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
(x^{B}_{g},y^{B}_{g},z^{B}_{g}). Using
these together with Equation 8, the tire's local
z-coordinate is given by:
z^{L}_{g} = z^{L}_{c} + C_{31}(x^{B}_{g}-x^{B}_{c}) + C_{32}(y^{B}_{g}-y^{B}_{c}) + C_{33}(z^{B}_{g}-z^{B}_{c}) | (77) |
Let z^{L}_{r} be the ground's local z-coordinate (the r means runway). If z^{L}_{g} > z^{L}_{r}, 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
displacement
.
This is found by replacing z^{L}_{g} in
Equation 77 with z^{L}_{r}, and solving for z^{B}_{g}, to
yield z^{B}_{g}', the actual position of the compressed gear where it
touches the ground. The difference between the uncompressed and
compressed tire locations is
.
Equation 78
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:
(80) |
F_{n} = F_{strut} / C_{33} | (81) |
More advanced models consider the tire deflection as well as the the strut deflection in calculating ground normal force.