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Tyre characteristics and modelling 297
{U S } 1
{Rw} 1
φ
GRF
O 1
W
{X sae } 1
{R PW } 1
{V } 1
P
{R P } 1
{Y sae } 1
{Z sae } 1
Fig. 5.54 Tyre geometry and kinematics. (This material has been reproduced from
the Proceedings of the Institution of Mechanical Engineers, K1 Vol. 214 ‘The
modelling and simulation of vehicle handling. Part 3: tyre modelling’, M.V. Blundell,
page 7, by permission of the Council of the Institution of Mechanical Engineers)
If the angular velocity vector of the wheel is denoted by {} 1 then the
velocity {V P } 1 of point P is given by:
{V P } 1 {V W } 1 {V PW } 1 (5.35)
where
{V PW } 1 {} 1 {R PW } 1 (5.36)
It is now possible to determine the components of {V P } 1 which act parallel
to the SAE co-ordinate system superimposed at P. The longitudinal slip
velocity V XC of point P is given by:
V XC {V P } 1 • {X SAE } 1 (5.37)
The lateral slip velocity V Y of point P is given by:
V Y {V P } 1 • {Y SAE } 1 (5.38)
The vertical velocity V Z at point P, which will be used to calculate the
damping force in the tyre, is given by:
V Z {V P } 1 • {Z SAE } 1 (5.39)
Considering the angular velocity vector of the wheel {} 1 in more detail
we can represent the vector as follows. The wheel develops a slip angle
which is measured about {Z SAE } 1 , a camber angle which is measured
about {X SAE } 1 and a spin angle which is measured about {U S } 1. The total
angular velocity vector of the wheel is the summation of all three motions
and is given by:
{} ˙{ ZSAE} ˙{ XSAE} ˙{ U s }
1 1 1 1
(5.40)