Calculating the Slip Ratio of a Vehicle Tire

I found this topic because I have the same problem, I'm trying to simulate the acceleration of a motorcycle. These are my consideration: When you apply a torque, an angular acceleration appear, that depends on the inertia of the wheels. This acceleration produce slip, this slip produce a torque that slow down the "free rotation" of the wheel, and so on:

If you write the equation of that wheel:

$M_{engine} - F(k) * r = I * dw/dt$

$dw = M_{engine}/I*dt - F(k)*r/I*dt$

Where $M_{engine}$ is the torque of the engine, $F(k)$ the tire force with $k$ the slip, $r$ the wheel radius, $I$ the wheel inertia and $dw/dt$ the angular acceleration.

I'm not sure this expression is analytically integrable, but of course numerically it is. Mine is a rude integration, you can use more sophisticated method, I write this one because I can check it's validity using a simple spreadsheet.

So, I use the simplified Pacejka expression, that one with only 4 parameters. Typical value for dry tarmac is:

B = 10

C = 1,9

D = 1

E = 0,97

So, at time = 0 set k = 0, the force will be 0. You can calculate $dw$ (with the equation I write before), using it to calculate $w_{i+1} = w_0 + dw$, then you can calculate $k$:

$k = (w_{i+1}*r-V)/(w_{i+1}*r)$

where $V$ is the actual speed of the vehicle. Now you have $k$, you can calculate $F$, and so the acceleration.

I try to simulate a motorcycle with a total mass of 300 kg, inertia of 0,4 $kg*m^2$, wheel radius of 0,3 m, with a starting speed of 20 m/s. I found that the higher speed will be obtained applying a torque of 899 Nm, and the vehicle reach 68.8 m/s after 5s.

If I apply 900 Nm, the wheel start to slip too much and loose traction: