Understanding Tire Forces in Bicycle Simulation: Physics and Code Explained

[xXNightEagleXx]

I'm creating a game, simple 2d game but that relies on individual tires forces simulation in a bicycle model. There are code out there but I'm not the kind of guy that just copy and paste. I like to understand and i love physics.

I started by reading this web page

It gave me a good starting point . I then found a copy of the book Milliken & Milliken Race car vehicle dynamics. Everything was fine and easy to understand until i reached the tire forces calculation and application.

First of all, I've read that
Ft - ∑ F_resistance = ma

which is pretty straightforward.

However speaking of tires in specific the driving tires receives this Drive torque from the shaft sometimes called
Tq_wheel = T_e * xg * xd * η

which is translate into traction force
F_t = T_wheel / r

So i can plugin this Ft in the first equation and i have a result

But both the link and book states that the longitudinal traction force in the tires is the result of a function of the slip ratio
SR = (ΩR/V) - 1

that gives us a normalized value that gets multiplied by F_z to obtain F_x (which in my understanding is the traction force F_t)

Since what push us forward is the friction force, like this video explain, then what the book says is not wrong (never doubted it)
F_t = μ(SR) * F_z

However this lead me to a math problem, to something that seems to be a contradiction. First of all, first equation says that F_t is directly related to Tq_wheel, but the other says that it is related to the friction coefficient and the load, in other words the friction force. Moreover, at t₀ with both V and Ω equal to 0, if i apply some Tq_wheel that rotates the tire at t₁ i'll still have SR = -1 (locked) because my V will still be 0...what am i missing here?

Trying to figure out what I'm missing, i thought that what happens here is that before the tire slips what matter is the F_fs
F_fs = mgμ (as static friction and not a function)

If that's the case then it means that initial instantaneous force is always maximum in proportion to the parameters, regardless you went full pedal or not...which seems something wrong...

Things get even stranger when i analyze front wheel (free wheel) where it creates a friction force in the opposite direction of the rear one. Using the F_fs equation above would mean again instantaneous force...

This video says that
F_fs = ma*1/2

That would make sense that the bigger the acceleration the bigger the friction force and the bigger the torque that rotates the front wheel but where does the coefficient of friction fit in this equation? I mean if i have 0 coefficient of friction then i have 0 torque.

As you can image at this point gets basically impossible to apply all of this.

In my head i thought that the code would be something like this (per tire)
(rear)
- compute Tq_wheel according to throttle pedal
- compute Tq_resistance (braking, rolling)
- compute F_t with dynamic weights using Tq_total (1st doubt here)
- compute F_lat = Coefficient of stiffness * Fz
- limit the total force to the amount of grip on the tire
(front)
- compute Tq_friction with dynamic weights
- compute Tq_resistance (braking, rolling)
- compute F_t using Tq_total
- compute F_lat = Coefficient of stiffness * Fz
- limit the total force to the amount of grip on the tire
(finally)
- Apply forces for each wheel
- Calculate wheels acceleration
- Integrate the wheels

I'll end for here because just clarifying above points will be extremely helpful and probably even make the pseudo more clear

Thanks in advance
W

 

[xXNightEagleXx]

I guess i have found the root of my doubts... slip ratio at low speed (v0 equal or close to 0)... that stupid thing brought me to a whole world of problem, that simply confused me. Now think about waking up at 6am, overthinking everything up to the point to messed up what was logic and well known...that's what happened to me...desperation to solve a problem up to the point where you believe that you could be wrong from the beginning.

That being said, everything makes sense now, even how the traction force (rear tires) and the inverse friction force (front tires) acts and how the car moves due to traction force.
Problem is that once V = 0 or close, obviously, the ForceTraction = mu(slipratio) * N gets impossible with the "standard" definition of the slip ratio since it will be undefinited.

I'm trying to figure out the best way to deal with low speed situation, maybe even using pajecka magic numbers.

If anyone has a suggestion about how to deal at low speed i'll be glad to hear it.

Thanks for everyone

 

[Tom.G]

While the driving force of the wheel is less than μ times the downward force on the wheel, the SR=0. That is the condition of normal street driving, i.e. gentle acceleration. The SR rises (the wheels spin) when the supplied engine torque exceeds μ times the downward force on the wheel (the sliding static friction).

See also this thread:
https://www.physicsforums.com/threads/why-do-tires-slip-easier-from-a-standstill-start.931807/