Lumped tire model : bristle deflection

[Deams]

Hi,
And thank you for your time! :)

I've been studying and coding for more than a month on a physics engine for car driving simulation.
Right now, I'm trying to change the whole tire model currently based on Pacejka for a more advanced one.
After quite a lot of reading on the topic, I decided to give a try at a modified brush tire model.
(for those who can, here is the paper : tandfonline)

My goal is by using predefined parameters and current simulation data to be able to calculate the longitudinal and lateral forces on each wheel.

From what I understand of the paper, this is partly done by considering the lumped state obtained by averaging the deflection variables z(ζ,t) of the bristles composing the surface patch.

Well I'm struggling with this notion of deflection variable. What is it? What does it represent? How to calculate it? This is the kind of questions I've been asking myself for the past 3 days.

If you can help me understand a bit better, thank you in advance. :)

Feel free to ask me for more information if you need it!

 

[Valerie Park]

The deflection variable z(ζ,t) represents the displacement of the individual bristles under the load. The bristles are modeled as linear springs, so the deflection for each bristle can be calculated by solving the equation of motion for the bristle. The equation is a second order differential equation, and can be solved numerically using standard numerical integration techniques. Once the equation of motion is solved, the displacement of the bristle can be calculated. This displacement is the deflection variable that you need to calculate in order to compute the lumped states.