> restart:  mylib :="C:\LIBS": libname := mylib,libname: with(MBsymba):

Variables

> x(t),y(t),z(t), center of mass coordinates (point G)
psi(t), yaw angle
phi(t), roll angle
theta(t); spin angle

Definition of Frames

frame for the wheel in vertical position with the z axis pointing downward

> TSAE := translate(x(t),y(t),z(t)) * rotate('Z',psi(t));

wheel frame (without spin rotation) where the x rotation represents camber angle

> TW := TSAE * rotate('X',phi(t));

wheel-fixed frame (spin rotation included) where the y rotation represents wheel spin

> TWS := TW * rotate('Y',theta(t));

Kinematics

wheel center

> G := origin(TW): show(G);

unit vector v (points from wheel center towards point of contact)

> v := make_VECTOR(TW,0,0,1): #show(v);

position of the contact point, where rho is the tire radius

> C := G + rho*v:
show(C);

sliding velocity of the contact point

> VC := velocity(C,'absolute'): #show(VC);

> VC := project(VC,TSAE): show(VC);

> VS := Xcomp(VC); forward velocity

> VN := Ycomp(VC); lateral velocity

rolling velocity (differentiating theta with respect to time multiplied by the wheel radius i.e. vertical component of the contact point position)

> VR := diff(theta(t),t)*evalm(Zcomp(C));

longitudinal slip (SAE definition: 1 plus the ratio of the rolling velocity over the contact point foward velocity)

> kappa := 1+VR/VS;

sideslip angle (SAE definition: arctangent of the negative contact point lateral velocity over the foward velocity)

> lambda := arctan(-VN/VS);