The theoretical dynamic longitudinal stability of a constant lift hydrofoil system

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ARCB1F

[b.v.

_heepsbouhj

NAVY DEPARTMENT

THE DAVID W. TAYLOR MODEL BASIN

WASHINGTON. 7, D.C.

THE THEORETICAL DYNAMIC LONGITUDIN&L STABIUTY OF A

CONSTANT-LIFT HYDROFOIL SYSTEM

Fr.derick H. Imlay

December 1954 _{Riport 925}

lechnische Hogeschc.

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T

_{TOTIC&L D!iANIC LONGITUDINAL STABILITY}

_{OF A}

CON$TNT-LXF! HYDROFOIL SYSTEM

by

Frederick H. Inilay

December 1951+

_{Report 925}

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The Theoretical Dynamic Longitudinal Btability of a Cnstañt-Lift Hydrofoil 8ystem

by

Frederick H. Imlay

ABSTRACT

A qualitative treatment is given for an Idealized constant-.ift system of hydrofoils that do not pierce the surface0 It

is shown that as a prerequisite for stability the center of itt

on the hydrofoil must move aft as the surface is approached. At submergences deeper than the order of one chord, where the

movement of the center of lift can be neglected, the system cannot be stable

I NTRODtJCTION

The use of hydrofoils for sustentation of water-borne craft has been advocated for many years and many principles of operation have been proposed, leading to a variety of distinctive hydrofoil

systems0 One of the systems proposed recently _{may bedescribed} as the "constant-lift" system0 The essential feature of this system Is that some mechanism Is employed that tends to keep the hydrodynamic

lift developed by the hydrofoils at a constant

value in spite of variation in angle of attack0 This paper' presénts a brief analysis of the stability characteristics such a system would have Inherently (without manual or automatic

control) If the constant-lift mechanism were ideal. No attempt is wade to discuss the response of such a system In a seaway.

'This report

Is the result of

_a

study made under Research task NR O6Olfl financed by the Office of Naval Research,

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EQUAT IONS OF MOTION

Following Reference 1*, and using the nOmenclature of Reference 2, the dimensional equations of motion _{of a} hydro-foil systemmay be written

m -m

dez

z

Z4eZ deòz+

a °

.2d2e

M

y

where Z.(t) and M(t) are disturbanceS terms.

EVALUATION OF DERIVATIVES

Were it not for the cönstant-lift mechAnism _{the partial} derivatives appearing in Equation

_Li]

could be evaluated by methods given in

Reference l

_{In evaluating these derivatives} with constant-lift it vili be assumed that the mechanism _is

ideal that _{S, that any change in angle of attack,o, of each}

hydroìoil instantly effects a rotation of the hydrofoil such

that .is returned tO its initial vàiue, _{This assption presents} the óonstant-lift system in its best light because it

_negl8ts

important lags that would be present

in any

_{actual system due} to the virtual and mass inertias of the hydrofoil and unsteady-lift effects0 For simplicity, the treatment vili be limited to hydrofoils that do not pierce thè surface0

Q pivot

dO M

/ Mist) (11 X2.

Figure 1 - Forces On Pitching Hy-drofoil

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3

In Figuré i assume that the constant-lift mechanism éxerte a constant torque Q on the hydrofoil0 _{With the hydrofoil} initial-ly at the equilibrium angle of attack °e (assumed small), _{a torce} is developed near the quarter chord. such that

[2]

00.

where x, .s the distance from the pivot point to the quarter chord (positive with the pivot behind the quarter chord)0 For a small incremental increase in angle of attack

=

[3]

There is no restraint to oppose A1, hence the hydrofoil will rotate until Ac. is reduced to zero0. The imposit±on of the constant torque Q is1 therefore, the necessary and sufficient

condition

for constant lift with

changing angle of attack.

For such a system

emphasis can be concentrated on the basic contribution of the constant-lift

action

to the stability by eliminating shallow hydrofoil submergences (less

than

the order of one chord) from consideration0 This restriction will eliminate the near-surface

effects so

the derivatives Z/z and

_VZ'D

will be insignif-icant0 It follows that the derivatives _{z/z9 and}

_M/2$

_{can then} also be igored0.. It will be assumed, therefore, that

Z

M

o

.

,

[5]

Normally

the principal contribution, to the remaining

dr1vatives, ci/q and Í?q _{results from the change in the} angle of attack at the hydrodynam.c center of the hydrofoil brought about by the pitching motion, and thIs change is

nullified by the

constant-lift mechanism0 A secondary óon-tribution to these derivatives results, however, from _an effective change In camber for the pitching hydroföil.

Referring to Figure 1, the changed effective camber associated. with a pitching hydrofoil produces 'an incremental Z force.

d

[6]

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Equation !6] :j obtaifled by eliminating the ang3»of-attaok

Contribution from Equation [19]of Reference 1.

This force has

a point of appiication. near the half chord; the

_assoCiated

incremental momeflt

about the hydrofoil, pivot is." = -x

_2U

qe

or

-. ,.- p' QC Z

AQ.2-

-x1-The hydrofoil will' rotate to nullify _{AQ2 by decreasing the} angle of attack by an amount such that

=

Xl

-

0625c

X 2U1.

This rotation will cause a rOdution in the _{incremental Z force,} from that given by Equation [6], to a final value of

= .2?5c. qc

Z

Thus

Z' 00125c2

Ufiless the midehord of the hydrofoil is located _directly below the

center of

gravity'9 the

incremental Z force given by

equatIon [9]

will produce a

pitching moment

_{about the center} of gravity0 In general9 thereforeq

the derivative M/q

will have a value other than zeros and the equations of motion

for the contantiIft system will be

dw

d9 IdeZ

ni -mU = _{+ Z t} = dt2 dt 'q M(.t) (7] [8] [.9] (10]

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mp

Upon making tise of the Sybo1 p tg 'rep'ré sent d1fferenti ationwith respect 'to time, the characteristic _equation

corresponding to Equations (11] can be written as the determinant

wbiòh expands tó

STABILITY OF THE MOTION

-(mU ₊

,2(fr2v.

_'

-.t

-

-The characteristic equation has two _{zero roöts and one root} with the value

-

1M

'p

Consequently two modes of the mOtion can never be other than

neutrally stable.0 _{The remaining mode will be unstable if M/q}

is positive0 Thus at best the system will be neutrally stable with respect to depth of submergence and pitch angle when _the hydrofoils are deeply submerged0

STABILITY AT SHALLOW SUBIRGENCE

The ear.lierc4mposed condition that. the submergence be greater than one chord will now be considered0 It is well known that as the hydrofoil approaches the surface, _{a logs of.} lift occurs at a given angle of attack and the slope .of the lift cwve is also rduced (see Reference l) If, at shallow Submergence, the. lift due to angle of attack continued to be developed at the quarter chord hOwever, the constant-lift

mechanist would simply produce a new equilibrium angle of attack and the conclusions regarding stability would be the same as

for deep submergence. .

xperimenta1 work done by At*sman at low Froude numbers indicates that the center of lift moves aft at shallow sub-mergen;e (Beférencé

₃₎

.' otchin and.Keldysch and Lavrentiev

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6

give theoretical developments that indicate _{a forward movement} of the lift cOnter as the surface is approached _under bigh-speed operation (see References # and _5)° _{The apparent} dis-agreement as to the direction of movement needs _{to be resolved,} but one *ay eXamine the near-surface stability in the light or the mOvement0

With the torque Q adjusted for constant lift at _some operating depth near the surface, the derivatives

Z/w and will be zero0 The derivative Z/z0 has a

value' other than zero only: if the lift center changes position with change in' operating depth; the sign of the derivative depends

on

the direction of movement0 _{Values of derivatives}

z/e and _{M/e depend on the direction of the moment}

and the geometry of the: system0 _{Practical limitations} _on _the geometry are such that M/ê will have the same 'sign as

Z/z0.

The equations

of motion are

m dw

_{- 'n}

TJde

_{= z03.1-..}

_e

Z.

+ z t

-2d2e_

-y '- z 6 +

The corresponding'charácteristjc equation is

ink

2'

' m M 3 2 2

y

P p -m

_i;

M Z

_'M

Z '

,M

Z

N Z

'

-o

-

- w

= o

dO?M

M

dtq4

t

A necessary requirement for stability is that al]. of

the

coefficients of equation [16]have' the same sign0 Therefore Z/z0 and M/Ø must 'be negative as a condition for stability. This conditiøn vil], be met only if the

center of lift moves

aft as' the surface is approached'0 ' If the results given 'in

References f. and gare valid, therefore, the constant.lift System will be unstable at high

'speeds0

In spite of a

[15]

[16] dz0

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7

favorable movement of the àenter of pressure at low speed

stability of the system is not

assured,

however, because

_bie

criterion for no

variation

in sign

_{of the coefficients of} equation [16]is a necessarr but not a sufficient _cQudition for stability0 _{Routh's discriminant (see Reference}

6)

would have to be evaluated in _{any given case in order to have} assurance of stability.

CONCLUDING REMARK8

This study indicates that the _{constant-lift system. will.} not be stable unless it operates with

_the

_{hydrofoils near} tie surface0 _{For stability the further}

requirement that the center of lift on the hydrofoil _{must move aft as the}

hydrofoil approaches the surface is _essential0 _{Without this}

_favorable

travel

_{of the center of pressure the constant-lift system will}

not be stable under any òircumstance.

These conclusions were reached under the assumption _that

the constant-lift mechanism

functions

perfectly. Any actual embodiment of the system will introduce _{resonances, lags and} dead zones caused by inertia, _{solid friction, et0 As}

in

any other dynamic system, such problems _{wil]. have a deleterious} effect on the system stability0

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8

REP!ES

L 'Imlay, frederick L,

"ti"

orebicai Motions of _drofoj1 Systems", lLACA Report *o _{9l8,May 9, l9f70}

2 _{Nomenciati'e for Tròating the}

Motion eta &bnerged _Body

Through a 71ui4' _{SNAME Technical and}

Research Bulletin

No0 i=5,

April

i4so

30 Ausman, John $tanley, "Pressure

_{Limitation on the Upper}

Surface

of a Hydrofoil",

_{University of California,}

Graduate Division,, .5 March l95f0

kf0

_{otchn,J. B,}

"1 theWave41akj

_{Resistance and Lift}

of Bodies Submerged

in Water", Transactions of the 'Conference on the Theory of _{Wave Resistance, u0s.s0,}

MOgow., l937 English _{translation by £L(T) Air} _Ministry, RoT,PG No0 _{66, March 1935. 5NAJ} _{Technical and Research}

Bulletin No0 l-8, August 1*52..

leldysch, w

V0, and Lavrentiev, L

_{A., "On the Motion of}

*fl Aerofoji tlndér the Surface of _{a Heavy Fluid, i.e., a} Liquid", paper 'to ZAHl, Moscow, _{1935. English translation} by Science Translation Service, _{3TB_5, November 19kf9.} louth, E Je "Advanced »igid Dyziaini,c&" _{Vol. II,}

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9

INITIAL

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Chief öf Naval Research, for dlstributionx

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