gliding angle 1 in 7,
and the average width
of the struts about 1
inch (the exact breadth
assumed depending on
the form and strength
of the section).
Then, since gliding
angle = ^- , every /
pounds of strut
\vright will give rise to 1 pound of resistance, in addition to
the aerodynamic resistance of the struts.f We can, therefore,
write T = 1- R, where T = thrust due to the struts, and
It is their aerodynamic resistance. Simplifying, we have
G = W + 7R, but, since this expression has a maximum value
for the least efficient strut, the reciprocal is here employed,
and multiplied by the constant 14300, giving C = = jp i 7^
The best strut under the conditions above specified is then
the one showing the highest value for C. The reason for
choosing this particular value for the multiplier is that it
makes C = 100 for the best strut of the first and largest series
which we shall consider.
If the speed of the machine for which the struts are being
selected is greater than 60 miles an hour, the resistance be-
comes of greater importance as compared with the weight,
and the merit factors for those sections which, although heavy,
offer very low resistances are relatively improved. If the glid-
ing angle is flatter than 1 in 7, a similar effect ensues.
elseif (getClientWidth() > 430)
On the other hand, if it becomes necessary to use struts
having a diameter of more than 1 inch or thereabouts, the ad-
vantage inclines toward the sections which have the greatest
strength for their weight, and the relative importance of re-
sistance is diminished, since, in similar sections, weight varies
as the square of the breadth and resistance only as the first
power. These effects are, however, of slight importance, and
would not be likely to change the merit factors enough to have
serious influence on the choice of a section in any given case.
The question of strength will be taken up more fully in
another section of the course. It will suffice to say here that
the strengths of two struts have been considered to be equal
when their moments of inertia about their longitudinal axes
Strut Sections Developed by Ogilvie
We may now proceed to the examination of definite data for
a number of series of struts, tested at various times and places.
The following figures are the result of experiments performed
at the N. P. L. at the suggestion of Alec Ogilvie, the sections
being illustrated in Fig. 1.
/ = moment of inertia for the section in question about its
longitudinal axis (inches* for a strut 1 inch wide).
R = resistance in pounds of 100 feet of strut 1 inch wide at
60 miles per hour.
W = weight in pounds of 100 feet of spruce strut 1 inch
b = width of strut whose strength will be equal to that of a
strut of section a, and 1 inch wide.
W = weight of 100 feet of spruce strut of width b.
C M = merit factor at 60 miles per hour.
t Relationships between weight and resistance on a glide will be
fully considered in Section 12.
Many very interesting conclusions can be drawn from this
table. In the first place, it is evidently of the utmost im-
portance to avoid rapid changes in curvature. Several sec-
tions, notably, e and I, although they appear to have a very
smooth outline, oppose a large resistance simply because the
transition from the entrance to the run is so abrupt that the
air-flow cannot follow its contour, and violent eddy-making
AERODYNAMICAL THEORY AND DATA
The good performance of several sections so formed indi-
cates that it may be wise actually to run the sides of the strut
parallel for some little distance, as illustrated by q and t.
This is counteracted, however, by the fact that skin-friction
increases in proportion to the " wetted surface " of the strut.
It is for this reason that the very longest sections did not give
such low resistances as those of more moderate form. This
matter of the ratio of length of section to width will be dis-
cussed more fully somewhat later, in connection with another
series of tests.
It will be seen, too, that the resistance is little affected by
the chopping off of a portion of the tail in such a manner as
to leave it straight across. Examples of this are furnished
by n, ( and i. This is due to the fact that it has not been pos-
sible in any strut yet designed to totally eliminate the region
of deadwater behind the strut. As will be evident from any
section of air-flow about a fair-shaped section, the lines of
flow always leave the contour of the strut some distance short
of the extreme rear. Since no changes made in the contour
within this region will have any decided effect on the re-
sistance, it avails nothing to go to the trouble and expense
involved in the attempt to construct a wooden strut running
out to a sharp point at the back.
Another Series of Struts Tested at the N. P. L.
At about this same time another series of struts was tested
at the same laboratory, the sections being those actually em-
FIG. 2. STRUT SECTIONS
TESTED AT N. P. L.
PIG. 3. N. P. L.
ployed in machines then existing. The outlines of the sections
tested are shown in Fig. 2, and the characteristics are given
Blerlot A 070
Bleriot B 107
Fmrma n 074
De Harilland. .052
B.F. 34 279
B.F. 35 238
of the symbols being the same as in the tables already given,
except that n = the ratio of the length to width of section.
Tims it is apparent that the best of these sections are inatc-
rinlly superior to the best of the sections tested by Ogilvie.
both in resistance and in merit factor. In Fig. 4 resistance of
UESISTAVCE AND FACTORS OP MEKIT FOR R. A.
100 feet of strut at 60 miles per hour, and merit factor at 60
miles per hour, are plotted against ratio of length to width.
As this ratio diminishes, the air-flow about the strut takes on
a very uncertain character, and the values when n is less than
2 are rather doubtful. Such extremely short sections as this
are also undesirable from the standpoint of lateral stability.
as will be shown in another section of the Course. On the
other hand, n may be considerably pi-eater than the absolute
optimum value without any great disadvantage, so it will be
well in general to employ a ratio of four, or even a slightly
higher figure. The photographs of Mow aliout strut sections.
reproduced in Fig. ~>. show clearly why such a procedure c-.-m
be safely adopted.
It will be seen that these figures simply supplement and
confirm the conclusions already deduced from the more exten-
sive and systematic investigations directed by Mr. Ogilvie.
TeU on Struts, Length to Width Varied
As a result of these and other tests. series of struts em-
bodying the best features of those already tried, and varying
only in the ratio of length of section to width, was made and
tested at the National Physical laboratory. Three rep re
live members of tin- series are shown in Fig. .'!. The table
below gives the characteristics of these struts, the meaning
Two Kiffcl Struts
Two struts of somewhat the same section as those just di-
eiisscd have recently l.een I, '-led by Kifl'el. and show remark-
ably low resistances. Their outlines arc uivcn in Fiir. (i. For
Xo. 1, having n equal to :!.L'.'I, If equals !i.7 pounds, while for
No. 2. with a somewhat sharper entry, i is L'.lKi and R is only
8.7 pounds. I'arl of this improvement oxer the best of the
Knglish tests, hoxvexei. i- undoubtedly due to the higher wind
speed which is secured in Kiffel's laboratory, the resistance
coefficient having a tendency to rise as the speed of test i^.
AERODYNAMICAL THEORY AND DATA
Effect of Length of Struts
We now turn our attention to the effect of the length of
the strut. While this point is less important than was gen-
erally supposed a few years ago, and while its effects are
largely determined by the nature of the surfaces in which the
strut terminates, the experimental results bearing on the mat-
ter should nevertheless be studied. For this data we are
indebted to Mr. Thurston, who has described his results in
the series of articles already cited. As the result of a great
nel would be exceedingly difficult to devise. The matter might
well be investigated in an outdoor, full-scale plant such as
that at St. Cyr.
Resistance of Inclined Struts
The only point which remains to be studied is the resistance
of struts which are not normal to the line of flight. Some
much more recent tests by Mr. Thurston have covered this
point, and show very surprising results. Struts of square,
rectangular, circular, and stream-line section were tested at
angles from to 90 degrees, and the effects of the ends of the
strut offering a direct resistance when inclined were overcome
by the use of the method of differences: that is, tests were
made first on a strut 34 inches long, and then on one 16
inches long, the difference of the figures obtained being equal
to the resistance of an 18-inch section of an infinite strut.
The ratio of the resistance of a strut inclined at various
ILLUSTRATING FLOW AROUND STRUTS
many experiments on manifold different types of strut, he
came to the conclusion that resistance for a strut with free
ends could best be expressed by the formula B = KltV 1 -
.0073fF 2 , where R is the resistance in pounds, I and *, re-
spectively, the length and thickness of the strut in feet, K a
constant, and V the speed in miles per hour.
It is evident from this equation that, even with the lowest
values of K yet obtained, the effects of length will be prac-
Fio. 6. Two EIFFEL STRUTS
tically negligible when the length is more than 50 times the
thickness, as it generally is. Since, in addition, the case of a
strut with free ends is one which never occurs in practise,
resistance may be considered as independent of length-thick-
ness ratio for all the purposes of design.
The form of air-flow about the wing may have very decided
effects on the resistance of interplane struts, but we have no
means of knowing how great these aro. and experiments cover-
ing this point and susceptible of performance in a wind tun-
INCMMATIOM Of BAH TO WIMO
FIG. 7. DATA FOR INCLINED STRUTS
angles to the resistance of a normal strut of like section and
equal projected length is plotted in Fig. 7. It will be seen
that the resistance at 30 degrees to the wind is less than one-
third of that at 90 degrees, and this large difference is by no
means accounted for by the difference in length of section
parallel to the wind. When a circular strut is placed at an
angle of 30 degrees to the wind, the section parallel thereto
is an ellipse having a length of twice its width, and the resist-
ance of an elliptical strut such as this, when placed normal.
is only 36 per cent less than that for a circular section.
About 45 per cent of the reduction due to inclination thus
remains unaccounted for.
Since, however, the curve of reduction is substantially a
sine curve, and is therefore very flat at the ends, there is
very little advantage to be gained from inclining a stream-
line strut unless it is inclined at least 30 degrees to the nor-
mal. This reduced resistance should, however, be kept in
mind as a point in favor of the staggered biplane. Eiffel
also made a few tests on struts inclined 30 degrees from the
normal, the results cheeking very well with Mr. Thurston's.
The Effect of Changing the DV Product for Struts
As was shown in Chapter 10, the resistance coefficient is
not an absolute constant, but is a function of VI), when-
V Is the speed and I) the diameter of the strut. The coeffi-
cient tends to decrease as VI) increases, but the change for
values of I'D (in foot/second units) above 6 is extremely small,
as Eiffel lias demonstrated. The tests made at thp National
Physical Laboratory have been made with a value of VD
equal to only 2.5. whereas, in an actual machine, this quantity
would never be likely to fall below 5, and is generally from
7 to 10.
i is AERODYNAMICAL THEORY AND DATA
We can therefore deduce from Kitlel's experiments that it References for Part I. Chapter 11
is safe to reduce the values for resistance here given (for the
N. P. L. tests) by about 25 per cent in applying them to a " strut*." FHoht. June is. 1012.
design. This indicates that, as was hinted above, the superior- -Aerodynamic RMMUM of struts. Bare, and wires." by A. v.
ity of Eiffel's strut sections is more apparent than real, and Thurston ; Aeronautical Journal, April and July, 191::.
that the best sections yet available are the N. P. L. sections Technical Reports of the British Advisory Committee on Aeronautics.
having fineness ratios of from 3.5 to 4.5. The correction 1911-12. 1912-13.
given here should be applied only to Struts of fairly good "The Resistance of Inclined Struts In a Uniform Air Curri'iit," by A.
section, as the value of VD has much less effect on those sec- i^ 1 " 1 "" " " Dd "' Tonnsteln ' Aeron <"" lcal """' Janu "5'-
tions for which the resistance is relativelv IUL-II. and in which
" Nouvelles Recbercues sur In Resistance de 1'Alr et 1'Avlatlon." by G.
there is more effect due to turbulence than to skin friction. Eiffel. (1914 edition.)
Resistance and Performance
It may be useful to restate the symbols which we employ
in considering performance curves, ascent and descent.
IT" = weight of the machine;
A = area of the wings.
i = angle of incidence of the wings.
L = lift.
K u = lift coefficient.
D = drag of wings.
K., = drag coefficient.
11 = resultant of lift and drag on the wings.
P = parasite or structural resistance of a machine.
Dt = total resistance or drag = T) -\- P.
R t = total resultant air force on a machine.
// = -propeller thrust.
6 = angle of flight path with the horizontal.
Structural and Wing Resistance for the British B.E.2
In Chapter 4, a problem was worked out on the sustentation
and resistance of wing surfaces, which in spite of some rough
MILES PER HOUR
FIG. 1. PERFORMANCE CURVES FOR THE B.E.2
assumptions, illustrated the main performance curves and cal-
culations employed. In Fig. 1 are shown curves for the Brit-
ish B. E. 2. It is not a particularly modern machine, but has
been worked out so thoroughly that it deserves particularly
The body or parasite resistance which includes the resis-
tance of the wing bracing, chassis, etc.. as well as the resistance
of the body proper, is taken as varying as T'" 2 and allowance
has been made for propeller slip stream velocity. The body
resistance is seen to play an unimportant part at low speeds.
But at about 53 miles per hour it becomes greater than the
plane or wing resistance, and at high speeds it. is almost twice
as great as the wing resistance. This emphasizes the imppr-
tance of minimizing the resistance for a high-speed machine.
However good a wing section itself may be, high structural
resistance will make high speeds impossible.
The plane resistance curve has a minimum value at about
65 miles per hour and increases on either side of this speed.
It is interesting to follow out how this increase in resist-
ance on either side occurs. At high speeds, the angles of
incidence and the drift coefficients are small but the speeds
are very great, and the increase in wing resistance is obvious.
At small speeds on the other hand the airplane is flying at
large angles of incidence to give the necessary sustentation and
the drift coefficients are large. The shape of the total re-
sistance curve follows from the summation of the two.
Theoretical Laws for Minimum Thrust and
From a theoretical treatment of the question, the following
interesting law has been derived :
Minimum thrust is required to overcome the resistance of an
airplane when Hie parasite resistance is equal to the drag of
For a proof of this law, reference to Chasseriaud and
Espitallier is appended. In the case we have selected, illus-
trated in Fig. 1, the structural air resistance and the wing
drag are equal at a speed of 53 miles an hour, while the
minimum resistance is at 49 miles per hour. The law does not
seem to be borne out by practice, though it may be occasion-
ally useful as a rough check.
The minimum horsepower required generally occurs at a
low speed, but not at the minimum speed; and its position
will vary for every machine. Another theoretically deduced
law states that:
Minimum horsepower is required irln'ii Hie machine is mov-
ing at a speed at which the wing resistance is three times the
" This law is often highly inaccurate, but may be useful.
Effective or Propeller Horsepower Available Curve
Typical curves for these are also illustrated in Fig. 1, and
are of the greatest interest to the designer. In establishing
such curves it is generally assumed that the engine is running
at the rated revolutions per minute and that in designing the
propeller the efficiency for this revolution per minute at every
airplane speed is known. Thus assuming an engine which
delivers 140 horsepower at an ail-plane speed of 80 miles an
hour, the propeller having an efficiency of 75 per cent at this
speed, the available horsepower will be
140 X 75
= 105 horsepower.
AERODYNAMICAL THEORY AND DATA
Since the power of a propeller is given by the product of
us thrust into the speed and the speed of the propeller is the
speed of the airplane, it follows that when the propeller is
delivering sufficient power, it is also delivering sufficient
thrust. Hence propeller horsepower available is sufficient for
all practical consideration, and propeller thrust curves need
not be included in a performance chart.
Minimum and Maximum Speed; Maximum Excess
Power; Best Climb; Descent
The maximum and minimum speeds of an airplane are gen-
erally given by the two points of intersection of the propeller
horsepower available and the total horsepower required. If
the machine is highly- powered, and the propeller efficient, the
two curves may not intersect at the speed at which the lift
becomes insufficient, and the airplane would climb at stalling
angle, unless the engine is considerably throttled down. The
climb decreases the angle of incidence, and checks stalling.
It is thus a decided advantage to have excess available power
at high angles.
It is a simple matter to deduce the speed of climb from the
excess power. This is absorbed in raising the machine.
Total weight X climb per second
The maximum excess power does not occur at the lowest
speed. To find it, we must measure the maximum ordinate
between the available propeller horsepower and the total re-
quired horsepower. In Fig. 1 this is to be found at 48 miles
per hour. The excess is 21 horsepower and the weight of the
machine is 1650 pounds.
21 X 550
= 7 feet per second or 420 feet per min-
ute. This is, however, only the initial rate of climb. As the
machine rises, the density of the air, the power of the engine,
and the climb gradually diminish.
In practice, the pilot need not know the change of in-
cidence that he produces to climb, although for a given ma-
chine it is an easy matter to calculate the correct angle from
the performance curves. In Dr. Hunsaker's words, " a care-
ful man moves his elevator slowly until he has placed him-
self on the desired trajectory." Part of the art of aviation is
to do this without exceeding safe limits, for obviously there is
a limit to the rate of climb the engine can handle. If the
machine is put on a climb too steep for the power of the ma-
chine, the speed is suddenly lost, the controls become ineffec-
tive, and the machine has stalled.
In descent, very analogous considerations obtain. The
pilot decreases his angle of incidence to a negative value. At
this angle the speed required for sustentation is beyond that
of the maximum, and the propeller horsepower is insufficient.
If D = deficiency in horsepower,
n Total weight X velocity of descent.
The machine descends and gains the required speed under tin-
action of gravity.
The Two Regions of Control. Control by I limiilin-
Consider the performance curves of the same machine, the
Hnti-h I'..K.'_' shown in Fig. 1. Suppose the machine to be
flown iit in degrees at the point J/ with the engine throttled.
so that there is equilibrium, and the power curve is as shown.
26 horsepower. The pilot wishing t<> rise will naturally in
crease his angle of incidence to say 12 degrees. He will thru
require 30 horsepower while the throttled engine will deliver
even less than the 2(j horsepower through the propeller. In-
stead of rising the machine will fall.
Suppose now that flying at the same point and under the
same conditions he wishes to descend, and decreases his angle
to 8 degrees. He will now have an excess of power of 3
horsepower as can be seen from the curves and will ascend
instead of descend. There is therefore a region of reverse
controls, known to French authors as the regime lent.
At the point M . when the pilot wishes to rise and in-
creases his angle of incidence, he does indeed obtain excess
power and rises. Here the controls are normal and the region
is known as regime rapide. For an inexperienced pilot the
regime lent is dangerous. Even if he knows the angle of in-
cidence at which he is working, he is likely to get into diffi-
With a flexible engine, an expert pilot can operate an air-
plane in the slow speed region by manipulation of the throttle
40 M 60 TO SO 90 100
MILES PEU HOUR
Fiu. 2. VARYING SPEED RANGE WITH ENGINE THROTTLED
alone. In Fig. 2 the propeller horsepower available is shown
with the engine throttled down to various speeds for a design
taken i'rom Dr. Hunsaker's pamphlet, to which reference is
appended. For each speed of the engine there is a different
maximum and minimum speed of the airplane, and a different
speed range. If the airplane is living at the minimum speed
in the regime lent region at a certain revolution per minute,
the pilot can by unthrottling his engine pass to a larger speed
!.-iii'_ r e, obtain excess power and climb without changing his
forward speed or angle of incidence. When an engine is
throttled the danger of reversed controls is still greater, lie-
cause the speed range becomes so very small. Kven the best
of pilots may mistake his position on the curve.
In French airplane contests, a premium has been placed on
low speeds, and the regime lent with throttling has been
largely and successfully used. Such operation does not seem
advisable for ordinary flying.
Variations in PropHIrr Horsepower (lunr-
We will now consider the possible variations in performance
by changing the design of the propeller from a high speed to