Between 1900 and 1905, the Wright brothers designed and built three
unpowered gliders and three
As part of the design process, they had to make some mathematical estimates
of the lift and drag of their vehicles.
The Wright brothers were bicycle mechanics
and had a good working knowledge of math and science. They knew about
of motion and about
They knew that they needed to generate enough
to overcome the
of their aircraft. They were also aware of mathematical equations
which could be used to predict the amount of
that an object would generate.
On this page we present the modern version of the drag equation.
The amount of drag generated by an object depends on a number of
factors, including the density
of the air, the
between the object and the
viscosity and compressibility of the air,
the size and
of the body, and the body's inclination
to the flow.
The drag equation states that drag (D)is equal to a
drag coefficient (Cd) times the density of the air (r) times half of the
square of the velocity
(V) times the wing area (A).
D = .5 * Cd * r * V^2 * A
In general, the dependence on body shape, inclination,
air viscosity, and compressibility is very complex.
One way to deal with complex dependencies is to characterize the
dependence by a single variable. For drag, this variable is called the
drag coefficient, designated "Cd".
At the time of the Wright brothers, the drag coefficient was usually referenced
to the drag of a flat plate of equal projected area.
On another page we show some typical
values for the drag coefficient.
For the Wright's aircraft, the basic drag coefficient was equal to about .045.
For given air conditions, shape, and
inclination of the object, we have to determine a value for Cd to
determine the drag. The drag coefficient is composed of two parts; a basic
drag coefficient which includes the effects of skin friction and shape (form),
and an additional drag coefficient related to the lift of the aircraft.
The additional source of drag is called the induced drag
and it is produced at the wing tips due to aircraft lift. Because of pressure
differences above and below the wing, the air on the bottom of the wing is
drawn onto the top near the wing tips. This creates a swirling flow
which changes the effective angle of attack along the wing and "induces"
a drag on the wing. The induced drag coefficient is equal to
the square of the lift coefficient (Cl) divided by the quantity: pi
(3.14159) times the aspect ratio (Ar) times an
efficiency factor (e). The
is the square of the
span divided by the wing area. For a
rectangular wing this reduces to the ratio of the span to the chord.
Long, slender, high aspect ratio wings have lower induced drag than
short, thick, low aspect ratio wings. Lifting line theory shows that
the optimum (lowest) induced drag occurs for an elliptic distribution
of lift from tip to tip. The efficiency factor (e) is equal to 1.0
for an elliptic distribution and is some value less than 1.0 for any
other lift distribution. For a rectangular planform, like the Wright brothers
wings, e = .7 . The total
drag coefficient is equal to the drag coefficient at zero lift (Cdo),
plus the induced drag coefficient.
Cd = Cd0 + Cl^2 / ( pi * Ar * e)
The Wright brothers learned about induced drag the hard way. Following
their first glider flights of
they knew that they had to increase the size of their wings to allow flight in
reasonable winds. For the
1901 aircraft they increased the chord of the wing but
kept the span nearly the same. This produced a wing with an aspect ratio
of 3.0 and high induced drag. The
brothers had made mathematical predictions of the performance of their aircraft.
But the 1901 aircraft did not meet their range predictions because of lower than
expected lift and higher than expected drag. During the winter, with the aid of
wind tunnel, they began to understand the role of
induced drag on their aircraft's poor performance. They then designed the
1902 aircraft wing to have a longer span and shorter chord
than the 1901 aircraft. The aspect ratio was changed to 6.0 with nearly the same
wing area. By doubling the aspect ratio, the brothers cut the induced drag in half.
The 1902 aircraft was able to meet their performance goals
and they were able to attain glides of over 650 feet.
Remember that determining the drag is
only a part of the design problem. You will find that a higher angle of attack
produces more lift, but it also produces more drag.
You will also find that increasing the wing area increases the lift.
But in the total design, increasing wing area also increases the weight and the drag.
Designers usually try to optimize the
lift to drag ratio.
This is an efficiency factor for the aircraft and inversely related to the
An aircraft with a high lift to drag ratio can glide a long distance
while losing only a little altitude.
The Wrights were aware that they needed both high lift and low drag.
You can view a short
of "Orville and Wilbur Wright" discussing the drag force
and how it affected the flight of their aircraft. The movie file can
be saved to your computer and viewed as a Podcast on your podcast player.
- Re-Living the Wright Way
- Beginner's Guide to Aeronautics
- NASA Home Page