VOLUME II

FLYING QUALITIES PHASE

CHAPTER5
LONGITUDINAL STATIC STABILITY

OCTOBER 1990
USAF TEST PILOT SCHOOL
EDWARDS AFBCA

mmi m

ViU..i.Xii i; l,'iwJu2'.y A

5.1

DEFINITION OF LONGITUDINAL STATIC STABILITY

Static

stability

equilibrium.

is

the

reaction

of

body

to

disturbance

from

To determine the static stability of a body, the body must be

initially disturbed

from its equilibrium state.

If,

when disturbed

from

equilibrium, the initial tendency of the body is to return to its original

equilibrium position,
stable.

the

body

displays

positive

static

stability

or

is

If the initial tendency of the body is to remain in the disturbed

position, the body is said to be neutrally stable.

However, should the body,

when disturbed, initially tend to continue to displace from equilibrium, the

body has negative static stability or is unstable.
Longitudinal

static

stability

determined in a similar manner.

or

"gust

stability"

of

an

aircraft

is

If an aircraft in equilibrium is momentarily

disturbed by a vertical gust, the resulting change in angle of attack causes

changes in lift coefficients on the aircraft (velocity is constant for this
time period).

The changes in lift coefficients produce additional aerodynamic

forces and moments in this disturbed position.

If the aerodynamic forces and

moments

to

created

condition,

tend

to

return

the

aircraft

the aircraft possesses positive

its

original

static stability or

Should the aircraft tend to remain in the disturbed position,

neutral stability.

undisturbed
is

stable.

it possesses

If the forces and moments tend to cause the aircraft to

diverge further from equilibrium, the aircraft possesses negative longitudinal

static stability or is unstable.

Pictorial examples of static stability as

related to the gust stability of an aircraft are shown in Figure 5.1.

=^

POSITIVE

FIGURE 5.1.

NEUTRAL

NEGATIVE

STATIC STABILITY AS RELATED TO GUST STABILITY OF AIRCRAFT

5.1

5.2

DEFINITIONS

Aerodynamic center. The point of action of the lift and drag forces such that
the value of the moment coefficient does not change with angle of attack.
Apparent stability. The value of dFs/dV
to as "speed stabi1ity".
Aerodynamic balancing.

about trim velocity.

Designing or tailoring (^

and

to

Also referred

either

in-

crease or decrease hinge moments and floating tendency.

Center of pressure. The point along the chord of an airfoil, or on an aircraft itself, where the lift and drag forces act, and there is no moment
produced.
Dynamic elevator balancing. Designing (^
(the floating moment coefficient)
a

to be small or zero.
Dynamic overbalancing. Designing^

to
a

be negative

(tail to rear aircraft

only).
Elevator effectiveness. The change in tail angle of attack per degree of
change in elevator deflection, T = dc^/dS. and equals -1.0 for the
all moving horizontal tail or "stabilizer".
Elevator power.

A control derivative.

Cm

-aVHl\T

Hinge moments.

The moment about the hinge line of a control surface.

Longitudinal static stability (or "gust" stability). The initial tendency of

an aircraft to return to trim when disturbed in pitch.
dCB/dCL
< u
for the airplane to be statically stable. The airplane must also be able
to trim at a useful positive CL.
Static elevator balancing. Balancing the elevator so that the q contribution
due to the weight of the surface is zero.
Stick-fixed neutral point. The eg location when dC/dC^ = 0 for the stickfixed airplane.
Stick-fixed stability. The
magnitude of dcydCL for
the stick-fixed
airplane.
Stick-fixed static margin. The distance, in percent MAC, between the eg and
the stick-fixed neutral point.
Stick force gradient. The value of dFB/dVe about trim velocity. Also referred to as "speed stability" and "apparent stability".

5.2

Stick-free neutral point. The eg position where dCm/dCL

free airplane.
Tail efficiency factor, h^.
wing dynamic pressure.
Tail volume coefficient.

5.3

VH

t ie

^/^w'

rat

^ f

for the

stick-

ta

^ dynamic pressure to

ltSt/CvSu

MAJOR ASSUMPTIONS
1.

Aerodynamic characteristics are linear |CL ,

2.
The
aircraft
Chapter 4 but = 0, $

is
in
= 0)

dCm/dCL,

Cn

etc.

steady,
straight
(not defined as in
, unaccelerated flight (q, p, r are all

zero).
3.

Power is at a constant setting.

4.

Jet engine thrust does not change with velocity or angle of

5.

The lift curve slope of the tail is very nearly the same
slope of the normal force curve.

6.

= CT
a

(dC /dC. ) is true for rigid

aircraft

at

attack.
as the

low Mach

when

thrust effects are small.

7.

X , Zv, V, and r^ do not vary with C:L

8.

C may be neglected since

the magnitude of Nw>

9.

Fighter-type aircraft and most low wing, large aircraft

very close to the top of the mean aerodynamic chord.

10.

it is 1/10 the magnitude of Cw and 1/100

Elevator effectiveness and elevator power are constant.

Symmetric Horizontal Tail (M
0).

5.3

have cg's

5.4

ANALYSIS OF LONGITUDINAL STATIC STABILITY

Longitudinal

equations

of

static

motion

of

stability
an

is

only

aircraft.

Of

special

the

six

case

for

equations

the
of

total

motion,

longitudinal static stability is concerned with only one, the pitch equation,
describing the aircraft's motion about the y axis.

Gy
The

fact

QIy - PR(I,'- Ix) + (P2 - R2)IX,

that

theory

pertains

to

an

(5.1)

aircraft

in

straight,

steady,

symmetrical flight with no unbalance of forces or moments permits longitudinal

static

stability motion

equations of motion.

to be

independent of

the

lateral

and directional

This is not an oversimplification since most aircraft

spend much of the flight under symmetric equilibrium conditions.

Furthermore,

the disturbance required for determination and the measure of the aircraft's
response takes place about the axis or in the longitudinal plane.

Under these

conditions, Equation 5.1 reduces to:

Gy

Since longitudinal static stability is concerned with resultant aircraft

pitching

moments caused

coefficients,

the

by momentary

primary

stability

changes in

angle

of attack

derivatives

become

Cn

or Cm
a

value

of either derivative is a

and lift
.
c

The

direct indication of the longitudinal static

stability of the particular aircraft.

To

determine

an

expression

for the

derivative

CB

, an

aircraft in

stabilized equilibrium flight with horizontal stabilizer control surface fixed

will be analyzed.

A moment equation will be determined from the forces and

moments acting on the aircraft.

moment coefficient form,

Once this equation is nondimensionalized, in

the derivative

This differential equation will be

an

with respect to CL will be taken.

expression for

Cm
c

directly

to

the

aircraft's

stability.
5.4

Individual

and

will relate

term

contributions

to

stability

will,

determining

the

in

turn,

be

analyzed.

flight

test

relationship

stability of an aircraft will be developed

for

followed by a

repeat of the entire analysis for an aircraft with a free control surface.
5.5

THE STICK FIXED STABILITY EQUATION

To

derive

the

longitudinal

aircraft in Figure 5.2.

pitching

moment

equation,

refer

to

the

Writing the moment equation using the sign convention

of pitch-up being a positive moment

RELATIVE
WIND

FIGURE 5.2.

AIRCRAFT PITCHING MOMENTS

5.5

0*

"cg

NMXu

CwZv

M.e

Mf

Ntlt

Ctht

MaCt

(5.2)

If an order of magnitude check is made, some of the terms can be

logically eliminated because of their relative size.

<J,t

Ct can be omitted since

NW

1DD

ID

is zero for a symmetrical airfoil horizontal stabilizer section.

ac

Rewriting the simplified equation

Meg

Nv Xv

CZ
ww

Kac

Mft

(5.3)

Ntt
1

It is convenient to express Equation 5.3 in nondimensional coefficient

form by dividing both sides of the equation by qvSvcu

eg

qSwCw

NX

qwSvCw

C Z

Mac

w w

qSwcw

5.6

M.f
+

"

q^c,

qvSwcw

N 1

* *

qwSv,cw

(5 4)

Substituting the following coefficients in Equation 5.4

eg
_ S c
q
w

ct
;g

total pitching moment coefficient about the eg

Cn

=
ac

c
q S c

wing aerodynamic pitching moment coefficient

C
=
f
f
q S c
CN

CK

=
t

Cc

fuselage aerodynamic pitching moment coefficient

wing aerodynamic normal force coefficient

N
t
q s
C

tail aerodynamic normal force coefficient

wing aerodynamic chordwise force coefficient

*w w

Equation 5.4 may now be written

cn

XV

Zw

=cN c +cc c -c

+c

NL
t t

=qSc

where the subscript w is dropped.

(Further equations,

will be with reference to the wing.)

To have the tail indicated in terms of a

coefficient, multiply and divide by qtSt

5.7

unless subscripted,

Substituting tail efficiency factor n^.

coefficient VH

"

cN

eg

c^/c^

and designating tail volume

ltSt/cS Equation 5.5 becomes

crCC
B
- c

BALANCE
EQUATION

M
n
NHX
eft

Equation 5.6 is referred to as the equilibrium equation in pitch.

(5.6)

If the

magnitudes of the individual terms in the above equation are adjusted to the
proper

value,

CID

the

aircraft

may

be

placed

in

equilibrium

flight where

eg

Taking the derivative of

Equation 5.6

with respect to

CL and

assuming

that Xw, Zw, VH and n^ do not vary with CL,

dC

cg - *.
3cBe"
:v

xw
-^

dcc
He"

zw
c:

dc

dc

*ac
He
/

WING

ac"
v

dC

Nt
3c~

(STABILITY!
v

,v

FUSELAGE

\
J

!EQUATI0N
<5-7>

TAIL

Equation 5.7 is the stability equation and is related to the stability derivative C

by the slope of the lift curve,

a.

Theoretically,

dC

a^r

dCr dC

a^Tdc:L

dC

=a

dc:L

dC

^ a ae:L

Equation 5.8 is only true for a rigid aircraft at low Mach

effects are small;

(5 8)

when thrust

however, this relationship does provide a useful index of

stability.

5.8

Equation 5.6 and Equation 5.7 determine the two criteria necessary for
longitudinal stability:

Criteria 1.

The "aircraft is balanced.

Criteria 2.

The aircraft is stable.

The first condition is satisfied if the pitching moment equation can be

forced to C

0 for useful positive values of C, .

eg

This condition is achieved by adjusting elevator deflection so that moments

about the center of gravity are zero
The

second condition is satisfied if

negative value.

From

Figure

5.3,

0).

Equation 5.7 or

negative value

necessary if the aircraft is to be stable.

attack increase

(i.e., M

Should

dC

for

/dC

Equation

has

5.7

is

gust cause an angle of

(and a corresponding increase in C ),

a negative C

be produced to return the aircraft to equilibrium, or Ct

0.

should
=s
The greater

the slope or the negative value, the more restoring moment is generated for an
increase in CL.
stability"

of

The slope of
the

aircraft.

dCB/dCL
(In

is a direct measure of the "gust

further

stability

equations,

subscript will be dropped for ease of notation).

AIRPLANE IN TRIM
AT A USEFUL CL

LESS
STABLE
MORE
STABLE
FIGURE 5.3.

STATIC STABILITY

5.9

the

e.g.

If the aircraft is retrinmed from one angle of attack to another,

basic stability of the aircraft or slope dCB/dCL

does not change.

Figure 5.4.

FIGURE 5.4.

STATIC STABILITY WITH TRIM CHANGE

5.10

the
Note

However, if moving the eg is changing the values of Xv or Zw, or if VH is

changed, the slope or stability of the aircraft is changed.

See Equation 5.7.

For no change in trim setting, the stability curve may shift as in Figure 5.5.

FIGURE 5.5.

STATIC STABILITY CHANGE WITH CG CHANGE

5.11

5.6

AIRCRAFT COMPONENT CONTRIBUTIONS TO THE STABILITY EQUATION

5.6.1

The Wing Contribution to Stability

The lift and drag are by definition always perpendicular and parallel to

the relative wind.

It is therefore inconvenient to use these forces to obtain

moments, for their arms to the center of gravity vary with angle of attack.
For this
whose

reason,

axes

remain

all

forces are

fixed with

resolved into normal and chordwise forces

the

aircraft

constant.

5.12

and whose

arms

are

therefore

/ l>- RESULTANT
Z*"1\
AERODYNAMIC
a /
11
FORCE
!\

RELATIVE
WIND

FIGURE 5.6.

WING CONTRIBUTION TO STABILITY

Assuming the wing lift to be the airplane lift and the wing's angle of
attack to be the airplane's angle of attack, the following relationship exists
between the normal and lift forces

(Figure 5.6)

L COS a + D sin a

(5.S)

D COS a - L Sin a

(5.10)

Therefore, the coefficients are similarly related

CN

CL cos a + CDsin a

CCD
= C cos a

The stability contributions, dC /dC

- C,1 sin a

and dCc/dCL, are obtained

5.13

(5.n:

(5.12)

(5.13)

dCN
He"

dCL
=

HT

da
cos a

sin a

ar

dCD
ar

sin a

.
ar

cos a

(5.14)
c

_
=

-- a
_
cos

n ggda sin
<- a
C
D

L
_i_ a
- g^ sin

- ^da cos a
C
L

Making an additional assumption that

Dp

dC

'he n

o
dC.

n AR e

and that Cn

is constant with chanaes in C,

p

'

2c
L

n AR e

If the angles of attack are small such that

cos a

1.0 and sin a

a,

Equations 5.13 and 5.14 become

<*c

ac:

FTe

acrB"a"cLar

Examining the above equation for relative magnitude,

CD

is on the order of 0.02 to 0.30

5.14

(5 16)

CL

usually ranges from 0.2 to 2.0

is small, < 0.2 radians

da

Be,L

is nearly constant at 0.2 radians

ii AR e

is on the order of 0.1

Making these substitutions, Equations 5.15 and 5.16 have magnitudes of

dC
g~ - 1 - 0.04 + 0.06 = 1.02 * 1.0

(5.17)
(5.18)

dcc
3- -

0.1 CL - 0.012 - 0.2 - 0.2CL

= -0.41
at CT

Li

2.0

ma x

The moment coefficient about the aerodynamic center is invariant with respect
to angle of attack (see definition of aerodynamic center). Therefore

dCsi

arr "
Rewriting the wing contribution of the Stability Equation,
dc

Equation 5.7,

= - 0.41

afc c
Lmax

WING

5.15

2Q

{5ig)

Fran Figure 5.6 when a increases,

chordwise force decreases.
these changes.

the normal

Equation 5.19

shows

force

the

increases and the

relative magnitude

of

The position of the eg above or below the aerodynamic center

(ac) has a auch smaller effect on stability than does the position of the eg
ahead of or behind the ac.
stabilizing.
stable

the

With eg ahead of the ac, the normal force is

From Equation 5.19, the more forward the eg location, the more
aircraft.

With the

eg below the ac,

the

chordwise

force

stabilizing since this force decreases as the angle of attack increases.

is
The

further the eg is located below the ac, the more stable the aircraft or the
more

negative

the value

of dCB/dCL.

The wing contribution

to

depends on the eg and the ac relationship shown in Figure 5.7.

DESTABILIZING

en

STABILIZING

MOST STABLE

FIGURE 5.7.

ac

DESTABILIZING

STABILIZING

CG EFFECT ON WING CONTRIBUTION TO STABILITY

5.16

stability

For a stable wing contribution to stability,

the aircraft would be designed

with a high wing aft of the center of gravity.

Fighter type aircraft and most low wing,

large aircraft have cg's very

close to the top of the mean aerodynamic chord.

Zw/c is on the order of 0.03.

For

these

neglected.

aircraft

the

chordwise

force

contribution

to

stability

can

be

The wing contribution then becomes

dC

(5 20)

c-

WING

5.6.2

The Fuselage Contribution to Stability

The fuselage contribution is difficult to separate from the wing terms

because it is strongly influenced by interference from the wing flow field.

Wind tunnel tests of the wing-body combination are used by airplane designers
to obtain information about the fuselage influence on stability.
A fuselage by itself is almost always destabilizing because the center of
pressure is usually ahead of the center of gravity.

The magnitude of the

'destabilization effects of the fuselage requires their consideration in the

equilibrium and stability equations.

In general, the effect of combining the

wing and fuselage results in the combination aerodynamic center being forward
of quarter-chord and the C

of the combination being more negative than the

tc

wing value alone.

dc

-sp- =
L

5.6.3

Positive quantity

FUSELAGE

The Tail Contribution to Stability

From equation 5.7, the tail contribution to stability was found to be

5.17

dC

HcT

*\

TAIL

For small angles of attack, the lift curve slope of the tail is very nearly
the same as the slope of the normal force curve.

dC

dCi
dot

*t

da

(5.22)

Therefore
C

(5.23)

tt

An expression for o^. in terms of CL is required before solving for dCN /dCL

FIGURE 5.8.

TAIL ANGLE OF ATTACK

5.18

From Figure 5.8

o

- i

+ i - e

(5.24)

Substituting Equation 5.24 into 5.23 and taking the derivative with respect to
CL, where au

dCL/do

da

de\

at

(1

de

1_\

(5

25)

upon factoring out l/aw

/.

de\

Substituting Equation 5.26 into 5.21, the expression for the tail contribution
becomes
dC_

Jl

_ !l

_ de\

vH

TAIL

The value of at/aw is very nearly constant.

These values are usually obtained

from experimental data.

The tail volume coefficient, VH, is a term determined by the geometry of
the aircraft.

To vary this term is to redesign the aircraft.

. l!l
H

(5.28)

CS

The further the tail is located aft of the eg (increase lt) or the greater the
tail surface area (St), the greater the tail volume coefficient (V), which
increases the tail contribution to stability.
5.19

The expression, r^ , is the ratio of the tail dynamic pressure to the wing
dynamic pressure and ry varies with the location of the tail with respect to
wing wake, prop slipstream, etc.

For power-off considerations, r^ varies from

0.65 to 0.95 due to boundary layer losses.

The term (1 - de/da) is an important factor in the stability contribution
of the tail.

Large positive values of de/da produce destabilizing effects by

reversing the sign of the

dC./dCL

term (1 - de/da) and consequently,

the sign of

.
Tail

For example,

at high angles of attack the F-104 experiences a sudden

increase in de/da.

The term (1 - de/da) goes negative causing the entire tail

contribution to be positive or destabilizing, resulting in aircraft pitchup.

The

stability of an aircraft

system.

is definitely influenced

by

the wing vortex

For this reason, the downwash variation with angle of attack should

be evaluated in the wind tunnel.

The

horizontal

stabilizer

provides

the

necessary

positive

stability

contribution (negative dCn/dCL) to offset the negative stability of the wingfuselage

combination

and

to make

the entire

aircraft

stable

and balanced

(Figure 5.9).
+
WING AND
FUSELAGE
WING

TAIL
FIGURE 5.9.

AIRCRAFT COMPONENT CONTRIBUTIONS TO STABILITY

5.20

The stability may be written as,

dC

dc

-T- v \
rut

5.6.4

(> -*)

:s.29i

The Power Contribution to Stability

The addition of a power plant to the aircraft may have a decided effect

on the equilibrium as well as the stability equations.

be quite complicated.
effects.

The overall effect may

This section will be a qualitative discussion of power

The actual end result of the power effects on trim and stability

should come from large scale wind tunnel models or actual flight tests.
5.6.4.1

Power Effects of Propeller Driven Aircraft - The power effects of a

propeller driven aircraft which influence the static longitudinal stability of

the aircraft are:
1.

Thrust effect - effect on stability from the thrust acting along the
propeller axis.

2.

Normal force effect - effect on stability from a force normal to the

thrust line and in the plane of the propeller.

3.

Indirect effects - power plant effects on the stability contribution

of other parts of the aircraft.

FIGURE 5.10.

PROPELLER THRUST AND NORMAL FORCE

5.21

Writing the moment equation for the power terms as

0*
MCQ

T 2,T + NP 1T

(5.30)

In coefficient form
c

*c

CT

ZT

n
+ C

c~

Np

**
~c

(5.31)

The direct power effect on the aircraft's stability equation is then

dC\ .

dCT

ZT

BC^/

a^

dC
+

^NP

*T

ac^

(5.32)

POWER

The

sign of dc/dCL

then depends on the

sign of

the derivatives

POWER

dCN /dCL anddCT/dCL.

p'

First consider the dCVdC,

derivative. If the speed varies at different
T
L
.flight conditions with throttle position held constant, then CT varies in a
manner that can be represented by dCT/dCL. The coefficient of thrust for a
reciprocating power plant varies with CL and propeller efficiency. Propeller
efficiency which is available from propeller performance estimates in the
manufacturer's data, decreases rapidly at high CL.
Coefficient of thrust
3/2
variation with CL is nonlinear fit varies with CL j with the derivative
large at low speeds. The combination of these two variations approximately
linearize CT versus CL (Figure 5.11). The sign of dCT/dCL is positive.

5.22

FIGURE 5.11.

COEFFICIENT OF THRUST CURVE FOR A

RECIPROCATING POWER PLANT WITH PROPELLER

The derivative dCN /dCL is positive since the normal propeller force
increases linearly witl? the local angle of attack of the propeller axis, o^.
The direct power effects are then destabilizing if the

eg is as shown in

Figure 5.10 where the power plant is ahead and below the eg.
power effects must also be considered in evaluating the
contribution

of

the

propeller

power

plant.

determine their quantitative magnitudes.

The indirect

overall

No attempt will

be

stability
made

to

However, their general influence on

the aircraft's stability and trim condition can be great.

5.6.4.1.1
the

propeller

Increase in angle of downwash, e.

increases

with

angle

of

attack

Since the normal force on

under

powered

flight,

the

slipstream is deflected downward, netting an increase in downwash on the tail.

The downwash in the slipstream will, increase more rapidly with the angle cf
attack than the downwash outside the slipstream.
positive increase with power.

The term

The derivative

(1 - dc/da)

dc/da

has a

in Equation 5.27 is

reduced causing the tail trim contribution to be less negative or less stable
than the power-off situation.
5.6.4.1.2
tail

is

Increase of r^

increased by the

i^/%)-

slipstream and h^

The dynamic pressure, q,., of the

is

greater

than unity.

From

equation 5.27, the increase of \ with an application of power increases the

tail contribution to stability.
Both slipstream effects mentioned above may be reduced by locating the

5.23

horizontal stabilizer high on the tail and out of the slipstream at operating
angles of attack.
5.6.4.2

Power effects of the turbojet/turbofan/ramjet.

The magnitude

of the power effects on jet powered aircraft are generally smaller than on
propeller driven aircraft.

By assuming that jet engine thrust does not change

with velocity or angle of attack, and by assuming constant power settings,

smaller power

effects would be expected than with a similar

reciprocating

engine aircraft.
There are three major contributions of a jet engine to the equilibrium
static longitudinal stability of the aircraft.

These are:

1.

Direct thrust effects.

2.

Normal force effects at the air duct inlet and at angular changes in
the duct.

3.

Indirect effects of induced flow at the tail.

The thrust and normal force contribution may be determined from Figure 5.12.

RW

FIGURE 5.12.

JET THRUST AND NORMAL FORCE

5.24

Writing the equation

"eg

^T

+t

*T

<5-33>

*T

or
Z +C
qic
T
NT
^
T

C eg "

(5 34

'

>

With the aircraft in unaccelerated flight, the dynamic pressure is a function

of lift coefficient.
W
~ CLS

(5.35)

Therefore,
Cm

= r,
r C,L + CN
W C

eg

(5.36)

If thrust is considered independent of speed, then

dC,

ac:L

wc"

acr
c"
1

(5 37)

The thrust contribution to stability then depends on whether the thrust line
is above or below the eg.
Locating the engine below the eg causes a
destabilizing influence.
The normal force contribution depends on the sign of the derivative
dCN /dCL.
The normal force 1^, is created at the air duct inlet to the
turbojet engine.
This force is created as a result of the momentum change of
the free stream which bends to flow along the duct axis. The magnitude of the

5.25

force is a function of the engine's airflow rate, Wa, and the angle o^ between
the local flow at the duct entrance

and the duct axis.

W
N,

U0 ^

(5.38)

With an increase in o^, Nj will increase, causing dCN/dCL to be positive.

The normal

force contribution will be destabilizing if the

ahead of the center of gravity.

inlet duct is

The magnitude of the destabilizing moment

will depend on the distance the inlet duct is ahead of the center of gravity.
For

jet

engine

to

definitely

contribute

to

positive

longitudinal

stability (dCm/dCL negative), the jet engine would be located above and behind
the center of gravity.
The indirect contribution of the jet unit to longitudinal stability is
the effect of the jet induced downwash at the horizontal tail.

This applies

to the situation where the jet exhaust passes under or over the horizontal
tail surface.
outward.
area.

The jet exhaust as it discharges from the tailpipe spreads

Turbulent mixing causes outer air to be drawn in towards the exhaust

Downwash at the tail may be affected.

The F-4 is a good example where

entrained air from the jet exhaust causes downwash angle at the horizontal
tail.
5.6.4.3

Power Effects of Rocket Aircraft.

Rocket powered aircraft such as

the Space Shuttle, and rocket augmented aircraft such as the C-130 with JATO
installed,

can

be

significantly

affected

magnitude of the rocket thrust involved.

longitudinally

depending

on

the

Since the rocket system carries its

oxidizer internally,

there is no inlet or incoming mass flow and no normal

force contribution.

The thrust contribution may be determined from Figure

5.13.

5.26

1
ZR

(EQUIVALENT)

FIGURE 5.13.

ROCKET THRUST EFFECTS

Writing the equation

<?

TRZR

WC9

or C

T Z

cg

R R

(5.39)

qSc"

Assuming that the rocket thrust is constant with changes in airspeed, the
dynamic pressure is a function of lift coefficient.

JL

therefore

TRZECT
-^

and
dc

TF Z R

Be

W C

(5.40)

POWER

From the

above

discussion,

it can be

seen

that

important in deciding the power effect on stability.

examined individually.

This

is

the

reason that

several

factors

Each aircraft must be

aircraft

are

stability in several configurations and at different power settings.

5.27

are

tested

for

5.7

THE NEUTRAL POINT

The

stick

fixed neutral

point

is

defined

as

the

center

of

gravity

position at which the aircraft displays neutral stability or where

dC /dCt

The symbol h is used for the center of gravity position where

eg

(5.41)

c
The stability equation for the powerless aircraft is

dC

dCm

de\

-^C "a"

>\

" 35/

(5,29)

rus

Looking at the relationship between eg and ac in Figure 5.14

FIGURE 5.14.
Xw
C~

CG AND AC RELATIONSHIP
(5.42)

h -

5.28

Substituting Equation 5.42 into Equation 5.29,

dC-

X +. dC
h - "Vc
"w*

3c[

ac^

a.
a
t

de\

^ \X " Hi)

,, ,,,

(5 43)

Fus

If we set

dCB/dCL

0, then h s hn and Equation 5.43 gives

.n

Xac dCin

^ a,t

T+

V
acr_
ir
H \ f-M)
~~LFUS
C

(5 44)

This is the eg location where the aircraft exhibits neutral static stability:
the neutral point.
Substituting Equation 5.44 back into Equation 5.43, the stick-fixed stability derivative in terms of eg position becomes
dC
= h - hn
(5.45)

Hc7

"

The stick-fixed static stability is equal to the distance between the eg

position and the neutral point in percent of

the mean aerodynamic

"Static Margin"

but is

refers

to the

same distance,

chord.

positive in sign for

a stable aircraft.
Static Margin

hn - h

(5.46)

It is the test pilot's responsibility to evaluate the aircraft's handling

qualities and to determine the acceptable static margin for the aircraft.
5.8

ELEVATOR POWER

For

an

aircraft

to

be

a usable flying machine, it must be stable and

5.29

balanced
C

throughout the useful CL range.

must

be zero.

Some

means

For trimmed, or equilibrium flight,

must be available for balancing the various

terms in Equation 5.47

X
c

eg

N"i

+ Cc

-c -cac

+c

*f -

(a

t ^

(5 47)

(Equation 5.47 is obtained by substituting Equation 5.23 into Equation 5.6.)

Several possibilities are available.

The

center

of gravity could be

moved fore and aft, or up and down, thus changing Xw/c or Zw/c.
would not only affect the equilibrium lift coefficient,
dCm/dCL in Equation 5.48.
dC

However, this

but would also change

This is undesirable.

dCM

dCr

dC

a.

,.\

<- as fa- c-*ar-i;

. M -*)
Fus
Equation 5.48
5.7.

is obtained by substituting Equation 5.26

<5-<>

into Equation

The pitching moment coefficient about the aerodynamic center could be

changed by effectively changing the camber of the wing by using trailing edge
flaps as is done in flying wing vehicles.
rear

aircraft,

trailing

edge

wing

flaps

pitching moment coefficient to zero.

On the conventional tail-to-the

are

ineffective

in

trimming

the

The combined use of trailing edge flaps

and trim from the tail may serve to reduce drag, as used on some sailplanes
and the F-5E.
The remaining solution is to change the angle of attack of the horizontal
tail to achieve a jCm

0) without a change to the basic aircraft

stability.
The control

means is

either an elevator

moving stabilizer (slab or stabilator).

on the stabilizer

The slab is used on most high speed

aircraft and is the most powerful means of longitudinal control.

5.30

or an all

Movement of the slab or elevator changes the effective angle of attack of

the horizontal stabilizer and, consequently, the lift on the horizontal tail.
This

in turn

changes

horizontal tail.

the moment

about

the

center

of

gravity due

to

the

It is of interest to know the amount of pitching moment

change associated with an increment of elevator deflection.

This may be

determined by differentiating Equation 5.47 with respect to 6e .

dC

da
e

=
o

-a, v n

(5.50)

This change in pitching moment coefficient with respect to elevator deflection

C

is referred to as "elevator power".

i

It indicates the capability of the

elevator to produce moments about the center of gravity.

The term doct/d6e in

Equation 5.49 is termed "elevator effectiveness" and is given the shorthand

notation x.

The elevator effectiveness may be considered as the equivalent

change in effective tail plane angle of attack per unit change in elevator
deflection.

The

relationship

between

elevator

effectiveness

effective angle of attack of the stabilizer is seen in Figure 5.15.

5.31

and

the

..+

t~ ANGLE ATTACK
OF TAIL

FIGURE 5.15.

CHANGE WITH EFFECTIVE ANGLE OF ATTACK

WITH ELEVATOR DEFLECTION

Elevator effectiveness is a design parameter and is determined from wind

tunnel tests.

Elevator effectiveness is a negative number for all tail-to-

the-rear aircraft.

The values range from zero to the limiting case of the all

moving stabilizer (slab) where x equals -1.

change plus

one

degree

for every minus

The tail angle of attack would

degree

the

slab moves.

For

the

elevator-stabilizer combination, the elevator effectiveness is a function of

the ratio of overall elevator area to the entire horizontal tail area.

5.9

ALTERNATE CONFIGURATIONS
Although

standard,

two

tail-to-the-rear
other

is

the

configurations

configuration

merit

some

normally

discussion.

perceived
The

as

tailless

aircraft, or flying wing, has been used in the past, and some modern desions
contemplate

the use of this concept.

The canard configuration has

also been used over the past several years with mixed results.

5.9.1

Flying Wing Theory

In order for a flying wing to be a usable aircraft, it must be balanced

(fly in equilibrium at a useful positive CL)

be analyzed as follows:
5.32

and be stable.

The problem may

For

the wing

in

FIGURE 5.16.

AFT CG FLYING WING

Figure

assuming that

5.16,

the

chordwise

force

acts

through the eg, the equilibrium in pitch may be written

eg

NX

- M

(5.51!

or in coefficient form

=9

"N

(5.52)

For controls fixed, the stability equation becomes

dC
<=S

dcT

dCH Xv
.3c~
c~
I*

5.33

(5.53)

Equations

5.52 and 5.53 show that the wing in Figure 5.16 is balanced and

unstable.

To make the wing stable, or dcydc^ negative, the center of gravity

must be ahead of the wing aerodynamic center.

now changes the signs
equations become

in

Equation

=9

dC
=9

HCT

5.51 .

- c,

Making this eg change, however,

The

equilibrium

and

(5.54)

dc.N xv
Hc^ c~

The wing is now stable but unbalanced.

stability

(5.55)

The balanced condition is possible

with a positive C
ac

Three methods of obtaining a positive C

are:
*c

1.

Use a negative camber airfoil section.

The positive C

will qive
ac

a flying wing that is stable and balanced (Figure 5.17).

RW

FIGURE 5.17.

NEGATIVE CAMBERED FLYING WING

5.34

This

type

of

wing

is

not

realistic

because

of

unsatisfactory

dynamic

characteristics, small eg range, and extremely low CL maximum capability.

2.

A reflexed airfoil section reduces the effect of camber by creating a

download near the trailing edge.

Similar results are possible with

an upward deflected flap on a symmetrical airfoil.

3.

A symmetrical airfoil section in combination with sweep and wingtip

washout (reduction in angle of incidence at the tip) will produce a
positive Cnm

by virtue of the aerodynamic couple produced between

ac

the downloaded tips and the normal lifting force.

Figure 5.18.

5.35

This is shown in

LIFT
SPANWISE LIFT
DISTRIBUTION

c~5&

FROM ABOVE:

VECTOR UP

FROM THE SIDE:

INNER
PANEL

-^e

TOTAL

H mac

-$
OUTER
PANEL

mac

i mac
RESULT: BALANCED
AND STABLE

FIGURE 5.18.

THE SWEPT AND TWISTED FLYING WING

Figure 5.19 shows idealized C.^ versus ^ for various wings in a control
fixed position.

Only two of the wings are capable of sustained flight.

5.36

UNSTABLE
AFTcfl

UNSTABLE
AFT eg

*- C.

NO SWEEP

SYMETRICAL

SWEEPBACK
WASHOUT

SYMETRICAL

UNSTABLE
AFT eg
UNSTABLE
AFT eg

*-CL

NO SWEEP

POSITIVE
CAMBER

FIGURE 5.19.

*- C,

NO SWEEP

VARIOUS FLYING WINGS

5.37

REFLEXED
TRAILING EDGE

5.9.2

The Canard Configuration

Serious work on aircraft with the canard configuration has been sporadic

from the tine the Wright brothers' design

evolved into the tail to the rear

airplanes of World War I, until the early 1970's.

canard
fighter.

airplanes

in

quantity

production

was

One of the first successful

the

Swedish

JA-37

"Viggen"

Other projects of significance were the XB-70, the Mirage Milan, and

the TU-144.

The future seems to indicate that we may see more of the canard

configuration, as evidenced by the X-29 Forward Swept Wing project.

WING AERO CENTER

RUTAN LONG EZE

FIGURE 5.20.
5.9.2.1

The Balance Equation.

BALANCE COMPARISON

From Figure 5.20 the balance equation can be

written as follows:

C
Meg

- -NX
+CZ
-Ma c
ww
ww

+ M-t + Ntt
1 + Ct h^ - Mc

5.38

(5.56)

Simplifying assumptions are

Ctht and K4C are small and may be neglected

M
c
c,"

C9

5-SvCw

Xw

Zw

- C

+ CIt

Ntlt

qtSt

Combining Terms:
cB

xu
-c.H ^
+ c
C

c"

C9

+ C,Ft

V,H hT

BALANCE
EQUATION

(5.56)

5.9.2.2
The Stability Equation.
Although the canard can be a balanced
configuration, it remains to be seen if it demonstrates static stability or
"gust stability". By taking the derivative with respect to CL, Equation 5.58
becomes
dC

dC

.0 N v
-^L.
HCT" " ~ ac;" _Lc

dC

-JiC
ac;
c

dC

"f +

a=-l
HC7

dC

\
-1
vH nn,
Hc7

STABILITY (5.59)
BQ^TJON

Equation 5.59 indicates that the normal (or lift) force of the wing now has a
stabilizing influence (negative in sign), and the canard term is destabilizing
due to its positive sign. It is obviously a misnomer to call the canard a
horizontal stabilizer, because in reality it is a "destabilizer"!
The degree
of instability must be overcome by the wing-fuselage combination in order for
the airplane to exhibit positive static stability dCB/dCL (negative in
sign). This is shown graphically in Figure 5.21.
5.39

FIGURE 5.21. CANARD EFFECTS CN

5 9 2.3
rM

dCB/dCL

jn a n^er simlar to the y a

-r-- ^h,H to Stability,

^tiThorisontal tail experiences a downwash field frc the *. rfth.

in,, the canard will see upwash ahead of the wing.

A, v^sh f.eld has a

^labilisin, effect on longitudinal stability because it aKes the tail ter

in the stability equation note positive.
The tail contribution fro. Equation 5.6! can be examined for the effects

of upwash, e'

dC,

acrv" *

d (ate^)

The tail angle of attack,

dcT"
V

do,.
V

- t

*t

ac:

can be expressed in terms of incidence and

upwash, as described in Figure 5.22.

5.40

FIGURE 5.22.

- it - e'

CANARD ANGLE OF ATTACK

a, - iv

(5.62)

Therefore
(5.63)

aw - i w + it + C

The tail contribution now becomes

d(aw - iw + it + C)

W\

ae;

, /l

/l

de'

da

dc' 1 \

5.41

Therefore,
dC

a,

L
L

It

is

extremely

important

to

note

v '

that

the

upwash

and

downwash

interaction between the canard and the wing are critical to the success of the
design.

The wing will see a downwash field from the canard over a portion of

the leading edge.

is

Aerodynamic tailoring and careful selection of the airfoil

required for the airplane to meet its design objectives at all canard

deflections

and

flap

settings

on

the

wing.

Designs

which

tend

to

be

tandem-wing become even more sensitive to upwash and downwash.

5.10

STABILITY CURVES
Figure 5.23 is a wind tunnel plot of CB versus CL for an aircraft tested

under two eg positions and two elevator positions.

Assuming

the

elevator

effectiveness

and

the

elevator

power

to

be

constant, equal elevator deflections will produce equal moments about the eg.
Points
C
m

A and

represent

the same elevator deflection corresponding to the

needed to maintain equilibrium. For an elevator deflection of 10, in the

eg

aft eg condition, the aircraft will fly in equilibrium or trim at point B.

the

eg

is

moved

forward with

no

change

equilibrium is now at A and at a new CL.

to

the

elevator

deflection

If
the

Note the increase in the stability

of the aircraft (greater negative slope of dCB/dCL).

For equilibrium at a lower CL
elevator is deflected to 5.

or at A without changing the eg,

The stability level of the aircraft has not

changed (same slope).

A cross plot of Figure 5.23 is elevator deflection versus CL for
This is shown in Figure 5.24.

the

Cn = 0.

The slopes of the eg curves are indicative of

the aircraft's stability.

5.42

NOSE
UP

^\>*"^"~ b~ +1

cn

^*

\^ V\A

5, +s

C,

\\^r" "

AFT CO

\\\\
\

NOSE
DOWN

FWDcg

FWDcg

FIGURE 5.23.

AFT CO

eg AND S# VARIATION OF STABILITY

FWDcg

AFT eg

FIGURE 5.24.

5.11

VERSUS Ct

FLIGHT TEST RELATIONSHIP

The stability equation previously derived cannot be

flight testing.

directly used

in

There is no aircraft instrumentation which will measure the

change in pitching moment coefficient with change in lift coefficient or angle

of attack.

Therefore, an expression involving parameters easily measurable in

flight is required.

This expression should relate directly to the stick-fixed

longitudinal static stability,

dCB/dCL, of the aircraft.

5.43

The external moment acting longitudinally on an aircraft is

M -

f (U, a, a, Q, S. )

Assuming that the aircraft is in equilibrium and in unaccelerated flight, then

M -

f (a, SJ

(5.65)

Therefore, using a Taylor series expansion,

AM

|M
3a

Aa+

|
3S#

as

(5.66]

and
C

Aa + C
a

AS
6

where
La

A5 e

a ao

5e - 6

assuming
ao

0
=

5.44

=5 e

=0

(5.67)

The elevator deflection required to maintain equilibrium is,

8.

Cn a
- - c^

(5.68)

Taking the derivative of Se with respect to CL

dC

dCB da
,

Ha~ HOT

dS e

Li

ca

(5.69)

In terms of the.static margin, the flight test relationship is,

f!i dcT

" "h = static margin

C
elevator power

(5>70)

The amount of elevator required to fly at equilibrium varies directly as the

amount of static stick-fixed stability and inversely as the amount of elevator
power.
5.12

LIMITATION TO DEGREE OF STABILITY

The degree of stability tolerable in an aircraft is determined by the

physical limits of the longitudinal control. The elevator power and amount of
elevator deflection is fixed once the aircraft has been designed.
If the
relationship between 8e required to maintain the aircraft in equilibrium
flight and CL -is linear, then the elevator deflection required to reach any CL
is,

. Zero

d8
HCTL

Lift

5.45

(5 71)
L

The

elevator

stop

deflection available.

determines

the

absolute

limit

of

the

elevator

Similarly, the elevator must be capable of bringing the

airplane into equilibrium at CLL

Max

Recalling Equation 5.69

dC

dS

e _

3cT "

HcT
I

(5.69)

&

Substituting Equation 5.69 into 5.71 and solving for dCB/dCL

corresponding

tOC

Wax

(5.72)
L

Max

Given a maximum CL required for landing approach, Equation 5.72 represents the
maximum stability possible, or defines the most forward eg position.

A eg

forward of this point prevents obtaining maximum CL with limit elevator.

If a pilot were to maintain CL

for the approach, the value of dCB/dCL

Max

corresponding to this CL

would be satisfactory. However, the pilot usually

Max

desires additional elevator deflection to compensate for gusts and to

flare the aircraft. This requirement then
dictates a dC /dCt
"/

less than the value required for CL

Mx

only.
*

In addition to maneuvering the aircraft in the landing flare, the pilot

must adjust for ground effect.

The ground imposes a boundary condition which

affects the downwash associated with the lifting action of the wing.

This

ground interference places the horizontal stabilizer at a reduced negative

angle of attack.

The equilibrium condition at the desired CL is disturbed.

5.46

To maintain the desired CL, the pilot must increase f>m to obtain the original
tail angle of attack.

The maximum stability dCB/dCL must be further reduced

to obtain additional S# to counteract the reduction in downwash.

The

three

conditions

that

limit

the

amount

of

static

stability or most forward eg position for landing are:

1.

The ability to land at high CL in ground effect.

2.

The ability to maneuver at landing CL (flare capability)

3.

The total elevator deflection available.

Figure 5.25 illustrates the limitation in dCm/dCL

'

Hri

y- 6. LIMIT
T6~ FOR GROUND EFFECT

"TAS, FOR*MANEUVERING AT C^

FIGURE 5.25.

LIMITATIONS ON dC /dC,

/'

5.47

Max

longitudinal

5.13

STICK-FREE STABILITY
The name stick-free stability comes from the era of reversible control

systems and is that variation related to the longitudinal stability which an

aircraft could possess if the longitudinal control surface were left free to
float

in

the

slipstream.

The

control

force variation with

change

in

airspeed is a flight test measure of this stability.

If an airplane had an elevator that would float in the slipstream when
the

controls were

free,

then

the

change

in

the

pressure

pattern on

stabilizer would cause a change in the stability level of the airplane.

change

in

the

tail

contribution

characteristics of the elevator.

would

be

function

of

the

the
The

floating

Stick-free stability depends on the elevator

hinge moments caused by aerodynamic forces which affect the total moment on
the elevator.
An airplane with an irreversible control system has very little tendency
for its elevator to float.

Yet the control forces presented to the pilot

during flight, even though artificially produced, appear to be the effects of

having

free

elevator.

If

the

control

feel

system

can

be

altered

artificially, then the pilot will see only good handling qualities and be able
to fly what would normally be an unsatisfactory flying machine.
Stick-free stability can be analyzed by considering the effect of freeing
the elevator of a tail-to-the-rear aircraft with a reversible control system.
In this case, the feel of stick-free stability would be indicated by the stick
forces required to maintain the airplane in equilibrium at some speed other
than trim.
The change in stability due to freeing the elevator is a function of the
floating characteristics of the elevator.
upon the elevator hinge moments.

The floating characteristics depend

These moments are created by the change in

pressure distribution over the elevator associated with changes in elevator

deflection and tail angle of attack.
The following analysis looks at the effect that pressure distribution has
on the elevator hinge moments, the floating characteristics of the elevator,
and the effects of freeing the elevator.
Previously,

an

expression

was

developed

5.48

to

measure

the

longitudinal

static

stability using

elevator

surface

deflection,

S#.

This

expression

represented a controls locked or stick-fixed flight test relationship where

the aircraft was stabilized at various lift coefficients and the elevator
deflections

were

then

measured

at

these

equilibrium values

of

CL.

The

stick-free flight test relationship will be developed in terms of stick force,

F , the most important longitudinal control parameter sensed by the pilot.
a reversible control system,

In

the motion of the cockpit longitudinal control

creates elevator control surface deflections which in turn create aerodynamic

hinge moments,

felt by the pilot

feedback from the control

as

control

forces.

There

surfaces to the cockpit control.

is

a direct

The

following

analysis assumes a simple reversible flight control system as shown in Figure

5.26.

(+)

POSITIVE STICK
DEFLECTION
(AFT)

F,
*

SIGN CONVENTION
NOSE-UP MOMENT

(+)

(PULL)

POSITIVE ELEVATOR
DEFLECTION (TEU)

5e (+)

BELL CRANK

FIGURE 5.26.

TAIL-TO-THE-REAR AIRCRAFT WITH A

REVERSIBLE CONTROL SYSTEM

A discussion of hinge moments and their effect on the pitching moment and
stability equations must necessarily precede analysis of the stick-free flight
test relationship.
5.49

5.13.1

Aerodynamic Hinge Moment

An aerodynamic

hinge moment

is

a moment

generated

about

the

control

surface as a consequence of surface deflection and angle of attack.

Figure

5.27 depicts the moment at the elevator hinge due to tail angle of attack
(6

0). Note the direction that the hinge moment would tend to rotate the

elevator if the stick were released.

HINGE
LINE

<*,{+)
He<+)

FIGURE 5.27.

HINGE MOMENT DUE TO TAIL ANGLE OF ATTACK

If the elevator control were released in this case, the hinge moment, H# would
cause the elevator to rotate trailing edge up (TEU).
was previously determined to be positive,
which,

if the

elevator

control were

5.50

Since the elevator TEU

a positive hinge moment is that

released would

cause the

elevator to

deflect TEU.

The general hinge moment equation may be expressed as

Ch Qt

Where S

(5.73)

is elevator surface area aft of the hinge line and c#

mean square chord of the elevator aft of the hinge line.

is the root

The hinge moments

due to elevator deflection, 6#, and tail angle of attack, o^ , will be analyzed
separately and each expressed in coefficient form.

5.13.2

Hinge Moment Due to Elevator Deflection

Figure 5.28 depicts the pressure distribution due to elevator deflection.

This condition assumes a,.

0.

The elevator is then deflected, 6t.

The

resultant force aft of the hinge line produces a hinge moment, H#, which is
due to elevator deflection.

HINGE
LINE

RW,
at = o

He(+)

FIGURE 5.28.
Given the
relationship

of

HINGE MOMENT DUE TO ELEVATOR DEFLECTION

sign convention specified earlier,

hinge

moment

coefficient

5.51

to

Figure

elevator

5.29

depicts the

deflection,

where

. (TEU)

(TED)

a, = o

FIGURE 5.29.

HINGE MOMENT COEFFICIENT DUE TO ELEVATOR DEFLECTION

st hinge nent curves are nonlinear at the extremes of elevator deflection

o' tail angle of attack.

The boundaries shown on Figure 5.29 signify that

only the linear portion of the curves is considered.

The usefulness of this

sunption will be apparent when the effect of elevator deflection ano tail
angle of attack are combined.
The slope of the curve

in

Figure

coefficient due to elevator deflection.

in the linear region.

The term

C,

5.29 is C^

It is negative in sign and constant

moment coefficient.
5 13 3 Hinge Moment Due to Tail Angle of Attack
Figure 5.30 depicts the pressure distribution due
This condition assumes S.

moment

is generally called the restoring"

6

attack.

the hinge

0.

5.52

to

tail

angle

of

The tail is placed at some angle of

attack.

As in the previous case, the lift distribution produces a resultant

force aft of the hinge line, which in turn generates a hinge moment.
term, C^

The

, is generally referred to as the "floating" moment coefficient.

HINGE
LINE

at{+)
He(+)

FIGURE 5.30.

HINGE MOMENT DUE TO TAIL ANGLE OF ATTACK

Figure 5.31 depicts the relationship of hinge moment coefficient to tail angle
of attack, where

Ch

1tS.c.

5.53

at{+)

(-)?

FIGURE 5.31.

5.13.4

HINGE MOMENT COEFFICIENT DUE TO TAIL ANGLE OF ATTACK

Combined Effects of Hinge Moments

Given the previous assumption of linearity,

the total aerodynamic hinge

moment coefficient for a given elevator deflection and tail angle of attack
may be expressed as

(5.74)
0

&

5.54

Figure 5.32 is a graphical depiction of the above relationship,

symmetrical tail so that (^

FIGURE 5.32.

assuming a

0.

COMBINED HINGE MOMENT COEFFICIENTS

The "e" and "t" subscripts on the restoring and floating hinge moment
coefficients are often dropped in the literature.

For the remainder of this

chapter:
Restoring Coefficient

(5.75)

5.55

Floating Coefficient

"t

- c,

(5.76)

Examining a floating elevator, it is seen that the total hinge moment

coefficient is a function of

elevator deflection, tail

angle of attack,

and

mass distribution.

If the elevator
there

may

f 5

"

< .' Of

(5.77)

>

is held at zero elevator deflection and zero angle of attack,

be some

residual aerodynamic hinge moment,

C^

If W

is

the

weight of the

elevator and

is the moment arm

between the

elevator eg

and elevator hinge line, then the total hinge moment is,

(5.78)
o

HINGE
LINE

FIGURE 5.33.

ELEVATOR MASS BALANCING REQUIREMENT

5.56

The weight effect is usually eliminated by mass balancing the elevator

(Figure 5.33).

Proper design of a symmetrical airfoil will cause C^

to

be

o
negligible.
When

the

elevator

assumes

its

equilibrium position,

the

total

hinge

moment will be zero and solving for the elevator deflection at this floating
position, which is shown in Figure 5.34

(5.79)

riot

The stability of the aircraft with the elevator free is going to be affected
by this floating position.
If the pilot desires to hold a new angle of attack from trim, he will
have

to deflect the

elevator

from this

floating position to the

position

desired.

DESIRED
POSITION
FLOATING
POSITION

ORIGINAL RW

ZERO
DEFLECTION

NEWRW

FIGURE 5.34.

ELEVATOR FLOAT POSITION

5.57

The

floating position will

required to use.

If the ratio

greatly affect the

C^/C^

forces the pilot

can be adjusted,

then

the

is

forces

required of the pilot can be controlled.

r
/ (^ is small, then the elevator will not float very far and the
If
stick-free "stability characteristics will
stick fixed.

But ^

must

be

deflection will be unreasonable.

small

or

be much the sane as those with the

the stick forces required to hold

The values of C,

and
a

by aerodynamic balance.

C,

can be controlled
*

Types of aerodynamic balancing will be covered in a

later section.
One additional method for altering hing,. moments is through the use of a trim
tab.

There are numerous tab types that will be discussed in a later section.

A typical tab installation is presented in Figure 5.35.

A\ SUCTION DUE TO
*'(-)"* TAB DEFLECTION

FIGURE 5.35.

ELEVATOR TRIM TAB

5.58

Deflecting the tab down will result in an upward force on the trailing
edge of the elevator.

This tends to make the control surface float up.

Thus

a down tab deflection (tail-to-the-rear) results in a nose up pitching moment

and is positive.

This results in a positive hinge moment, and the slope of

control hinge moment versus tab deflection must be positive.

The hinge moment contribution from the trim tab is thus,

and continuing with our assumption of linearity,

the control hinge moment

coefficient equation becomes,

(5.80)
0

for a mass balanced elevator.

(5.81)

5.14

THE STICK-FREE STABILITY EQUATION

The stick-free stability may be considered the summation of

stick-fixed stability and the contribution to stability of freeing
elevator.

nay-

" suss5.59

5i:r

the
the

The stability contribution of the free elevator depends upon the elevator
floating position.

Equation 5.83 relates to this position

5_

--^

*riot

a,

(5.83)

S,

Substituting for o^ from Equation 5.24

- - cT

(a

- " *

** "e)

(5 84)

'

Taking the derivative of Equation 5.84 with respect to CL'

d& e

dC,

^*_ I/i-5s'
da
C.

(5.85)

V a.

Substituting the expression for elevator power, (Equation 5.50) into Equation
5.69 and combining with Equation 5.85.

<V

dc

BcT

(5 50)

H \ ^

'

<

a,:*.*

Fra*
El.v

5.60

(i-&)(^

(5 86)

Substituting Equation 5.86 and Equation 5.29 into Equation 5.82,

the

stick-free stability becomes

dC

dC

3^ i- *aj
StickFt

a.

. v ,

(5 87)

i \ \ (* - X " ^)

ru

The difference between stick-fixed and stick-free stability is the multiplier

in Equation 5.87
1-TC
/C
, called the "free elevator factor" which
a.'

is

designated

F.

The

magnitude

and

sign

magnitudes of T and the ratio of (^ /C^

a

floating tendency has a small

(^

elevator

and

has

stick-free
large

/ C^

depends on

.An

elevator

with

giving

of F around unity.

value

are

practically

contribution

of

the

only

tendency (ratio

of

the same.

C^ / (^
a

stability

the relative
slight

stability

floating

a ^

Stick-fixed

of

If

large),

the
the

horizontal tail is reduced (dCB/dCL

is

Stiek-

rr

less negative). For instance, a ratio of C^/C^

a

-2 and a T of -.5, the

floating elevator can eliminate the whole tail contribution to stability.

Generally, freeing the elevator causes a destabilizing effect. With elevator
free to float, the aircraft is less stable.
The stick-free neutral point, h/, is that eg position at
which
dC /dCt
is zero.
Continuing as in the stick-fixed case, the stick-free
Stick-

rr

neutral point is,

X

dC
C

-L

a,
a
Pus

5.61

. \

and
dC_
h-h

(5.89)

Stickrra*

The stick-free static margin is defined as

Static Margin
5.15

h^ - h

(5.90)

FREE CftN&RP STABILITY

While it is not the intent of this paragraph to go into stick-free

stability aspects of the canard, it is useful :-.o present a summary of the

effects of

freeing the

elevator.

Remember

that

the

tail

term will

be

multiplied by the free elevator factor F

1 - T
\

As F becomes less than unity, the tail (canard) contribution to stability

becomes less positive, making the airplane more stable. In turbulence, stick
free, the nose tends to fall slightly from an up gust, resulting in a sort of
load alleviation or ride smoothing characteristic (reversible control system).
If this characteristic is desired in a tail to- the rear aircraft,
easily

be

obtained

by

dynamic

overbalancing

as

described

in

it can

Paragraph

5.17.6.4, page 5.84.

Table 5.1 compares the differences in stability derivatives and control
terms between the canard and tail to the rear aircraft.

5.62

TABLE 5.1
STABILITY AND CONTROL DERIVATIVE COMPARISON

Tail-to-Rear

Canard

(-)

(-)

(+)

(+)

(-)

(-)

(+)

(-)

et

c,
5.16

STICK-FREE FLIGHT TEST RELATIONSHIP

As was done

for

stick-fixed stability,

a flight test

required that vail relate measurable flight test

free stability of the aircraft, dCa/dCL
Stickrrtt

5.63

parameters

relationship is
with the

stick-

J*

FIGURE 5.36.
^

, , .^

G, - f(a, b, c, d, ,

ELEVATOR-STICK GEARING

stick deflected with a stick force F,.

The control

iot
9earing to
e
j"J^--*
^
*-*
*
*
"
rt 5 IT ^elevator deflects and the aerodynamic pressure proouces aa
r/e .L" at I elevator that exactly balances the _* P-duced by the
pilot with force F,.
F.l.

- - Gx H.

1 is included with the gearing, the stick force becomes

If the length 1,
(5.91)
r.

- 6H.

5.64

The hinge moment H# may be written

H#

q, q S, c,

(5.92)

-GC, q S. c.

(5.93)

Equation 5.91 then becomes

Fs

Substituting

&

and using
d6
5

. Ztro

HcrI. CL

(5 71)

Lift

and
S:

"

w ~ *w

\ "

(5.24)

With no small amount of algebraic manipulation, Equation 5.93 may be written

C

Fs

- Aq (B + C,

5T *x

where
A -

- GS# c#

5.65

^. dC
^

"i

\
\

stick-y
rr.. '

(5.94)

Zero

Lift

Writing Equation 5.94 as a function of airspeed and substituting

unaccelerated flight, CLq W/S and using equivalent airspeed, V#,

Fs

1/2 Po

/
A(B
\

T>h,6
6T)-A|^

+ Ch

J
'

dC
^

(5.95)
Stickrrt

Simplifying Equation 5.95 by combining constant terms,

Fs - 1^ V 2 + Rj
K,
"i

contains

which determines

trim speed.

for

K,
^

(5.96)
contains dC/dC.L

StickFr

Equation 5.96 gives a relationship between an in-flight measurement of stick

force gradient and stick-free stability. The equation is plotted in Figure
5.37. -

5.66

PULL

FIGURE 5.37.

STICK FORCE VERSUS AIRSPEED

The plot is made up of a constant force springing from.the stability term plus
a variable force proportional to the velocity squared, introduced through
constants and the tab term q
5T. Equation 5.96 introduces the interesting
o

fact that the stick-force variation with airspeed is apparently dependent on

the first term only and independent in general of the stability level.

That

is, the slope of the Fs versus V is not a direct function of

dC /dC
. If the derivative of Equation 5.96 is taken with respect to V,
Stick-

the second

term

containing

level

trim

tab

and

the stability drops out.

setting,

stick

force

For constant stability

gradient is a function

of trim

airspeed.

dv~

Po V.

(f

5.67

\ *r)

(5.97)

However, dF /dV is a function of the stability term if the trim tab

setting, S,,, is adjusted to trim at the original trim airspeed after a change
in stability level, e.g., movement of the aircraft eg.

The tab setting, &T,

in Equation 5.95 should be adjusted to obtain

at the original

F,

trim

velocity.

"

(VTri.

dc[

(5.98)

^
Stickrrtt

This new value of 6

for F

0 is then substituted into Equation 5.97 so

dF.

a^i.

dC

(vTri'

mi

(5.99)

)
Stickrr*t

Thus, it appears that if an aircraft is flown at two eg locations and

dF /dV
s'

Tlii

is determined at the same trim speed each time, then one could

extrapolate or

interpolate

to determine the

stick-free neutral point hn.

Unfortunately, if there is a significant amount of friction in' the control

system, it is impossible to precisely determine this trim speed. In order to
investigate briefly the effects of friction on the longitudinal control
system, suppose that the aircraft represented in Figure 5.38 is perfectly
trimmed

at Vx

i.e.,

(\

S.

and

iT

St].

If

the

used to decrease or increase airspeed with no change to the trim

elevator

is

setting, the

friction in the control system will prevent the elevator from returning all
the way back to 6

when the controls are released.

5.68

The aircraft will return

PULL

F.

FIGURE 5.38.
only to V2 or V3.

CONTROL SYSTEM FRICTION

With the trim tab at $T

, the aircraft

is

content to fly

at any speed between V2 and V3.

The more friction that exists in the

system,

the wider this speed range becomes. If you refer to the flight test methods
section of this chapter, you will find that the FB versus V plot shown in
Figure 5.38 matches the data plotted in Figure 5.65
is

the

(pg 5.102).

In theory,

of

d*",/^.
siope
the parabola formed by Equation 5.96.
With that
portion of the parabola from V 0 to Vstall removed, Figures 5.37 and 5.38
predict flight test data quite accurately.
Therefore, if there is a significant amount of friction in the control
system, it becomes impossible to say that there is one exact speed for which

5.69

the aircraft is trimmed.

predicting the

Equation 5.99 is something less than perfect for

stick-free neutral point of an aircraft.

To

reduce

the

undesirable effect of friction in the control system, a different approach is

made to Equation 5.94.
If Equation 5.94 is divided by the dynamic pressure, q, then,

dC
gc-

^LSI,

Fs/q A

(B + C,

&T)
&

-g
n.
&

tS. 100)
.
Stickrrtt

Differentiating with respect to CL,

d(Fs/q)

dC,
(5.101)

Stickrrt

or
d(F /q)

-ar- -

dCB

f (

(5 102)

-TzrL stick>
. ,

mi

Trim velocity is now eliminated from consideration and the prediction of

stick-free neutral point h^ is exact.
A plot of (dFs/q)/dCL
versus eg
position may be extrapolated to obtain hn.
5.17

APPARENT STICK-FREE STABILITY

Speed stability or stick force

gradient

dFB/dV, in most

reflect the actual stick-free stability dCB/dCL

of
Stick-

5.70

an

cases does not

aircraft.

In

fact, this apparent stability dFg/dV, may be quite different from the actual
stability of the aircraft.
marginal f dC /dC,
^

Where the actual stability of the

aircraft may be

small), or even unstable ( dCB/dCL

Stick-

'

positive),
Stick-

'

Tv

r:<

apparent stability of dFB/dV

acceptable.
In flight, the

may be such as to make the aircraft quite

test pilot feels and evaluates the apparent

stability of the aircraft and not the actual stability, dCB/dCL

Stick-

The apparent stability dFs/dV is affected by:

1.

Changes in dCB/dCL
Stick-

2.

Aerodynamic balancing

3.

Downsprings, bob weights, etc.

The apparent stability

of the stick force gradient through a

speed increases if dC /dC

is made more negative.

The

given trim

constant

K2

of

StickFr

Equation 5.96 is made more positive and in order for the stick force to
continue to pass through the desired trim speed, a more positive tab selection
is required.

An aircraft operating at a given eg with a

shown in Figure 5.39, Line 1.

5.71

tab

setting

6T

is

FIGURE 5.39.

EFFECT CK APPARENT STABILITY

is increased by moving the eg forward, then Kj

If dC /dCt

(which

is a

Stick

in Equation 5.95) becomes more positive or increases,

function of dC /dC.
Stickrr

and the equation becomes

(5.103)

*i

This equation plots as Line 2 in Figure 5.39.

tab setting 5T

The aircraft with no change

operates on Line 2 and is trimmed to V2.

Stick forces at

in
all

airspeeds have increased.

dC /dC

At this juncture,

has increased,

although

the actual

stability

there has been little effect on the stick force

Stickrrii

gradient or apparent stability.

about the same.)

(The slopes

of

Line

and Line

So as to retrim to the original trim airspeed Vx,

applies additional nose up tab to ^ .

.

5.72

being

the pilot

The aircraft is now operating on line

3.
The stick force gradient through \ has increased because of an increase
in the 1^ term in Equation 5.96. The apparent stability dFs/dV has increased.
Aerodynamic balancing of the control surface affects apparent stability
in the same manner as eg movement. This is a design means of controlling the
hinge moment coefficients, (^ and (^ . The primary reason for aerodynamic
a

balancing is to increase or reduce the hinge moments and, in turn, to control

stick forces. Changing (^ affects the stick forces as seen in Equation 5.100.
o

In addition to the influence on hinge moments, aerodynamic balancing affects

the real and apparent stability of the aircraft. Assuming that the restoring
hinge moment coefficient C^ is increased by an appropriate aerodynamically
o

balanced control

surface,

ratio of q, /(^

the

in Equation 5.87 is de-

creased.

--'.* (i-$(i-*e-)

HOL

W 11
ru

Stick

&

The combined increase in dCm/dCL

and C^ ,
StickFr*a

(5 87)

increases the K2

term in Equation 5.96 since

_AW

Ch

dC

"h

(5.104)

Stickr r

Figure 5.39 shows the effect of increased Kj. The apparent stability is not
affected by the increase in Kj if the aircraft stabilizes at v2. However,
once the aircraft is retrimmed to the original airspeed vt by increasing the
tab setting to ST , the apparent stability is increased.
2

5.73

5.17.1 Set-Back Hinge

Perhaps the simplest method of reducing the aerodynamic hinge moments is
to move the hinge line rearward.

The hinge moment is reduced because

the

moment arm between the elevator lift and the elevator hinge line is reduced.
(One may arrive at the same conclusion by arguing this part of the elevator
lift acting behind the hinge line has been reduced, while that in front of the
hinge line has been increased.)
value of both (^

and C^ .
a

the

sign

The net result is a reduction in the absolute

In fact, if the hinge line is set back far enough,

of both

derivatives can be changed.

A set-back hinge is

shown in

Figure 5.40.

nK

X
LARGE

^L.

s=
FIGURE 5.40.

SMALL

SET-BACK HINGE

5.17.2 Overhang Balance

This method is simply a special case of set-back hinge in which the

5.74

elevator

is

designed

so

that

when

the

leading

edge

protrudes

into

the

airstream, the local velocity is increased significantly, causing an increase

in negative pressure at that point.
hinge

moment which

This

negative

pressure peak

creates a

opposes the normal restoring hinge moment, reducing C

.
i

Figure 5.41 shows an elevator with an overhang balance.

NEGATIVE PEAK PRESSURE

HELPING
MOMENT

FIGURE 5.41.
5.17.3

OVERHANG BALANCE

Horn Balance

The horn balance works on the same principle as the set-back hinge; i.e.,
to reduce hinge moments by increasing the area forward of the hinge line.
horn balance,
Cjj

and Cjj .

especially the unshielded horn,

This

arrangement shown in

is

very effective in reducing

Figure 5.42 is also a handy way of

at

improving the mass balance of the control surface.

5.75

The

UNSHIELDED

FIGURE 5.42.

HORN BALANCE

5.17.4 Internal Balance or Internal Seal

The internal seal allows the negative pressure on the upper surface and
the positive pressure on the lower surface to act on an internally sealed
surface forward of the hinge line in such a way that a helping moment is
createdV again opposing the normal hinge moments.
As a result, the absolute
values of C^
airspeeds,

and C^
"but

are both reduced.

is' sometimes

This method is good at high indicated

troublesome

shows an elevator with an internal seal.

5.76

at

high

Mach.

Figure

5.43

LOW
PRESSURE
HELPING
MOMENT

HIGH
PRESSURE

FIGURE 5.43.

5.17.5

INTERNAL SEAL

Elevator Modifications

Bevel Angle on Top or on Bottom of the Stabilizer.

This device which

causes flow separation on one side, but not on the other, reduces the absolute
values of C^

and C^ .
at

&

Trailing Edge Strips.

nation

with a

balance

This device increases both C^

tab,

trailing

positive (^ , but still a low (^ .

a

edge

strips

and C^ .

produce

In combivery

high

This results in what is called a favorable

"Response Effect," (i.e., it takes a lower control force to hold a deflection

than was originally required to produce it).
Convex Trailing Edge. This type surface

produces a more negative C^ ,

O

but tends as well to produce a dangerous short period oscillation.

5.17.6 Tabs
A tab is simply a small flap which has been placed on the trailing edge
of a larger one.
The tab greatly modifies the flap hinge moments, but has
only a small effect on the lift of the surface or the entire airfoil. Tabs in
general are designed to modify stick forces, and therefore C^ ,
9

affect C^

very much.

5.77

but will not

5.17.6.1

Trim Tab.

A trim tab is a tab which is controlled independently of

the normal elevator control by means of a wheel or electric motor.

The
purpose of the trim tab is to alter the elevator hinge moment and in doing so
drive the stick force to zero for a given flight condition.
A properly
designed trim tab should be allowed to do this throughout the flight envelope
and across the allowable eg range. The trim tab reduces stick forces to zero
primarily by changing the elevator hinge moment at the elevator deflection
required for trim.

This is illustrated in Figure 5.44.

5.78

^-iM
FIGURE 5.44.

5.17.6.2

Balance Tab.

T.

TRIM TAB

A balance tab is a single tab,

not a part of the

longitudinal control system, which is mechanically geared to the elevator so

that a certain elevator deflection produces a given tab deflection.
tab is geared to move in the same direction as the surface,
leading tab.
tab.

it is called a

If it moves in the opposite direction, it is called a lagging

The purpose of the balance tab is

and stick

If the

force (lagging tab)

effectiveness.
effectiveness at

usually to reduce

at the price

of

the

certain

hinge moments

loss

in control

Sometimes, however, a leading tab is used to increase control

the price of increased

stick

forces.

The

leading tab may

also be used for the express purpose of increasing control force.

Thus (^
o

may be increased or decreased, while (^

remains unaffected.

If the linkage

shown in Figure 5.45 is made so that the length may be varied by the pilot,
then the tab may also serve as a trimming device.
5.79

LAGGING
TAB

FIGURE 5.45.
5.17.6.3

Servo or Control Tab.

BALANCE TAB

The servo tab is linked directly to the

aircraft longitudinal control system in such a manner that the pilot moves the
tab and^the tab moves the elevator, which is free to float.
elevator

hinge moment due to elevator deflection just balances out the hinge

moments due to c^. and 6T.

hinge moment or C^

.
e

5.17.6.4

The summation of

Spring Tab.

The

Again C^

stick forces are

now a

function of

the tab

is not affected.

A spring tab is a lagging balance tab which is geared

in such a way that the pilot receives the most help from the tab at high
speeds where he needs it the most;
pressure.

i.e., the gearing is a function of dynamic

The spring tab is shown in Figure 5.46.

5.80

HINGE OF
FREE ARM

FIGURE 5.46.

SPRING TAB

The basic principles of its operation are:

1.

An increase in dynamic pressure causes an increase in hinge moment

and stick force for a given control deflection.

2.

The increased stick force causes an increased spring deflection and,

therefore, an increased tab deflection.

3.

The increased tab deflection causes a decrease in stick force. Thus

an increased proportion of the hinge moment is taken by the tab.

4.

Therefore, the spring tab is a geared balance tab where the gearing
is a function of dynamic pressure.

5.

Thus, the stick forces are more nearly constant over the speed range
of the aircraft.
That is, the stick force required to produce a
given deflection at 300 knots is still greater than at 150 knots,
but not by as much as before. Note that the pilot cannot tell what
is causing the forces he feels at the stick. This appears to change
"speed stability," but in fact changes actual stability or

dc./aq,.
6.

After reaching full spring or tab deflection the balancing feature

is lost and the pilot must supply the full force necessary for
further deflection. (This acts as a safety feature.)
5.81

A plot comparing the relative effects of the various balances on hinge

moment parameters is given in Figure 5.47 below.

The point indicated by the

circle represents the values of a typical plain control surface.

The various

lines radiating from that point indicate the manner in which the hinge moment
parameters are changed by addition of various kinds of balances.

Figure 5.48

is also a summary of the effect of various types of balances on hinge moment

coefficients.

Ch

_f / ELEVATOR "FLOPS'
(DEQ) S AGAINST THE STOP

004

PLAIN
SURFACE

f ^ LAGGING
7~^" BALANCE TAB
INTERNAL SEAL

^e^f

C5 (DEG)

.004
ROUND-NOSE
OVERHANG

ELLIPTICAL-NOSE
OVERHANG

UNSHIELDED
HORN

-.008

FIGURE 5.47.

TYPICAL'HINGE MOMENT COEFFICIENT VALUES

5.82

NOMENCLATURE

SIGN
NORMALLY (+)

SIGN
ALWAYS (-)

SET-BACK HINGE

REDUCED

REDUCED

OVERHANG

REDUCED

REDUCED

UNSHIELDED HORN

REDUCED

REDUCED

INTERNALSEAL

REDUCED

REDUCED

BEVEL ANGLE STRIPS

REDUCED

REDUCED

TRAILING EDGE STRIPS

INCREASED

INCREASED

CONVEX TRAILING EDGE

NO CHANGE

INCREASED

NO CHANGE

INCREASED
OR
DECREASED

Id^"' INCREASED

TABS

les,
TOP
VIEW

1G>
4t *

DECREASED
LAGGING BALANCE TAB

NO CHANGE

DECREASED

LEADING BALANCE TAB

NO CHANGE

INCREASED

lO=t

^
BLOW DOWN TAB OR
SPRING TAB

NO CHANGE

FIGURE 5.48.

INCREASED,
DECREASES
WITH "q"

&

METHODS OF CHANGINS AERODYNAMIC HINGE

MOMENT COEFFICIENT MAGNITUDES
(TAIL-TO-THE-REAR AIRCRAFT)

5.83

In
of C

summary,
and (^

aerodynamic

during the aircraft

decrease hinge moments.

affects

the

dynamic

control

is

"tailoring"

the

values

of

It is a method of controlling stick forces and

balancing

is

often

defined

as

making

In the literature,
C^

small or just

Note from Figure 5.47 that the addition of an unshielded

horn balance changes C.

exactly zero,

the

design phase in order to increase or

real and apparent stability of the aircraft.

slightly positive.

the same.
Making C^

balancing

without affecting C^

aircraft's

negative

is

stick-fixed

very much.
and

defined as overbalancing.

If c;

stick-free
If

is

made

stability

is made

are

slightly

negative, then the aircraft is more stable stick-free than stick-fixed.

Early

British flying quality specifications permitted an aircraft to be unstable

stick-fixed as long as stick-free stability was maintained.
Overbalancing
increases stick forces.
Because of the very low force gradients in most modern aircraft at the
aft center of gravity, improvements in stick-free longitudinal stability are
obtained by devices which produce a constant pull force on the stick
independent of airspeed which allows a more noseup tab setting and steeper
stick force gradients.

Two devices for increasing the stick force gradients

are the downspring and bobweight.

Both effectively increase the apparent

stability of the aircraft.

5.17.7 Downspring
A virtually constant stick force may be incorporated into the control
system by using a downspring or bungee which tends to pull the top of the
stick forward. From Figure 5.49 the force required to counteract the spring
is
F
"Downspring

Ti
2

- K3

(5.106)
Downspring

If the spring is a long one, the tension in it will be increased only slightly
as the top moves rearward and can be considered to be constant.
The equation with the downspring in the control system becomes,
5.84

h<
As shown in Figure

5.39,

(5.107)

K+ h Dowaapring

the apparent stability will

increase when the

aircraft is once again retrimmed by increasing the tab setting.

downspring

increases apparent

Note that the

stability, but does not affect

; no change to Kj).
stability of the aircraft (dCB/dCL

the actual

Stickrti

&T
TTAO

"/"'

-F.

TENSIONS CONSTANT = T

-MmwewnmmetjtJWMs- <j-

0.

U
FIGURE 5.49.

DOWNSPRING

5.17.8 Bobweight
Another ethod of introducing a nearly constant stick force is by placing
a

bobweight in

5.50).

the control system

which causes a constant moment (Figure

The force to counteract the bobweight is,

(5.108)

-. nW
Bobwight

5.85

Like the downspring, the bobweight increases the stick force throughout the
airspeed range and, at increased tab settings, the apparent stability or stick
force gradient.
aircraft

The bobweight has no effect on the actual stability of the

fdC /dC
V

^
Stick-'
rtti

T-H-F.

TTA ^h7TTTr

nW

FIGURE 5.50.

BOBWEIGHT

A spring may be used as an "upspring," and a bobweight nay be placed on the

opposite side of the stick in the control system.
illustrated in Figure 5.51.

Those configurations are

In this configuration, the stick force would be

decreased, and the apparent stability also decreased.

emphasized

It

should again be

that regardless of spring or bobweight configuration, there is no

effect on the actual stability (dC/dC^

*

j of the aircraft.
Stick-'
rrti

Further use of these control system devices will be discussed in Chapter 6,

Maneuvering Flight.

5.86

T-^F.

TT

Ir

-F.

-Art W

T77TT

^r-JimmmirT = CONSTANT

FIGURE 5.51.

ALTERNATE SPRING & BOBWEIGHT C0NFIGURATIC3NS

To examine the effect of the stick force gradient dFs/dV on Equation

5.102 and to find h' , Equation 5.94 is rewritten with a control system device

dC

Ch

Fs

- AqCB + q
b

Fs/q A(B

6T) -ACLq ^- ^

+ Ch

1*

&

AC.L
6T) --

*3
Stickrrtt

(5.109)
Dtvic

5
w/s

dC

BcT

(5.110)

Stickrra

d(F,/q)
-v

^T~

K
4

dC,

* StickFree

5.87

W/S

(5.111)

The eg location at which (dFs/q)/dCL goes to zero will not be the true h

when

a device such as a spring or a bobweight is included.

5.18

HIGH SPEED LONGITUDINAL STATIC STABILITY

The effects of high speed (transonic and supersonic)

on longitudinal

static stability can be analyzed in the same manner as that done for subsonic
speeds.

However, the assumptions that were made for incompressible flow are

no longer valid.
Compressibility effects both the longitudinal static stability, dCB/dCL
(gust stability) and speed stability, dFs/dV.

The gust stability depends

mainly

wing, fuselage, and tail in

on the contributions to stability

of the

the stability equation below during transonic and supersonic flight.

*>
. - + _-v^-)
aCT

Fus

5.18.1 The Wing Contribution

in subsonic flow the aerodynamic center is at the quarter chord.

At

transonic speed, flow separation occurs behind the shock formations causing
the aerodynamic center to move forward of the quarter chord position.
The
immediate effect is a reduction in stability since X/c increases.
As speed increases further the shock moves off the surface and the wing
recovers lift. The aerodynamic center moves aft towards the 50% chord
position. There is a sudden increase in the wing's contribution to stability
since X/c is reduced.
The extent of the aerodynamic center shift depends greatly on the aspect
ratio

of the aircraft.

The shift is least for low aspect ratio

aircraft.

Among the planforms, the rectangular wing has the largest shift for a given
aspect ratio, whereas the triangular wing has the least (Figure 5.52).

5.88

^-AAR-2^/\
.6

AR = 4_
.4

c
.2
0

//

i
V

^-AR = 2

2
MACH

FIGURE 5.52.

AC VARIATION WITH MACH

5.18.2 The Fuselage Contribution

in supersonic flow, the fuselage center of pressure moves forward causing
a positive increase in the fuselage dC./dCL or a destabilizing influence on
the fuselage term.

Variation

with

Mach

is

usually

small and will be

ignored.
5.18.3

The Tail Contribution

The tail contribution to stability depends on the variation of lift curve

slopes, a and at, plus downwash e with Mach during transonic and supersonic
flight.

It is expressed as:

5.89

(-at/a) VH r^

(1 - de/do)

During subsonic flight at/aw remains approximately constant.

The slope
of the lift curve, aw varies as shown in Figure 5.53. This variation of a in
the transonic speed range is a function of geometry (i.e.,
thickness, camber, and sweep).

aspect ratio,

at varies in a similar manner.

Limiting

further discussion to aircraft designed for transonic flight or aircraft which

employ airfoil shapes..with small thickness to chord ratios, then aw and a
increase slightly in the transonic regime.

For all airfoil shapes, the values

of aw and at decrease as the airspeed increases supersonically.

12.0

8.0

(RAD"1)
4.0
RECTANGULAR
WING

DELTA WING

FIGURE 5.53.

LIFT CURVE SLOPE VARIATION WITH MACH

5.90

The tail contribution is further affected by the variation in downwash,

e, with Mach

increase.

The downwash at the tail is a result of the vortex

system associated with the lifting wing.

A sudden loss of downwash occurs

transonically with a resulting increase in tail angle of attack.

The effect

requires the pilot to apply additional up elevator with increasing airspeed to

maintain

altitude.

instability.

This

(Speed

additional

stability

will

up
be

elevator
covered

contributes

more

to

thoroughly

speed
later.)

Typical downwash variation with Mach is seen in Figure 5.54.

THIN SECTION

FIGURE 5.54.

TYPICAL DOWNWASH VARIATION WITH MACH

The variation of de/d with Mach greatly influences the aircraft's gust
stability dC /dCL.

Recalling from subsonic aerodynamics,

114.6^
e

It AR

and

de

114.6aw
n~R~

Since the downwash angle behind the wing is directly proportional to the lift
coefficient of the wing, it is apparent that the value of the derivative de/da
is a function of aw.

The general trend of ds/da is an initial increase with

5.91

Mach starting

at

subsonic

approaches zero.
the

aircraft.

speeds.

Above Mach 1.0, de/da decreases and

This variation depends on the particular wing geometry of

As

shown in Figure 5.55,

de/da may dip

sections where considerable flow separation occurs.

for

thicker wing

Again, de/da is very much

dependent upon a.

0.8

STRAIGHT
i
TAPERED PLANFORM

de_
da

0.4
DELTA
PLANFORM

3.0

FIGURE 5.55.

DOHNWASH DERIVATIVE VS MACH

For an aircraft designed for high speed flight, the variation of de/da
with increasing Mach results in a slight destabilizing effect in the transonic
regime and contributes to increased stability in the supersonic speed regime;
therefore, the overall tail contribution to stability is difficult to predict.
5.92

A loss of stabilizer effectiveness is experienced in supersonic flight as

it becomes a less efficient lifting surface.
The elevator power, C
,
increases as airspeed
effectiveness <decreases.

approaches
Typical

Mach 1 .

variations in

Beyond
Ca

Mach

with Mach
6

in Figure 5.56.
ONE PIECE
HORIZONTAL TAIL

"*.

TYPICAL SWEPT WING

-c

FIGURE 5.56.

MACH VARIATIONS ON C

5.93

1,

AND C

elevator
are shown

The overall effect of transonic and supersonic flight on gust stability

or dC /dCt is also shown in Figure 5.56.
Static longitudinal stability
increases'supersonically. The speed stability of the aircraft is affected as
veil. The pitching moment coefficient equation developed in Chapter 4 can be
written,

,5 112)

C--C^c|
k
+ C*VTT+C^+C
m *n,
^^o
o
a.

<J

2U

Assuming no pitch rates, Equation 5.112 can be written

C - C &+C TT + C

(5.113)

All three of the stability

Equation 5.113 are functions of Mach.

derivatives

in

The elevator deflection required to

trim as an aircraft accelerates from subsonic to supersonic flight depends on

how these derivatives vary with Mach.
For supersonic aircraft, speed
stability is provided entirely by the artificial feel system.
usually depends on how S. varies with Mach.

However, it

A reversal of elevator deflection

with increasing airspeed usually requires a relaxation of forward pressure or

even a pull force to maintain altitude or prevent a nose down pitch tendency.
Elevator deflection versus Mach curves for several supersonic aircraft
are shown in Figure 5.57.

The important point from this figure is that

supersonically dS /dCL is no longer a valid indication of gust stability. All

of the aircraft "shown in Figure 5.57 are more stable supersonically than
subsonically, if you were to look purely at neutral points or values of the
stability derivative CB .

5.94

MACH
0.6

0.8

'FIGURE 5.57.

Whether the

STABILIZER DEFLECTION VS MACH FOR

SEVERAL SUPERSONIC AIRCRAFT

speed instability or

reversal

in elevator deflections and

stick forces are objectionable depends on many factors such as magnitude of

variation, length of time required to transverse the region of instability,
control system characteristics, and conditions of flight.
In the F-100C, a pull of 14 pounds was required when accelerating from
Mach 0.87 to 1.0.

The test pilot described this trim change as disconcerting

while attempting to maneuver the aircraft in this region and recommended that
a "q" or Mach sensing device be installed to eliminate this characteristic.
Consequently,

mechanism

was

incorporated

to

automatically

change

the

artificial feel gradient as the aircraft accelerates through the transonic

range.

Also, the longitudinal trim is automatically changed in this region by

the use of a "Mach Trimmer."

5.95

F-104 test pilots stated that F-104 transonic trim changes required an
aft stick movement with increasing speed and a forward stick movement when
decreasing speed but described this speed instability as acceptable.
F-106 pilots stated that the Mach 1.0 to 1.1 region is characterized by a
moderate

trim

change

to

avoid

large

variations

in

altitude

during

accelerations. Minor speed instabilities were not unsatisfactory.

T-38 test pilots described the transonic trim change as being hardly
perceptible.
Aircraft design considerations are influenced by the stability aspects of
high speed flight.

It is desirable to design an aircraft where trim changes

through transonic speeds are small.

incidence

or

A tapered wing without camber, twist, or

a low aspect ratio wing and tail provide values of Xv/c, aw,

at, and de/da which vary minimally with Mach.

gives negligible variation of CB

with

An

Mach and

all-moving tail
maximum

(slab)

control effec-

tiveness.

full power, irreversible control system is necessary to counter-

act the erratic changes in pressure distribution which affect C^ and C^.

5.19

LONGITUDINAL STATIC STABILITY FLIGHT TESTS

The purpose of these flight tests is to determine the longitudinal static

stability characteristics of an aircraft.

stability,

These characteristics include gust

speed stability, and friction/breakout.

Trim change tests will

also be discussed.
An aircraft is said to be statically stable longitudinally (positive gust
stability) if the moments created when the aircraft is disturbed from trimmed
flight tend to return the aircraft to the condition from which it was
disturbed.
Longitudinal stability theory shows the flight test relationships
for stick-fixed and stick-free gust

stability, d^/dq, to be

5.96

uo

stick-fixed:

-&r- -

dC

.1

- =c,- -&
dCl
0

stick-free:

d(Fs/M
/q)
-
dC
,

-m

-A\
-A
*_

(5.69)

'stick
rix.d

dC

_m

(5.101)
Stick
rtM

Stick force (Fs), elevator deflection (S#), equivalent velocity (V#) and
gross weight (W) are the parameters measured to solve the above equations.
When dS /dC is zero, an aircraft has neutral stick-fixed longitudinal static
stability.
As d5#/dCL increases, the stability of the aircraft increases.
The same statements about stick-free longitudinal static stability can be made
with respect to d(F /q)/dCL. The neutral point is the eg location which
gives neutral stability, stick-fixed or stick-free. These neutral points are
determined by flight testing at two or more eg locations, and extrapolating
the curves of dS#/dCL and d(Fs/q)/dCL versus eg to zero.
The neutral point so determined is valid for the trim altitude and
airspeed at which the data were taken and may vary considerably at other trim
conditions. A typical variation of neutral point with Mach is shown in Figure
5.63.

5.97

AFT

hn (9b mac)

FWD
MACH

1.0

5.63.

STICK-FIXED NEUTRAL POINT VERSUS MACH

The use of the neutral point theory to define gust stability is therefore time
consuming.
This is especially true for aircraft that have a large airspeed
envelope and aeroelastic effects.
Speed stability is the variation in control stick forces with airspeed
changes.

Positive

stability requires

that

increased aft stick force be

required with decreasing airspeed and vice versa.

stability but may be

considerably different

stability augmentation systems.

due

It is related to gust
to

artificial

feel

and

Speed stability is the longitudinal static

stability characteristic most apparent to the pilot, and it therefore receives

the greatest emphasis.
Flight-path stability is defined as the variation in flight-path angle
when the airspeed is changed by use of the elevator alone.

Flight-path

stability applies only to the power approach flight phase and is basically
determined by aircraft performance characteristics.
Positive flight-path
stability ensures that the aircraft will not develop large changes in rate of
descent when corrections are made to the flight-path with the throttle fixed.
The exact limts are prescribed in MIL-STD-1797A, paragraph A.3.1.2

5.98

An aircraft likely to encounter difficulty in meeting these limits would be

one whose power approach airspeed was far up on the "backside" of the power
required curve.
airspeed,

A corrective action might be to increase the power approach

thereby

placing

it

on

flatter

portion

of

the

curve

or

by

installing an automatic throttle.

5.19.1 Military Specification Requirements

The iyi>4 version of MIL-F-8785, the specification which preceeded the
current MIL-STD-1797A, established longitudinal stability requirements in
terms of neutral point. This criteria is still valid for testing aircraft
with reversible control systems, and many aircraft with irreversible control
systems which are not highly augmented. MIL-STD-1797A is not written in terms
of neutral point. Instead, section A.4.1 specifies longitudinal stability
with respect to speed and
pitch attitude.
speed

range

The requirements of this section are relaxed in the transonic

except

for

transonic operation.

those

aircraft

which

are

designed

for

prolonged

As technology progresses, highly augmented aircraft and

aircraft with fly-by-wire control systems may be designed with neutral speed
stability.

The F-15,

speed stability.

F-16, and F-20 are examples of aircraft with neutral

For these aircraft, the program manager may require a mil

spec written specifically for the aircraft and control system involved.
5.19.2

Flight Test Methods

There

are

deceleration)
5.19.2.1

two

used

general

test

to determine

Stabilized Method.

methods

either

(stabilized

speed

and

acceleration/

stability or neutral points.

This method is used for aircraft with a small

airspeed range in the cruise flight phase and virtually all aircraft in the
power approach, landing or takeoff flight phases.

Propeller type aircraft are

normally tested by this method because of the effects on the elevator control
power

caused

by

thrust

changes.

It

involves

data

taken

at

stabilized

airspeeds at the trim throttle setting with the airspeed maintained constant
by a

rate

of descent

or

climb.

As

long as

the

altitude does

not vary

excessively (typically + 1,000 ft) this method gives good results, but it is
time consuming.
5.99

The aircraft is trimmed carefully at the desired altitude and airspeed,

and a trim shot is recorded.

Without moving the throttle or trim setting, the

pilot changes aircraft pitch attitude to achieve a lower or higher airspeed

(typically in increments of 10 knots) and maintains that airspeed.
Aircraft with both reversible and irreversible hydro-mechanical control
systems exhibit varying degrees of friction and breakout force about trim,
the breakout

force

is the force

required to begin a tiny movement of the

stick. ' This initial movement will not cause an aircraft motion as observed on
the windscreen.

The friction force is that additional amount of pilot-applied

force required to produce the first tiny movement of the elevator.

These

small forces, friction and breakout, combine to form what is generally termed
the "friction band."
Since the pilot has usually moved the control stick fore and aft through
the friction band, he must determine which side of the friction band he is on
before recording the test point data.

The elevator position for this airspeed

will not vary, but stick force varies relative to the instantaneous position
within the friction band at the time the data is taken.

Therefore, the pilot

should (assuming an initial reduction in airspeed from the trim condition)

increase force carefully until the nose starts to rise.
frozen at

this

point

(where

any

increase

in

stick

The stick should be

force will

result

elevator movement and thus nose rise) and data recorded.

The same technique

should be used for all other airspeed points below trim.

For airspeed above

trim airspeed, the same technique is used although now the stick is frozen at
a point where any increase in push force will result in nose drop.
5 19 2.2 Acceleration/Deceleration Method.
This method is commonly used for
aircraft that have a large airspeed envelope.
testing.

It is less time consuming than the stabilized method but introduces

thrust effects.
maintains
airspeed.

It is always used for transonic

the

The U.S. Navy uses the acceleration/deceleration method but

throttle

setting

constant

and

varies

altitude

to

change

The Navy method minimizes thrust effects but introduces altitude

effects.

5.100

The sense trim shot is token as in the stabilized method to establish trim
conditions.

MIL-STD-1797requires that the aircraft exhibit positive speed

stability only within 50 knots or 15 percent of the trim airspeed, whichever

is the less.

This requires very little power change to traverse this band and

maintain level flight unless the trim airspeed is near the back side of the
thrust required curve.

Before the 1968 revision to MIL-F-8785, the flight

test technique commonly used to get acceleration/deceleration data was full

military power or idle, covering the entire airspeed envelope.

Unfortunately,

this technique cannot be used to determine the requirements under the current
standard with the non-linearities that usually exist in the control system.
Therefore a series of trim points must be selected to cover the envelope with
a typical plot (friction and breakout excluded) shown in Figure 5.64.

TEST
ALTITUDE
SERVICE
ENVELOPE

FIGURE 5.64.

SPEED STABILITY DATA

5.101

The aost practical method of taking data is to note the power setting
required for trim and then either decrease or increase power to overshoot the
data band limits slightly.

Then turn on the instrumentation and reset trim

power, and a slow acceleration or deceleration will occur back towards the
trim point.

The

data will

be

deceleration with trim power set.

valid

only during

the

acceleration

or

A small percent change in the trim power

setting, may be required to obtain a reasonable acceleration or deceleration

without introducing gross power effects.

The points near the trim airspeed

point will be difficult to obtain but they are not of great importance since
they will probably be obscured by the control system breakout and friction
(Figure 5.65).

ACCEL

PULL

DECEL

FRICTION AND
BREAKOUT BAND
F,(lb)

PUSH -:

nGURE 5.65.

ACrFT.KRATION/DErTT.KRATION DATA
ONE.TRIM SPEED, eg, ALTITUDE

5.102

Throughout
control

is

reversed

the

stick

during

acceleration or deceleration,

force.
the

It

is

test run.

important
A

that

the primary parameter

the

friction band not

slight change in

altitude is

to
be

preferable

(i.e. to let the aircraft climb slightly throughout an acceleration) to avoid

the

tendency to

reverse

the

stick

force by over-rotating

the

nose.

The

opposite is advisable during the deceleration.

There
transonic

is
area

operation.
determine

relaxation
unless

the

in the

requirement

aircraft

is

for

designed

speed
for

stability in

continued

the

transonic

The best way to define where the transonic range occurs is to

the

point where

MIL-STD-1797A allows

the

Fs

versus V goes unstable.

specified maximum instability.

The

In this area,
purpose

of

the

transonic longitudinal static stability flight test in the transonic area is

to determine the degree of instability.
The transonic area flight test begins with a trim shot at some high
subsonic

airspeed.

The

power

is

increased

to

maximum

thrust

and

an

acceleration is begun.
It is important that a stable gradient be established before entering the
transonic area.

Once the first sensation of instability is felt by the pilot,

his primary control parameter changes from stick force to attitude.

From this

point until the aircraft is supersonic, the true altitude should be held as
closely as possible.

This is because the unstable stick force being measured

will be in error if a climb or descent occurs.

A radar altimeter output on an

over-water flight or keeping a flight path on the horizon are precise ways to
hold constant altitude, but.if'these are not available, the pilot will have to
use the outside references to maintain level flight.
Once the aircraft goes supersonic,

the test pilot should again concern

himself with not reversing the friction band and with establishing a stable
gradient.

The acceleration should be continued to the limit of the service

envelope to test for supersonic speed stability.

The supersonic data will

also have to be shown at 15% of the trim airspeed, so several trim shots may
be required.
careful

A deceleration from VMX to subsonic speed should be made with a

reduction

in power to decelerate

supersonically and transonically.

The criteria for decelerating through the transonic region are the same as for

5.103

the acceleration.

Power reductions during this deceleration will have to be

done carefully to minimize thrust effects and still decelerate past the Mach
drag rise point to a stable subsonic gradient.
5.19.3 Fli^it-Path Stability
Flight-path stability is
landing

qualities.

It

is

criterion applied to power

primarily

determined

by

the

approach and
performance

characteristics of the aircraft and related to stability and control only

because it places another requirement on handling qualities.
one way to look at flight-path stability.

The following is

Thrust required curves are shown

for two aircraft with the recommended final approach speed marked in Figure
5.66.

Ul

cc

5
o
ui
cc

f-

CC
X

A~

om,<PA>

FIGURE 5.66.

THRUST REQUIRED VS VELOCITY

(TWO AIRCRAFT)

5.104

If both aircraft A and B are located on the glidepath shown in Figure 5.67,
their relative flight-path stability can be shown.

POSITION 1

PRECISION
GLIDEPATH

POSITION 2

FIGURE 5.67.

AIRCRAFT ON PRECISION APPROACH

At Position 1 the aircraft are in stable flight above the glidepath, but
below

the

recommended

final

approach

speed.

If

Aircraft A

is

in

this

position, the pilot can nose the aircraft over and descend to glidepath while
the airspeed increases.

Because the thrust required curve is flat at this

point, the rate of descent at this higher airspeed is about the same as before
the correction, so he does not need to change throttle setting to maintain the
glidepath.
differently.

Aircraft B,

under the same conditions,

will have to be

flown

If the pilot noses the aircraft over, the airspeed will increase

to the recommended airspeed as the" glidepath is reached.

5.105

The rate of descent

at this power setting is less than it was before so the pilot will go above
glidepath if he maintains this airspeed.
At Position 2 the aircraft are in stable flight below the glidepath but
above the recommended airspeed.

Aircraft A can be pulled up to the glidepath

and maintained on the glidepath with little or no throttle change. Aircraft B

will develop a greater rate of descent once the airspeed decreases while
coming up to glidepath and will fall below the glidepath again.
If the aircraft are in Position 1 with the airspeed

higher

than

recommended instead of lower, the same situation will develop when correcting
back to flight-path, but the required pilot compensation is increased.

In all

cases Aircraft A has better flight-path stability than Aircraft B.

As
mentioned earlier in this chapter, aircraft which have unsatisfactory
flight-path stability can be improved by increasing the recommended final
approach airspeed or by adding an automatic throttle.
Another way of looking at flight-path stability is by investigating the
difficulty that a pilot has in maintaining glidepath even when using the
throttles.

This problem is seen in large aircraft for which the time lag in

pitching the aircraft to a new pitch attitude is quite long.

In these

instances, incorporation of direct lift allows the pilot to correct the

glidepath without pitching the aircraft. Direct lift control will also affect
the influence of performance on flight-path stability.
5.19.4 Trim Change Tests
The purpose of this test is to determine the control- force changes associated
with normal configuration changes, trim system failure, or transfer to
alternate control systems in relation to specified limits.
determined that no undesirable flight characteristics

It must also be
accompany these

configuration changes.
Pitching moments on aircraft are normally associated
with changes in the condition of any of the following: landing gear, flaps,
speed brakes,

power,

jettisonable device.

bomb bay doors,

rocket and missile

doors,

or

any

The magnitude of the change in control forces resulting

from these pitching moments is limited by MIL-STD-1797A, and it is the

responsibility of the testing organization to determine if the actual forces
are within the specified limits.
5.106

The pitching moment resulting from a given configuration change will

normally vary with airspeed, altitude, eg loading, and initial configuration
of the aircraft.

The control forces resulting from the pitching moment will

further depend on the aircraft parameters being held constant during the
configuration change.

These factors should be kept in mind when determining

the conditions under which the given configuration change should be tested.
Even

though

the

specification

lists

the

altitude,

airspeed,

initial

conditions, and parameter to be held constant for most configuration changes,

some variations may be necessary on a specific aircraft to provide information
on the most adverse conditions encountered in operational use of the aircraft.
The altitude and airspeed should be selected as indicated in the specifications or for the most adverse conditions. In general, the trim change will be
greatest at the highest airspeed and the lowest altitude.
The effect of eg
location is not so readily apparent and usually has a different effect for
each configuration change.

A forward loading may cause the greatest trim

change for one configuration change, and an aft loading may be most severe for
another, using the build up approach, a mid eg loading is normally selected
since rapid movement of the eg in flight will probably not be possible. If a
specific trim change appears marginal at this loading, it may be necessary to
test it at other eg loadings to determine its acceptability.
Selection of the initial aircraft configurations will depend on the
anticipated normal operational use of the aircraft. The conditions given in
the specifications will normally be sufficient and can always be used as a
guide, but again variations may be necessary for specific aircraft. The same
holds true for selection of the aircraft parameter to hold constant during the
change. The parameter that the pilot would normally want to hold constant in
operational use of the aircraft is the one that should be selected.
Therefore, if the requirements of MIL-STD-1797A do not appear logical or
complete, then a more appropriate test should be added or substituted.
In addition to the conditions outlined above, it may be necessary to test
for

some

configuration

changes

that

5.107

could

logically

be

accomplished

simultaneously. The force changes might be additive and could be objectionably large. For example, on a go-around, power may be applied and the landing
gear retracted at the same time.
If the trim changes associated with each
configuration change are appreciable and in the same direction, the combined
changes should definitely be investigated.

The specifications require that no

objectionable buffet or undesirable flight chracteristics be associated with

normal trim changes.

Some buffet is normal with some configuration changes,

e.g., gear extension, however, it would be considered if this buffet tended to

mask the buffet associated with stall warning.

The input of the pilot is the

best measure of what actually constitutes "objectionable," but anything that

would interfere with normal use of the aircraft would be considered objectionable.

The same is true for "undesirable flight characteristics."

An example

would be a strong nose-down pitching moment associated with gear or flap

retraction after take-off.
The standard also sets limits on the trim changes resulting from transfer
to an alternate control system.

The limits vary with the type of alternate

system and the configuration and speed at the time of transfer but in no case
may a dangerous flight condition result.
A good example of this is the
transfer to manual reversion in the A-10.
the pilot

to

study the operation of the

It will probably be necessary for

control

system and methods

of

effecting transfer in order to determine the conditions most likely to cause

an unacceptable trim change upon transferring from one system to the other.
As in all

flight testing,

a thorough knowledge of the aircraft and the

objectives of the test will improve the quality and increase the value of the
test results.

5.108

PROBLEMS
5.1

In Subsonic Aerodynamics the following approximation was developed for

the Balance Equation

"eg

where C

+ CL (eg - ac) + CB

"ac

tail

is the total stability contribution of the tail.

"tail

(a)

Sketch the location of the forces, moments,

and eg required to

balance an airplane using the above equation.

(b)

Using the data shown below, what contribution is required from the
tail to balance the airplane?
C

-0.12

Cn

tail

(c)

ac

0.25 (25%)

eg

0.188 (18.8%)

CL

0.5

If this airplane were a fixed hang-glider and a CL

1.3 were
required to flare and land, how far aft must the eg be shifted to
obtain the landing CL without changing the tail contribution?

(d)

If a 4% margin were desired between max aft eg

and the

aerodynamic

center for safety considerations, how much will the tail

contribution have to increase for the landing problem presented in
(c)?
(e)

List four ways of increasing the tail contribution to stability.

5.109

5.2

Given the aircraft configurations shown below, write the Balance and

Stability Equations for the thrust contributions to stability.

Sketch

forces

and which

involved

and

state

which

effects

are

stabilizing

the
are

destabilizing.
(a)

5.3

DC-10

(b)

Britten-Norman Trislander

(a)

Are

the

two expressions below derived

for

the

tail-to-the-rear

aircraft valid for the canard aircraft configuration?

c?

HC:

h - h

Static margin
5.110

- h

(b)

Derive

an expression

configuration.

(c)

for

for

the

canard aircraft

Determine its sign.

What is the expression for elevator effectiveness

aircraft configuration?

5.4

elevator power

for the canard

Determine its sign.

During wind tunnel testing, the following data were recorded:

Ci.

(Steg)

(Seg)

h - 0.20

h - 0.25

(eg)
h - 0.30

0.2

-2

-3

-4

0.6

-2

1.0

10

Elevator limits are + 20

A.

Find the stick-fixed static margin for h = 0.20

B.

Find the numerical value for elevator power.

C.

Find the most forward eg permissible if it is desired to be able to

stabilize out of ground effect at a C

Ljnax

5.5

1.0.

Given the flight test data below from the aircraft which was wind tunnel

tested in Problem 5.4, answer the following questions:

5.111

A.
Find the
conservative?

ai

rcraft

neutral

point.

Was

the

wind

tunnel

data

eg @ 10% mac
25 -

20-

(DEG}

- C,

TRIM

B.

What is the flight test determined value of C^

If the lift curve slope is determined to be 0.1, vhat is the flight

test determined value of C,

(per deg) at a eg of 25%?

5.112

5.6

Given geometric data for the canard design in Problem 5.5, calculate an
estimate for elevator power.
HINT:

Assume:

- *

n,

a^

HT

wind tunnel value

l\ - 1.0
Given:
Slab canard
1,
ST
c
S
5.7

- 14ft
. 10ft2
-

7ft
200ft2

The Forward Swept Wing (FWS) technology aircraft designed by North

American is shown below.
The aerodynamic load is shared by the two
"wings"

with the

aircraft weight.

forward wing designed to carry about 30% of the

For

this

design

condition,

answer

the

following

multiple choice questions by circling the number of correct answer(s).

RW

5.113

A.

B.

At the eg location marked FWD the aircraft:

(1)
(2)

Can be balanced and is stable.

Can be balanced and is stable only if the eg is ahead of the neutral

(3)

point.
Can be balanced and is unstable.

(4)

Cannot be balanced.

At the eg location marked MID the aircraft:

(1)

Can be balanced and is stable.

(2)

Can be balanced and is stable only if the eg is ahead of the neutral

(3)

point.
Can be balanced and is unstable.

(4)

Cannot be balanced.

At the eg location marked AFT the aircraft:

(1) Can be balanced and is stable.
(2) Can be balanced and is stable only if the eg is ahead of the neutral
(3)

point.
Can'be balanced and is unstable.

(4)

Cannot be balanced.

For this design the sign of control power is:

(1)

Negative.

(2)
(3)

Positive.
Dependent on eg location.

5.114

5.8

Given the Boeing 747/Space Shuttle Columbia combination as shown below,

is

the

total

shuttle

orbiter

wing

contribution

to

stabilizing or destabilizing if the eg is located as

explain the reason for the answer given.

SHUTTLE a.c. @ 50% iiMAC

747 a.c. @ 25% MAC

5.115

the

combination

shown?

Briefly

5.9

Given below is wind tunnel data for the YF-16.

.10-

WINGAND
FUSELAGE

.05-

TOTAL
AIRCRAFT
.35

r-cL

-.05

-.10-

A.

Answer the following questions YES or NO:

Is the total aircraft stable?
Is the wing-fuselage combination stable?
Is the tail contribution stabilizing?
5.116

B.

How mich larger (in percent) would the horizontal stabilizer have to be
to give the F-16 a static margin of 2% at a eg of 35% MAC?

Assume all

other variables remain constant.

5.10 Given below is a C

versus CT curve for a rectangular flying wing from

wind tunnel tests and a desired TOTAL AIRCRAFT trim curve.

0.1 -i

"0.15

o-

cL
TOTAL
AIRCRAFT

-0.1

A.

FLYING
WING

Does the flying wing need a TAIL or CANARD added or can it attain ths
required TOTAL aircraft stability level by ELEVON deflection?

Is the TOTAL aircraft stable?

C.

Is CB

positive or negative?
m

D.

Does the flying wing have a symmetric wing section?

5.117

E.

What is the flying wing's neutral point?

F.

Is C

G.

What is the static margin for this trim condition?

positive or negative?

5.11 Given the flight test data shown below, show how to obtain the neutral
point(s).

Label the two cg's tested as FWD and AFT.

5.

5.118

5.12 Given the curve shown below, show the effect of:
A.

Shifting the eg FWD and retrimning to trim velocity.

B.

Increasing C^

C.

Adding a downspring and retrimning to trim velocity.

D.

Adding a bobweight and retrimning to trim velocity.

and retrimning to trim velocity.

-Ve

5.13

Which changes in Problem 5.12 affect:

apparent stability

5.119

actual stability

5.14

Read the question and answer true (T) or false (F).

if a body disturbed from equilibrium remains
position it is statically unstable.

in

the

disturbed

F ~ Longitudinal static stability and "gust stability"

thing.

F -

Static longitudinal stability

longitudinal stability.

Aircraft response in the X-Z plane (about the Y axis) usually cannot
be considered as independent of the lateral directional motions.

Although

is

a. direct

is

prerequisite

indication

of

are the same

for

longitudinal

dynamic

static

stability, C>

is relatively unimportant.
c

At the USAF Test Pilot School elevator TEU and nose pitching up are
positive in sign by convention. CAREFUL.

Tail efficiency factor and tail volume coefficient are not normally
considered constant.

With the eg forward, an aircraft is more stable and maneuverable.

There are no well defined static stability criteria.

F To call a canard surface a horizontal stabilizer is a misnomer.

The most stable wing contribution to stability results from a low

wing forward of the eg.

With the eg aft, an aircraft is less maneuverable and more stable.

A thrust line below the eg is destabilizing for either a prop or

turbojet.

The normal force contribution of either a prop or turbojet is

destabilizing if the prop or inlet is aft of the aircraft eg.

A positive value of downwash derivatives causes a tail-to the-rear

aircraft to be less stable if de/da is less than 1.0 than it would
be if dc/do were equal to zero.

Verifying adequate stability and maneuverability at established eg

limits is a legitimate flight test function.

An aircraft is balanced if it is forced to a negative value of Cm

CG
for some useful positive value of CL.
5.120

An aircraft is considered stable if dCB

The slope of the C

/dCL is positive.

versus CT curve of an aircraft is a direct

eg

measure of "gust stability."

T

Aircraft center of gravity position is only of secondary importance

when discussing longitudinal static stability.

Due to the advance control system technology such as fly-by-wire, a

basic knowledge of the requirements for natural aircraft stability
is of little use to a sophisticated USAF test pilot.

Most contractors would encourage an answer of true (T), to the above

question.

For an aircraft with a large vertical eg travel, the chordwise force

contribution of the wing to stability probably cannot be neglected.

FWD and AFT eg limits are often determined from flight test.

A canard is a hoax.

The value of stick-fixed static stability is equal to eg minus hn in

percent MAC.

The stick-fixed static margin is stick-fixed stability with the sign

reversed.

A slab tail (or stabilizer) is a more powerful longitudinal control

than a two-piece elevator.

Elevator effectiveness and power are the same thing.

Elevator
effectiveness
configuration.

Static
margin
configuration.

Elevator power is positive in sign for either a tail-to-the-rear or

a canard configuration.

Upwash causes a canard to be more destabilizing.

The main effect on longitudinal stability when accelerating to

supersonic flight is caused by the shift in wing ac from 25% MAC to
about 50% MAC.

The elevator of a reversible control system is normally statically

balanced.

is

is

negative

5.121

negative

for

in

sign , for

statically

canard

stable

canard

is

always

negative

and

is

known

as the "restoring" moment

coefficient.
(^
is always

negative

and

is

known

as the "floating" moment

coefficient.
With no pilot applied force a reversible elevator will "float" until
hinge moments are zero.

F:

A free elevator factor of one results

stick-free stability being the same.

Generally, freeing the elevator is destabilizing (tail-tc-the-rear).

Speed stability and stick force gradient about trim are the same.

Speed stability and apparent stability are the same.

Speed stability and stick-free stability are the same.

Dynamic control balancing is making

in

the

small

stick-fixed

or

just

and

slightly

positive.
eg movement affects real and apparent stability (after retrimming).

Aerodynamic balancing affects real and apparent stability (after

retrimming).

Fr

Downsprings and bobweights affect real and apparent stability (after

retrimming).

In general, an aircraft becomes more stable supersonically which is

characterized by an AFT shift in the neutral point.

Even though an aircraft is more stable supersonically (neutral point

further AFT) it may have a speed instability.

The neutral point of the entire aircraft

aerodynamic center of the wing by itself.

Aerodynamic balancing is "adjusting" or "tailoring"

Neutral point is a constant for a given configuration and is never a

function of CL.

The eg location where dF /dV - 0 is the actual stick-free neutral

point regardless of control system devices.

5.122

is

analogous
C^

to

the

and C^ .

The effects of elevator weight on hinge moment coefficient are

normally eliminated by static balance.

A positive 8# is a deflection causing a nose-up pitching moment.

A positive q is one defined as deflecting or trying to deflect the

elevator in a negative direction.

C^

(^

is normally negative (tail-to-the-rear) .' .

is always negative (tail-to-the-rear and canard)

C^

and Cjj
a

are under control of the aircraft designer and can be

i

varied to "tailor" stick-free stability characteristics.

T

C^

and C^
a

are identical.
a

Cjj

dFs/dV does not necessarily reflect actual stick-free stability

characteristics.

There are many ways

characteristics.

A bobweight can only be used to increase stick-force gradient.

CB

and (^

are not the same.

to

alter

an

aircraft's

cannot be determined from flight test.

5.123

speed

stability

ANSWERS

5.1.

b.

+0.151

"t.ii

c.

eg

d.

0.226 or 22.6%
-

+.172, 14% inc.

tail

5.4.

5.5.

A.

SM

B.

C.

A.

hn

B.

C.

0.15
-

.01/deg or .57/rad

10%
-

30%
-

0.01/deg

-0.005/deg

5.6.

5.9.

B.

5.10.

0.01/deg

50% larger

E.

hn

0.25

G.

SM

0.10

5.124

- /. ."

Bibliography
1.

Perkins, C. D., and Hage, R. E., Airplane Performance Stability and

Control, John Wiley & Sons, Inc., New York NY, 1949.

2.

Phillips, W. H., Appreciation and Prediction of Flying Qualities, NACA

Report 927, National Advisory Committee for Aeronautics, Washington DC,
1949.

3.

Etkin, B., Dynamics of Atmospheric Flight, John Wiley & Sons, Inc.,
New York NY, 1972.

A.

Kolk, W. R., Modern Flight Dynamics, Prentice-Hall, Inc., Englewood

Cliffs NJ, 1961.

5.

Perkins, CD., Editor, Flight Testing, Volume II, Stability and Control,
North Atlantic Treaty Organization, Pergamon Press, New York NY, 1962.

6.

Anon., Military Standard, Flying Qualities of Piloted Aircraft, MILSTD-1797A, Department of Defense, Washington DC 20402, 30 Jan 90.

5.125