f d y n a m i c s t a b i l i t y analysis. T h r o u g h the calculation and the numerical s i m u l a t i o n conducted i n the present s t u d y , an u s e f u l m e t h o d t o design the h i g h p e r f o r m a n c e h y d r o f o i l saihng boat was developed.
6, Conclusion
As a succession o f Part 1, the m e t h o d o f d y n a m i c s t a b i l i t y analysis f o r a h y d r o f o i l saihng boat was pre-sented. Since the m o t i o n s o f the boat i n l o n g i t u d i n a l and lateral dnections a f f e c t e d each other f o r the sail-ing boat, the d y n a m i c stabihty was examined w i t h ac-c o u n t i n g f o r their i n t e r a ac-c t i o n . S t a b i h t y derivatives f o r saü were calculated b y p a r t i a l l y d i f f e r e n t i a t i n g the aerodynamic forces acting o n i t , } a l t h o u g h those f o r the h y d r o f o i l s were calculated using the procedure p r o -posed b y K a p l a n et al.
T h e d y n a m i c stability analysis was p e r f o r m e d i n ah o f the p o s s i b l é e q u i h b r i u m sailing state w h i c h had been presented i n Part 1. The results o f analysis i n d i c a t e d t h a t , at higher v e l o c i t y the boat f e l l i n t o a static i n -s t a b i l i t y or divergence w i t h l i m i t i n g the m a x i m u m at-tainable boat v e l o c i t y . I t was c l a r i f i e d t h a t the unstable state has occurred w h e n the submerged l e n g t h o f w i n d w a r d f o i l became smaUer than about 0.1 m , al-t h o u g h al-the e q u i h b r i u m sal-taal-te c o u l d be o b al-t a i n e d u n al-t i l the l e n g t h becomes t o zero, i.e. t a k i n g o f f f r o m water surface, f o r the conventional p e r f o r m a n c e p r e d i c t i o n w i t h o u t i n c l u d i n g the stability analysis. T h u s i t be-canie apparent t h a t the conventional p r e d i c t i o n gave the over-estimated value f o r the m a x i m u m v e l o c i t y . I t was also c l a r i f i e d that the sail w o r k e d as the stabihzer o f the boat at higher w i n d v e l o c i t y i n c o n j u n c -t i o n w i -t h i-ts original r o l e as -the -t h r u s -t i n g device. T h e v a h d i t y o f the results o f dynamic stability analysis was c o n f i r m e d b y the numerical s i m u l a t i o n using non hnear equations. T h r o u g h o u t the present study one o f the u s e f u l methods f o r designing the h i g h p e r f o r m a n c e h y d r o f o i l saihng boat was p r o v i d e d w i t h adequateness o f the calculation i n the stability ana-lysis. N o m e n c l a t u r e c F 8 H area o f sah
drag c o e f f i c i e n t s o f h y d r o f o i l and sail l i f t c o e f f i c i e n t s o f h y d r o f o i l and sail c h o r d length o f h y d r o f o i l
drag forces acting o n h y d r o f o i l and sail h y d r o d y n a m i c f o r c e acting i n n o r m a l d i r e c t i o n t o the f o i l panel
acceleration due t o gravity
v e r t i c a l height o f C.G. f r o m water surface level I K.M.N L.L, m P.Q.R
u,v,w
UA-VA.WA X , Y , Z r 6 e X Pa' Pwmoments o f inertia about x, y and z axes p r o d u c t o f i n e r t i a about z and x axes u n i t o f complex n u m b e r (= V — 1 ) moments o f roU, p i t c h and y a w h f t forces acting o n h y d r o f o i l and sail length o f r o t a t i n g a r m o f h y d r o f o i l (see Figure A l )
length o f submerged p a r t o f h y d r o f o i l mass o f boat and crew
angular velocities i n r o l l , p i t c h and y a w perpendicular distance between C.G. and dihedral h y d r o f o i l (see Figure A l )
velocity components o f C.G. along x, y and z axes
velocity components o f apparent w i n d along X, y and z axes
true w i n d v e l o c i t y
apparent w i n d v e l o c i t y i^^Uj + Vj+Wj ) boat v e l o c i t y (= ^U^-fV^+W^ ) f o r c e components along x, y and z axes attack angle i n the plane o f i n c i d e n t f l o w o f h y d r o f o i l
attack angle o f sail angle o f leeway
angle between apparent w i n d v e l o c i t y and centerline o f boat
dihedral angle o f h y d r o f o i l
angle between true w i n d v e l o c i t y and centerline o f boat
angle o f r u d d e r
t r i m angle o f sail (angle between b o o m and centerline o f b o a t )
angle between perpendicular line t o the dihedral h y d r o f o i l and connecting hne f r o m C.G. t o the t i p o f the f o i l (see Figure A l )
angle between perpendicular line t o the dihedral h y d r o f o i l and connecting line f r o m C.G. t o the surface-piercing p o i n t o f the f o i l (see Figure A l )
r o o t o f characteristic e q u a t i o n densities o f air and water
angles o f heel, p i t c h and y a w (Euler angles) (subscripts) = 1; f r o n t starboard 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; rear h y d r o f o i l = 4 ; rudder = 5; sail center o f submerged p a r t o f h y d r o f o i l surface-piercing p o i n t o f h y d r o f o i l
References
1. Kaplan, P., Hu, P.N. and Tsakonas, S., 'Methods for estimat-ing the longitudinal.and lateral dynamic stability of hydro-foil craft', Stevens Institute of Technology E.T.T. Report No. 691, 1958.
2. Masuyama, Y., 'Stability analysis and prediction of perform-ance for a hydrofoil sailing boat (Part 1) Equilibrium sailing state analysis'. International Shipbuilding Progress, 1986, Vol. 33, No. 384.
3. Masuyama, Y., 'Motion of a hydrofoil system in waves', Journal of the Kansai Society of Naval Architects, Japan, No. 196, 1985 (in Japanese).
A P P E N D I X Derivatives of hydrofoils
I n the case o f a surface-piercing dihedral f o i l ( / = 1, 2 ) , the symbols such as I p., 1^., Rp, and f . are d e f i n e d as shown i n Figure A l , where the submerged length o f the f o i l , Ip^, and r o t a t i n g arm length, , are expressed as f o l l o w s :
/^,. = i ? ^ ( t a n f . - t a n f „ ) .
The values o f b o t h (9C^/3a:,^) and ( a q / a a , J are determined using the calculated values shown i n the Figures A 3 and A 4 o f Part 1. A l l o f the derivatives o f h y d r o f o i l s are hsted as f o h o w s , where the t o p o f minus-plus sign is apphed t o the f r o n t starboard f o i l and the b o t t o m t o the p o r t .
port P 1 starboard
Figure A l . Definition of symbols for surface-piercing dihedral foils (/= 1,2). hi Rp(tani. + tani^)/2 \ i = - P.VBCDihi<^i ' ( ^ - = 1 - 4 ) Pui = - P . . V s C j p f y , ( / = 1 ~ 4 ) 3 ^ \ Pw Ve-^Fi^i c, Li nl i 3 C , 1 = 0
11
- " 1
smr,. , ( ( = 1 - 4 ) sinr,. , (/ = 1 - 4 ) c o s r . , ( / = 1 ~ 4 ) c o s r . , (/ = 1 ~ 4 ) I., : then 2 '-~P.VB<^i 93a„
a
'Li hi-ia =
3)a
= 4 ) 9 C 4 = 0 , 1 f^'^L Ppi = + 4 P>v ^B^^i ^ + ^ | - ( t a n 2 f , - t a n ^ f ^ ) , (r = 1,2)a =3)
( A - 1 ) (A-2) ( A - 3 ) (A-4) (A-5) (A-6) ( A - 7 ) ( A - 8 )2 q . z . -9 ^ 901 + Cjj. x . c o s r . , ( z = l ~ 4 ) , ( ; = 1 ~ 4 ) where
: due t o change o f submerged area
c<pi ^ 2
f
9C„\
Rp '- i _ p vi l„.c - , — - • : due t o change o f d e p t h o f submergence + B Fi j ƒ 2cosf; t h e n - 1 r / 2
I
^Di , ^Fi f?^D\1
R-F + — p V„c.{ +— 2 ^ ' I cosf,. 2 \bh j i \ cosr,. = 0 , , ( ^ = 1 , 2 ) ( / = 3 , 4 ) 0 ' Cu ^^Fi i^C, cosf,. 2 \ 9 / 2 / / cosf. = 0 , , ( ; - = i , 2 ) ( / = 3 , 4 ) where ^tei'^-Xcei X . tei X ctiii „
v^C c — ^ t h e n 2 ^ ' " I Isinr,.] 2 \ 9 / 2 ei Isinr^.l 2 [bit9 q
2 /'w^i?'^-/'^/! " ^ j , ^ ; ' 2 Ismr^.l 2 \ 9 / j y , 1 9C, • hi - + J/2 X i 2"'^ •s''' isinr.l 2 \ 9/2 2 ^ ' " ^ ^ " i9 / J /
'
1 X,..- = ^vi — »2j.sinr. ,: due t o change o f submerged area
: due t o change o f d e p t h o f submergence
( / = 1 , 2 , 4 )
a =
3 ) (;•= 1 , 2 , 4 ) (/ = 3 ) ( i = 1 , 2 , 4 ) ( / = 3 ) ( i = 1 , 2 , 4 )a
= 3 ) ( r = l ~ 4 ) (r = 1 - 4 ) ( A - 9 ) ( A - 1 0 ) ( A - 1 1 ) ( A - 1 2 ) ( A - 1 3 ) ( A - I 4 ) ( A - 1 5 ) ( A - 1 6 ) ( A - 1 7 ) ( A - 1 8 )F,.,.= -m,.cosr,.
Fj,i = ^Tn.Rp{tan^.+ tan f J / 2 ,
= 0 ,
= m.z. ,
where
and k. was taken to be 0.9 f o r calculation.
( / = 1 ~ 4 ) ( 2 = 1 , 2 ) (^ = 3)
