Methods for estimating the longitudinal and lateral dynamic stability of hydrofoil craft

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Lab

y.

Scheepsbouwkun

Technische Hogeschool

Deift

DAIpSON

LABORATORY

}ETHODS FOR ETIMA.TING THE LONGITUDINAL

AND LATERAL i)YNANIC STABILITY OF HYDROFOIL CRAFF

by Paii]. Kaplan Pung N. Hu and Stavros Tsakonas Report No. 691

Nayi9S8

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EXPERIMEÑTAL TOWING TANK

STEVENS INSTITUTE OF.TECHNOLOGY HOBOKEN, NEW JERSEY

ETHODS FOR ESTfl'ATING THE LONGITUDINAL

AND LATERAL DYNAMIC STABILITY Of' HYDROFOIL CFAFT

by Paul Kaplan Pung .N. J!ú and Stavros Tsakonas. PREPARED UNDER 'OFÇ ICE OP NAVAL RESEARCH

CONTRACT NO. 263(14), TASK NO. NR062-195

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TkBLE ÒF CONTENTS

StJNNM

INTDUCTION

PART I: LONGITUDINAL STABILITY IN SM0Oi WATER' Assumptions

Equilibrium Conditions Equations of Motion

The Force and Moment Derivatives

Contribution Due to Longitudinal Velocity Perturbation u Contribution Due to Vertical Velocity Perturbation w Contribution Due to Pitch Angle e

Page iii i 14 .14 S S

6

7 7

8 R-691

Contribution Due to Submergence Change h . .

Contribution Due to Pitching Velocity q Contribution Due to Vertical Acceleration , Contribution Due to Pitching Acceleration

lo

10 11

12

Dimensionless Forms (Stability Derivatives) 12

Hydrodynanic Coefficients and Their Derivatives is

Determination of Longitudinal Dypamic Stability 2].

PART II: I6.TERPLL STABILITY IN SMOOTE WATER 25

Assumptions 25

Equilibrium Conditions 25

Equations of Motion 26

The Force and Moment Derivatives 26

Contribution Due to Sideslip Velocity y 27

Contribution Due to

Ron

Angle

₀

3

Contribution Due to Rolling Velocity p 31

Contribution Due to Yawing Velocity r

38

Contribution Due to Sideslip Acceleration 140 Contribution Due to

Rolling

Acceleration 143 Contribution Due to Yawing Acceleration 146

Dimensionless Forms (Stability Derivatives) 147 Hydrodynamic Coefficients for Lateral Stability

53

Determination of Latera]. Dynamic Stability

55

CONCLUDING REMARKS 57

e

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-il-T&BLE OF CONTENTS

(Cntinued)

a Page REFERENCES ....

NONCITURE

6]. FIGURES 63

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5UMMA

The equat'ions of motion of tandem hydrofoil craft in both longitudi-nai. and lateral 'modes, each mode having three degrees of freedom, are &erived on the basis of the linear theory of small distur'bances. Various stab?lity dérivatives appearing in the equations are determined in terms of the lift and drag forces acting at the foils, together with their derivatives with respect to angle of attack and depth. Various theoretical and experimental methods for determining these hydrodynamic quantities are discussedin the.

report. Stability criteria for both the longitudinal and lateral dynamic stability of.hydrofoil craft also are presented. The results may be used for estimating the dynamic stability of hydrofoil craft while in the'prelim-mary design stage, and it is expected that the equations are sufficiently accurte to indicate the trends in the degree of dynamic stability as changes

are made in the design configuration.

.

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INT1DUCTION

Development of and interest in hydrofoil craft has increased great]7 during the past ten years because hydrofoil vessels have a higher efficiency,

i.e., higher lift-drag ratio,and a better performance in a seaway than other types of ships in the same speed and size range. A hydrofoil craft is ac-tually an ordinary ship hull supported clear of the water surface by

under-

-water wings, known as hydrofoils, which support the craft by the dynamic

lift generated by the foils due to the flow of fluid around them. Essentially, a hydrofoil is an underwater wing and functions in the water in the same way as an airfoil functions in air. Certain differences, however, exist between

hydrofoils and airfoils. A hydrofoil operates in proximity to a free

sui'-face and, hence, certain effects due to the presence of the sursui'-face manifest

themselves. The forces acting on the hydrofoil, i.e., the lift and drag forces, are different from those of an airfoil traveling at the same speed

and orientation. This is evident since the hydrofoil, while moving near the surface, creates surface waves which carry away eneri. and hence are related to the lift and drag acting upon the hydrofoil. This drag, known as wave drag, becomes small as the speed increases to the practical range of present-. day hydrofoils (see Wu1). Hence it may not be considered too serious. The

lift at high speed is affected only slightly by the presence of the free sur-face, and this is a known effect which can be considered in preliminary static

calcuiatione. However, in considering the dynamics of hydrofoil craft, the presence of the free surface playa an all important part.

The similarity of a hydrofoil to an airfoil appears to make it rather simple to compare the operation of one device to that of the other in ita

own particular media. An analysis of the dynamics of hydrofoil craft should follow that of an aircraft, with particular modification

being

made because

of the action of the free surface in modifying the forces. The hydrofoil

craft is

a

particularly appealing waterborne craft from the point of view of théoretical analysis, since all of the forces may be said to be fairly

"localieed8, i.e.,

they can

be assumed to be acting at the center of pressure

of the

foil

(modified by are' particular

influence or torees acting on the

supporting struts),

The limitation. of the hydrofoil craft

with regard to operation near.

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-69].

-2-the free surface Is due to -2-the following possibility. A hydrofoil craft. has only a limited distance, either up or down, in which it can traverse

in recovering from waves or from any other arbitrary disturbance, before it either broaches the surface with its foils or has a crash landing on its

'hull. _{On the other hand, when an airplane departs from a straight path}

be-cause of wind gusts, control surface deflections, etc., the aircraft has

"freedoin to returi-i to its original equilibrium course without _{any definite}

damage or danger having occurred. In view of this, hydrofoil boats must therefore take care since small departures from their ordinary

straight-line equilibriuni course may cause severe, instability. Other dangers

asso-ciated with hydrofoil boats include the possibility of cavitation

and.,ven-tilation, i.e.,surface aeration, which can occur due to the low pressure associated with reduced hydrostatic head and high speed

Two conflicting requirements exist for hydrofoil craft operating in

a seaway. _{In small waves, the hydrofoil boat should disregard the} _waves

and move straight through them a if they were. not present. _{In larger waves,}

the hydrofoil boat should track the waves, i.e., follow the waves. These,

two modes of motion for the two conflicting extreme.

_{wave conditon, Le.,}

different frequencies of encounter, are part of the general dynamic re-sponse required of a suitùle hydrofoil craft. _{Since this is actually a} response to a sinusoidal forcing function, and depends upon the natural frequency and damping characteristics of the hydrofoil _{craft in free}

os-cillation, it. is essential that something be known of the stability and

re-sponse characteristics of the craft with respect to arbitrary sxnall.tran-. sient disturbances. _{However, since the problem of wave response is a}

com-plicated one, it will not be studied in any detail within the _{present report.} It Is obvious (in the mathematical sense) that any hydrofoil craft which is to operate in waves must first be stable in smooth water, since the response in waves, according to the assumptions of a linear theory (which will be utilized herein), is the sum of the smooth water transient response and the particular steady-state solution to the wave-forcing functions. _{In addition,} *

The forces acting on the foil system in these flow regimes are verr different from those in the case of fully wetted flow and, _{for systems designed to}

operate in fully wetted flow, the sudden onset of cavitation (not _incipient) or ventilation during a disturbed motion usually will result in either

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in the practical sense, the craft should be stable in smooth water operation before being subjected to the additional disturbing influence of a seaway.

A number ol' reports have been written on the subject of the dynamic

stability of hydrofoil craft. Imlay2, Amster3, Hugh and xapianb, and Cannons, to name a few, bave derived equations of motion for both longitudinal and

lateral st.bility of particular types of hydrofoil craft. In each case the

equations were derived differently., the hydrodynainic coefficients were

deter-mined in different ways, and the equations were applicable only to certain

specific configurations. It is the intention of the present report to derive

a consistent set of equations that will be applicable to any type of

hydro-foil craft employing a tandem system, and which will provide information on methods for determining the necessary hydrodynamic parameters.

The present report deals with the longitidina1 and lateral stability of both full submerged and surface-piercing hydrofoil systems in smooth water.

Both lorigitudinal and lateral stability are considered in the linear sense,

each with three de'ees of freedom, and discussion is given of both theoret-ical and experimental means for determining the neçesaary hydrodynamic

deriv-atives. In all cases the hydrofoil craft are considered to be in a tandem

arrangement having, in general, different physical foil configurations

for-ward and aft. It is intended that this report will serve as a

convenient

reference work for those engaged in the evaluation of the dynamic

longitu-dinah and lateral stability of hydrof oil craft in the preliminary design

stage.

-The study waé conducted at the Experimental Towing Tank, Stevens In-stitute of Technolo'. under Office of Naval Research Contract No. Nonr 263(lli),

Task No. NR O62-,9.

E-691

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-3--69I

-PART I

LONGITUDINAL STABILflT IN SMOOTH WATER

Ass'uinptions

The analysis of longitudinal stability of a hydrofoil craft is

limit-ed to the motions of a nncavitating tandem hydrofoil configuration in the

plane of symmetry.. 'It is assumed that ¡each craft has two hydrofóils, one at the front and the other at the rear,' each having, in general, different

planform characteristics. The hull is supported on the foils by a pair of

vextical struts on each hydrofoil. However, any other arrangement of the tandem hydrofoil configuration is subject to the same type of analysis, and the methods may be used for foils with dihedral as well as for flat hydrofoils.

The water surface is assumed to be smooth, except for the disturbances due to the motions of the craft which may alter the surface conditions in the immediate neighborhood of the foils.

The basic method of analysis is the linear theory of small disturbances in which ail products of small quantities are neglected. The hydrodynamic lift and drag force coefficients are considered as function of the angle of attack, a , the speed, .V , and of the depth, h , below the water surface.

These coefficients and their derivatives are evaluated at the undisturbed (equilibrium) position of the hydrofoil system, according to the

require-ments of the linear stability theòry. _{The nómenclature used in this report} is the standard Society of Naval Architects and Narine Engineers Nomenclature

for Treating the Motion ofa Submerged Body Through a Fluid _(Reference

6),

with additional terms being introduced to take into account the free surface

influence. . . .

The effect of the thrust variation with speed and depth of submergence

is assumed to be negligible. _{Furthermore, the thrust is assumed to act}

through the center of gravity in the plane of symmetry, making ari angle E' with the x-axis. _{Any other position of the thrust with respect to the}

hydrocraft can be handled in a similar fashion by specifying the distance of its point of application to the center of gravity, and its ángular

orienta-tion with respect to the coordinate axes.

Since the strut effects are small compared to those of the hydrofoil system, their Influence will be neglected in the analysis of longitudinal

stability. _{The buoyant forces of the submerged foil system also are}

neglect-ed as being small relative to the hydrodynamic forces. _{The downwash effect}

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When a hydrofoil craft is in its indisturbed condition, it moves in equilibrium along the x-axis at a definite depth, at zero trim, and with a constant forward speed V . It is evident that in this static equilibrium.

state the following conditions must be. satisfied:

) The sum of the forces in the x- and

z-directionà

must be equal to

zero, i.e.,

D + DR = T cos E'

LF +LW-Tain'

b) The moment about the center of gravity de to the torcee acting

on

the system must be equa:L to zero, i,e,,

Equations of Motion

The hydvofoil öonflguraton, with the forces acting on it in the dis-turbed condition associated with the velocities u and w , the pitch angle

9, and the angular velocity q , is shown

in the 8ktoh: on pagò

.6

The equations of motion may be written, after expanding the resultant forces in a linear Taylor series and

neglecting

all

higher order small

quantities, in the following form:

-Foroe8 along the x-axis;

m1Xuu+Xww+Xee+XqQ+Xhh.

Forces along the z-axis:

m(* -Vq)

Zu+ Zw+ z9e +Zqq+ Zhh+Z,,*+Z

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-due to the presence of the front foil is asumed negligible for this análysia. The basis for this and other assumptions will be discussed in a later section which analyzes in detail the various hydrodyrami parameters and the methods used for their determination.

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-691

6-horizontal hydrocra.í' t reference axis

L+AL

+ ADR

q,4

Sketch 1

The Force and Moment Derivatives

Force nd moment derivatives may be ealuated by means of superposing the individual effects of small perturbations.

r. ç'. r.. -r.r((r.r r raç- ç'fp.ç r' -''C rr'

"

cccoC cc Oj.c.c r»r Ç. -r ÇrrÇrÇ.

Moments about the y-axis:

Mu Mw + M06 +Mqq+ Mhh + M+

()

In addition to these equations there are two other equations necessary, which are the result of kinematic conditions. These equations are

q=

ê

(6)

which equates pitch velocity to the time rate pf change of pitch angle, and

- ve

(7)

which relates the true vertical velocity (with respect to the fixed inertial

reference frame) to the velocity components referred to the moving axis

(12)

Contribution Due to Longitudinal Velocity Perturbation u

When the velocity, V , is perturbed by the quantity u , the increment of drag for both foils is, according to linear theory

+ u)(A,c0 + AR%) -

A%)

PV(A% + A1C)u = 2(DF + DR)

.

(8)

It follows that

xU =

_pV(Ä,% + ARCD)

= _2(DF + DR)/V

(9)

Similarly the increment of lift is

pV(AFC + ARC)u = 2(LF + LR) {

V(AC

P 1h P

lì

Q

V

r

T&F 1' A

TRjJ

W

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-7-zu S.PV(AFC + ARCJ) = _2(LF + LR)/V

(u)

from Equations

(8)

and (io) it is evident that the increments of drag and lift are proportional to the equilibrium values of drag and lift. Since in the equilibrium condition .the moment about the y-axis is zero. [see

Equation (2)], this results in

M

O (12)

u.

Contribution Due to Vertical Velocity Perturbation w

A downward velocity, w , causes an increment in the angle of attack. The magnitude of this increment is at each foil, and components of lift and drag are resolved along the x- and z-axes. The increment along the x-axis may be found as

(13)

691

8-and similarly along the z-axis he increment is

r

òc

-v(Ac

_{+ AR%) +}

V [A

_{+ AR (-a)ajJ'}

These'lead to

X E V(A C ₊

_ARCL)

V[A

W&F

+ AR

w

2

FLF

oc

Z = -

_{V(AF% + A) -}

.V[AF

+ AR ()Rj

Increments of the forces along the x- and z-axes result in a pitching moment about the y-axis, which leads to

=

_VA[C

+ - . +

(.L)R](2

Xp)

.

()F]xF

VAR

R

(17)

Contribution Due to Pitch Angle

e

A pich angle,

e

,

causes a component of the weight,

-we,

to be

resolved along the x-axis, and also results in changes in submergence of

the front and rear foils. These changes in submergence cause changes in the lift and drag forces, which are functions of depth below the free sur-face, and also cause a change in the wetted area for the case of dihedral

hydrfoia.

This resulta n aìerations of the forces ;ct{ng oñ the i hedral foils.

The creznent in

drag due

to the

_change

f

imergencc associated

with the pitch angie

is

. V24, ' V4 (18)

and that due to the

change of wetted area Thr ihedral hydrotols*

j

*

Those terms containing cot r

should be discarded in the

oase of fat foils.

Ç fl%çflç'

-'rrrç»'-nr'nr-r,::n'.r-n,-:

(]J4)

(is)

.

(16)

)acr

+ VA.[C1 _-

_()F]dF

₊ .

VA4C,

:.. () d

.

(14)

Theu reaulta lead to

ÔCD a '

V2A(-),

_Xp

V2AR(..)R( e. x,)

PV!OCCD,OOt4p.

u

_{PV2OR( I'}

zv)%RootI'g

and

X?

v2AR,,CI

--

PV2ØR(j

-

,)Ccot.E

The incremente of drag and lift for the front and rear toile ¡'inuit in a

pithing moment, whioh givee

I

Me a

v2[A,,

X?

2o,c,0Doott]dT

-

y2

[ÀR4)n(e.

x) f

_20R(

8'

_xF)cootrR]

_v2{

dOL

Ap(aT)p Xp + 2xCootr,j X?

I

(22)

R-691

-9-v2c

(Ar.

_{2cFCotrF xFO)}

-+ V2 [A ₊

_2oRootrR(4.

x)e]

-

V2CJAJ

-pvc,xF%ootrFe

+

pv2oR( e - xF)%cotrRe

(19)

where

r ie the dihedral angle.

Similarly, the increment of litt due to

both effeoto te.

òo

òc

-

_{vA1() xe +}

_v2AR(.)R(I

(15)

-691 lo

-q

ÒCD

- PV(AFCDdF + ARC,dR) + .

v[AF()F

xp_AR()R(l_

XF)] ,(27)

-

v2[AR()R(t

_{xF) + 2cR(l}

_xF)ccotrR](-

_XF) (23)

Contribution Due to Submergence Change h

The increments in drag and lift due to changes in submergence, h ,

are similar to those in the preceding paragraph, and hence the required force and moment derivatives are

=

_.v2AF.)F

-:V2cDcotr'

- PV2CRCDC0trR (21).

Zh

2V AF(_)F

-- PV2CFCICOtFF -

PV2CRCCOWR

(a5)

=

2()

+

2cFcotrFJdF

- + 2cR%cotrRJ dR + ₊

_2cYccotrF]xF

-

v[ARL)R

+ 2cRcLRcotrR] e - Xp)

(26)

Contribution Due to Pitching Velocity q

A pitching velocity, q , results in linear velocity perturbations along both the x- and z:axes for the front and rear foils. The perturba-tion along the x-axis is an increment in the forward velocity of qd at

the forward foil and at the rear foil. The velocity perturbation along the z-axis results in a change of qx/V in the angle of attack at the forward foil and a change of q(l - xF)/V in.the angle.of attack atthe rear

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r

òc

Z

_{= ipV(AFCdF + ARCdR) + V}

-

AR(-)R(2

-

XF)] , (28) T òC Mq =

-

PL2cD dF -

xA+ V2Cd ()FxFcF

-

_y

+

-

x)I

ARdR - V +

é

-

X)

AR

(t

-

XF) . (29)

Contribution Due to Vertical Acceleration v

The effect of the vertical acceleration, i , may be expressed in terms

of the virtual mass of the hydrofoils in the vertical direàtion (along the

z-axis). Because of the similarity between an elliptic disc and the planform of an actual hydrofoil, the theoretical calculations made for an elliptic disc can be used :t0 determine the virtual mas8 of the hydrofoils, as in

Ref-erence 14, which can be written as

kp,R P (30)

where k is the inertia factor given by Munk7. Thus, the resultant force along the z-axis due to the downward acceleration

j5

n

.'

n

E

AFCFW -

E

and the pitching moment due to the forces açting on the front and rear foils ie (31) p

_{AFCCF -}

kR p .

ARCR(2_

XF . (32) R-691

-u-- r

C It follows that i

Z. - -p

_{(kA1c + IcAo)}

_'I"

(17)

691

12

-Contribution Due to Pitching Aòceleration ₄

A pitching acceleration, , results in a vertical acceleration (along the z-axis) of for the front foil and

(2

_-

_XF) for the

Dirnens ionle ss Forms (Stabi liti Derivatives)

The motion équations, and hence the force and moment derivatives, are made diensionles by dividing Eiations (3) and (Lj.) by p/2.AV2, arid Equation

(5)

by p/2.AEV2,

where A

is the total projected submerged area of the

hyth'b-foils in the equilibrium condition, i.e., A ₌ _{+ AR .} Thus, the dimensiòn-less forms of the equations of motion are

m'ii! = XIur

+ X'w' + xe + Xq' + Xh'

u

w

- qt)

= Z'u' + Z'w' + ze + Z'qt + Zh' + (38) w

₄

n'4' = M'w' + M'O + M'q' + Mh' + MJï' +

M4'

y

e

q

and the kinematic equations become

q' and Ç cL

d8

c Lr. _4 V d

r

(39)

rear foil. Therefore,

the saine manner

z. = . q and M. = -p q

the coefficients Z and M

as Z. and M. leading to

w

w - kARcR( e- XF)J

[kc,xF2

_{+ kRAcR(}

I

2) may be evaluated .n

(35)

(36) ç1 (Lj.l)

In Equations 37 through

bi,

the time derivatives are replaced by derivatives with respect to s = Vt/2, dimensionless léngs traveled.

(18)

follows:

A A Xlii X, q

z,

_u

z,

w A

F,D

òc_ A ô =

- T 1TIF - r

ARdR

' A A R

r LTLR

A rA OC A Oc

i

X OC

AF CF

(L)

- A

(i

-

) J4R +

c1ootr

A OC h

- T

T p

CCotrR

A òC

- T

r R

-rA o i

F,

LT "1F +

3 AR _D

T

AF 0%

xF

AR _D Xe

r (T)F 7 - r

- 7)

CCF

_{CR(l_Xp)}

_w

+ 2

A

%.cotrF - 2

A

%RR

v2

0Ft

2

T- %FF T

1

x

ÒC AR

Li-

¿

-

t

\

_¡L

Â

c/

cl

2 -f CcotrF - 2

_CJCotrR

A d

fA ÒC A

x,

OC

q

i-7

TZ L1

1ri

FT-z-R

AC

_AR0R

Z=-. (%r-+kRr7)

'.

I

A CFIXF

AR CR(i_

)

L'FT2

kRj

(12)

(1i3) I (145) (146) (148) Ç 149)

'(50)

3(51)

(52)

R-691

-13-,

(3,)

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691

w M M'

*

=

N' =

_q

AF XF

T 7

I

AF d,

r C

1AxòC1.

CFXF

t t

a7F

+ a

A

x

dc

Lr 2'T3R

[AFxF ÒCL

CFXF

[rT(5ET)F2

A rA

(i- T)FT)R + a

F CD) [A

Òh' F

[RÒCD

[T

(ar)R

rA _AC

,

L. + A

ac.

- [r

r R

T[2cDF

[2CD

T +

I

Ac

ni

F =

LkF T

-M' = -

₂

/

dOL + Òc0

ct

+ 2

..

CDCotfR.1

:

+ 2 + ccotrR

d

_XFi4dF

à Cb

XA dR

c:JJz

dc

-J +.

+ 2

CJ.coti'F]T

cR(- XF)

A T

--iP

FCF

S_AR CR(

L-

XF)

A, cx2

A cR( XF

+kRf

A X

r-/

ò

i

-:

(1 -

[CDR,-AR dR [

T T

-r

CD

CotrF

7 cR(lxF)

'dR

+2

_A

..%R

Rj

CcotFR]

[2c4

]

XF

¿

2]

(

,

A'f

AL

òCL/.xF1 À. XF

7

(Sr?) X

£

(56)

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Hydrodynamic Coefficients and Their Derivatives

The dimensionless stability derivatives above are dependent upon the hydrodynamic lift and drag coefficients of each foil, _CL and CD , their derivatives with respect to angle of attack and depth, and upon certain physical size characteristics of the foil configuration. It was assumed,

throughout the derivation of the equations of motion, that the variation of the lift and dag coefficients themselves with respect to forward speed is

negligible. This assumption will be shown to hold true at all speeds in which a practical hydrofoil craft operates. In addition, it was assumed

that the hydrodynamic moments of the foils themselves., about their

hydro-dynamic centers, may be neglected in comparison to the moments about the craft cenEer of gravity due to the localized forces acting at the foils.

In order to determine the lift and drag coefficients and their rates of thange with respect to angle of attack and depth for a particular hydro-foil form, the simplest and most positive approach is that of direct

ex-perimental measurement. A foil, having an appropriate aspect ratio, should be tested at the design speed, over a range of angle of attack that includes the angle of attack value required for equilîbrium of the entire craft, and

over a range of depths from less than to greater than the design depth. Both lift and drag should be measured, and from this data, the lift and drag

coefficients may be pl6tted against angle of attack and depth. The values

of the lift.and drag coefficients at the design conditions, as well as their

derivatives ÒCT/òa .,

òC/òa

, dC1/dh , and

ÒC/òh

, are obtained from

this data. On the basis of experimental data reported by Wadlin, et. al.8, there will be no speed dependence, i.e., no dependence on a Froude number,

_p

for values of the Froude number based on chord, _F = v/.jE, greater than

II <'Ç

about five. Since most high-speed hydrofoil craft operate in this range, the Froude effects on the lift and drag coefficients should be negligible.

The procedure outlined in the, previous paragraph is applicable to te forward foil of a tandem configuration, but is not directly applicable to the rear foil,'since the rear foil is situated within the downwash field created by the lift associated with the equilibrium angle of attack of the

forward foil. Thus, for the static value of the rear foil lift coefficient in the equilibrium condition, the downwash angle appears to be required.

R-691

(21)

-ls-H69l

16

-However, with a definite design configuration, the foil angles of attack

are known. This knowledge, together with experimental knowledge of the f

or-ward foil lift coefficient and the fact that the total lift should equal the weight at the design speed and submergence, allows the rear foil lift

coefficient to be found.

The drag coefficient of the rear foil also is affected by the dowriwash,

and is difficult to assess accurately. However, since drag is not as impor-tant as lift for considerations of longitudinal stability, any errors made in the rear foil drag determination should not affect stability to any great

degree. One possible way of determining the rear foil drag coefficient with-out actual model tests of a tandem configuration at the appropriate scaléd speed, depth, etc. is to find the value of the drag corresponding to the

rear foil trim angle of. attack, multiply the difference between this value and the assumed profile drag coefficient of the f oil section by the square of

the ratio of the lift coefficient with downwash to the lift coefficient with-out downwash (free stream value from tests), and then add the profile drag

coefficient. A more accurate method, of course, is direct experimental

meas-urernent of both the lift and drag coefficients of the rear foil when a mode]. of the coniplete tandem configuration is run at the necessary equilibrium conditions. In many cases, however, the values of the lift and drag coeffi-cients at the equilibrium condition usually are available as part of the preliminary design, and the stability anaiysls can proceéd using these values

whenever necessary.

As far as the derivatives of the. lift and drag coefficients with respect to angle of attack at the rear f öl]. are concerned, their values may be

obtained .experiinentally in the same manner as was done for the forward foil.

Since the downwash angle rate of change with respect to angle of attack, dE/da , was neglected in the stability analysis, there will be no. essential

difference betwen the methods for determining _ÒCL/òcx for the forward and

rear foils. In addition, the methods for determining the depth derivatives òCL/òh and ÒCD/Òh at the rear foil will be the same as that technique used for the forward foil,

The values of the lift and drag coefficients of

a hydfçil travel!ng

near the f'e surface dUfer from those of a

foil in an infinite fluid becaue

(22)

/1/y

where CL is dynamic value)

of the effects of "image" flows and the formation of gravity waves. The

foil suffers a decrease in lift, as compared to the infinite fluid case, be-cause of the reduced virtual mass associated with the foil and the flow modification due to the wave formation. A number of theoretical

investig-tions of the lift of hydrofoils have been carried out, but the most useful and valid result is the three-dimensional solution given by Wu1. Wu's theory

originally was derived for flat hydrofoils, and is represented in a rather

complicated f ôrm. 'In place of this cumbersome expression, an empirical

approx-iination was proposed by Tinney9 and was found to be applicable todihedral and flat hydrofoils, provided that the concept of mean submergence is adopted. The expression for the lift coefficient given by Tinney is

C = C (1 -

O.22e1

hm/Q)

L L

(60)

the value of the

and h is the

in

thé case of a surface piercing

h

b.

-'tanr

The value of _CL , which is the three-dimensional lift coefficient in an infinite fluid, is found for a particular hydrofoil section by correcting

the two-dimensional section lift coefficient (given in. MACA

Report

821i., for

example) for aspect ratio, i.e., by dividing by (1 + 2/.).

The variation of the lift coefficient with respect to angle of attack

then is

òC ÒC

L ,

= -)(l -

0.L22e_1

h/C)

₍₆₂₎

where (ÒCL/ô) is the aerodyhamic (infinite fluid) value for the particular

foil configuration. The derivative ÒCL/òht , for the case of flat foils,

is given by

C,

\\\

_ÒCL

- 0.6]JCL

(63)

A

7

.t S -r

Gi9

lift coefficient at infinite depth (ehe aero-mean submergence of the foil, which -- for dihedral hydrofoil -- may be expressed by

(61)

R-691

(23)

--691 18

-For the case of dihedral surface-piercing foils this derivative is oc

= 0.307.

CL_eR hm/C

,

(61)

since the change of mean submergence due to a depth change h of the entire

craft is Ah/2 . AU of these results [Equations

(60)

through (6h.)] are only

true for the case of high speed. At high speed, the free surface acts as a surface of constant pressure, similar to the jet boundary of an open jet wind

tunnel, and the lift on the. submerged foil is essentially the same as the

lift on the lower wing of a biplane configuration with separation given by

twice the submergenóe. The. speed beyond which any Froude effects aré present

The drag coefficient, which is composed of profile drag, induced drag and wave drag, is a complicated function, which usually is evaluated by numer-ical integration. However, for the high-speed condition, Breslin10 has succeeded

in simplifying the expression for the noriviscous portion of the drag coefficient,

-i.e., the sum of induced and wave drag coefficients. This simplified expression is represented by

CD+CD =CL[

+y

(X)Kc]

O

i

w

where o(X) is the biplane function, X 2h/b is the submergence-semi-span

ratio, 6 is the aerodynamic planform correction factor, y(A) is a function found from an integral, and which is graphed by Breslin10, and K0 2

The formula given by Equation

_(6S)

has been found to give good agreement with experimental data for flat hydrofoils, when the total drag .coefficientis represented by

(6S)

CD C + CD + p 1 W + 2 Ii +-()) + 5 _{... '()}

_KcJ

,

()

LR

p

where GD is the profile drag coefficient found from aerodynamic section

datá (assuming no dependence upon depth of submergnce).

No simplified theoretical expression similar to Equation

₍₆₆₎

for flat foils exists for dihedral hydrofoil drag coefficients. _Tinney9 _attempted to find a useful expression, but it did not agree with experiment and was

(24)

R-691' 19

-based, in part, on certain extreme assumptions, such as use of two-dimensional wave drag expressions. An attempt at a rational analytical expression for

the dra of dihedral hydrofoils, based upon a three-dimensional .hydrodmamic theory similar to that of Breslin10,i.e., use of an appropriate doublet dis-tribution, was made by Kaplan11. However, no numerical evaluations were made of the integral representations and, hence, these results are of no

signif-icance for the present report. Since the drag coefficient cannot be evaluated

with great accuracy, and due to its small influence relative to the lift

co-efficient in longitudinal stability studies, the drag coco-efficients for

dihe-dra]. foils will be calculated on the basis of the same formulas as was done f or flat foils, i.e., using the mean submergence as the depth parameter.

Since the drag coefficient is based upon the projected area of the foil, and

profile drag is dependent upon the actual wetted. area, the profile drag

co-efficient of a dihedral foil is

CD w (CD )

eecr

, (67)

p

'pIwO

where (C ) is taken from aerodynamic section drag data, on the assumption

ro

that there is no depth effect on, the profile drag. Thus, the total drag co-efficient Óf a dihedral, surface-piercing hydrofoil is approximated by

0D

(CD)

seer + CL2

{1 +) + 8

+v().) x0]

(68)

where X

ia the ratio of the mean submergence to the projected semi-span.

The hydrodynamic derivatives ÒC,/Ò and _ÔCD/Òht'

are found by

1ff-erentiating the expressions for _CD given in Equations (66) and (68). Thus,

2 CLT

1

+O

+ 8 +v(X) K0o] (69)

on the assumption that the profile drag coefficient does ñot vary appreciably with angle of attack.

If

the

foil

section

is one where

there is an appreciable

variation

of

pÑfile drag with angle of attack at the equilibrium angle, this

value can be. added to the va]. us given by Equation (69). The derivati+e

ò%/rhit ,

obtained from differentiating the approximate expression for _CD is a complicated expression, dependent upon the derivatives of oX) and

y0)

(25)

69l

20

-and must be obtained from the graphs in Reference 10. It is expressed, for the case of flat foils, by

òC òC

l+0(X)+6+y())KCl

= 0L { .

The downwash

the rear foil of a

angular setting of

dowrxwash angle E it. tends to reduce

I O)i.)

+ v'().)K

I. IL

oJ'

('70) where b is the fòil span. This quantity will be small since the first term

is positive and the second term is negative. For dihedral surface-piercing foils, the value of ÒCD/òht will be half that obtained in Equation (70)

since the mean submergence is used in the definition of the depth parameter. More detailed information on the hydrodynamic coefficients for different foil sections can be found in the Gibbs and Cox Hydrofoil Han dbook, but the formulas outlined above are still sufficient for estimating the coefficients

to be used in a dynamic stability analysis.

angle E is of importance since the. angle of attack at tandem 'hydrofoil system is the différence between the the rear foil (relative to zero iif:t angle) and the mean

Thus, a -E', where the sense of E

i8

positive when the anglé of attack at the rear foil.

The downwash angle resulta from the vertical velocity field due to

the, vortex system representing the forward foil, the biplane image vortex

system, and the generated wave system. A three-dimensiänal theoretical eval-uation of E (from which dE/da could be easily calculated) was carried out by Kaplan, Breslin and JacobsU, and the complex formulas therein could be used for determination of E and dE/da . The quantity dE/da is used, in certain longitudinal stability analyses of hydrofoil craft, euch as Reference

L, to correct the rear foil derivatives with respect. to angle of attack by

multiplying by the quantity (i - dc/da). Another place .it appears is in the

rear foil lift force and the craft pitching moment, where the mechanism for

its influence is the fact that there' is a time lag between the time that the

àngle of attack changes at the front foil and the time when the downwash reaches the reär foil, i.e., the lift at the rear foil at time t is

in-fluenced by the downwash created by the front foil at a time t -

L Iv

.

The

(26)

unsteady flow phenomenon, and is the only manifestation of unsteady effects considered in the usual quasi-steady aircraft dynamic stability analysis, aside from virtual mass effects (which are only important for the hydrofoil

case, i.e., in a dense' fluid medium such as water).

Calculations of the dynamic longitudinal stability of hydrofoil craft were carried out by Imiay2, where the only influence of downwash conidered was the modification achieved by multiplying the rear foil angle derivatives

by (1 - dE/da). The resùlts of these calculations èhowed that stability

hardly was inf].uenced.by the value of dE/da , and, hence, this term is neglected in the present stability analysis. Further support fòr neglect

of the dowrnwash influence was given by the results of calculations performed

at Massachus3tts Institute of Technolor (Reference ]J), where the total

influence of downwash (changing the angle at the rear foi]. and also the time

lag effect) was considered in a manner somewhat different than the usual method, i.e, the effects of time lags and unsteadiness were considered. In this case, the results indicated that the total effect of downwash on stability

was negligible. The usual methOd of including dowmiaèh (e. g. the work of

Hugh and Kaplan,) indicated that the downwash la effects were significant but this was proved erroneous by the results of the MIT calculations.

Determination of Longitudinal Dynamic Stability

Since the equations of motion are linear1 and their solutiQns are linear functions, the latter nay be expressed in the form of exponentiaÏs,

i.e.,

-ç-..

aj8

u'

¿ue

,

w' - ¿w1e

ç_'a-a

71

qID

_L±e

_,

_h'-

¿hie

i i.

Substituting these expressions into the differential equations of motion, Equations (37) through (Zi), they reduce to a system of five algebraic

equations for the

Qj ,

the values of which determine the stability of the

motion. In order for nontriviàl solutions to exist the determinant of the algebraic equations must be equal to zero, i.e.,

e

I.-R-691

-

(27)

f

m'o.

-u

-Z'

o

This results in a quintic equation for the which. is written as

aç.S

+ + a3c +

a2o2

+ a10-1 + a

ao O

(73)

jo

The necessary and sufficient condition that the solutions be stable (real parts of are less. than zero), is given by means of the H,urwitz criteria.

or Routh' s discrintlnant (Reference

iS),

which requires that

i) all the

&j) O

and simultaneously, 2) a1 a0 a3 a2 >0; a1 a3 o o

a3

a2 a1 a0

a5

a

a3

a2 a

In many cases, the effect of surge may be neglected äs usually is done In aircraft longitudinal stability analyses and the x-equation [Equation (37)] as well as áll terms involving u' are dropped from the equations

of mötion, . This resulting system of four equations leads to a quartic quation in O , after evaluating the stability determinant given by

-xt

w

(m' - Z')o - (M' + Mç.) Zt

w

Z1

e

-M -Z'o-i (n' --x t q - (m'+Zt) q M) o-. - M' -Zt h -M (72) o o i o -1 i o

(28)

(m' - Z')a. - Z' -z'

1. W e

-o.

i

The quartic equation may be represented by

ao.1i0

and the stability criteria _{are that all} _a1. _O

arid a a i o a2 >0; a1 o a2 -Zc-1 _{- (m' + z)} _-z1 (nj' -M')a. ¿ M' -N.' y i q i o o o-i O 3 >0; a, a . O J. o a3 a2 a1 a0 O a a3 a2 o

When these

conditions

are satisfied simultaneously, the values of

will result in stable motion and all disturbances are damped out.

in certain stability investigations of particular hydrofoil _craft employing flat foils, the depth variation of the lift and drag forces _was ào small that terms of this type were negleoted, i.e., derivatives with respect to h' and e _{were deleted from the equations, and the resulting}

system of equations were the _{same as standard airoraf t stability equations,} After neglecting the influence of surge, the stability equation becomes _a

quadratic _{from which the conditions for stability can be easi].ydetermjned.}

Another simplification used in longitudinal stability investigations was the neglect of all drag forces, for both flat

_foils

_{and dihedral}

surface-piercing hydrofoils. _{Mo definite statement can be made as to the relative}

influence of.any of these simplifications on the degree of _{stability indicated} by the resulting solutions,

since they were carried out only for particular configuratjon, _{Therefore, it appears that}

al]. the terms required for

=0

(7S) R-691 23 -(76) >'0 (77) -(M' + M'o w -1

(29)

691

14 -.

evaluation of all dimensionless stability derivatives should be used, as the most general treatment will then result. On the basis of aircraft motion

experience,

as

well as solutions

of

some hydrofoil equations (both including

and neglecting the

urge degree of freedom), the most acceptable approximation

to be used in simplifying the equations of motion would be the neglect of.

surge

and the reduction of the polynomial stability equation to a quartic.

In summary, the hydrodynamic lift and drag coefficients and their

derivatives with respect to angle of attack and depth for both the forward

and rear foils should be found first. These quantities should be inserted

into the dimensionless stability derivatives defined by Equations (Ii2) through (S9), and the resulting stability equation (quintic or quartic.) determined. The criteria of stability (Hürwitz criteria) are tested for

the particular stability equation. This determines whether or not th

partic-ular configurationis stable or unstable. If it is deired, the equations themselves may be solved fairly simply by analytical means or by use of a computing machine, in order to determine the actual transient motion of the

(30)

PART II

LATERAL STABILITY IN SMOOTH WATER Assumptions

The analysis of lateral stability in smooth water is limited to the motions of a noncavitating tandem hydrofoil system in the transverse plane due to freedom in sideslip, yawing, and roiling. Since small asymmetric motions do not result in arty symmetric disturbances, there will be no effect on the forces and moments associated with longitudinal motion (X,Z,M) due to the lateral motions, and also vice-versa, Therefore, the equations of longitudinal and lateral stability are, not coupled to each other when the linear theory of small displacements is used. However, a more refined non-linear treatment would consider the coupling of longitudinal and lateral

motions', but that is beyond the scope or purpose of the present paper.

The analysis is based on the assumption of a smooth water surface, and the presence of the supporting struts is taken Into account since their

influence, on certain láteral derivatives is of significant valu&. The linear

theory for small perturbations is utilized throughout. Downwash arid sidewash

effects are neglected because of the difficulty involved in evaluating them and also because of their small influence on hydrofoil stability. The

nomenclature used here is the same as that used in Part I of this report. Another assumption is that the struts for the case of dihedral surface'. piercing foil configurations have only small submerged portions, if any, and their effects may be neglected in comparison with those of the foils

themselves. In certain cases, this assumption may not be valid, but the nature of the interferences in those examples probably will act to invali-date any of the results obtained by use of the techniques outlined in this

report.

Equilibrium Conditions

At the static equilibrium condition, the hydrofoi3 craft moves

for-ward with a unifòrm. velocity, V , in the x-direction with its foils sub. merged at a definite depth and with the reference axes set at zero yaw and

roll angles. Tà achieve this condition, the following requiremeñts must

be satisfied:

R-691

(31)

--691 26

-force and moment coefficients may be considered to arise from the

contributions:

the generai effect of the foils, regardless of' their

con-figuration (flat or dihedral), the effects of the struts,

an additional effect of the foils, due to dihedral, above and beyond:effect No. (i).

The sum of the forces in the y-direction must eqia1 zero,

The resultant moments about both the x- arid z-axes due to

the forceà acting on the

foils

and struts must equal zero.

Equations of-Motion

Whená hydrofoil craft is disturbed by the perturbation quantities

y,

,

0 , p ,

,

j, r , and as shown in Figures 1, 2 and 3 on pages 63 , 6h. and

6S ,

the

following

equations

of

motion may be

found by

linearizing the accelerations and expanding the corresponding resultant force and moments

into linear Taylor seiies in the neighborhood of the equilibrium positions

m(r + Vr)

= Yv + Y# +

p+ Yr + Y.- + Y

+ Y (78)

= K,v + + Kp + Krr +

Kr +

+ (79)

y +N#

+ Nr +

+ N.2' (80)

In addition there are two kinematic relations,

p=

(81).

and

r=*

(82)

There are no derivatives with respect to the yaw angle, j', as no hydro-dynamic reactions cari occur because

of

orientation relative to an inertial

-frame. This holds as long as no change in the relative position

of

thé bod

and the velocity vector at any point on the body takes place. The Force and Moment Derivatives

The

following

(32)

Since the effects of the struts in the case of dihedral foils are neglected in comparison with those of the flat foils, it is easy to see that effects

(i) and (2) furnish the coefficients for flat foil configurations, while (i) and (3) provide those for the dihedral foil craft. Therefore, by using the subscripts 1,2 and 3 for the effects (i), (2) and (3), respectively, a

co-'efficient, e.g. Y, , may be expressed as

y =(y) +(y')

y

v] v2

'

for flat foi]. craft, and

YV =

vi

+ (7)3

(r= o)

for dihedral foil craft. Consequently, the evaluation of the force and moment derivatives will be carried out by considering effects ]., 2 and 3

separately.

The values of the force and moment derivatives found in the following sections only may be viewed as relatively rough approximations, since theor-etical methòds for determining the lateral stability derivatives for air-planes do not yield very reliable results (see' Campbell and MciCinney6). In view of the e,ffects due to the presence of the free surface, it appears that the results for hydrofoils will be even less satisfactory. Therefore, no elaborate treatment will be made, and only the quantities having a major effect on the partrcuiar derivatives will be found. The

contribution

of' other effects which are neglected will be

mentioned,

but no precise

oval-uation of their contribution will be carried out..

Gontribution

Due t Sidealip Velocity i'

(i) A sideslip velocity, y of the craft results in an equal aideslip velocity for each of the foils, which introduces a component' of the drag

along .the y-axis. As shown in Sketch 2 on the following page, the component df y along the panel is y cosr,, so that the drag component along the y-axis

is - D _cos2r. Therefore,

w 4

.pV(AC

oos2r _{+ ARCD oosrR)}

where, in the' case of flat foils, r o The rolling and yawing moments

(83)

R-691

(33)

-691

28

-and

D

-Pv[xF4)F

XF - XR()B

(1_xF)]

where

_(L/òa)

is the strut side force rate with angle of attack, i is

the projected submerged area of a strut, d - h/2 , and it is assumed

that there are two equal struts both forward and a.t' t.

Sketch 2

due to this effect result in

(K)1 =

_{Pv[AF% 400s2rF}

- ARCD dRcos2rR_I (81&) and

4

Pv[AFCDiCFcOS2rF - ARCD

(1

-

xF)cosrR} . (8S)

(2) The sideslip angle v/V at the struts results in lift forc8s,

lee.,, side forces, on the struts Ìnthe negative y-directiOn. This gives

v2

= _PV[xFI(7)F + IR(T_)R]

v2

PV[IF()F

dF

1R(J)R

dR]

(3) For dihedral foils, the velocity y can be resolved into a component y cosf along each panel and a component y sinr normal to it.

(86)

(8?)

(34)

The normal component causes an increment of + v/V sinl' in the angles of

attack for the foils, where the plus sign holds for the starboard parl and

the minus sign for the port.

These increments of angle of attack, give rise

to a pair of lift fOrces acting normally to each panel ás shown in Sketch

3 below.

Sketch

3

The increment of the normal lift is calculated as

Oc

¿C.

sinf

pV A' ()sinÇv

(89)

where

A'

is the actual area of a panel and

a

_n

is the angle of attack

1?

measured in a plane normal to the panel.

Purser and Campbell

have shown

that

a , the angle of attack in the vertical plane, is related to

a

by

a

_cosf

=a

₍₉₀₎

and using the projected area. of the foils as a reference, the total

Y-cornpo-nent for both foils is

Oc oc

-

PV[AF(-)F

tanF + AR(.-)R

tan2rRi

V

(9i)

This leads to a yawing moment equal to

Oc

-

PV[AF(7)piXF

tan

- AR()R(_

xF)tan2fR]

y

(92)

R-69l

(35)

-691.

-and a rolling moment given by

r _oC _òC

PY{AF()FF

tanl'F + AR()RdR tan

The vertical componente

of

A L fori a rolling couple about the x-axis equal

to

+ AR()Rtar1r

bROOsrRl

R'

2J

+ AR(..)RbR

einrRI

r

oc

4

ØV[Ay(._)taflfp

-* PV[AF()Fbp.

.(93)

where the center ôf pressure of the forces is assumed to be at _{the middle} of each panl in the derivation of Equations (93) and (9l).

A roll couple due to the vertical component of the drag forces _{on each} panel also is present, but is neglected because it is small in comparison to the other contributions to the total rolling moment due to sideslip. _Another effect that is neglected in this analysis is the mutual intörferenòe between the two panels of a dihedral hydrofoil. _{This interference results in different} loading, distributions when equal and opposite changes of angle of attack occur

on each panel (see Purser and Campbell17).

When the results of Equations (91) through (9L) _{are combined, the} fol-lowing expressions are farmed:

r

v3

=

PVLAF()F tan2

=

pVfoc

k(.rL)F(F,tan2rF -

sinF)

2 bR

+

Aá(-)R(R

tan FR

-

T

si.nrR)] ,

v3

4pv[AF(.)pxF tan2FF

-

AR(-)R(1u

xF)tan2fR]

(97)

(9L1.)

(96)

(36)

Contribution Due to Roll Angle Ø

(i) A roll angle Ø results in a component of the weight which has

a magnitude of wØ along the y-axis. Associated with this roll angle is an increment of + * bØ cos

_F

in the average submergence of the starboard

and port panels ôf the foils (or half panels for flat foils), respectively.

This results in differences in the lift and drag of each panel, and there-fore results in rolling and yawing moments as shown in Sketch L below.

{There is no Y-force except for dihedral foils, and hence it is considered only in (3)] water surface Sketch 14 - Since òC dC

AL

pV2 ()

seoFAh

pV2Ab (.j)Ø

the rolling couple due to the vertical componeñts of these litt forcés

on both foils is

òc.

-

pv2[AFbF2t)Fcoa2rF

ARbR2(.)RCOs21R]

₀

(9e)

(99)

As nntioned earlier, the y-component of AL only

is

associated with the

dihedral angle and will, be considered in (3), together with

the rolling

and

R-691

(37)

-91

)

yawing moments that occur due to this effect.

The change in the drag force on each panel is

given

by

ÒCD

AD

= pV2

(-)Ah

=

pVÄb (-)cosrØ

(loo)

whiôh results in a

yawing

couple for both foils equal to

i.

pV2[A,bF2()Fcos2FF

+

ARbR2(.)Rcos2rR]

₀

(loi)

Combining all of the above expressions (98 to loi) with the resolved compo-neñt of the weight leads to

(Ye) i ( 102) and 1

Øi

= Oc )Fcos2rF +

ARbR2()Rcos2ÇR]

(N ) =

pv2íAFbF24)Fcos2rF

+

ARbR2()Rcoa2rR]

01

32

As

far

as the struts

are

concerned, a rol]. angle

₀

resulta in

increments of wetted area for each strtit. of magnitude

+ .72 .ZØ.

.

Changes

in drag

occurs

but these

only give

a yawing couple and make rio coflrbution

to either the side

force

or roll moment.. The ya4ng moment, however, is

small and Is

neglected.

Hence, (T0)2

Ø2

o (loS)

The roL.

angle

₀

causes

ari inorenient

of + bc/2.cotrcosrø

n

the

wetted, area of the folla, where the plus

sign holds

for

the starboard

panel

and

the minus sign for the

port.

Since the normai foroe.aoting on a panel is

pVC, seo

f.

AA

_{*pV2CLbc cotf 0}

*

_{PV2CLA csorØ,(.O6)}

the roll couple is

(38)

-

_{pv2[AFccotrF(}

_{cosrF) + ARCL cotrR(T ¿osrR)]}

=

_.Pv2[AFbFccotrFcosrF

_{+ ARbRCLRc0tFR}

C0SrR] Ø

.

(lo?)

The total y-force due to the lift increment

AL

on the foils is

_PV2[AFC

+ ARCLIJØ

(108)

which is independent of dihedral (excluding the presence

of a function of

the dihedral angle in the definition of the

projected areas).

When the

thrust acts parallel to the x-axis, the value of the

y-force given by Equation

(108) is

-wØ

, accordìng to

Equation (i), and it may be seen that this term

will cancel the contribution of the term

(Y0)1 found in Equation (102) in

the case of dihedral surface-piercing foils.

This y-force value results in

a roll moment given by

pV

[yFcL

₊

ARCL]Ø

(109)

and a yawing moment given by

_1.PV2[AFxFC

-

AR(l

XF)CL]Ø

..

(110)

As shown in Equations (98) and (99), the lift increments

ALA, caused

by changes in the average submergence of the half panels

(for dihedral foils

only), also result in a y-force, which, in turn, produces roll and yawing

moments.

In addition, the drag increments arising from, changes in the

wetted

areas of the foils produce a yawing moment.

These contributions also are

neglected as small quantities relative to those given in Equations

(io?)

through (110).

Combining Equations (io?) through (110) leads to

(Y)3

_pv2[AFc

+

ARCL]

(K0)3 a

.

v2{AFcL [F

- bF cotrFcosrF

+ ARCL [LIaR

-

bRcotrRcosrR]}

(112)

R-691

(39)

--69l

314-and

(N0)3

4PV21AFxFC_ AR(l_ xF)cLi

Contribution Due to Rolling Velocity p

(i) A rolling velocity p about the xaxis may be resolved, for the foils, into a sideslip velocity -pd and. a rolling velocity p about the

axis which is parallel to the x-axis and passes through the center of the flat foils or the vertex of the dihedral foils as shown in Sketch

be-low.

dD(2)

.10 V

A(dL)

n Sketch

₅

As shown in the sketch, when the foil has a rolling velocity _p about axis 01 , the strip ds has an increment of + ps/V in the angle óf attack normal to each panel, where the plus sign holds for the starboard panel and the minus sign for the port, These angle increments result in a drag

ccnpo-nent .dD.ps/V norma]. to the panel and an increment in the normal lift dL Thé value of the increment in dL may be found and inteated over the

total span, but this complicatea procedure is removed by making the assumption that the center of pressure on each panel is at the center of the panel span.

Thus

(40)

and

D = pV

_%

pVAb CD

for each panel. The rolling couple arising from these forces on the front and rear foils then is

-

_pv{

secfF + CD COSfF] (.

COSr)

dC

ARbR secrR + CD COSfR1 ( C0SIR)}

r ÒC

r òc

pv{AFbF2

I

FD

cos2fFj

+ARbR2

L

RCDR

cos2rJp

(116)

The y-components of these forces, and hence the roll and yawing moments contributed by them, only are associated with the dihedral effect, and will be considered in (3).

The contributions due to the effective sideslip velocity -pci are

found by using the results of Equations (83) to (S), which yield a y-force

equal to

PV(AC

dF COS2fF + ARCD dR cOsrR)p (117)

a roilmoment equal to

dIi. cos _{+ ARCD} _{dR2 COSfR)p} 14PV(AFCD

2 2

and a yawing moment equal to

PV[AFCDXFdFI cosrF

- ARCS

(t- XF)cIR

Combining all of the above expressions leads to

PV(ACFd.

CO521'F _{+ ARCDRdR} oos2FR)

J

(41)

69l

p2

PV[IF(F

+ -pV and

(N)2

₌ _Pv[xF(

_{XFF - AR)R(}

2- x

to the effective sideslip velocity -p , at the midpoint of the. submerged. strut span. According to the results of Equations (86) to (88), the following

are obtained:

J

.

Certain othe.r terms, whose contribution is small compared to those

given by Equations (23) to (12S), have been neglected. These terms arise from the components of the strut drag and also from the contribition of the small submerged portion of the strut which is regarded.as rotating with an angular velocity p about the midpoint of the submerged span.

(3) The increments of. lift and drag that act normal to.the panels

of a dihedral foil due to the effective rolling about the axis 01 result

in a y-force, whiôh is given by

-*PVAb

[(a) sec2f

₊ sinÇ

for a single fOil. The total y-force for the present configuration then is

36 -and

(K)1

(N)1

(2) +

8Ad2

= . PV[AFCD The main =

_pv{AFbF2

C0S2ITF XFdF effect of dc + + ARbR , )dR cosrR] p on

(L)

RC0 LB] (121) (122) is due

[(L)

+ 8ARCD dR2 cos2rRJ cos _{- ARCD}

(j-the rolling velocity

.

(42)

The contribution of the dihedral effect for an effective sideslip velocity -pd is evaluated according to the results obtained in EquationB

(91) thròugh (91.), leading to a y-force equal to

r

tan rF + ARdR(.-)R tanrR j p a rolling moment equal to

oc

*PV

_{AF()F[bF.

sinrF - 2

F tan2rFj

oc

+

ARdR(7)4bR

sinrR - 2 tan 1'R

J }

pV

[AFxF

tan

r. -

_{AR()R( j- x.)d}

tan2r Combining Equations (126) through (131) leads to

(129)

(130)

(Ui)'

tanf

F + CDFF 8iflT'pJ (T ) - _{pV {AF [(7)F(bF tanrF seCFF}

p3

1ACL

(bR tarira seorR -. 2dR tari2fR)+CDbR

ainrJ}

,(132)

r OC

(K)3

s

pVAF()F[bF

tanrF secrF +bF siflrF - 2dF

tan2rF}

r

R-691

-y?-+ ARbR sec + CD

]siflrR

J P

(126)

The y-force also produces a roll moment

+

given by

PV{AIb

[(L)

_secr'F

CDF]F

+ ARbR

secrR +

CDJsinrR}

' (127)

and a yawing moment equal to ÒC

* PV{AFbFxp

[F

sec +

cD]sinrF

-

_{ARbR(J_ XF)[(7)R}

+ CDR]

R} p

(128)

(43)

8-and

+ AFCD _F SlflfF + ARCD sinrR C

r

+ AR(.-)R J tanl'R secfR + bRdR sinfR - 2d

tan2rR] },(133)

dc

(N ) ₌

_{_'PV{AF(7)FXF[bF}

p3

tanrF .secfF - 2dF tan2rF]

+ AFcDbF SiflfF - ARCD bR(, - XF)

siflrR

oc

-

AR(-)R('1_

xF)[bR tanl'R secrR - 2d, tan2rR]}. (131)

Contribution Due to Yawing Velocity r

(.1) A yawing velocity about the z-axis is resolved into a sideslip velocity _xF.r for the front foil, a sideslip velocity -(t - xF)r for the rear foil, and a yawing velocity r for each foil about a vertical axis

through its own geometric center.

Considering the yawing about its own axis, the resultant forward, velocity at the center of pressure of each panel of a foil changes by

.+rcosr ,where the plus sign holds forthe port panel and the minus

sign for the starboard.. Associated with this change in velocity are increments in the lift and drag of each panel, given by

ALpVCAV and ADpVCDAV

where

AV = + r cosF

The lift increments result in a rollmoment given by

.

pv[tc.

CO5TF (

+ ARCbR cosrR

( cosrR)] r

= _pV

_{[AFcbp2 cos2rF}

₊

_ARCbR2

(44)

R-691

-39-and the drag increments lead to a yawing moment

given by

_Pv[AFcDbF2

COS2Ç ₊ _ARCO _bR2 cos2fR] r (137)

- The contribution of the effective sideslip velocities at the front

and rear

foils, which

are

_{xFr and -(J-}

x1)r , respectively, are

eval-(2) The main effect of the yawing velocity r on the struts ariseB from the effective sideslip velocities xr at the forward struts and

uated as

in

Equations

PV(ACx

COS

(83) to

(85)

and result

in

a y-force equal to

rF - ARCD(

X)

cosrR)r

(338)

a roll moment equaÏ to

PV(AFCDXFdF COS

_-

_ARCD C

-

XF)dR cosrR) r , (139)

4PV(AFCD.X72 COS2IF

+ ARCO

(Z Xp)2 cos2r) r.

(3140)

Combining the above expressions results in

(

4pV

[AFcDxFcos2rF

- ARCO

(1 -

xp) cos2rR]

(3141)

pVA ooa2rF

[cb2

+

_8CxpdF]

+ AR

[CbR? :

(/- X)d ])

(3142)

and

iípVAFCflcoa24 [bF2

+ 8xF2J

(45)

91

-( I - xir at the rear struts. Following the procedures in Equations

(86)

to (88)

results in the following:

and and

r2

(Nr)2 1.

F7FF

- XF)

I

ò ò

r2

= pvLIF (.7)PXFaF - - XF)

oc

AR)R(.i XF)(aR

tan2fR

sinL').]

oc

-

pV[A

_()pcF2

tafl2FF

+ AR(r)R(

- XF)2 taxi

Contribution Due to Sideslip Acceleration t''

A yawing moment contribution arising from the increments of strut drag has been neglected, since it is small relative to the quantity in

Equation (]16).

(3) The contribution of the dihedral effect for the effective

side-slip velocities (xFr at the forward foil and -(

L-

x)r at the rear

foil) is found by the same analysis used in Equations (91) through (9I).

This leads to

=

-Pv[AF( tan2îF -

AR(-)R( /-

xF)tan2ra] (]J47)

b

r 3 = pV

- -

_sinr)

(Th8)

I(]1i9)

(i) Obviously, a sideslip acceleration will not produce any

force. or moment on a flat foil, but will have a reaction on a dihedral foil due to the dihedral effect alone. Therefore,

(46)

R-691

A sideslip acceleration r resülts in a side force on the struts

due to the virtual mass effect. As mentioned in Part I of this report [see

Equation (30)] , the virtual mass of a strut is evluated as

,

where Is the virtual mass coefficient of the submerged portion of the

strut. Thus, the total y-force for the front and rear struts is equal to

-P

_{+ kRRCR}

which results in a roll moment equal to

P +

RRRaR]3

(152)

-

- YRR2 -

CF)] . . (153) Therefore,

r2

[FFCF+ RRCR]

(1SLL) = .4. and

_RRCR(/

J (156)

A sideslip acceleration may be resolved into two componente, as shown in Sketch 6. The component normal to the panel, ' sinI , pro-duces a normal force on each panel due to the virtual mass effect. The virtual mass of each half panel is