For simplicity, assume that the strut is parallel to the airplane's body z-axis, and that the landing surface is level. If the tire has made contact, its z-coordinate in local axes would be greater than the z-coordinate of the ground (remember, the z-axis points down).
The tire's coordinates in body axes are
these together with Equation 8, the tire's local
z-coordinate is given by:
Let zLr be the ground's local z-coordinate (the r means runway). If zLg > zLr, then the tire touches the ground. Of course, the tire cannot actually be located below the ground; the ground compresses the tire and landing gear strut.
The compressive force in the strut depends on the total gear
This is found by replacing zLg in
Equation 77 with zLr, and solving for zBg, to
yield zBg', the actual position of the compressed gear where it
touches the ground. The difference between the uncompressed and
compressed tire locations is
gives the formula for this.
A simple model for landing gear displacement force is to assume a
linear damped elastic strut, and ignore the tire compression. Then,
the compressive force in the strut is:
|Fn = Fstrut / C33
More advanced models consider the tire deflection as well as the the strut deflection in calculating ground normal force.