Longitudinal stability of surface piercing hydrofoil systems for water based aircraft

LONGITUDINAL ST.ABILTTY

OF SURFACE-P]RCING HYDROFOIL SYSIE

FOR WA1ER-BASED AIRCRAFT

by

Gerard Fridsma

Prepered Under

Office of Naval Research

Contract No. Nonr 263-20

Task No. NR-062-012

(DL PROJT NO. KH-2O].)

Report No. 732

Approved by

October 199

Daniel Savitslcy

'S U M MA BY

A theoretical investigation was conducted on tandem surface-piercing hydrofoil configurations to define quantitatively the importance of geomet-nc and hydrodynamic parameters which influegeomet-nce a system's longitudinal sta-bility, in smooth water. The hydrofoil system was analyzed on the basis of

the linear theory of small disturbances without the presence of controls. A wide range of parameters including lift curve slope, dihedral, foil spacing,

load, speed, equilibrium lift coefficient, and chord and area distributions was investigated to determine their effecton the regions of stability. The results of this analysis are put in the form of design charts which illus-trate the regions of stability and instability. These regions are bounded by an upper limit curve which is indicative of oscillatory divergent motions and a lower limit curve indicative of a continuously divergent motion. The results presented herein are applicable to hydrofoil equipped aircraft aswel.l as hy drofoil equipped seacraft.

Within the scope of this report the following conclusions are reached:

Variations in the vertical center of gravity position or introduction of aerodynamic quantities do not make an appreciable change in the size of the stability regions.

At a given foil spacing and front foil chord, increasing the speed _or decreasing the total foil load extends the stable operating. region.

At a given total foil load and speed, the stable operating range is increased as the foil spacing or front foil chord is increased.

Increasing the lift curve slope, dihedral, and dimensionless mass and momentof inertia all tend to lessen the available stable operating region.

Page

Introduction 1

Nomenclature

General Method of Analysis Assumptions.

TABLE OF CONTENTS

6 7 Equilibrium Conditions 7 Equations of Motion 9

Solution of Stability Equations 10

Parametric Variations 12

Results and Discussion 14

Stability Boundaries 14

Effect of Foil Coefficient, C 17

Effect of Center of Gravity Position,

_xe/a

. 17

Effect of chord Ratio, _CF/CR 17

Effect of Lift Curve Slope, dç/da 18

Effect of Dihedral Angle, F. 18

Effect of.Mass and Pitch Moment of Inertia _...18

Concluding Remarks 19

References 20

Appendix I (General Form of:the Dimensionless Stability Derivatives) 21

Appendix II (Simplified Hydrodynamic Stability Derivatives) 23

Appendix III (Coefficients of the Characteristic Equation) 24

Appendix IV (Dimensionless Aerodynamic Stability Derivatives) 25

Appendix V (Divergent Boundary) 26

INTRODUCTION

The improvement of the rough water operational capabilities of water-based aircraft has been a constant challenge to the designer of seaplanes. Although modifications to hull forms have resulted in significant. improve-ments in rough water performance, the possibility of further improveimprove-ments by the use of auxiliary alighting gear such as hydrofoils or hydroskis is well recognized. Considerable research efforts have been directed to a study of hydroski elemeits and, as a result, the hydroski has been successfully ap-plied as landing gear for seaplanes. The application of hydrofoils as alight-ing gear offers further promise for improved performance in that

the lift-drag ratios of hydrofoils are superior to those of either the hull or hydroski and

the surface-piercing hydrofoil has the design potential of achieving reduced landing impact loads and reduced pitch-up and rebound motions. (See Brown')

Hydrofoil alighting gear, however, present hydrodynamic problems which are not normally encountered by the hull or hydroski. Among these are. the effects of the free water surface, ventilation of the lifting surface, and cavitation. Many of these problems have been and continue to be the subject of extensive research analysis. The longitudinal dynamic stability of hydro-foil systems, particularly as applied to water-based aircraft, has not, how-ever., been studied in great detail. Imlay2 investigated the dynamic stabili-ty of only a limited number of hydrofoil configurations but did not define all equilibrium cohditions necessary for steady-state operation. Thus, a com-plete picture has not yet been made of the effects on the longitudinal sta-bility of various independent geometric and hydrodynamic parameters..

The purpose of this paper is to present the results of an analytical parametric study of the variables which affect the smooth water longitudinal dynamic stability of tandem hydrofoil systems. The analysis is for the sur-face-piercing hydrofoil system since this arrangement is most suitable for application to water-based aircraft. A wide range of longitudinal center of gravity positions is investigated to be representative of conventional, Ca-nard and intermediate (tandem) hydrofoil systems.. For a number of these ar-rangements, systematic variations are made in the hydrofoil lift curve slope, equilibrium lift coefficient, dihedral angle, speed, load, chord, and foi.l area distribution in order to evaluate their effect on the longitudinal dy-namic stability. A stability evaluation is also made of a foil system having

aerodynamic derivatives included. The results of solutions obtained with an analog computer and by analytic techniques are presented in simplified

sum-iiary plots which permit a rapid evaluation of the longitudinal dynamic sta-ility of a given hydrofoil system.

The equations of motion presented herein _{are applicable to both fully} vetted and fully ventilated type flows, _{so long as operation is entirely} iithin either of these flow regimes. Due to super-cavitated oii sections, the latter situation is of particular interest because separated flow of the

surface is induced at much lower speeds and angles of attack. Utiliz-ing:this _{type of foil on high-speed seaplanes minimizes the problems}

of the udden loss of lift and flow instability _{associated with cavitation of} con-ventional foil sections (see Benson and King3), _{thus allowing'stable}

opera-ion over a greater speed _{range. Although the emphasis of the study reported} terein is towards water-based aircraft, the summary plots presented are also applicable to a study of the stability of surface-piercing hydrofoil sea-craft.

This study was conducted at the Davidson Laboratory, Stevens Institute f Technology, under Office of Naval Research Contract Nonr 263(20),.DL Pro-ject No. KH-2081.

NOMENC

TUBE

The nomenclatureused in this report, with few exceptions, is the stan-dard form used by the Society of Naval Architects and Marine Engineers in treating the motion of a submerged body through a fluid. In addition, certain quantities which are characteristic when dealing with hydrofoil systems are defined below. Should any conflict arise as to the quantitybeingdefined in two ways the listing found herein is the proper definition.

SYMBOLS

A A = 2hc cot F , horizontal projection of submerged area of nydrofoil

a,b coefficients

c mean hydrofoil chord

D C0 C = , drag coefficient %PAV2 CL CL L , lift coefficient Y2pAV2

d vertical distance from x-axis to center of pressure of foil D drag force

h _{submergence of foil below smooth water surface; also used as vertical} displacement of foil from equilibrium

h h' dimensionless vertical displacement

I, _{hydrofoil craft pitch moment of inertia}

k virtual mass coefficient

distance between forward and rear foils

L lift force

m hydrofoil craft mass

m m , dimensionless mass

pA

M M , dimensionless pitching moment derivative with respect to

%pAV2 depth 21

n' n' , dimension]ess moment of inertia

pA.3

horizontal perturbation velocity

dimensionless horizontal perturbation velocity

V equilibrium velocity of center of gravity, CG

w vertical perturbation velocity

w' _{dimensionless vertical perturbation velocity}

V.

static gross weight of hydrofoil craft

X = , dimensionless longitudinal force derivative with re-zpAV2 spect to depth

XF distance from G to fOrward foil

Z , dimensionless vertical force derivative with respect to

YzoAV2 depth

GREEK LETTERS

x. hydrofoil incidence measured from zero lift

rdihedral angle

load supported by hydrofoils pitch angle

water density

IP air density

roots of the characteristic equation, i

= 1.2,...

PARAMETERS

AR

foil area ratio

t time

foil chord ratio

CR

CL

]ift curve slope

--A

C, C, = , non-dimensional foil coefficient useful in describing

PC F2 V2 the hydrofoil system's dynamic stability

XF

longitudinal center of gravity position, CG

SUBSCRIPTS AND SUPERSCRIPTS

F front foil

R rear foil

GENERAL METHOD OF ANALYSIS

The general method of analysis consists of making certain assumptions about the hydrofoil system which is then assigned a set of differential equa-tions that will adequately describe the moequa-tions of the uncontrolled configur-aion when it is disturbed from its'equilibrium position. Within the assump-tions of this report, a set 'of linear coupled differential equaassump-tions is pre-sented having constant coefficients. Two conditions then should be incorporat!

eJi in the design of the hydrofoil configuration in order to assure acceptable h,rdrodynamic operation; namely satisfactory static balance and dynamic sta-bility. A satisfactory static balance or steady-state equilibrium position is obtained when the unbalanced weight, L , (gross weight less aerodynamic lift)

distributed entirely between the front and rear foils such that 0 _q w 0 a1nd there are zero net forces and moments about the center of gravity. A hy-drofoil system having static balance does not necessarily insure dynamic sta-bility. A dynamically stable system is defined to be one whose motions return to its operating steady-state condition when once disturbed from equilibrium. F'rom the linear theory used to derive the equations of motion, the motions and hence dynamic stability are found to be dependent upon the basic

hydro-dynamic force and moment derivatives which define the coefficients of the

differential equations.

1 Although there have been'other contributions made todescribe the

sta-Iility of tandem hydrofoil systems in terms of the basic hydrodynamic deriva-tives (see Ref. 4 to 8), for reasons oE simplicity the equations of motion and

?nytica stability derivatives used in this report are those developed by

Kaplan, et.al.7. The derivation of these equations is based on the application the linear theory of small disturbances and result in a final set of cou-pled linear differential equations of motions. After simplifying these equa-ions and stability derivatives, and applying them to specific hydrofoil

con-figurations which have freedom of motion in the longitudinal vertical plane, parameters such as dCL/da , dihedral angle, foil spacing, speed, chord, area

iistribution, and lift coefficient were varied and their effect on the regions of longitudinal dynamic stability was noted. Aerodynamic derivatives were also inclu,Ied as a possible additional parameter. In all cases, the equilibrium "position or static balance was defined, Twomethods of determining the dynamic

stability were employed:

1. Hurwitz c.riteria or Routh's discriminant, and

if 2. an analog computer which presented agraphical solution to

the linear differential equations. This latter method was most effective in defining the degree of damping and type of stability present in a given hydrofoil system.

The. basic assumptions and considerations inherent in the development of the stability equations in Ref. 7 are summarized below for the sake of com-pleteness of presentation.

ASSUMPTIONS

The motions of the disturbed hydrofoil system are small so that the products of small quantities are neglected when they appear in the equations of motion.

The, free water surface is smooth, i.e., the hydrodynamic effects of waves and perturbations due to the motion of the hydrofoils are neglected.

The hydrodynamic lift and drag coefficients together with

their derivatives, i.e.,

CD/a,

CD/h,

are evaluated at the equilibrium position, according to the requirements of the linear theory.

The power plant thrust is constant and acts through the. center of gravity making an angle., E' , with the x-axis.

Support strut effects, buoyant forces and the hydrodyna-mic moments of the foils about their center of pressure are smal] compared to the hydrodynamic effects and hence are neglected.

The equilibrium rear foil lift coefficient is defined as that developed by this foil when operating in the downwash of the front foil.

The rate of change of downwash with respect to angle of attack of the front foil is taken to be zero.

EQUILIBRIUM CONDITIONS

When a hydrofoil seaplane is running in its equilibrium position it moves at constant velocity along the ac-axis at zero trim. (See Sketch 1) If the Static hydrodynamic moments generated by the hydrofoil drag forces are assumed to be balanced by the aerodynamic pitching moment provided by the horizontal stabilizer of the seaplane, and the thrust Jine makes a small an-gle with the x-axis, then since the sum of the forces and moments about the OG are zero, the fo]lowing must be true:

DF + DR = T

LF + L A

and

LF XF -

LR(J-

xF) = 0

AERO MOMENT HYDROFOIL

CRAFTI_.

REFERENCE AXIS _CG a

J

I,

L SKETCH I

HYDROFOIL CRAFT IN EQUILIBRIUM POSITION

FREE WATER SURFACE HYDROFOIL CRAFT REFERENCE AXIS Da+ AERO MOMENT z SKETCH 2

HYDROFOIL CRAFT IN DISTURBED CONDITION

X + HORIZONTAL FREE WATER SURFACE CD AF + CD Aft

I

(4)

Y2pV2 C

A +C

A -. LF P B A

(5)

4oV2 and CL _AF

₂

XF

(6)

CL Aft X_F

.ornbining Eq. 5 and 6 gives

A

2

x 1

(7)

CF1/V2

£

and A XF ₁

C:

_LB

__ -

(8)

pV2

.L A8

EQUATIONS OF MOTION

The force and moment equations of the tandem hydrofoil system made di-mension]ess by '/oAV2 and '/2pAV2 , respectively, are given by'Kap].an, et.al.7 as

1'

VS

+ Xw' + X6 + X,q' + Xh'

(9)

mu

I I

m'(i'

-

q')

Zu' + Zw' + Z8 + Zq' + Zh' + Zi'

Z'

and (10)

n' = Mw' + M9 + Mq' + Mh' +

+

and the kinematic equations follow as

and

These five linear difeferential equations

(9

through 13) with constant coefficients are, on the basis of the previous assumptions, the equations of motion of the hydrofoil system in the disturbed condition. (See Sketch 2) The dynamic stability is defined by the solutions to the above equations and

is directly related. to the values of the constant coefficients or dimension-less stability derivatives. Since the motions are perturbations from the equilibrium position, the hydrodynamic stability derivatives are evaluated at this position and are functions of the 'geometry (foil spacing, chord, dihe-dral, etc.), CD, CL, and their derivatives with respect to angle of attack and depth of submergence. The stability derivatives for the surface-pierc-ing tandem hydrofoil configuration, asgivenby Kaplan, et..a,lT, are presented in Appendix I.

In order to convert the equations and stability derivatives to a more wieldy form, several additional conditions will now be applied.

The. influence of surge will be neglected. Based on exper-ience with aircraft motion and existing specific solutions of hydrofoil equations, the effect of surge does, not ap-preciably change the longitudinal stability.

The virtual mass terms and dragcoefficientsare neglected in the expressions for the stability derivatives (given by Kaplan, et.a.11), since these terms are. small compared with other quantities such as the lift coefficients

ap-pearing in these equations. .

Similarly the change in lift and drag coefficient.with depth of submergence is small for depths greater than one

A = AF

+ AR which ja 'the total projected submerged area of both foils.

q' (12)

chord and therefore will be neglected where they appear in the expressions for the hydrodynamic stability deriv-atives. Experimental resu]ts obtained on surface-piercing

dihedral foils have demonstrated the applicability of

this assumption. (See Brown9, and Benson and Land'°)

In addition, the hydrofoil system will be taken, for simplicity, to be such that the front and rear foils have the same dihedral and similar lift

urve slopes.

m'i' = Z'w' + Zh' + (m' +

+ Z9

(14)

and

n' =

+ Mh' + Mq' + M&

, (15)

wltile the kinematic equations are given by Eq. 12 and 13. The previous as-sImptions together with the static balance conditions have simplified the siabiIity derivatives to an extent where only the most significant terms are pesent. Thus only eight stability derivatives are required for a solution of the above equations, and they are presented in Appendix II.

SOLUTION OF STABILITY EQUATIONS

For a satisfactory hydrofoil system the static balance requirements of Eq'L 5 and 6 must be satisfied along with the dynamic stability which is pre-ditable from Eq. 12 through 15. Equations 5 and 6 can be solved algebra-ically. The results for Eq. 6 are presented graphically in Fig. 1. The sou-tins of Eq. 12 through 15, although not complicated, require special tech-nicues. It was found during the course of this study that two teèhniques haj special advantages.

The first technique was to obtain solutions with an analog computer whifh yielded graphical plots of the actual motions under specified initial con1itions; A schematic diagram is presented in Fig. 2 indicating how the differential equations of motion were inserted into the analog computet. Typical time histories of the heave motion obtained with the analog computer are\Presentel in Fig. 3. It is seen that a range of solutions were obtained ranging from highly divergent and unstable oscillations to stable and highly

damped oscillations. Fora given set of initial conditions and hydrofoil

georietry, the resultant disturbed motion of the system could be characterized by ne of the eight typical motions shown in Fig. 3. If the motions were of the \Types 1,. 7, or 8, they are unstable; if of Type 2, the motions are

neu-tra1\ly stable; and if Types 3, 4, 5, and 6, they are stable.

The second technique for the solution of the dynamic stabi]ity equa-Under these simplifying assumptions, the equations of motion become

tions is the "classical" treatment of solving linear differential equations; namely by expressing the unknowns in exponential form and reducing the ori-ginal equations to a system of algebraic ones. This procedure is reviewed by Kaplan, et.al.7, and can be found in most text books on differential equa-tions. (See Uspensky") The characteristic equation resulting from this treat-ment has' the form:

a4cr4 + a3cr3 + a2o-2 + a.10-. + ₀ (16)

The criteria for the motion to be stable requires that all the coef-ficients, a4 to a0 , and certain Hurwitz determinants be positive. This

analytical technique defines two boundaries, between stable and unstable mo-tion. (See Sponder'2) One is a boundary defining the limit of unstable os-cil.lations and is given by

a1(a2a3 - a1a4) - 80832 0 (17)

The other boundary linedefines the limit of divergent instability and is

given' by

(18)

The, values of a are functions of the stability derivatives, their express-ions being presented in Appendixill. In the present study the analytical

ex-pression represented by Eq. 18 is used to define the divergent boundary

(lower limit line). Because of the' complexity of the formulation defined by Eq.; 17, the analytical definition of the unstable oscillatory 'boundary (Upper

limit line) was only carried out for a small, number of' hydrofoil systems. For the most part, the upper limit line was defined from the results of the

ana-log computer.

Figure 4 presents a typical summary of the stability boundaries for a specified hydrofoil, system as the area ratios and lift coefficients of the front and rear foils are varied while' maintaining static balance. The ordi-nate and abscissa of this plot represent the front and rear foil equilibrium

lift coefficients, while the parametric lines represent the corresponding area ratios of the front to rear foils. Spotted on this plot are the eight typical characteristic motions' as illustrated in Fig. 3. It is noted that 'points to the' left of the upper limit line in Fig. 4 represent highly

un-damped oscillatory motions; points' to the lower right of and below the lower limit, line represent highly divergent'motions; while intermediate points represent damped stable motions. Charts similar to that o'f Fig. 4 were pre-pared for many hydrofoil systems based on the results of the analog computer and Eq. 17 and 18. These. will be discussed in detail in' subsequent sections of this report. Figures 3 and 4 have been discussed hereto illustr.ate'the form of the results obtained with the analog cqmputer.

PARAMETRIC VARIATIONS

Analytical studies were made on the regions of dynamic stability to

1etermine

the effect of the basic geometric and hydrodynamic parameters of hydrofoil systems. The parameters investigated and the extent of their

van-tion are summarized in Table I.

TABLE I

It will be noted that the range of dCL/da extends from that of the fully attached hydrofoil flow, 6.0, to the case of supercavitated flow, 1.5.

T)e

center of gravity positions are such as toincludethe conventional, tan-dm and canard systems. A foil coefficient, C , is introduced herein as a

function of the total foil load, spacing between foils, chord of the front foil, and speed. This coefficient appears insixof the eight basic stability derivatives (see Appendix II) and, as will be subsequently disclosed, was fund to be an excellent collapsing factor for the computed stability re-suits.

Since the longitudinal stability analysis is directed toward a hydro-foil equipped seaplane, certain aerodynamic derivatives must be included to

mhe

the evaluation complete. Surge being neglected, these aerodynamic de-rivatives appear as changes in the vertical force and moment due to

pertur-bkions

in the angle of attack and pitch velocity. Since the notation in the ecuations of motion has been defined relative to or at the

Q,

the aerodyna-micand hydrodynamic effects can be combined to give a general picture of tIiose quantities affecting the stability of the hdrofoil configuration. The

PARAMETER SYMBOL RANGE OF VARIATION

dC dC

Hydrofoil Lift Slope , per radian 1.5

4-da

6.0 da

Hydrofoil Dihedral Angle

r ,

deg. 15 .

r

450

Hydrofoil Lift Coefficient _CL .05

CL .30

Center of Gravity Position .09 .75

Hydrofoil Area Ratio

Be-tween Front and Rear Foil AR

.1 < <

-

AR -15 A Foil Coefficient C .018 5 C 5 .150 2

PcF2V

I

Hydrofoil Chord Ratio

.L

I

.L

3.

dimensionless aerodynamic stability derivatives are presented inAppendix IV.

Various combinations of the above parameters were used to represent various physical hydrofoil systems. For each system, the stability deriva-tives -- as given in Appendix II -- were calculated, and from these the sta-bilitycharacteristics were established using the analog computer or analy-tical techniques. Hence, the final stability characteristics are established in terms of basic physical quantities whose' values for _{any hydrofoil system} are known to the designer early in the progress of a design.

HESULTS AND DISCUSSION

The results of the stability computations performed on the analog com-puter and by the analytical techniques previously discussed are plotted in Fig. 4 to 13. These plots define the stability boundaries for specified hy-drofoil systems in terms of the area ratios and operating lift coefficients of the front and rear hydrofoils. Figure4 presents the stability limits for a conventionalsurface-piercing hydrofoil configuration. Figure 5 is for the case when the center ofgravityof. the hydrofoil system is moved progressively. aft so that the hydrofoils form a canard system and includes variation in C8

from .0178 to .15. The stability boundaries plotted in Fig. 6 are the re-suits for the hydrofoil system when the chord ratio _cF/cR is varied from 1 to 3 and C is varied from .0178 to .10. Figures 7, 8 and 9 are essentially similar to Fig. 6 except for moving the G to positions of .25, .50, and .75, respectively. The effect on the stability limits resulting from variation in dCt/da is presented in Fig. 10, while the effect qf variation in dihedral

ngle is shown in Fig. ii. The dimensionless mass and moment of inertia ef-fects are plotted in Fig. 12 and 13, respectively. An important factor to keep in mind as this discussion proceeds is the distinction between the ac-tual stability at a point and the regiorrs of stability. In the following sections the discussion

will

be concerned primarily with the parametric ef-flect. on the regions

of stability.

_{As an orientation to interpreting the} plotted results, the stability boundaries will first be discussed.

STAB IL I TV BOUNDAR I ES

The stability boundaries for the conventional system _{(as defined on} pge 11) are plotted in Fig. 4 as a function of _CLF and

CL.

The straight lines radiating from the origin of the plot indicate the area ratio _AF/AR required to attain static balance for arbitrary combinations pf _CLF and Superimposed on this plot are the two boundaries which delineate the various stability areas. The dynamically unstable areas are characterized by either oscillatory divergent motions or continuously divergent motions. Areas between the limit lines are representative of dynamically stable operating cbnditions with the lightest damped hydrofoil system appearing _{to the left of} the area and progressively higher damped _{systems appearing toward the right} just above the lower limit line. The actual motions corresponding to points

O

through ® spotted on Fig. 4 are illustrated by the time history plots (1obtained with the analog computer) given in Fig 3.

Analytical expressions for the upper and lower limit stability bound-aiieswere given in Eq. 17 and 18, respectively. Considering the mathemati-cl condition for the lower limit (a0 = 0) , and upon substitution of the

stability derivatives given in Appendix 11 and the relations given in Eq. 5 and 6, the following simple formulation for the lower limit line results in

aC

_+bCL

+°

F F Solving for CL F C CL F = 2 - a

where the positive sign of the radical is used to insure that the lift coef-ficient always will be positive. The quantities b/2a and c/a are defined in Appendix V and are functions of the various geometric and hydrodynamic quantities. An examinatio,n of the ratios b/2a and c/a indicates that the c/a term predominates. The ratio b/2a , although small, cannot be

neglect-ed. However, as a rough first approximation, Eq. 20 can be written:

CL

Substituting the values for c/a given in Appendix V leads to

(AF\2

CF ,t

-C. _X

(AF\2

_£

:c) 2- XF

Applying the condition for static balance, _CLR can value of _CIF by Eq. 6. Hence, for a given hydrofoil line of stability is given by a combination of Eq. 6

(21)

(22)

be re]ated to the above system, the lower limit and 20. This lower limit line was calculated from Eq. 20 for the various hydrofoil systems described in Fig. 4 through 13 and is shown plotted on these figures. The analog com-puter was also used to establish this lower limit line, and it was found to be in excellent agreement with that developed analytically.

Besides proving the limits, the time histories obtained from the ana-log computer were useful in showing the degree of damping present in a hy-drofoil system. The analytical technique as developed so far wiLl only

indi-cate whether a system is stable or not. With further analysis it is possible to obtain the roots of Eq. 16 and derive the expressions for the actual mo-tions.

A further examination of the lower limit equation reveals that this stability boundary is independent of the mass and moment of inertia of the hydrofoil system. Furthermore the limit does not depend upon the q

deriva-CL

tives. This result suggests that the lower limit is independent of the rotary derivatives or pitch damping terms. Of the two ratios which determine'Eq. 20 for the divergent boundary, b/2a is the only quantity which is proportional to the vertical center of gravity position and which changes when the aero-dynamic terms are introduced. Since b/2a is small but not neg]igible com-pared to c/a , it can be concluded that the vertical center of gravity

po-ition and the aerodynamic quantities have only a small effect on the regions of stability. That the vertical center of gravity effect is small has been verified by Imlay2. Because of this fact, the vertical 1X position was fixed at a value of d/Z = 1/3 . The aerodynamic effects are small for a number of

reasons, viz.,

The ratio of densities between water and air is approxi-mately 800:1 which more than compensates for the

differ-ence in size or lift coefficient between the airfoil and hydrofoil operating at pre-take-off speeds.

Since the lower divergent limit is independent of the

pitch damping terms, only the aerodynamic quantities M and Z can have an effect.

The stability equations for aircraft.motions would be

quite similar to those for a hydrofoil system i'-fthe heave dependent terms, which are characteristic for a body

oper-ating near a free surface, are neglected. It is these

surface-piercing effects which are so influential in de-termining the regions of stability. Were it not for the fact that the h and 0 derivatives are non-zero, the upper oscillatory boundary would disappear.

The upper limit stability line appearing in Fig. 4 is analytically

defined by Eq. 17. It is an oscillatory boundary which is a function of all the stability derivatives as well as the system's dimensionless mass and

mo-ment of inerita. Equation 17 is a most unwieldy equation which cannot be

simplified into a tractable form such as was done for the lower limit equa-tion. Consequently, the upper limit line was established using the analog cmputer. This upper limit line was also analytically evaluated for seleêted

hydrofoil systems and was found to be in good agreement with that obtained through the use of the analog computer.

It, should be noted from Fig. 4 that there are wide areas of dynamic stability available to the designer. For this conventional hydrofoil system, operating both foils at lift coefficients greater than .10 should insure dy-namic stability if the system is designed to be statically balanced. Figure 4 is also useful in indicating the degree of stability inherent within a hy-drofoil system. It may not always be desirable to design a system which is dynamically "stiff" since this condition may penalize the maneuverability of

the craft. Hence by properly selecting combinations of CLF and CLa given in Fig. 4, various degrees of dynamic stability can be designed into the hy-drofoi) system.

EFFECT OF FOIL COEFFICIENT, C8

The effect of increasing the foil coefficient from a value of C8 .0178 is illustrated in Fig. 5 for various locations of the G

(xF/)

, and in Fig. 6, 7, 8 and 9 for various chord ratios

_(CF/CR)

at xf/ values of .0909, 0.25, 0.50 and 0.75, respectively. In all cases it is evident that an inérease in C8 reduces the stable operating region. For small values of the lower limit line is most effected by changes in C, , while at

large valuesof xF/ the upper limit line is predominately effected by C8 changes. From the definition of C8

= A/pcV2

, the following conclusions

are reached:

At a given foil spacing and front foil chord, the stable operating range is increased as the speed is increased,, or the load decreased. At a given load and speed, the stable operating

range

is

increased as the spacing between foils is increased or as the front foil chord is

in-creased. .

EFFECT OF CENTER OF GRAVITY POSITION,

xF/i

The effect of the center of gravity position, XF/2. , on the regions of dynamic stability is demonstrated in Fig. 5 where stability limits are plotted for four center of gravity positions. It is seen that, if _CLF. and CL are left constant as the center of gravity is moved aft, there is a de-crease in .damping causing the hydrofoil system to develop larger oscillatory motions as the G moves aft. It was noted prom Fig. 5 that, for the conven-tional system (XF/_

.09)

, wide areas of

highly damped

stable operation

are available, whereas for the canard system (xF/J = .75) the reverse is true, i.e., only small areas of

highly damped

stable operation are available.

EFFECT OF CHORD RATIO, _CF/CR

Figures 6, 7, 8 and 9 show the effect of chord ratio, cF/cR , pn the stability regions of hydrofoil systems varying from the conventional to the canard type. It is seen that increasing the chord ratio has an appreciable effect on the upper limit by shifting the boundary to the right. This shift

is more pronounced on the canard configuration than on the conventional one. The lower divergent limit is affected by the chord ratio only at the lower values of

_"Ar/AR

. At increasing area ratios the limit approaches an

dCL/da is the same for both the front and rear foils. The effect of lift curve slope on the regions of stability is shown in Fig. _{[0 where, for} various chord ratios, center of gravity positions and foil coefficients, the areas of dynamic stability are increased with decreasing _{dCL/da . For a} supercavitating foil system having a lift-curve slope almost onefourth that of a fully wetted foil, the area of stable operation is wider than that of a non-cavitated foil system.

EFFECT OF DIHEDRAL ANGLE, F

The effect of foil dihedral on the regions of dynamic stability is ii-ltstrated in Fig. 11 for several chord ratios, center of gravity positions ard foil coefficients. It is evident that there is a pronounced dihedral effect on the areas of stable operation. Reducing the dihedral ang]e from 45 _{to 15° raises the upper limit and reduces the lower limit lines such that}

th stable operating region increases by significant proportions. The

extra-polation to zero dihedral cannot be made since, in the derivation of the

stability derivatives, a finite dihedral angle was assumed.

EFFECT OF MASS AND PITCH MOMENT OF INERTIA

In Fig. 12 and 13 can be found the effect of the dimensionless mass and moment of inertia on. the stability regions for various center of gravity positions. As mentioned previously, he lower limit is independent of the mass and moment of inertia of the system and therefore remains fixed. The upper oscillatory limit moves to the right with increasingmass and moment of inertia, thus decreasing the available stable operating region. The _amount of hift is, however, dependent on the center of gravity position. It is seen that a varIation in the mass causes an appreciable change in the _{upper limit}

only for X positions greater than _XF/Z .50 . The moment of inertia has

CONCLUDING REMARKS

Based upon the, assumptions, analysis, and results of this report, the following, concluding remarks are made:

As a result of the analytical technique presented herein, two bound-aries were obtained which delineated the regions of stable motion. The di-vergent boundary or lower limit has been collapsed into the useful mathe-matical expression

where the ratios b/2a and c/a are functions of the known geometric quan-tities inherent in the hydrofoil configuration.

The design charts described herein were obtained from a systematic variation of various geometric and hydrodynamic parameters after utilizing both the analytical technique and analog computer to determine the dynamic

stability. The general conclusions found are summarized below.

Variations in the vertical center of gravity position or introduction of the aerodynamic quantities do not make an appreciable change in the size of the stability regions.

At a given foil spacing and front foil chord, increasing the speed or decreasing the load extends the stable operating region.

At a given foil load and speed, the stable operating region is

in-icreased as the foil spacing or front foil chord is increased.

Increasing the lift-curve slope, dihedral and dimensionless mass, and moment of inertia tends to narrow down the available stable operating region.

Ventilated, Dihedral Hydrofoils", DL Report 731, July 1959.

Imlay, F.H.: "Theoretical Motions of Hydrofoil Systems", NACA Report 918, 1948.

Benson, J.M., and King, D.A.: "Preliminary Tests to Determine the Dy-namic Stability Characteristics of Various Hydrofoil Systems for Sea-.planes and Surface Boats", NACA RBNo. 3KOE, November 1943.

Hugh, W.C., Jr., and Kaplan, P.: "Theoretical Analysis of the Longi-tudinal Stability of a Tandem Hydrofoil System in Smooth Water", ETF Report 479, July 1953.

Amster, W.H.: "An Investigation of the Stability of Hydrofoil Craft", Joshua Hendy Corporation Report, Hydrofoil Studies, June 1950.

6L Cannon, B.H., Jr.: "Performance of Hydrofoil Systems", Thesis for Doc-tor of Science Degree, Massachusetts Institute of Technology, 1950.

7. Kaplan, P., Hu, P.N.,and Tsakonas, S.: "Methods for EstimatngtheLon-gitudinal and Lateral Dynamic Stability of Hydrofoil Craft",ETf Re-port 691, May 1958..

8 Arnold, 1., and Slutsky, S.: "Application of Supercavitating Hydro-foils to High Performance Seaplanes", Part V, Gruen Applied Science Laboratories, Inc., Technical Report 67, July 1958.

Brown, P.W.: "The Force Characteristics of Surface-Piercing

Fully

Ven-tilated Dihedral. Hydrofbils", ETF Report 698, October 1958.

Benson, J.M., and Land, N.S.: "An Investigation of Hydrofoils in the NACA Tank; I - Effect of Dihedral and Depth of Submersion", NACA,Ad-vance Confidential Report, September 1942.

Uspensky, J.: _{"Theory o.f Equations", McGraw-Hill Book Co., Inc., New} York, Toronto, London, 1948.

Sponder, E.: "On the Representation of the Stability Region in Oscil-lation Problems With the Aid of the Hurwitz Determinants", _NACA Tech-nical Memorandum 1348, August 1952.

x

x,

XI =

-

±.E

/C0\

Aft

fc(\

h A 3h') F - -2 _{CD cot 1F} 2 C0 cot XI = -2(.1 +

±

) XF

(CD\

!I (1 XF) ic0\ 1 q

\

_A A' DF A / D L A

7

F A

-r

R] =

-2(C

\A LF _A

Lft/

rAF

fC1\

AF

XF fCL\

(1 Xf /CL\ -2

Cft(1

_x)

CL

cot

A F

fCL\

Aft

(cL\

- 2

h')F

TiT)R

A 7,

'h

q

z

z:

_q + A 0F AF /C0\ X A

/dC0

(1

-FJR

+2

GENERAL FORM OF THE DIMENSIONLESS

STABILITY

DERIVATIVES'

-2 d_F

c

+

!

c

+

AT LF

AR

LR)

LA

= 2 CFXF A C F cot F

/

Ac

_{F F} F

_{A 2}

rk AF CFXF

2[F.A

/2 Aft A C

)

rAF

fC0\

2

A2

Also presented by Kep1sn

_et.sl.7

c(R

!

c(/

- xF)

APPENDIX I

xF) A C0

cot

w AV2 AR (1 XF)

PCL\1

A

7 \a)Rj

Aft (C1\ 1 + 2

L!

A CLF

cot

1'F CL F

cot

2 C

cot

-A Lft V

\A

LF _{A H]}

(CD

A

M9 = M M q

=±!i IC

A A'

[

D. -+ 77 +

(L'Y'

_-

_{(1 -}

_E) [CD

a/Fj

A

(1+±[

(R'

A 2

L LF

_\3a./FJ

A A7 _L LR

\a/a

+2

M =EAF XF

_fCD\

A /

h1

F A CDF Fj

J

(1 ...i)

(i"

+ 2

cR(J-x_F) C

cotr

\h'IR

A DR

LA

1f(J\

_2-C

cotr1

[ A /

F A LF Fj

FAR (1

XF) (CL\

+ 2

c(2

- xF)

cot

_rRl

(1

T

A 1

XF)l

AR dR

\alIl

A' J A A' d

C\

7

(-

u

2

j

A

(1

211k

-i

k

''

CR(

- Xf)

]

2[

F A 2 R

AF CF4

_{+ kR__}

cR(2

-/3

xF)1

+

\

a / B XI. A' 2 _CDF

'cot

rjT

ldF

d_R 2 _CD

cot

A' C_F' 2 _CL Cot

r1i

_FJ A' 2 C

cot

(1

A LR

('

1 AF dF

+

[2c

dF

(CL'

xF1 AF

XI.

A 7

LF

₂

\'ôaIF

P J A 2

+r±

(L'

+

L A

\h'/F

IAR (CL

L A

\oh'/a

I M q

.,

cotF

F C,

2-cot

r

a

-APPENDIX II

SIMPLIFIED HYDRODYNAMIC STABILITY DERIVATIVES

1 cft AF CF Aft

C2

L XF XF -

1.07'

--XF _XF £ 2- XF 1 (XF CL d / Aft

\1 -)'

xF L[. L1 (1 F

C2

LF A ft'\ A F) Aft _{/- XF} X. )2]

It .g aumed t.t

C0/a

1.07 CL

CL/a

C2

AR /- XF1 AF

XF j

X _CL

f

z

cot F

1 +

1+

Cfi 4F_{CF Aft}

-C.

e..

_XF Aft

1+-i

F CLF d

/

XFCL

q - _X. ,' a (1 _A F) Cft

Af

A-1 + CFAR XF

a2

APPENDIX III

COEFFICIENTS OF THE CHARACTERISTIC EQUATION

a4

mn

a3 m

-+ M'Z'

qw

-

+ Z) - Zn

-q

where

DIMENSIONLESS AERODYNAMIC STABILITY DERIVATIVES

-

[('

::! + (!:.\

A1

L

)

A a )

7]

-A p

M

'J \ A

P

7)

-stabilizer area wing area

A

tail length (distance from C to center of pressure of stabilizer)

s,w subscripts which refer to the stabilizer and wing respectively

APPENDIX V

DIVERGENT BOUNDABY

The divergent boundary or lower limit is expressed in mathematical form as where CF

2

XF C8

_'\¼;)

_- CR XF cot r

-

)

i-Xe

c _{AF XF} b

/b2

c

+ I

--2a\J(2a)

a b r

L

= - 1/211 _-

1.07

-2a _L d cot

1+

CFAR i

XF cR AF XF cF AR

FOR A TANDEM HYDROFOIL CONFIGURATION

(SEE EQUATION 6)

dii'

I.0__

'Ail

,pr

I

A

AUIS

-20 25 9.09

:.

0I

XF

U

- _XF

£

U- III

.0I .02 .03 04 .05 06 01,0609 .2 .3 .4 .5 .6 .1 .6 .9I 0 2 3 4 5 6 7 8 9 CL

FOR SOLVING EQUATIONS

12

THROUGH

15

w

S

I. C.

ANALOG COMPUTER SCHEMATIC

-é I

-w.

fTh,

+ Z z rn

-J

+100 +10 +10 L C. TO RECORDER

-h

_-e

(THESE MOTIONS CORRESPOND TO POINTS ® TO

IN FIG.4)

£

-Anti,

h

-

-T

T'VV

UNSTABLE OSCILLATION ALMOST NEUTRAL STABILITY

A

&

V

STABLE-LiGHTLY DAMPED OSCILLATION STABLE-DAMPED OSCILLATION

L

STABLE-DAMPED OSCILLATION STABLE-HIGHLY DAMPED OSCILLATION

©

UPPER LIMIT UNSTABLE OSCILLATORY REGION STABLE LIGHTLY DAMPED OSCILLATORY REGION STABLE

HIGHLY DAMPED REGION .20 .18 .16

d 14

0

U-

I-z

0

u .12

U-0

I-

_z

.IO U. U. Ui

0

Q I.- U--J IA -J C) .06 .04 .02 00 AF/AR:I ₂

I

/

NOTE: ACTUAL MOTIONS AT NUMBERED

3 POINTS ARE SHOWN IN FIGURE 3

X =3.16 4 XI, UNSTABLE DIVERGENT REGION

- Zr

.02 .04 .06 .08 .10

CLRLIFT COEFFICIENT OF REAR FOIL

-LOWER LIMIT 10

.12 .14 .16

.3 CLç .2 1,5 , E 3O , CF/CR 1 , m' 1 , n = 1 .4 _.4 .2 0 0 .3 0 .2 .3 .3 .2 0 .4 _.4 .3 CLF .2 CLR .2 .3 3.18 = Ics=.I0 I / I ₁₀ I , I

/

I /

/

/.10 I / 1 I 4,1 / .0178

f

1

1/

f_L_ "-.._

,/

/

/

/,Ø5,

.oe

,I8 -IIJ

//

_/

-Ill,,,

'

1!1,''/,

,.

x

111111111£ I 11,11th

1

cs,I.

A /

_-I.73Z

/ AR / 2 .0178 I / I F / _// I / / /

I C.I8

/

_/

3

,'

I4///___

1I#'

'/,

. .017e

,"''

-5 25

cs.,0

.5 - -- .0178 /

I,;

Ar 1.0 /,/ Cs .I5 1.5

-.50

-- .0178 ,I

/

,

-.4 / , -/ / / -/

I

-/

C

-

.10 ... '.5 .2 .3 0 CLR _{.2 .3}

.2 0 1.5 ,

X/

= .0909 , m' 1 , ; 1 .4 .4 0 .3 cso.1 5

/

A

I

/

I'

,

,, -IS

I-.11.1 1.1111 11111 CR 2 I AR B II 44

V

,c.io

,,l0

I III

/.''

Ill

r"Tr

/

,__.-7,

IS

-,

!tt1'I,I-L1,f

Cp

- : 2

CR

II liii LII

2 II

C..I0

A

/

/

I-C51LL.l0 ' 5 -/ I I / I /

,

I / ,

-,

/ 10

-

I V 0I1B IS 20 __f__$j _Jr - --- -

-Ill,

= CR

0

.2 CLR .2 .3 .3 .2 .3 CL .2 0 0 CLR _.2 .3

CL/a

= 1.5 , = .25 , m' = 1 n, 1 .4 .4 .3 CLF .2 0 .4 .3 .2 0 .3 CL .2 0 0 .2 .3 2

/

,C8. Jo -- .0178' / /

/

/

,

,4

- I

1fi

,

cs.Io I, R

,/IIllII

Cs .

,,,;

/

C.IO

/

-CsO

//

,

,

_.--. - A -// / I/It//Ill -I /

--I

/ :, /f /

,_-_

/

-

-/

--

-

--,

--Ill,

II

sill

I I

r

__J.. CR

II

---3

lilt!

9 0 .2 .3 CLR .2 .3

.4

.3

CL .2 = 1.5 f 30° ,

xF/J

.50 , m' = 1 , n, .4 .3 CLF .2 0 CLR .3 .2 0 0 .2 .3

--

I!

Il

II

.3

'I

ii

.5 / / / / A A

-I

'I

II

I

II

I

I

I

/ csc.10 / I / / / / /

/

,

/

,l.o ,I.2

,

-- I I / / / I I I , / / / / / / / /

'

f

_/

,

/

/

/

/

/

,

,,cs.,0

,

,

-/

,

-,

-.0178_

.50 -C$10 .5

/

-.0175

I

I'

\

-/ ,,/

'A

.411 1.5 2.0 .10 / -

/

/

,

1.0 - /

/'

/

/

cs..Io ___

A 2.0

0

.3 .2 CLR CLR _.2

.3

.4 0 = 1.5 = 30° ,

x/

= .75 , m' = 1 , n .4 0 0 .3 .2 0 .2 .3 cs

Io,i

/ .2 / / / / / / 4F. R

r447

f /

/

/

/

/

/, / /

/

//

/

--.Tio

-I / I / .2-I / / -/ I I / S / /

.333

"I//I

::--

_-ri

C5.i0

/

/ .2 / -- .0170 / / / / / / / / / / . / / / / .3

/

/

-\

_\i

/ - /

III

/

/

/

/

/

/ // A R --..7 cs .0175 1.1 .3 CLF .2 .3 .2 .3 0 .2 0 CLR .2 .3

.4

.3

.2

0

.4

.3

CL,, 2

0

0

0 r = 3O , rn' = 1 , = 1 .4 CL8 CL8 .2 .2 3 .3 .10 0178 .3 CLF 0178 .2 0 .4 Cs .IO .3 .2 0 0 0 CL8 CL8 .2 .2 .3 .3 -I / I A

----I.73

/

2.8 a

-.

I I -CS.I0' - / I

II.

₇

/

/ 3

/

/

/

3 p

,

'1.5

I//I'

_///

'

5 : / àCL 6

//

/

'

a 3.l6

//'

C j .

/75

1#j

.io 1.5.'

,:z.

yp-,

_/

/ /

,/

--'1/

'

'I,''

!'

Q9O9,f:I

. .5.' ,cs_.I0

--"r6

/

₄₄

_/'

/Q

3

/7

1:25,

.E:2 A

-i.-

4.47 8 / / / 6 / e àC.L

6

.10 A

/j

-

1.

W / A /

.ÔCL 10

WI, /

_'p

,r

Jill',' ,

11,1.11'

,

- _ - I-F CF

--:.O9O9,:2

CR

C1/a1.5,m' =i,nl

.4 ₂ .4 .4

0

0

.4 .3 CL, .2 0

C;.t0

/

-.3.I6

I\

\

(

r 4,50'

-15° 30°45° / /

/

I / // /10 II

/

:300 /

/

,

C.I0

/

/

/

A"

,/-,/JJI1,II

'501'L

I

III iS° ,300 Ill, /1

/'

r.45

3 4 5, C57.10 4 4R I

\\

\,ii

)

15° -300 . ,/ A / /

,

.Ill

,/

r_ 30°

.'

_c3io

--CF

\

," .2.45 5° I / I 300 45° .

A,

-,45° . / /

/

/

3 .: P_I/i

,1'::::::

I

Ii

i-i

liii

jILl

I LII jul

R

.3

.2 .3 CL, .2 .2 .3 0 .2

.3

CLR 0 .2 .3 .2

.3

.3 CLF .2 0

.4 CL, 0 .4 .3 0 = 1.5 , 1' , CF/CR - 1 C = 0.10 .4 .3 CL .2 0 1J 7I

;;;;;_. m

I

,//,

25

-f-3

2

/

,/

//,///

!I

I,///

7, 10

..

1

,/

XE A

-AR

AI7S

-. - .75 A AR

nt

4/f/V

íA

j5

0 .2 .3 0 CLR .2 0 .2 .3 .2 .3 .4 .3 CLF .2 0

0

0 0

CL/a

1.5 , F , cF/cR = I , C 0.10 m' .4

.3

CL, .2 0 .4 .3 CL, .2

0

.3

1'

1

1,1,1,11

III

x

Sill 111

AF g

11.51

I,,.

i / 2

,/

-1111

.25 AR

_i,It4

fI1r/

4 .5-? fl'y

,,

_ - - -

-V

A\

ç.2

,

- .4

,,

-F.75

.4

.3

CL, .2 .4

.3

CL, .2 .3 CLR _.2 CL _.2

0

.2

.3

CLR _.2

.3

25

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WASFUNOTON 25, D.C.

PROF. RB. COUCH

DEPARTMENT OF NAVAL ARCHITECTURE

UNIVERSITY OF MICIT!GAN ANN ARBOR, MICHIGAN

bIRECTOR

INSTITUTE OF MATHEMATICAL SCIENCES NEW YORK UNIVERSITY

25 WAVERLY PLACE NEW YORK 3. NEW YORK

PROF. Wi. PIERSON, JR.

OCEANOGRAPHY DEPAItTMENT

NEW YORK UNIVE1SITY UNIVERSITY HEIGHTS NEW YORK 53, NEW YORK

DIRECTOR

INSTITUTE FOR FLUID DYNAMICS AND APPLIED MATBEMATICS UNIVERSITY OF MARYLAND

COLLEGE PARK, MARYLAND DIRECTOR

SCRIPPS INSTITUTION OF OCEANOGRAPHY UNIVERSITY OF CALIFORNIA

LA JOLLA, CALIFORNIA

1 DIRECTOR

WOODS HOLE OCEANOGRAPHIC INSTITUTE WOODS HOLE, MASSACHUSETTS

2 DIRECTOR

ST.. ANTHONY FALLS HYDRAULIC LABORATORY

UNIVERSITY OF MINNESOTA MINNEAPOLIS 14, MINNESOTA

ADMINISTRATOR

WEBB INSTITUTE OF NAVAL ARCHITECTURE

CRESCENT BEACH ROAD

GLEN COVE, LONG ISLAND, NEW YORK

AITH: TECHNICAL LIBRARY PROF. W.R. SEARS

GRADUATE SCHOOL OF AERONAUTICAL ENG INEERING

CORNELL UNIVERSITY ITHACA, NEW YORK

DR. J. KOTIK

TECHNICAL RESEARCH GROUP

17 UNION SQUARE WEST NEW YORK 3. NEW YORK DR. J.M. ROBERTSON

THEORETICAL AND APPLIED MECHANICS

DEPARTMENT

COLLEGE OF ENGINEERING

UNIVERSITY OF ILLINOIS URBANA, ILLINOIS PROF. i.E. VENNARD

CIVIL ENGINEERING DEPARTMENT

STANFORD UNI VERS I IT

STANFORD. CALIFORNIA

DIRECTOR

DAVIDSON LABORATORY

STEVENS INSTITUTE OF TECHNOLOGY 711 HUDSON STREET

HOBOKEN, NEW JERSEY -5

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