Stability analysis and prediction of performance for a hydrofoil sailing boat Part 2: Dynamic stability analysis

VPCH1EF

Lab.

_{Technische Hogeschaol}

y. Scheepbouwkunde

STABILITY ANALYSIS AND PREDICTION OF PERFORMANCE FOR A OFOIL SAILING BOAT

PART 2 DYNAMIC STABILITY ANALYSIS

Yutaka Masuyama

Kanazawa Institute of Technology

SUMMARY

A dynamic stability analysis for a hydrofoil sailing boat

has been carried out on the boat with two surface-piercing dihedral front foils and an inverted "T" rear foil. The small-disturbance theory was applied to the analysis with accounting

for the interaction between the longitudinal and lateral motions.

Stability derivatives were calculated for the hydrofoils and the

sail. As a result of the analysis, it was clarified that there

was a critical sailing state for the maximum attainable boat

velocity which was lower than that of the conventional

predictions for the hydrofoil sailing boat. When the velocity exceeded the maximum velocity determined as the critical sailing state, the boat fell into static instability or divergence.

i INTRODUCTION

r

is essential to conduct a dynamic stability analysis, because the

boat is balancing with dynamic lift force on the hydrofoils.

Although the dynamic stability analysis has already been

conducted by Kaplan et al? for a powered hydrofoil craft, there

is _{little information about the analysis available for the case}

of sailing boat. The motion of the powered craft can be analyzed through classifying into longitudinal and lateral directions because the craft is symmetrical with respect to the _centerline

with advancing straightforward at equilibrium state. However,

for the case of the sailing boat the dynamic stability _analysis

must be conducted with accounting for the interaction of the

motions in a longitudinal

_and

_{lateral directions.}

2)

In the present paper as a succession of the part i which presented an estimation method of the equilibrium state _{and the}

high _{velocity sailing characteristics with foil-born mode, the}

dynamic stability for a hydrofoil sailing boat is _{examined. For} the examination, the stability derivatives _{for sail were obtained}

by partially differentiating the aerodynamic _{forces acting on it,}

and _{those for the hydrofoils were calculated using the procedure} proposed by Kaplan et al? The results of analysis were then

confirmed by the numerical simulation conducted _{using non-linear}

equations of motion.

2 _{LINEARIZED EQUATIONS OF MOTION AND CHARACTERISTIC EQUATION}

The _{motion of hydrofoil sailing boat referring to the body}

axes is described by the Euler's equation _{as follows:}

m(LQW-RV)

X,

f RU - PW) Y

m(W4 PV -QU)

Z

III k- í

k+PQ -QR( !

) K 1yy

I( R2 P2)

RP( 1zz M

i k-

(t'- QR)

- PQ( I- J, )

N.,

The Euler angle rates are also given by

0

P+(Qsinø+Rcosø)tanø,

e

Qcosø--Rsinø,

1(Qs:nO-1-Rcosø)sec&

From the results of equilibrium sailing state analysis, it was

indicated that the angles of pitch, heel and leeway were

relatively small, especially in pitch. Thus the perturbed equations of motion can be reduced from the Euler's equation as

follows:

m (ü+qw,-rv0)

4X-mg

m (ü+r uo- Pw,)

_4Y+mgcos'bo,

rn (th±pv,-qu0)

iZ-mgçbsinc/i0,

Irr blzr 7

4K

=4M,

1zz t Izr j'

4N.

In the equation the symbols with zero suffix indicate the values

in equilibrium state, and the symbols without suffix for

velocities (u,v and w), angular velocities (p,q and _{r) and angles} and O ) indicate the deviations from their equilibrium

respectively.

Since the sailing boat runs usually accomodating with

sideslip and heel, causing that aerodynamic force on sail directs

differently to the advancing course, the deviations in any

directions affect simultaneously on both motions in longitudinal

and lateral directions. From this aspect, it is required to

carry out the stability analysis using all of the formulae in

Eq.(3). The motion around z axis, however, would not be affect

for the dynamic stability analysis, although it should be treated in the course stability analysis. Thus the last formula in

Eq.(3) and the terms of r and i' are eliminated. As

consequence, the linearized equations of motion can be expressed

by the following Eq.(4).

m(ú-qw0)

4XmgO,

m (f3 pub)

4Y±mgcb

m(th+pvo_qu0)4Zmgct4)o,

4K,

I0q

4M.

The angular _{velocities are also reduced to çt p and q.} The velocity of the C.G. in the true vertical direction is then

expressed as follows:

w - u0 O + y0

(5) The perturbed force along x axis is expressed in terms of derivatives as

4X-Xu+Xvv+x+xww+xthw+x+xP+x.

₍₆₎

+ XeO + XqO+ XO+ X,, h

(4)

-: u, eAt,

e',

u V1 e2', w w, e",

O

O e,

h -- h, e2'

Substituting these expressions into Eq.(7) and cancelling throughout by et , we obtain the six simultaneous equations:

(mA -

X)

u, - (XA + X,,) V, - (X,, A +

_X)

w -

(XA2 + Xp 1 +X) I'i

+ { -XA +

(mwo-Xq) 2+

(mg-Xe) } O,-X h,

O,

(Y 2

YqA+ Y6) OlYh h, = O,

-Zu,- (Z;A+Z,)v,+ { (m-Z)A-z}_{w,+ { -ZA2±(mvo-Zp) A + (mg0-Z) } i}

_{ZA2+(muo+Zg)A+Zo}0I_Zhh,O,

-Kku,- (K;A+K,) y,- (K,A+K) w,+{

_(KA2+Kq1±Ke)OI_KAhIO,

-M u, - (M;A +1v!,) y, - (M,A ±M) w,

- (MA2 +MA +) (/),

+ { (¡y,, M) 22 Mq A -M6 } 0,- MA h, = O,

-w,-v0,+u00,+A h,

O

(7)

(9)

zIY, 4Z, ¿1K and 4M are also expressed by similar formulae as

Eq.(6). Substituting these expressions into Eqs.(4) and (5), we can rewrite the linearized equations of motion as follows:

,nù+mwoO+mgô Xu--XVX;V-XwW-Xsi,W

xX-

xb

X80-X0-X0Xh

O,

mv - mw0

- mgçb - Y, u -

Y,

v-Y; b - Yw - Yb

-

Yp _ Y Y6 0 Y O

Y 0 -

Yk h O,

mw + mVO4 mu0O + mg'0l'

_Zu_Z,V_ZZwWZc,W

Z_ZpZZe0ZqÛ

Z0-ZhO,

- K u - K, u

K,

i' -

w - K,, t

-KK-K KoOKq0K0K=0,

Ô M, u M, u - M, b -

Mw -

M,th

-M' -M -M

-

0- MQO -MO-Mh

= O, - W + - V0 = O

Since the equations of motion are linear, _{their solutions can be} expressed as follows:

}

r

Now we have the stability determinant as,

above equation.

A' + ¡7 A7 +

f6 16 + f A +f4 A

+f3 A3+f2 A2+f, A+f0 -

0 ₍₁₁₎

The necessary and sufficient condition to make the solutions stable _{is determined by either applying the Routh's discriminant} method to the coefficients of Eq.(11), or solving Eq.(11)

to find directly the roots without positive real _parts.

3 STABILITY DERIVATIVES

3.1 _{Derivatives of Hydrofoil}

The _{results of equilibrium sailing state analysis indicated}

6

mA-Xe

-

XA - X -X,,A -X,, -X622 - X,,2 - X XA2 + (mw0Xq)A

-+ (mg-Xe)

-

(m-)A-y

-Y,,A-Y

-Y6A2-(mwo±Yp)A -A2-}A-Y

- (mg+ Y)

-Ya,

Z62 -Z

(m-Z)A-Z

-Z,A2 + (mv0-Zp)A

-ZA2 -

(mu0f Zh

+ (mqo-4)

-z8

₀

₍₁₀₎

-K

K-K

_-K,-K

(I-Kt)A2-KpA

K22KqAK8

-KA

--M

-MA-44

-MthA-M

-M.A2-MA-M

(J-M)A2-MA

-MA -M8

0 0 -1

-y0

_{u0 1}

that the leeway angle was relatively small. _{Hence we may follow}

the _{calculation procedure of Kaplan et al'?}

for the stability derivatives of each hydrofoil. _{Formulae for the derivatives}

are

described _{in the Appendix using the}

hydrodynamic forces. The derivatives of the force _{acting in normal direction to}

the

hydrofoil panel, F3, , are resolved into y and z directions as

F,

r,

(i- 1--4)

T _{11 COS} ï',

(12)

where _{[' is a dihedral angle of the foil}

and jz u, y, w, p, q, ,

, h,

ì,

w,

and ¿. Consequently, _{the derivatives of foil}

system for the forces X, Y and Z are obtained as,

x1=x,

Y,==>y,j,

(13)

The _{derivatives for the moments K and M with respect}

to u,

y, w, p, q, sr, _{j and}

¿i are calculated as

K =

(z,,

y, - y, _,)

M,= (X1 , - Z11 X,)

and considering the _{difference of application point of}

perturbed forces _{which are caused by the change of both submerged area and} depth of submergence, _{the moments with respect to}

,

and h are

also calculated as

K, y, - Y,j z,) + _{(ze,, y,1}

-

_Y1

M1 =

(x1,

z -

z,

x) +

_(xi,1

_{z,, - Z,,, z,,)}

} (14)

}

3.2 Derivatives of Sail

Since we may take both '' and to be zero in the equation

for the velocity components of the relative wind for the sail

which is described in part 1, the equation can be simplified as

follows:

(JA = U + U5 COS T

VA = V + (iST Sin 7- COs 0,

W = W - UST Sin T.. sin ø

Among these components we may neglect the value of _{WA , because of}

its relative smallness and its irrelevancy with the sail

performance. Then the relative wind velocity, U , and relative

wind direction, ₍₃ , _{are expressed respectively as follows:}

= u+

V2

= (U+U5

cos r)2 + (V+U- sin r7 cos

Ø)2

= sin' (VA/USA)

V + UST sin Yr COS 0

tan () = tan

_{(U+UST Cos YT}

The aerodynamic forces, X and Y , are calculated by the

following relationships:

where

X5= _{P0U1, C5A/2}

YS= PaU _Cy5A,r/2

8

} (19)

cxS= Cils

cos 8A+ CL$ sin ß

(20)

Cils Sfl 11A CLS COS 13A

}

The derivative of X with respect to u is

a x5 a x a x a u a x5 BA

-

au -

U5,, OU

aBA au

Upon partially differentiating both sides of Eqs.(17) and (18) by

U, we have

OUSA

ÔU -

UA COSBA

OBA V4 SÍ71BA

OU

U2

USA

Substitution of Eqs.(22) and (23) into Eq.(21) yields

ac55

Xus = Po USA

(2

C5 cos

-8 BA sin BA)

Furthermore, from Eq.(20) we have

ac5

ac

cos BA+ COS Sifl BA

aBA

a CLX

a BA B + CLS COS BA

6CDS acLS

aBA a9A SU2 BA COSCOSBA _011A COSBA+ CLS Sifl BA

Finally substituting Eqs.(20) and (25) into Eq.(24), we obtain

X5 POuSAAS{_2cQ5cos2BA+

+C)

COS _BASÍfl BA

a CLS

- ( a

_BA + CO5) sin2 BA

Similarly, the derivatives of Y with respect to u is given as

a_C5 = _{Pa USA A5 { - 2 CLS COS2} BA ( BA

-

Cos) cos BA St BA ( +

Cu) Sin2

BA

L

The derivatives of Xsand with respect to y are obtained in a similar way as follows:

(aCDS aCLS

Xr5 = p USA A5 { - _a

ßA

C1)

COS2 í

+ (

- c)

cos _{13A Sifl IA}

+ _{2CLS Sjfl2 lIA}

a c

Y,S= _{ßaUsAs {_}

(S

± c)

cos2

ß-

aj,

+ _CLS) cos lIA Sin lIA

_2Cstn2lIA}

The

(Cs

_/òßA)

and (C5 /ø) in the above equations

correspond

to the slope of the lines representing the relationships of C and C versus attack angle of the sail indicated in Fig. 6 of part 1.

The derivatives of X and Y with respect to p and q are

obtained as follows:

xps=xv5z5,

Yps--Yvszs (30)

XqsXus Z YqE =YuS Zs (31)

where z5 (negative value) is coordinate for C.E. of the sail. In the same manner the derivatives of moments Ks and M5 are given as follows:

K5 = -

Y,45 zs, M5= X1s Zs ₍₃₂₎ Krs = - Yv5 Zs. Mrs = X5 zs , ₍₃₃₎ K5 = Y5

Z,

M5 = - X, z , (34) Kqs = - Y5 M45 =X5 : (35)

lo

The derivatives of Z are assumed to zero because of

negligible influence of the sail force on the motion in the

direction. Consequently, summing up the derivatives of

hydrofoils and sail, we obtain the total stability derivatives

of the boat, where the derivatives of aerodynamic force on the

hull were not taken into account because they are considerably smaller than those for the sail.

4 RESULTS OF STABILITY ANALYSIS

The dynamic stability analysis was performed in all of the

possible equilibrium sailing state as a function of sail trim

angle, E. . Table i shows an example of the calculated derivatives

for the hydrofoil and the sail in the dimensional value for the

case of UST= lOm/s, ITZ 90° and E. 25° . The submerged length of

front starboard and port foils are 0.30 and 0.79m, respectively.

The derivatives of the front port and rear foil are dominant among those of hydrofoils, because these foils have larger submerged area than the others. It also can be seen that the derivatives of the sail occupy the large portion to those for the

force X and the moment K.

The Eq.(l1) are solved using Bairstow's method. The

examples of the root locus diagrams for the cases of U51- 10, 14

and 18m/s at E 90° are shown in Figs. 1, 2 and 3 respectively. The simultaneous movements of the eight roots are represented

with decreasing the E. , and the corresponding variations of

sailing state parameters have already been illustrated in Figs.

low, the real parts of all roots are negative and the movements

of them are relatively small. With decreasing the E the

movements of the roots become large gradually, and then one of

them on the real axis becomes positive at 20°, 30° and 38° for _UST

= 10, 14 and 18m/s, respectively. This means that, regardless of

wind velocity, the boat falls into a static instability or divergence at small E which yields high velocity. As mentioned in part 1, it should be noted that the boat becomes unstable when the submerged length of starboard (windward) foil, _Jr-, , becomes

smaller than about 0.lm, although the equilibrium equations can be solved until _LF, becomes to zero, i.e. taking off from water

surface.

Figure 4 is polar diagram which compares the boat velocity

at the limit for stable sailing determined by the present

analysis (solid curve) with the highest velocity derived by the

equilibrium equations (dot-dash-curve) for UST 10, 14 and 18m/s. Each shift of the solid curve from the dot-dash-one is shown by the shaded zone where the boat is actually unstable although the

equilibrium equations give some solutions. It appears that the conventional performance prediction through the equilibrium equations results in the over-estimation by 5 to 10% for the

maximum attainable velocity. Dotted curve in the figure shows

the stability limit derived by the calculation excluding sail

derivatives for each wind velocity. Comparing the curve with that for including sail derivatives (solid curve), the latter is shifted to higher velocity side than the former by about 5% for the cases of U3-1-= 14 and 18m/s although both curves almost coincide for

UT=

lOm/s. According to the result of analysis

presented in Fig. 4, the sail has the roles not only as the

thrusting device but as a stabilizer at high wind velocity.

5 NUMERICAL SIMULATION USING NON-LINEAR EQUATIONS OF MOTION

In order to confirm the validity of both linearization of equations of motion and calculation of the derivatives described in the previous sections, the motion of the boat was simulated numerically using non-linear equations at the same condition in

which the dynamic stability analysis was conducted. The

simulation was carried out under the five-degree-of-freedom

(with O ) with fixing the rudder angle to the equilibrium

state value by applying the Runge-Kutta-Gill method. The forces and moments for Eq.(1) are calculated by the following procedure.

First, the length and depth of submerged part of hydrofoils at

any moments are determined from the contemporary attitude of the

boat, Then, velocity components for the center of submerged part

of each foil and C.E. of the sail are calculated as

U = U +

Qz, -

Ry, = V + Rx, - Pz1 = W + Py, - Qx1

Since attack angle of each part can be obtained from these velocity components, we have the forces and moments using the

formulae described in part 1. Where, the added mass effect of

the foil was also included for the motion normal to the foil

panel. The applicability of the calculation method for the

hydrofoil system was confirmed through towing tank test including

regular wave conditions.

The simulation results for E- 30° , 25° and 20° under the

wind condition of UCT lOm/s and 900 _{are shown in Figs. 5, 6}

and 7 respectively. For roll disturbance of p _{-0.5rad/s at}

t 5s, the time dependent fluctuations of , , and U of the

boat were illustrated in the figures. In accordance with the

results of dynamic stability analysis, these parameters

representing the motions of the boat fluctuated without

divergence for 30° and 25°, whereas for 20° they varied

abruptly with followed by the immediate capsize of the boat.

In Fig. 5, for 30° , the fluctuations damped rapidly and

the boat exhibits high stability. Since the nearest complex eigenvalues to the imaginary axis indicated in Fig. i are -0.887± 2.672j, the period and the time to half amplitude of the fluctuation are 2.35 and 0.78s respectively. These values agree well with the simulation results.

Figure 6, for= 25°, indicates that the initial fluctuation

damps moderately, but the fluctuation continues with constant

amplitude. Since it is apparent from the simulations with

various values of rudder angle that the amplitude of the

fluctuation is sensitively affected whereas the period of the

fluctuation is unaffected by the angle, the continued fluctuation observed in Fig. 6 can be considered as so-called self-excited vibration due to the deviation of the value of rudder angle from

its equilibrium one. The occurrence of the

self-excited-vibration-like fluctuation is a signal of approaching the

critical sailing state. The nearest complex eigenvalues to the

imaginary axis for this case are -0.496± 2.890j, giving the

period of fluctuation of 2.17s, although the period derived by the simulation is 1.4s. This discrepancy seems to be caused by the enlarged non-linear effect due to decrease in the foil

submerged length which is used for the calculation of

hydrodynamic force.

The simulation was also conducted at the condition without initial disturbance to clarify the instability of the boat for

£= 20° as shown in Fig. 8. At this value of , the boat tends to

heel to leeward side and capsizes without disturbance as same as the case of Fig. 7. The and change monotonously in Fig. 8, which apparently corresponds to the static instability determined

by the dynamic stability analysis. The eigenvalue on the real

axis is 0.767 in this case, giving the time to double amplitude of 0.90s, which agrees well with the simulation result in Fig. 8. Consequently, the numerical simulation with the non-linear equations resulted in the confirmation of the validity of dynamic stability analysis. Through the calculation and the numerical simulation conducted in the present study, an useful method to design the high performance hydrofoil sailing boat was developed.

6 CONCLUSION

As a succession of part 1, the method of dynamic stability analysis for a hydrofoil sailing boat was presented. Since the

motions of the boat in longitudinal and lateral directions affected each other for the sailing boat, the dynamic stability

was examined with accounting for their interaction. Stability

r

the aerodynamic forces acting on it, although those for the

hydrofoils were calculated using the procedure proposed by Kaplan

et al.

The dynamic stability analysis was performed in all of the

possible equilibrium sailing state which had been presented in

the part 1. The results of analysis indicated that, at higher

velocity the boat fell into a static instability or divergence

with limiting the maximum attainable boat velocity. It was

clarified that the unstable state was occurred when the submerged length of windward foil became smaller than about O.lm, although the equilibrium state could be obtained until the length becomes to zero, i.e. taking off from water surface, for the conventional performance prediction without incl'uding the stability analysis. Thus it became apparent that the conventional prediction gave the over-estimated value for the maximum velocity. It was also clarified that the sail worked as the stabilizer of the boat at higher wind velocity in conjunction with its original role as the

thrusting device. The validity of the results of dynamic

stability analysis was confirmed by the numerical simulation using non-linear equations. Throughout the present study one of the useful methods for designing the high performance _hydrofoil sailing boat was provided with adequateness of the calculation in the stability analysis.

NOMENCLATURE

As sail area

CD, CDS drag coefficients of hydrofoil and sail

CL, CLS

lift coefficients of hydrofoil and sail

c chord length of hydrofoil

D, Ds drag forces acting on hydrofoil and sail

F hydrodinamic force acting in normal direction to the

foil panel

g acceleration due to gravity

H vertical height of C.G. from water surface level

I ,I ,I _{moments of inertia about x, y and z axes}

I product of inertia about z and x axes

j unit of complex number (zIi)

M, N moments of roll, pitch and yaw

Ls lift forces acting on hydrofoil and sail

l _{length of rotating arm of hydrofoil (see Fig. Al)}

'F

length of submerged part of hydrofoil

m mass of boat and crew

P, Q, R angular velocities in roll, pitch and yaw

perpendicular distance between C.G. and dihedral hydrofoil (see Fig. Al)

U, V, W _{velocity components of C.G. along x, y and z axes}

UA VA ,WA velocity components of apparent wind along x, y and z

axes

UST true wind velocity

U apparent wind velocity (U+Vs+WA2)

VB boat velocity

(JU2+V2+W2)

attack angle in the plane of incident flow of hydrofoil attack angle of sail

angle of leeway

angle between apparent wind velocity and centerline of

boat

P

dihedral angle of hydrofoil

T1 angle between true wind velocity and centerline of boat angle of rudder

E. trim angle of sail (angle between boom and centrline of boat)

angle between perpendicular line to the dihedral

hydro-o

foil and connecting line from C.G. to the tip of the foil (see Fig.A1)

angle between perpendicular line to the dihedral hydro-foil and connecting line from C.G. to the surface-piercing point of the foil (see Fig. Al)

root of characteristic equation density of air and water

c, 9,iT _{angles of heel, pitch and yaw (Euler angles)}

(subscripts)

i l; front starboard hydrofoil

2; front port hydrofoil rear hydrofoil

4; rudder =5; sail

C center of submerged part of hydrofoil

REFERENCES

Kaplan,P., Hu,P.N. and Tsakonas,S.: Methods for Estimating the

Longitudinal and Lateral Dynamic Stability of Hydrofoil Craft, Stevens Institute of Technology E.T.T. Report No.691, 1958. Masuyama,Y.: Stability Analysis and Prediction of Performance

for a Hydrofoil Sailing Boat (Part 1) Equilibrium Sailing State Analysis, International Shipbuilding Progress,

Masuyama,Y. Motion of a Hydrofoil System in Waves, Jour. of the Kansai Soc. of Naval Architects, Japan, No.196, 1985. (in Japanese)

APPENDIX DERIVATIVES OF HYDROFOILS

In the case of a surface-piercing dihedral foil (i1,2), the

symbols such as ..4.,

2,i,

R,

and are defined as shown in

Fig. Al, where the submerged length of the foil, _JFL , and

rotating arm length, J, are expressed as follows:

IF:

RF (tan C, - tan C0)

IA:

R (tan C

Ian Co) z 2

The values of both ('dCD/&L,) and (CL/di%p1,) are determined using

the calculated values shown in the Figs. A3 and A4 of the part 1.

All of the derivatives of hydrofoils are listed as followings,

where the top of minus-plus sigìi is applied to the front

starboard foil and the bottom to the port.

20

I OC - p,,V8c, { ( aa)' + C01I lF Z = X1,+ X0S1 where, 1 F X,4,= T _P,V,Cffic, 000 = ; p _{y: i c} (---2 cos C1 then C0

IF, fCO\

R-= _{LÇ C,} + C,

(i = 4)

Xg,

_PVßIc,[2Cz,_((0CD),_C,}X.

_cosi',],

(i=I-4

(A-9)

a

pq,

-

PvV8I, C, [2Cr, z,f(L),

+C01}

cosr]

(i = I-4)

(A-lo)

due to change of submerged area

due to change of depth of submergence

- _{0k' V8 C0}

IFt Ci

(i= 1-4)

(A- 1)

=

-x. =

x,

ß., V8 C, c,

- -- p

va ¡, cj

((4Q-),

- --

p,,V81,, C1

-

s.1/a ¡p, c, {

(jQ),

(j

I - 4 )

- c, }

s1

+ c,,} smi',,

- c

}

i

(i

(i

(i

1 4)

I-4)

I 4 )

(A-2) 3) 4) 5)

F,, = -

P,,, V ¡Fi c, _{ + C,,, } COS [' , ( i 1 4 )

(A- 6)

= p V8 c _{( (} C0),

- cj I ¡.

i _{: then} X, + o. V8c, _{{ (),CLIJR2, (Ian2} C -C,} ¡, z, Ian2C0),(:=l.2)

(i=3)

(i

4)

(A- 7)

ow V8 c,

{ (Ç)1

-O CL

i'D' = ßwV8Cj _{{ + CO3} R (Ian2 C, - Ian2 C ),( i = 1.2)}

-

o,

(i

3)

(A- 8)

L4 C, _{ 1,., 8 CL R

= T

cosC,

(a

h

"

cosC, then X8 p,,V2 _Ci ¡ C01 +

'1sinI

(i=3. 4) XII = X.18, + Xc,,p where,

XII, =

-= -- p,,V2i

xl p.,, V82 C, C, r', ace, =

pv

(a

),,,

F6, = CL, 1F, 2 '9CL

=2

pV8/

c, '9h ) , Coi +

x, =

_-

_{

sin f

9C0

= - -- p..v

h I CIÍ

F= -

.,VJ Ci

_{siniI +}

'9CL =

---- P,.VJIFIcI('9h),

22

(i=

1. 2)

due to change of submerged area

due to change of depth of submergence

(SCI)

₍₁₌₁

4) 2.

(i =3)

'5'CL

(=

I. 2. 4)

(i =3)

--

(SCD

_(i

_{= I.}

2. 4)

ô h

(i =3)

(i

= 1

(i

= 3) 2 4) 5h

)},

S _{C0 X,} " S _h1' 2 } (A-12)

where

= k, p, _I, c. / 4 , (i i 4)

and k was taken to be 0.9 for calculation.

( EriD)

x,. = x. =

X,, = = o (i = I -4) (A-17)

r,

ssn[',

F=-rcosfl ,

=

T n,

R, (tan + tan C,) / 2 (i = 1-4)

(i=1-4)

(i = I 2) (A-18) (A-19)

=0,

= m1 z (i=3) (i = 4) (A-20) Fi = ?fl Xi

r

(i = I 4) (A-21

List of Table and Figures

Table 1 Example of stability derivatives of hydrofoils and sail for the case of Us1: lOm/s, T1 900 _{and F} 250

(dimensional value)

Fig. i Root locus diagram for the case of U- lOm/s and T 90°

(dimensional value)

Fig. 2 Root locus diagram for the case of UST: 14m/s and 90'

(dimensional value)

Fig. 3 Root locus diagram for the case of U5-1: 18m/s and : 90°

(dimensional value)

Fig. 4 Polar diagram comparing the boat velocity at the limit for stable sailing with the highest velocity derived by the equilibrium equations

Fig. 5 Numerical simulation result showing the fluctuations of

roll, pitch and velocity for roll disturbance ( E 30°)

Fig. G Numerical simulation result showing the fluctuations of roll, pitch and velocity for roll disturbance ( 25

Fig. 7 Numerical simulation result showing the fluctuations of roll, pitch and velocity for roll disturbance ( a: 20°)

Fig. 8 Numerical simulation result showing the fluctuations of

roll, pitch, velocity and leeway angle without initial disturbance ( E 20°

Fig. Al Definition of symbols for surface-piercing dihedral foils (j: 1, 2)

Table 1 _{Example of stability derivatives of hydrofoils and} sail for the case of _{UST lOm/s, y= 90° and 25°}

(dimensional value)

St. Port Rear Rudder Sail Total

J

(i=1 ) (i=2) (i=3) (i=4) (i5) Dimension

u -1.59 -4.89 -4.23 -1.20 -9.48 -21.39 kgfs y 0.20 -4.09 0.0 0.74 29.08 25.93 m w 0.23 4.88 -0.57 0.0 -- 4.54 p 0.16 -4.96 0.0 -1.31 55.26 49.15 kqfs q -3.21 -12.48 -9.25 -2.11 18.02 -9.03 rad XJ -103.4 131.4 0.0 0.0 -- 28.0 kg! 0 42.1 48.6 0.0 -81.7 -- 9.0 rad h -52.0 -60.0 0.0 -35.2 -- -147.2 kqf/m 0.0 0.0 0.0 0.0 -- 0.0 kgf-s2 0.0 0.0 0.0 0.0 -- 0.0 m 0.0 0.0 0.0 0.0 -- 0.0 kgfs2 0.0 0.0 0.0 0.0 -- 0.0 rad u -6.6 36.3 0.0 6.4 5.7 41.8 kgfs y -26.7 -149.7 0.0 -56.7 -41.9 -275.0 ni w -31.8 178.4 0.0 0.0 -- 146.6 p -22.1 -181.6 0.0 99.9 -79.7 -183.5 kgfs q 13.1 -81.2 0.0 11.2 -10.9 -67.8 rad Y1 -505. -1172. 0. 0. -- -1677. kg! 6 196. -412. 0. 599. -- 383. rad h -241. 508. 0. 258. -- 525. kgf/m ' -0.46 -1.25 0.0 -0.88 -- -2.59 kgf-s2 i -0.56 1.49 _0.0 0.0 -- 0.93 ni -0.39 -1.51 0.0 1.55 -- -0.35 kgf-s2 0.45 -1.20 0.0 0.0 -- -0.75 rad u -7.9 -43.2 -1.4 0.0 -- -52.5 kgfs y -31.8 178.4 0.0 0.0 -- 146.6 m w -37.9 -212.7 -515.9 0.0 -- -766.5 p -26.4 216.5 0.0 0.0 -- 190.1 kgf.s q 16. 97. -1199. 0. -- -1086. rad Z1 -601. 1397. 0. 0. -- _{796. kgf} 0 233. 490. -67. 0. -- 656. rad h -288. -605. -29. 0. -- -922. kgf/m ' -0.56 1.49 0.0 _0.0 -- 0.93 kgfs2 r -0.66 -1.77 -4.16 0.0 -- -6.59 ni -0.46 1.80 0.0 0.0 -- 1.34 kgfs2 0.54 1.43 -9.64 0.0 -- -7.67 rad u -5.5 44.0 0.0 -11.2 10.9 38.2 kgf.s y -22.1 -181.6 0.0 99.9 -79.7 -183.5 w -26.4 216.5 _{0.0 0.0} --190.1 p -18.3 -220.3 0.0 -176.3 -151.4 -566.3 kgf.s.m q 10.9 -98.5 0.0 -19.8 -20.7 -128.1 rad K.1 -518. -2022. 0. 0. -- -2540. kgfm 0 203. -721. 0. 0. -- -518. rad h -250. 890. 0. 0. -- 640. kg! " -0.39 -1.51 0.0 1.55 -- -0.35 kgf.s2 -0.46 1.80 0.0 0.0 -- 1.34 -0.32 -1.83 0.0 -2.74 -- -4.89 kgf.sm q 0.37 -1.46 0.0 0.0 -- -1.09 rad u 3.4 26.5 -11.2 -2.1 18.0 34.6 kgf-s y 26.1 -151.7 0.0 1.3 -55.3 -179.6 w 31. 181. -1198. 0. -- -986. p 22. -184. 0. -2. -105. -269. kgf.s.m q -19. -100. -2800. -4. -34. -2957. rad M.1 300. -936. 0. 0. -- -636. kgf.m 0 -113. -325. -155. 0. -- -593. rad h 139. 401. -67. 0. -- 473. kg! 0.45 -1.20 0.0 0.0 -- -0.75 kg!. s2 0.54 1.43 -9.64 0.0 -- -7.67 0.37 -1.46 0.0 0.0 -- -1.09 kgf.sm -0.44 -1.16 -22.36 0.0 -- -23.96 rad

-20

U5TlO mA

C T 9O0

-15

-10

-5

Real axis

Fia.

i

Root locus diagram for the case of

iOn/s and 900 (dimensional value) X

50

O

25°

o

20°

UST14

fl). C VT

g0

X 500 o 350 0

o

30

-25

-20

-15

-10

t t u f 0 4

airnensionai vaiue

Fig.

2

root locus diagram for the case of UST= l4ri/s and

900

Fig.

3

Ioot locus diagram for the case of

UST= l8m/s and

90°

20

VB rn/s

lo

lo

20 150°

stable sailing limit

(including sail derivatives)

do.

(excluding sait derivatives)

highest velocity by the

equilibrium equations

unstable region

90°

Fig. 4 _{Polar diagrari coraparing}

the boat velocity at the 1irit for stable sailing with the highest velocity derived by the equilibrium equations

USI

I O -U51 10 rn,,5

Yî

90 C 30° u F I I I I 20 -5 10 1 3

sec 40

Fig. 5

Numerical simulation result showing the fluctuations

of

roll, pitch and velocity for roll disturbance

(E=

300)

m5

initial roll disturbance

Po

-0.5 rad/sec (at 5 sec

10-m/5 deg

-ich

-4 ¡'5 PS PS

I \ J

\ I

initial roll disturbance

Po

-o.s rad/sec (at 5 sec

G

\f'JV\î'JV

Fig.

6

Nuiierical simulation result showing the fluctuations of roll, pitch and velocity for roll disturbance (= 25°)

-z:: sec 40 30 UST 1O rn/s

'YT9O°

C 25°

¡ Ï I 10 20 2S 3G

35SeC4O

Fig. 7

Numerical simulation result showing the fluctuations of roll, pitch and velocity for roll disturbance

(E= 200)

Uî :10

/5

Y :90°

C :20°

rn/S 1 U

initial roll disturbance

Po : -0.5 rad/sec (at 5 sec)

d eg

-rn/S deg

40-

20-o

-U511O rn/S T 9O0 without

initial disturbance

c I I 0 2 4

6

8

10

12

14

16 secl8

Fig. O

p

\/ \\

/ / /

.A1

port

starboard

Fig. Al

Definition of symbols for surface-niercing dihedral foils (i= 1,