Weight is the force
generated by the gravitational attraction of the earth on the
airplane.
Each part of the aircraft has a unique weight and mass,
and for some problems it
is important to know the distribution. But for total aircraft
maneuvering, we only need to be concerned with the total weight and
the location of the center of gravity. The
center of gravity
is the average location of the mass of any object.
How do engineers determine the weight of an airplane which they are designing?
An airplane is a combination of many parts; the
wings,
engines,
fuselage, and
tail,
plus the payload and the fuel.
Each part has a weight associated with it which the engineer can
estimate, or calculate, using Newton's
weight equation:
w = m * g
where w is the weight, m is the mass, and
g is the gravitational constant which is 32.2 ft/square
sec in English units and 9.8 meters/square sec in metric units.
The mass of an individual component can be calculated if we know
the size of the component and its chemical composition.
Every material (iron, plastic, aluminum, gasoline, etc.) has a unique
density. Density r is defined to be the
mass divided by the volume v:
r = m / v
If we can calculate the
volume
m = r * v
The total weight W of the aircraft is simply the sum of
the weight of all of the individual components.
W = w(fuselage) + w(wing) + w(engines) + w(payload) + w(fuel) + ...
To generalize, if we have a total of
"n" discrete components, the weight of the aircraft is
the sum of the individual i component weights with
the index i going from 1 to n.
The greek mathematical symbol sigma is used by mathematicians to denote this
addition. (Sigma is a zig-zag symbol with the index designation being
placed below the bottom bar, the total number of additions placed over
the top bar, and the variable to be summed placed to the right of the sigma
with each component designated by the index.)
W = SUM(i=1 to i=n) [wi]
This equation
says that the weight of the airplane is equal to the sum of the
weight of "n" discrete parts.
What if the parts are not discrete? What if we had a continuous
change of mass from front to back?
The continuous change can be
computed using integral calculus. The sigma designation is changed
to the integral "S" shaped symbol to denote a continuous variation.
W = S w(x) dx
The discrete weight is replaced with w(x) which indicates that the
weight is some
function
of distance x. If we are given the form of the
function, there are methods to solve the integration. If we don't
know the actual functional form, we can still numerically integrate
the equation using a spread sheet by dividing the distance up into a
number of small distance segments and determining the average value
of the weight over that small segment, then summing up the
value.
Activities:
Guided Tours
-
Aircraft Weight:
-
Fuselage:
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