Ground Normal Force

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 + C₃₁(x^B_g-x^B_c) + C₃₂(y^B_g-y^B_c) + C₃₃(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 $\Delta z^B_g$. 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 $\Delta z^B_g$. Equation 78 gives the formula for this.

$$\Delta z^B_g = z^B_g - {z^B_g}' = z^B_g - \frac{z^L_r - z^L_c + C_{31}(x^B_g-x^B_c) - C_{32}(y^B_g-y^B_c)}{C_{33}}$$ (78)

Because the strut is damped, the force also depends on the rate of landing gear displacement. Taking the time derivative of Equation 78, the only non-constant term on the right side is z_c/C₃₃. Therefore, the displacement rate is given by:

$$\Delta \dot z^B_g = \frac{\dot z^L_c}{C_{33}}$$ (79)

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:

$$F_{strut} = K(\Delta z^B_g) + C(\Delta \dot z^B_g),$$ (80)

where K and C are the spring and damping constants for the strut. The strut force acts along the axis of strut; however, the force exerted by the ground acts vertically upward. The value of the total ground normal force is the value where the component along the strut axis is the strut force. That is,

F_n = F_strut / C₃₃ (81)

More advanced models consider the tire deflection as well as the the strut deflection in calculating ground normal force.