In cruising flight,
an airplane can sustain a constant speed and level flight in
which the forces are all in
equilibrium.
As shown on the slide,
the weight is balanced by the
lift, and the drag
is balanced by the thrust. However, if the
forces become unbalanced, the aircraft
will move in the direction of the greater force. We can compute the
acceleration (a), velocity (V), and final
displacement (X)
of the aircraft using Newton's Second
Law of Motion. On this slide, we will consider only the
horizontal (X-direction) motion, but simlar equations could be
developed for the vertical and side-to-side motion as well.
If the mass (m) of the aircraft remains a constant
we can use the familiar form of Newton's second law to solve for the acceleration:
F = m * a
a = F / m
We have to determine the
mass of the aircraft from the weight. The force (F) will be the
difference between the opposing forces (thrust minus drag).
If the force also remains constant, the basic equations of motion can
be solved. For a constant force and constant mass, the
acceleration remains constant. The velocity (V) at any time (t)
is the acceleration (a) times the time plus the initial velocity (Vo).
V = a * t + Vo
Similarly, the location (X) at any time (t) is given by 1/2 the
acceleration times the time squared, plus the initial location (Xo),
plus the initial velocity times the time.
X = .5 * a * t^2 + Vo * t + Xo
Note that these equations can be used only if the mass and the
force (and the
acceleration) are constant. The mass of an aircraft remains fairly
constant during cruise since the only loss is for the fuel which is consumed.
Fuel mass is normally a small percentage of the mass of an aircraft.
However for aircraft, the lift and drag forces
are themselves functions of the square of the
velocity. So we can assume a constant force for only a very small
amount of time. To solve the actual equations of motion for an
aircraft, we must use calculus and integrate the equations of motion.
The integration can be performed analytically or numerically.
Navigation..
- Re-Living the Wright Way
- Beginner's Guide to Aeronautics
- NASA Home Page
- http://www.nasa.gov
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