Friction Force

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:

  

$$u^L_g \cos\Psi + u^L_g \sin\Psi$$ u^G_g = (85)

$$-v^L_g \sin\Psi + v^L_g \cos\Psi$$ v^G_g = (86)

The ground-path coordinate system is like local coordinates in that the z-axis points vertically down. However, in ground-path axes, the x-axis points in the direction of travel along the ground (while remaining in the horizontal plane). The x- and y-components of the tire's velocity in ground-path coordinates are, in fact, the forward and sideward components.

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.

Sideward Force.

First, v_g, the sideward groundspeed component, is compared to a threshold velocity, V_k. If |v_g|>V_k, then a kinetic coefficient of friction applies. If $\vert v_g\vert\le V_k$, the static coefficient of friction applies $^{\ref{ref:larcsim}}$. The friction force in the sideward direction is given by Equation 87.

$$F_s = \left\{ \begin{array}{ll} \displaystyle -\frac{v_g}{V_k... ...n}(v_g) \mu_k F_n, & \vert v_g\vert \le V_k \end{array}\right.$$ (87)

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.

Forward Force.

The forward force calculation resembles to the sideward force calculation. One complication seen with forward force calculations is the possibility of braking.

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 ( $\delta_b=0$) to full pressure ( $\delta_b=1$).

  

$$V_{k,roll} + \delta_b(V_{k,slide}-V_{k,roll})$$ V_k = (88)

$$\mu_k \mu_{k,roll} + \delta_b(\mu_{k,slide}-\mu_{k,roll})$$ = (89)

Using these definitions of V_k and $\mu_k$, the forward friction force is as listed in Equation 90.

$$F_f = \left\{ \begin{array}{ll} \displaystyle -\frac{u_g}{V_k... ...n}(u_g) \mu_k F_n, & \vert u_g\vert \le V_k \end{array}\right.$$ (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:

$$F_f \cos\Psi - F_s \sin\Psi$$ X^L_G = (91)

$$F_f \sin\Psi + F_s \cos\Psi$$ 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)