Stability in a body stabilized by fins and suspended from an airplane


Material Information

Stability in a body stabilized by fins and suspended from an airplane
Alternate Title:
NACA wartime reports
Physical Description:
23, 9 p. : ill. ; 28 cm.
Phillips, William H
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:


Subjects / Keywords:
Air-speed indicators   ( lcsh )
Aerodynamics -- Research   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Summary: A theoretical investigation has been made of the oscillations performed by suspended bodies of the type commonly used for trailing airspeed heads ans similar towed devices. The primary purpose of the investigation was to design an instrument that will remain stable as it is drawn up to a support underneath an airplane without attention on the part of the pilot. Flight tests of a model airspeed head were made to supplement the theoretical study. Unstable oscillations of the body at short cable lengths were predicted by the theory, but the rate of increase of amplitude of these oscillations was very small. In flight tests, more violent types of instability were believed to be caused by unsteady or nonuniform air flow in the region where the cable was lowered from the airplane. No practical method was found to provide large damping of the oscillations at short cable lengths, but the degree of stability present in a suitably designed suspended body was shown to be satisfactory if the body was lowered into a uniform air stream.
Includes bibliographic references (p. 23).
Statement of Responsibility:
W.H. Phillips.
General Note:
"Report no. L-28."
General Note:
"Originally issued April 1944 as Advance Restricted Report L4D18."
General Note:
"Report date April 1944."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003617553
oclc - 71341084
sobekcm - AA00006298_00001
System ID:

Full Text


April 1944 as
Advance Restricted Report L4D18

By W. H. Phillips

Langley Memorial Aeronautical Laboratory
Langley Field, Va.


NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
advance research results Lo an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.



Lylb 9


Digitized by the Internet Archive
in 2011, with funding from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation






By W. H. Phillips


A theoretical investigation has been made of the
oscillations pnerfor'.ed by suspended bodies of the type
co.mmonly used for traili,ar airspeed heads and similar
towed devices. The ori'mairy p'rpFos of the investiga-
tion v'as to design an instrLunirit that will remain stable
as it is d swnL up to a support u nd rneath an airplane
without attention on the part of the nilot. Flight
tests of a model airspeed head were .iade to supplement
the theoretical study. Unstable oscillations of the
body at short cable lengths were predicted by the theory,
but the rate of increase o' amplitude of these oscilla-
tions was very small. In flight tests, more violent
types of instability vere believed to te caused by
unsteady or nonuniform air flow in the region where the
cable was lowered from tlih airplane. iTo practical
method was found to provide large dampinE of the oscil-
lations at short cable lengths, but the degree of
stability present in a suitably designed suspended
body was shown to be satisfactory if the body was
lowered into a uniform air stream.


Suspended devices that consist of heavy streamline.
bodies stabilized by fins have be-n used in the past
for various purposes. A frequent application of this
type of device is the suspended airspeed head used for
the accurate measurement of airplane speed (reference 1).
Certain difficulties have been encountered in the use of
these instruments because of unstable oscillations of
the cable and suspended body. One co:,rnon type of
instability has been a tendency of the instrument to
swing violently back and forth and from side to side as
it was drawn up close to the airplane. Because of this


I'..CA AFR No. L4D18

tendency, these instruments need considerable attention
in handling and usually require the services of a person
other than the pilot. Another type of instability has
been an oscillation of the body and x.hipping action of
the cable when the body was being towed at the full
length of the cable. This motion has occurred only
when the instrument was lowered from certain airplanes.

The present investigation was undertaken in an
effort to develop a type of trailing airspeed head that
could be lowered from and drawn up to a support under-
neath the airplane without close attention. This
requirement necessitates that unstable oscillations of
the instrument be avoided at any cable length.

A study of both lateral and longitudinal oscilla-
tions of the body and cable system was made in refer-
ence 2. This study was based on the assumption that
the damping of the motion due to air forces on the
cable could be neglected. The present investigation
shows that this ass-umption leads to :erroneous conclusions
with regard to the boundaries of stability.


m mass of towed body

y lateral displacement

Cy coefficient of side force on vertical tail

Yc lateral displacement of cable element
I angle of yaw

N yawing moment

P angle of sideslip

V forward velocity

I tail length

Y lateral force

C moment of inertia about vertical axis
/dCa sloe of tail lift curve (negative
a slope of tail lift curve (negative) y--dy


S vertical-tail area

p air density
Z '-:rtical distance between airplane and body

z v'er t!ic l distance

g icceleration of Cravitv

D effec-iiv drag, also, operator indicating differ-;ntia-
tion with resooct to time

D' nccndirrmensional oDerator (DT)

D dcira:c 0 body

Dc total cable drag

C dra- coefficie.-nt of bod 7 ---
Db (A3)

CD eciui'salent lag croeffi'cient cf .cb!ile
CT LN, z J ((

CD .effecti ve drag cos'fficient D' + Dt

X Ltc-rizorital distAnce between airplane and body
cd d-rag coeff'LciLeent of cable ;:er unit vertical
c /cabl0 drcae
height ----
\ R",I'wcz /

i relative-densit-7 coefficient ( 7)

R ratio of tail l -ntr, to vertical distance below
airplane (1/1)

C1 miom-nt-of-irertia factor (k/L)2










radius of ;:rL .-t-ion


time unit =I

coefficients of quartic

cable diameter

angle of cable with horizontal

fraction of cable side force applied to body

side force on body

side force on cable

period of oscillation

number of cycles to damp to one-half

( m o3
S1 6Y

y. -V

N 6N
y nt

1 6Y
N -1

A dot over a symbol indicates the first derivative
of the quantity with respect to time t and two dots
over a symbol indicates the second derivative.


Because of the axial c-Tin-.rtry of a trailing airspeed
head, its longitudinal and lateral motions occur

NACA ARE No. L4D18 5

independently. These types of motion miay, therefore,
be treated separately. The lateral motion of the
instrument will be analyzed in considerable detail
because this mode of motion is theoretically .nost
likely to become unstable.

Lateral Oscillations

Mathematical treatment is possible only for the
case of small oscillations, fo' which the forces acting
on the body var, linarly with the displacemmnts and
angular velocities. The instrument will swinrr from
side to side like a pendulum but, for small amplitudes,
its motion may be considered to take place in a hori-
zontal plare. A restoring fcrce depending on the
cable length under consideration '.vill be assumed to act
through ;-he oivot point.

The subsequent analysis indicates that tne drag
force on the cablEs -Cid todt hts an important influence
on the damipinr cf the oscillations. In practice,
al.nost all the drag acts si, the cable. For purposes
of naalysis, however, an e fectivs drag force due to
the cable will be assumed to act on the body at its
center of :ravity. The relation between this effective
drag force and the characteristics of the cable will
be discs-.sed later.

Th3 nctation used in considering the lateral motion
is shown in figure 1. The equations of motion with
respect to a fixe d system of axes are as follows:

yop 6Y 6Y
op- y
0 7 o 6*

In order to s'imlify the notation, let D = d
and define the stability derivatives

1 6Y

1 CS


and define the other derivatives similarly. These
equations may be solved by the usual procedure of
setting the determinant of the coefficients equal to
zero. This determinant may be expanded to give the

aD4 + bD3 + cD2 + dD + e = 0 (1)


a = 1

b = -.. -

c = N + T v k+

d = -NpY- + Y-,

e = -YyNp

In order to find the nature of the motion from
equation (1), it is necessary to evaluate the stability
derivatives in terms of the dimensions and aerodynamic
characteristics of the. instrument. Tn setting up a
simplified form for the stability equation, it is
sufficiently accurate to assume that aerodynamic forces
other than drag forces will act only on the vertical
fin of the instrument. The derivation of the expression
for YP is given as an example:


= pa V2S
a -

-Y a S2

Y3 m op
a-V S

n 2

IN.CA AER No. L4D8 '

The other aerodynamic derivatives may be determined in
a similar manner. Only the derivatives related to the
forces exerted by the cable require special consideration.
The Y-force caused by a lateral displacement of the body
is found by assumiing that the body and cable system,
when _viewed front, the front, deflects as a simple
pendiilu. (fig. '). The restoring force c;ue to a small
deflection y of the body suspended a vertical dis-
tance Z below the -irlplane is

Y = --HE

1 6y
Y =
y m y


The derivative Y- is found fraon- the drag force
acting on the body andd cable. A drag force acting on
the body will have a component of side force as shown
in figure 3. Thus

Y = ,( + W)

r Pv23

= CDb -iVS (2)
67 .'

The component of side force due to the bodr is

1 Y

Db -M

Inasmuch as the drag of the cable ordinarily far
exceeds the drag of the body, the value of the deriva-
tive Y:, will be principally determined by the cable

8 NACA ARR 1o. L4D18

drag. The method of calculating an equivalent drag
coefficient CDc to take into account the effect of the
cable is given in the appendix. The derivative Y-
is then given as follows:

b Y DDb c
C D m
The value of the coefficient CDC may be determined as
a function of the ratio of horizontal length to height
of the cable X/Z from figure 4.

All the aerod-nri-irc derivatives have been evaluated
in terms of the dimensions of the body and cable system.
In order to reduce the number of variables, it is con-
venient to express these derivatives in terms of non-
dimensional ratios of the quantities involved, which are
given as follows:

Froude number, y2
F = -

Relative-density coefficient,

Ratio of tail length to vertical distance of body below
point of support,

Moment-of-inertia factor,
Cl 2

Time is expressed in terms of the time unit T =
When the derivatives are expressed in terms of these
variables, the stability quartic becomes

aD'4 + bD'3 + cD'2 + dD' + e = 0




D' = DT


a = 1

b --i a + CD

a L.R aD

ae =--- aI
d Cl C l


The stability of the towed airspeed head may be
determined by substituting numerical values in the
formulas for the coefficients and factoring the quartic.
The two quadratic factors dettermirne the period and
damping of two modes of oscillation. One quadratic
factor yields values of the rer'od and damping very
close to those obtained with a simple pendulum having
a length equal to the vertical distance of the towed
body below the airplane and damping equal to that
supplied by the drag force. The other quadratic
factor gives an oscillation that has values of period
and damping very close to those of the body rotating
as a weather vane about a verticall axis through its
center of gravity. The weather-rane oscillation
generally has a short period and is always rapidly
damped. The damping of the pendulum oscillation is,
however, very slow at short cable lengths, because
the cable drag is small.

The coupling between the two modes of oscillation
introduces the possibility of instability of the pendulum
oscillation. In order to find the conditions for
instability, the coefficients of the quartic may be
substituted in Routh's discriminant, which states that
the motion will be stable if the coefficients satisfy
the relation

(bc ad)d b2e >0 (4)

10 NACA ARR No. L4D18

When the values of the coefficients (equation (3))
are substituted in formula (4), Routh's discriminant
be c homes

1 CDiR2 aRCD acDR DRC2 ap2R
-- R + -" -
F2 a / C2 C1 C1 C1

212CDR 2 2 \ a+CD aCD2 apCD
- i- CD>R + aC + +. D --2
Cl C12 C12 Cl

aCD2 CD3
C Ci
C1 1

The expression is given in this form merely for the sake
of completeness. In practice, a great simplification
may be made, with negligible loss of accuracy, by neg-
electing the small term C in coefficient c, for-
mula (3). The simplified form of the discriminant is
2CD a -a
C C CI a D a
R V L 2aCD 1

The minus sign before the expression CD a gives
one condition for stability

< 1 (5)
-a 1+

and the plus sign before the same expression gives
another condition for stability

F C1(CD a) (
R- > C (6)
Boundaries of stability are plotted in figure 5. It is
seen that below a certain small value of the param-
eter F/RP, given by formula (5), the motion is stable
for all values of the drag coefficient. As F/RL is

NACA aRE No. L4D18

increased above this value, the motion is unstable until
the boundary of stabilit'r given by formula (6) is reached.
The motion then becomes stable again at all higher values
of F/RV.

Examples have been worked out from the general
boundaries of stability (fig. 5) to show the variation
of stability of an actual airspeed head as the cable
length is changed. The characteristics of the airspeed
head and cable used in the calculations are as follows:

m, slug ......................................... 0.466
b, foot .................................. ......... 0.45
S, square foot .................................. 0125
k, foot .............. .......................... 0.416
Aspect ratio .................................... 2.25
a, per radian .................................. -2.10
Cl .................. ........................... 0.855
Wc, inch ................................... ... 0.375
Cable weight, pound per foot ................... 0.05
p, slug per cubic foot ......................... 0.00233

The cable length is plotted against the effective
drag coefficient CD = C + CDc in figure 6. The
method for determining this curve is given in the apoendix.
The boundaries of stability for this particular case are
plotted in the same figure in order that the region of
instability may be found. As the body is lowered from
the airplane, it will be stable for a very short distance
and will then become unstable until the upper boundary
of stability is reached. The upper boundary of stability
occurs when the body is 2.8 feet below the point of support
at an airspeed of 200 feet per second, or 6.3 feet below
at 100 feet per second. For all greater cable lengths,
the body will be stable. When thu body is drawn up to
the airplane, it will again pass through the unstable

The period and degree of damping of the oscillation
at various cable lengths for the airspeed head having the
characteristics previously given have been calculated
by substituting numerical values in formula (3) and are
given in the following table for an airspeed of 100 feet
per second:


Distance below Pendulum W.- eather-vane
airplane, oscillation oscillation
(ft) P P N
(sec) N (sec) l4/2.
4.0 2.22 23.4 to double amplitude 1.08 3.87
6.3 2.86 '(neutrally.stable) 1.07 4.10
10.0 3.51 20.0 toone-half amplitude 1.06 4.28
20.0 4.95 4.95 to one-half amplitude 1.07 4.34

From these calculations it is seen that the damping or rate
of divergence of the pendulum oscillation is very small
for cable lengths some distance on either side of the
stability boundary. For the longest cable length, how-
ever, the oscillation damps to one-half amplitude fairly

The boundaries of stability determined theoretically
are in good qualitative agreement with the observed
behavior of the NACA trailing airspeed head. Actually,
there is no sharply defined boundary of stability because
the oscillation is only slightly damped after the body
has been lowered some distance into the stable region.
As will be explained later, disturbing influences not
taken into account in the theory may cause an unstable
oscillation of the body when it would theoretically
have a slightly damped oscillation.

The boundaries of stability shown in figure 5 indicate
that, when the drag coefficient is zero, the body will be
unstable at all values of cable length greater than that
corresponding to the lower stability boundary. For very
small values of the drag coefficient, such as would be
obtained by neglecting the cable drag, the theory
indicates that the body will be unstable over a large
range of values of the cable length. The results of
reference 2, in which the damping effect of air forces
on the cable is neglected, are therefore believed to be
in error.

Investigation of Modifications to Improve Stability

In order to investigate the changes that might be
made to improve the stability of a conventional type of
airspeed head, it is convenient to express the condition
for stability (formula (6)) in the following form, where

;TACA aRR No. L4D18

the nondimensional expressions have been removed by
substituting the dimensional quantities that they

1 k2/g2 k2 ,/,2
1 > k2/2 + !(7)
mg- -aV2S D
Z 2

where k is the radius of gyration about an axis
through the pivot point and D is the effective drag
obtained by multiplying C = CD + CD by V2S.
D D Db 2VS.

The curves of figure 5 show that the region of
instability for a conventional type of towed body can
never be entirely eliminated. The following changes
would tend to restrict the unstable region to a region
closer to the airplane-

(a) Decrease in weight mg.

(b) Increase in drag

(c) Increase in area and aspect ratio of fin

(d) Decrease in ratio of radius of gyration to
tail length k/l

The first two changes are impractical because they
interfere with the usefulness of the instrument as an
airspeed measuring device. The second two chances,
however, provide practical methods of imp-ro"ement. For
example, the greatest distance below the airplane at
which unstable oscillations occur in the example
previously given could be decreased from 6.Z feet to
4.2 feet by doubling the tail length withoutt increasing
the radius of gyracion. This chan-e could be accomplished
by mounting a light set of fins on a boom behind the
instrument. Formula (7) indicates that increasing the
speed will restrict the unstable region to shorter cable
lengths. Once the oscillation becomes unstable, however,
it will probably increase in amplitude faster at higher
airspeeds. It may be advantageous, therefore, to raise
and lower the body at low flying speeds.

The use of special devices to improve the stability
will now be considered. It has been found by the writer
that the two modes of oscillation given by a quartic

14 NACA ARR No. L4D18

will damp to one-half amnplitude in the same time if the
coefficients satisfy the relation

b3 bc + d = 0
S 2
Such a condition will give the optimum use of damping
in the system. Through examination of the coefficients
of the quartic, for-mula (1), it is found that this
relation may be satisfied by greatly increasing the
damping in yaw N or by reducing the directional
stability t almost to zero. Physically, a condition
is thus reached at which the body remains approximately
parallel to the average direction of flight as it swings
from side to side instead of turning into the relative
wind. Forces are thereby brought into play to damp
out the pe ."luia oscillation.

AIe foregoing method of obtaining stability may
also be explained in terms of the stability boundaries
plotted in figure 5. In the small stable region below
the Icwver boundary of stability, the pendulum oscil-
lation is damped out by the mechanism just described.
By reiuci". the directional stability and increasing
the daring in yaw, the lower boundary of stability is
raised to higher values of F/Ry. It is theortically
possible; by using s".cial devices that arbitrarily
increase the damping in yaw or reduce the directional
stability, to raise this stability boundary so that
the unstable region is eliminated. It will be noted
that ? ,bisi method of improving stability is different
in principle from the one described following for-
mula (7)o The method based on formula (7) consisted
in extending the stable region by lowering the upper
boundary of stability The method now being considered
consists in raising the lower stability boundary.

It has been found impossible, in practice, to
reduce the directional stability of a conventional
towed body to the extremely small value required.
The bo.-- of the instrument is generally unstable ind
some fin area is required to give neutral directional
stability. Any small cha-ige in the characteristics
of the body due to Reynolds number or due to small
changes in shape would be sufficient to mako it either
directionally unstable or too stable to obtain damping
by vi:-u.te of its low directional stability. The
alternative, greatly increasirnj the damping in yaw,


miirht be accomplished by operating the rudder of the
instrument by means of a gyroscopic element to cause
the rudee' to deflect an amount proportional to the
yawing velocity. The complication introduced by such
a mechanism would probably make the method impracticable.

Another method of increasing the damping in yaw
and at the same time reducing the directional stability
is to use two fins, one at the front and one at the rear
of the body. Calculations show that the directional
stability must be reduced to a very small value (approxi-
mately 4 percent of the stability contributed by the
rear fin) in order to avoid the unstable oscillations.
A moderate decrease in directional stability, even when
combined with a dampinr in yaw of 20 times that for a
conventional body, will not avoid the unstable oscilla-
tions. If the required small directional stability
could be obtained, any slight misalinement of the front
and rear fins would cause the body to trim at a high
lift coefficient. This condition would cause the body
to fly out to one side and would also make it undesirable
as an airspeed measuring device.

Longitudinal Oscillations

The longitudinal notion of a towed body has been
treated theoretically in reference 2. This analysis
neglected the damping of the motion contributed by air
forces on the cable. The boundaries of stability
calculated in reference 2 are therefore believed to be

In practice, the fore-and-aft pendulum motion of
the body has never: been observed to become unstable at
long cables lengths. It is noted that the effect of
the cable could be taken into account as an equivalent
drag coefficient, as it was for the lateral oscillations.
If a value of drag coefficient of the correct order of
magnitude is substituted in the relations presented in
reference 2, the pendulum oscillations may be shown to
be well damped at long cable lengths.

At short cable lengths and moderate speeds, the
body hangs approximately vertically below the point
of support; therefore, very little coupling exists
between fore-and-aft movement of the body and pitching
motion. The oscillation is simply a pendulum motion

16 N-CA ARR No. L4D18

with damping supplied by the cable drag. Inasmuch as
this cable drag is small at short cable lengths, the
oscillation, though theoretically stable, is slowly
damped and may become unstable if disturbing influences
are present.

Ti: analysis of reference 2 shows that other modes
of longitudinal oscillation involving bowing of the cable
and pitching of the body are theoretically possible, but
such oscillations have never Le.n observed in practice.
It is believed that the dra on the cable rr-vents these
oscillations from becoming unstable.


Flight tests were made of an approximately 1-scale
model about : yrnamicallly~ sl:, 1 i1. to--t-e--.CA trailing
airspeed head suspended from a Stinson SR-b& airplane.
A drawing of the model is shown in figure 7. In order
to simulate pulling the head up to a su i:ort under the
airplane, the cord was run through an eyelet on the cabin

The instrument was stable when towed on the end of
a 75-foot cable at speeds between 0O and 150 miles per
hour. Lateral and fore-and-aft oscillations da:-'ed out
in a small number of cycles. It should be noted that
the corresponding cable lengths on a full-scale towed
airspeed head, twice the size of the one tested, would
be twice as great. The corresoc ,din speeds would be
\/2 times as great in order to maintain the same value
of the Froude number F = V2/ L.

When the model was drawn up to about 3 feet from
the airplane, it was sufficiently stable at 80 miles per
hour.- Unstable oscillations did not start while the
body was left in this -osition for about a -inute. This
behavior does not necessarily indicate -"at the oscilla-
tions would have damped out once they had started. The
theory shows that a large number of oscillations is
required to double amplitude; the body might, therefore,
have to be towed for a considerable length of time
before oscillations would become notice'cl9e. Unfortunately,
no means were available to start an oscillation.

As the s:pe-d was increased, the motion became less
stable, until at 95 miles per hour increasing oscillations
occurred. As predicted by the theory, both the


fore-and-aft and lateral oscillations had periods close
to the period of a simple pendulum. The lateral and
fore-and-aft oscillations inevitably combine.s to cause
the instrument to travel in an elliptical orbit. The
direction of rotation was such that the instrument
swung back as it came closest to the fuselage. Proba-
bly the increased velocity near the fu.selage fed energy
into the motion with each oscillation and caused a
greater rate of increase in the amplitude than would
have been predicted by the theory.

Several modifications of the model were tried in an
effort to improve the stability. Two modifications
appeared to improve the stability of the pendulun oscil-
lation at short cable lengths. One of these changes
consisted in shifting the pivot point rearward 1/2 inch,
and the other consisted in equioping the model with a
hinged rudder with weight behind the hinge line and
viscous damping. These changes prevented the oscilla-
tion from appearing spontaneously as the speed was
gradually increased from 80 to 140 miles per hour. A
theoretical study indicates that these changes should
have only secondary effects o.- stability. These tests
are not considered to be a conclusive der.monrtration of
the stability: of the body because it is not known whether
oscillations would have damerd ou.t once they were started.

Various other modifications that were tried resulted
in unstable short-period oscillations of the body. These
tests were made at a speed of 80 miles per hour. A
forward shift of the pivot ooint caused a pitching oscil-
lation. This motion was believed to be the result of
elasticity of the cable and mass unbalance of the body
and was similar in nature Lo flutter. A freely hinged
rudder with weight behind the hinge line caused a short-
period yawing oscillation. The use of an asymmetrical
vertical fin, extending only below the body, caused a
short-period rocking motion of the body.

Another type of instability has been encountered on
a few occasions when the full-size l.aCa airspeed head
was lowered at the full length of the cable (approxi-
mately 200 ft). In one case in which this motion was
observed, the airspeed hed ed was lowered from the door
of a twin-engine low-wing cabin monoplane. The head
was steady at speeds below 150 miles per hour, but at
this speed oscillations of about Z-foot wave length -in


the cable originated at the airpla ne and traveled down
to the body. As the speed was increased to 165 miles
per hour, the oscillations became very violent and
caused a pitching motion of the body. The whipping
action at the lower end of the cable eventually caused
it to break. A metal sphere was later towed from the
same airplane and the oscillations occurred as before.
The oscillation was therefore not related to the aero-
dynamic characteristics of the body. It was believed
to be caused by the action of unsteady air flow from
the wing-fuselage juncture on the tow cable. The same
airspeed head has been used without difficulty at much
higher speeds on other airplanes.

Several relatively light, large-size towed bodies
have been tested in flight. The pendulum oscillation
of these bodies has never been known to become unstable,
even when the body was raised or lowered from the air-
plane quite slo-wly. This behavior is in agreement
with the theoretical prediction. These bodies have a
small value of u compared with that of the towed
airsps-.1 head; the unstable region at normal flying
s'-eds is therefore very small.


The theoretical and experimental investigations
have shown that the pendulum motion of a towed body may
become unstable when the body is drawn up close to the
airplane. The theory shows that the instability is
not serious because the amplitude of the oscillations
increases very slowly. The maximum cable length at
which unstable oscillations can occur may be reduced
by reduction of the ratio of radius of -:yration to tail
length of the body and by increase of the fin area and
aspect ratio.

More violent instability of the pendulum oscilla-
tion than would be predicted by the theory, as well as
other types of instability, may be introduced by
unsteadiness or lack of uniformity of the air flow
in the region where the body is lowered from the air-
plane. Inasmuch as no practical method has been found
to provide large damping of the pendulum oscillations
when the body is close to the airplane, it is desirable
to lower the body from a point where it is not subjected


to these disturbing infl-lunces. IA suitable location
would probably be on the plan of sy-rmetry of a t.jin-
sn.rine airplane, or on the wing of a single-pngine
airplane at a point outside the slipstream. It also
appears desirable to lower and raise the body at low
flying speed, because the unstable oscillations then
increase in amplitue t very slowly. If these precautions
are taken, it should bo possible to lower a towed body
without attention on the part of the pilot. The only
possibility for unstable oscillations to develop would
be if th3 body wer'i left fori long periods suspen-ed
only a few feet below the airplane. Oscillations of
the system at large cable 1ineths ars racpidly damped
because of the cable dr-ag.


1. A theoretical study of the- motion of a suspended
body stabilized Ly fins showed that it had two modes of
lateral oscillation with the following characteristics:

(a) Weather-vane oscillation

The weather-vane mode of oscillation was
rapidly damped and had a period about equal to that
of the instrument oscillatinrr as a weather vane about
a vertical axis through its center of gravity.

(b) Pendulum oscillation

The period of the pendulum mode of
oscillation was about the sime as that of a simple
pendulumn of length equal to the vertical distance
of the body below the airplane. The oscillation
was damped by the cable dra~ at large cable lengths
but was unscable at short cable lengths. The rate
of increase of amplitude in the unstable region was
very small. It was found that the unstable region
could be restricted. to short cable len'tbs at normal
airplane seeds by keening the radius of gyration
of the body small and increasing the fin area,
aspect ratio, and tail length.

2. In flight tests, more violent instability of
the pendulum motion was enco',i.ntered than would have
been expected from the theory and other tyoes of

20 NACA ARR No. L4D18

instability occasionally occurred. These conditions
were attributed to the action of unsteady air flow on
the cable. It is believed that unsatisfactory behavior
of a towed suspended body can be avoided by lowering and
raising the body at low flying speeds from a point on
the airplane where the air flow is uniform.

Langley Memorial Aeronautical Laboratory,
National Advisory Committee for Aeronautics,
Langley Field, Va.




The drag of each cable element of height dz is

dDc = Cd ;'c EV2dz
c -

where cde is the drag coefficient of the cable per unit
vertical height. The variation of this drag coefficient
with inclination of the caole has been obtained from the
data of reference 3 and is presented in figure R. If
the assumption is made that the cable remains straight
when viewed from the front, each cable element has a
lateral velocity proportional to its distance below the
airplane. The side force acting on each cable element
d70 = dD c
ScW, z dz
dc 2 2 z

The total side force is

z z

c= w V Cdc

This side force has been determined by graphical
integration for cables with various values of X/Z, the
ratio of horizontal length to height. The cable form
was assumed to be that of one-quarter of a sine wave,
as shown in figure 9(a). Although the shape of the
actual cable may deviate somewhat from a sine curve,
the error in the calculated side force will be small.

The location of the resultant side force may also
be determined graphically as the center of gravity of
the area representing the side-force distribution. IT
the inertia of the cable is neglected, the lateral force


will be balanced by reactions on the body and on the
point of support; the magnitude of the reactions will
depend on the position of the resultant side force on
the cable. As shown by figure 9, most of the side
force on the cable is transmitted to the body. Let
the fraction of the total side force that is applied
to the body be K. The side force applied to the
body is then

Yb = KY

f1 11
dYb K Zwc/V Cdc d
d7 S 2'

By comparison with formula (2), the quantity in brackets
may be substituted as an equivalent drag coefficient in
the formula for Y The relations may be summarized
as follows:

S= (CDb + CDc) m


KWcZ z
CDc ZI cd c dZ/

The value of CDc may be determined from figure 4 for
cables of various values of the ratio X/Z.

In order to determine the variation of CDc with
cable length as the body is drawn up to the airplane, it
is necessary to know how the ratio X/Z changes with
cable length. Typical examples have been worked out
for two airspeeds for the airspeed head and cable
previously described. The shape of the cable was


determined from consideration of the forces acting on
the cable elements, obtained from reference 3. The
cable shape for each speed is shown in figure 10. Any
point on the cable may be considered as a point of
suspension. The variation of tha ratio X/Z as the
cable is drawn in or let out may therefore be determined
graphically from this figure.


1. Thompson, F. L.: The Neasurement of Kir Speed of
Airplanes. TN No. 616, NACA, 1937.

2. Glauert, H.: The Stability of a Body Towed by a
Light Wire. I. & hI. No. 1312, British A.R.C.,

3. McLeod, A. R.: On the Action of Wind on Flexible
Cables, with Applications to Cables Towed blow
Aeroplanes, and Balloon Cables. R. & .i. No. 554,
British A.C.a., 1918..


Figs. 1,72









4 t



.a o
"O r
.$^ P
^ or ,
u D



Fig. 3 NACA ARR No. L4DI8

Figure 3.- Top view of trailing airspeed head showing component
of side force due to drag.



^*t "P" -! ia 'i *(+t '*'-'-1- "g.tct W : ^ SiLtL ^ :ri~~~tjt~- ^: ^::-: -'- f *'^ ^

-4 -
.r .4 .r .&.. ... ...r .


VW~:thtt~tflT1Zr wtrmt41rw-Qnltztlmm+~~..trtrn4 ttt~tA.~Iict


..a ~ :::.? .it-::: : .- ^. .H t
Tt 5'-L~r~ T tir-TtofpE -F::!^:!?- -i+: t; Ia: l-"+: ;: I-t-^dr: ~ ---


+ri i
-.4; .| .; . ...... -----

4- IE4-^ 4-f^ ^ 4- 1 glit IN :1|

4ll i|l|

:7~"tF~FF~~Fi~. .F'~i~ .....
at . .; il s ^ t' t;t^ ;i ^ .

Fillli ililii llllllll llllill+lii 2l l7-l 44+i

Fig. 4




t t ++ __ -at


~f- I,;

i. -:

- ] -I-
-- .-- ] .. t "_ I-

I- 4---------d--- -

14 V I2L t- 1. .4: 1


2~ 7t~i-~

S: i i- -I : *:i- 4 -t- --- .- 4

F -: ---- 4-3
A T T 1 7 g

i.. i L f .. I \ ... .. -. "-

-- ---- ---- ----

t :-^ i---A,::i-j
S_ _1 1*


j A.

4-1 Li


]i I -
Value of a is -2.10 per radian.
-: -- I I .l t: 1

4 .

4 ....J i ,1 I. ,r II I .

-ig re B I ,- a towed boy--.

Value o -i is p
-~ ~ ,' -/ .-1 Iz_ L... _t. _' .i .. 'L --"~l

Fig. 5

--"Ii- '
!_ i..

' I



.... .... .. ..

r .

.Th.... .-.....

(a) Airspeed, 100 feet per second.

F .;F- +eTe o ve..a .is-tanes t

a||d boCy tt ive i ili| | irsee head an|| I abi| le
-se i ns
l ii Id I

4-, A I f p s

and body that give instability for airspeed head and cable
used in calculations.

used in calculations.

Figs. 6a, b

Fig. 7 NACA ARR No. L4D18

1.----- ^ ^


4 z1


Sa a

4 4.

5 U

( o

^ ^.^ ,





^::;:s^ i tI::^fit:gj5.,^:^I;^lI)-:t!-||ttfitj
*i-1 .' -
.44 4: 4-=
~ ~ ~ .. .... -H ..4 4 .

~ c~f~-~*'+ -r:~-c-P
-L 4 -

.... ...

f-r L4-uSrj-

Fr -,

4 7

ttj'i. r-rrm dr-,t Ii-K~~~ 4- -4-- :--

.1 .- -s

-~~ ~ I .Ii4 I -
#- -I -T .~ .-

f .-;_- ----_-! -f i K ,

F PGtX' L4Ptij:3

Figure 8.- Variation of cable drag coefficient per unit vertical
height with inclination. (Data taken from reference 3.)

Fig. 8


't I4- I. ...

-^ ------


--4- T--^

4 tL ti- 7r;+ti 7
HThT t 77#

F. T.-2trT

4-- 4i|ji^ |^|y ^S -

1^^^I^^II^^^ ^ ^I l ||| ^^1


f;^F I': I
^^^^^^-w^^p^|^^^47| 7|:|i

g~gl^^lS~iTiS IglS

.0 0 C

* II

o L


o *

0 0

o4 *
.f 0
8 ^
" ? ..a
*. a
Bl 0
I t '

%, fl"


Fig. 9


Fig. 10

J: IL f ',
:4 I 'T
n.L:aItIilS~tf^^^^i' 1^^^^2!^ 14 I *4

r"-4 T

if.ItV+- iT. r -~ -..i jn ,'
"' -LI t I t t

-t -,+ ... ...
Z4. t4

i ..L i I I' iii-+: :lli .-'-" i :--, i '.. ^ '

++ +.: ,+ ,+ If~ ~":-I Y : .i

( A d re AION

ir4 '-i t AS .. .||
.I .V+- ;;-+- .^ ^ ''E .E ,
U_ -7 .q +

(U) Alrspeed, 100 feet per second, NATIONAL ADVISORY

T- 1 |.|.

-K| 1|' ... | ..lI. i|^ +| -
U + 1 l | ii .-i _

I: l
IT 1l iB
4 -4. T- T- If 4LI:~YY- -~a LJ1-l~i~~1T~~~f T r :lr~l 1--* :tr,:


;iN4 RUF

iV --44
..... ,m-+' i +"+

(b) Airspeed, 200 feet per second.
Figure 10.- Cable shape for two values of airspeedor a body and
cable that have the characteristics used in the calculations.



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iIIllll111 Il IIII
3 1262 08104 987 5

P.O. SOX 117011
GC NESVILLE, FL 32611-7011 USA