2 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 that of the longitudinal motion and unless strongly damped, the

^2

SMITHSONIAN MISCELLANEOUS COLLECTIONS

VOL. 62

NO. 5 STABILITY OF

AEROPLANES— IIUXSAKEK AND OTHERS

₇₃ In a similar

manner,

the rolling

moment,

duetoside slip, orrestor- ing

moment,

such as is given by high fins or raised

wing

tips, should be largeto avoid" spiral" instability. In the present case, however,

we

wish to

make Lv

small.

Likewise the natural

banking

due to spin in

yaw we

wish small for "spiral ^'' stability, but

we now

wish to

have

^this coefficientlarge.

The

conflicting nature of the requirements for stability is here

shown by

the use of rather drastic simplifications in the

more

exact formulze.

For

the analysis of stability the exact formulae are easily applied,

and

the present

approximate forms

are

deduced

onlyin order to tracetheefifect on the

motion

of suchchangesas the designer

may

be

tempted

to

make

ona machine.

It is believed that

an

excessive dihedral angle

upwards

is not a cure-all for stability problems. Indeed, in practice, aeroplanes with a large dihedral angle for the

wings have

been

found

so violent in their

motion under

certain circumstances that theaverage pilothas a firm prejudice against the use of such a

wing

arrangement.

That

this prejudice has

some

physical basis has been

shown

here.

A

dihedral angle

machine

is not likelyto run into a " spiral dive,'' but

it is very likely tobeunstable on

what we may term

a ^"

Dutch

roll,"

from

analogy toa

well-known

figure of fancy skating.

We may

imagine an aeroplane to

yaw

to the right accidentally.

Due

to

Lr and Lv

^{the aeroplane}

banks

in a

manner

proper for the turn, but the roll is retarded

by

the large

damping

due to Ly.

The

turn is assisted

by

the increased drift

on

^the lower

wing

due to A';,,

and were

^itnot for the

much

discussed ^"weather

helm

^"given

by

Nv, the aeroplane

would

run off on a right turn.

However, Nv

tends to turntheaeroplanebacktohercourse. If

Nv

besufficient,the

machine

will

swing

backtoher course

and

the

bank

will flatten out.

But

since the

moment

of inertia in

yaw

isconsiderable, the

machine

will

swing

pasther course

and

startona turntotheleft. This swingingtoright

and

left of her course is

accompanied by

rolling

outward and some

side slipping.

The

analogytoa ^"

Dutch

roll" on skatesis obvious. If the skater lean too far out he

may

^fall,

and

^if the aeroplaneroll too far

on

the side swings it

may happen

that the

motion

will

become

unstable. If the airbe gustyit isverylikelythat such an aeroplane

may

be caught onthe roll

by

a sidegust

and

capsized.

The

^"

Dutch

roll " in ordinary aeroplanes

(which

are ^" spirally

''

unstable) is not likely to be present, since there is

no

dihedral

and

a large rudder.

The

average pilot

would much

prefer to deal with a

74 SMITHSONIAN MISCELLANEOUS COLLECTIONS

VOL. 62

machine which

tended to

swing down

into a " spiral dive

"if

leftto itself becausethereis

no

oscillation of rapidperiod involved.

The

production of a laterally stable aeroplane is attendant with

many

compromises,

and

itcannot be too stronglyinsisted

upon

thata freak type designed to be ^"very stable^" is likely to be rapid

and

violent initsmotion,

and

evenif stableagainst a " spiraldive" tobe frankly unstable against the "

Dutch

roll."

One may

inquire

whether

a

machine made

directionallyneutralcan be

made

^stable. In the notationhereused

Nv would

be approximately zero.

The

condition that " spiral" instability be not present is:

Lv/Nv>Lr/Nr.

But

for

Nv

zero,

we need

only

make Lv

^{slightly positive to insure}

stability in this motion.

Lv may

be

made

positive

by

^a very slight preponderance of fin surface abovethe center of gravity, raised

wing

tips, etc.

However,

in the

approximate

criterion for stabiHty inthe ^"

Dutch

roll,"

we have

-N^^/Lv>Np/Lp,

and

for A^• zero,the

motion

is clearly unstable unless the

magnitude

of the neglected terms is greater than

Np/Lp, which

is unlikely.

Replacing neglected terms in C„,

we

obtain as a

more

nearly exact expression

:

(C,

_ £A _ ^{L, /N,} _NA_y_Kl ^N,

^..

[b,

dJ-KcALp lJ

^{^}

KIL,

If

we make

A^„ very small as in the case

under

analysis, the last

term

vanishes as well as the second,

and we have

as a condition for

C E

r,""

-^

^positive

:

Substitutingnumerical values for the derivatives, for theslow-speed condition,

we

find

and

L^Np__ 160x57 ^ _

Q.6

Kc^Lp 48.6x224

^.

^^

^•

The

slow-speed

motion

would,therefore,be veryunstableif

Nv were

zero. Consideration of the

magnitude

of the derivatives leads us to the conclusion that in

any

aeroplane, if A''^ be

made

very small, the

NO. 5 STABILITY

OF AEROPLANES— HUNSAKER AND OTHERS

₇₅ motion called "

Dutch

roll " will probably be unstable at low speeds

where Np becomes

great.

For

highspeed,ifbothA^",,

and Np

arezero,thelateralmotion should be stableregardless of the

magnitude

of the other derivatives.

With

the

yawing moment due

to rolling as

measured by Np

increas- ing

from

zeroat high speedto

+57

^at

^low

^{speed, it}

^{would seem}

^that,

atthe

maximum

speed,

any

reasonable aeroplane willbe stable so far as the^"

Dutch

roll" _isconcerned,butat

low

speed it

may become

un- stable in this particularmotion.

In general, for high speed, considering the

two

possible kinds of lateral instability, it is believed that very slight modifications in fin

disposition will suffice to render

any

ordinary aeroplane laterally stable. Likewise, at high speed, longitudinal stability is easily secured.

At low

speed, thelongitudinal motion tends to

become

un- stableas well asone or the other kind of lateral motion.