A tire is designed to roll easily in one direction, while sliding with
difficulty in the perpendicular direction. It makes sense to resolve
the friction force into a forward rolling friction and a sideward
sliding friction. This paper assumes that the plane of the tires is
parallel to the body *x*-*z*-plane (i.e., no steering).

Because friction always acts in a direction opposite the velocity, the
calculation of friction force requires the velocity of the tires
resolved into those directions. The velocity of the tire in body
coordinates is the velocity of the CG plus contributions due to
angular rates:

u^{B}_{g} |
= | u + q(z^{B}_{g}' - z^{B}_{c}) - r(y^{B}_{g} - y^{B}_{c}) |
(82) |

v^{B}_{g} |
= | v + r(x^{B}_{g} - x^{B}_{c}) - p(z^{B}_{g}' - z^{B}_{c}) |
(83) |

w^{B}_{g} |
= | w + p(y^{B}_{g} - y^{B}_{c}) - q(x^{B}_{g} - x^{B}_{c}) |
(84) |

These can be transformed to local coordinates with the airplane direction cosine matrix. Then, the tire velocity in local coordinates are transformed into the ground-path coordinate system by Equations 85 and 86:

The ground-path coordinate system is like local coordinates in that the

For the rest of the section, the superscript on *u*^{G}_{g} and *v*^{G}_{g} is
dropped; all tire velocities are in ground-path coordinates.

Both the static and kinetic coefficients are functions of both the tire and landing surface. Note that the force of static friction is proportional to the negative velocity.

There is one factor which has been neglected above. A rolling tire has a relieving effect on the sideward friction, meaning that if the tire is rolling, the sideward friction force of a tire will be somewhat lower than calculated. The method of calculating this effect is not very transparent. Reference 19 presents a more sophisticated method for calculating side force of a rolling tire.

Braking changes both the threshold velocity and the kinetic friction
coefficient. We assume the effect varies linearly with brake
pressure. Equations 88 and 89 give
formulas for the threshold velocity and kinetic friction coefficient
under different braking conditions, where the brake pressure varies
from no brake pressure (
)
to full pressure (
).

Using these definitions of *V*_{k} and ,
the forward
friction force is as listed in Equation 90.

The forward friction, sideward friction, and ground normal forces are
the *x*-, *y*-, and *z*-components of force in the ground-path
coordinate system. (*F*_{n} could have been written *Z*^{G}_{G}.) The
forces are transformed first into local coordinates:

X^{L}_{G} |
= | (91) | |

Y^{L}_{G} |
= | (92) | |

Z^{L}_{G} |
= | F_{n} |
(93) |

Then, the components of force are transformed from local to body coordinates. The moments are obtained from multiplying the force components in body coordinates by the appropriate moment arms:

L_{G} |
= | Y_{G}(z^{B}_{g}' - z^{B}_{c}) - Z_{G}(y^{B}_{g} - y^{B}_{c}) |
(94) |

M_{G} |
= | Z_{G}(x^{B}_{g} - x^{B}_{c}) - X_{G}(z^{B}_{g}' - z^{B}_{c}) |
(95) |

N_{G} |
= | X_{G}(y^{B}_{g} - y^{B}_{c}) - Y_{G}(x^{B}_{g} - x^{B}_{c}) |
(96) |