# Wheel rolling resistance

I think that some of you know what is rolling resistance. It is force that works in opposite direction to roll and is caused due to deformation of tire. It also affects car wheels. There is one problem with it. I have found 2 equations around:

• $F_\mathrm{roll} = -V_\mathrm{car} * C_\mathrm{rr}$

• $F_\mathrm{roll} = W_\mathrm{load} * \sin\alpha * C_\mathrm{rr}$

However, none of them seems to be right one. First one seems to be incomplete, inaccurate, but second one doesn't depend on speed making car shake when being in standstill.
I need to know what is accurate(but still game-compatible, not too heavy) equation for rolling resistance.

Before we dig into practical equations, I'll just say that tires are surprisingly complex in their behavior. There are a variety of equations that try to fit the experimental data reasonably well. Hard to say that any one is right or wrong.

That having been said, the first equation is pretty inconsistent with the everything that I've ever seen. Weight on the wheel should almost certainly be included and rolling resistance rarely approaches 0 at 0 velocity in the literature. Two forms cited by Gillespie [ 1] are:

$$\frac{F_{r}}{W} = f_r = 0.1(1+V/100)$$

for low speeds and

$$\frac{F_{r}}{W} = f_r = f_o + 3.24 f_s(V/100)^{2.5}$$

over a broader range of speeds. Were F_r, W, V and f_i are rolling resistance force, weight on the tire velocity and empirical constants. Units are lbf, lbf and mph (sorry!).

The second term looks like an expression for increased rolling resistance when steering/slipping. $\alpha$ would be the slip angle of the tire. Something like that would be added to either of the above equation. To give you a sense of the what this looks like, check out this graph. Also for Gillespie: Lastly, with the above equations, you'll probably still have an issue with jittering around $V=0$ in a numerical simulation. Instead of using simply switching direction of force with the sign of velocity ($F_r*sign(V)$), try multiplying the force by $tahn(\beta V)$ instead, to smooth out the transition ($\beta$ is a parameter to control how sharp the transition is).

[ 1]Thomas, D. Gillespie. "Fundamentals of vehicle dynamics." Society of Automotive Engineering Inc (1992): 168-193.

• Few comments. First equation in my question is oftenly found in car physics tutorials for games(e.g. Marco Monster, Brian Beckman), while the second term is from Wikipedia. $W_\mathrm{load} * \sin\alpha$ is the normalforce and $C_\mathrm{rr}$ is the rolling resistance coefficient. Propably must ask how to calculate the coefficient. Because in 0 speed it should be also 0, otherwise car would start to just move when in standstill. – Adrians Netlis Mar 6 '16 at 6:00
• So the $V*C_{rr}$ does seem to be common in game physics, but doesn't match reality very well. Friction exists even if an object isn't moving. An object at rest will resist applied force, remaining stationary, until that force exceed the static friction force. Implementing this is more tricky numerically, hence I recommend something like $F_{rr}=W C_{rr} tanh(\beta V)$. With a high $\beta$, $F_{rr}$ will remain nearly constant (realistic) except right near $V=0$ (numerically stable). Best of both worlds. – Dan Mar 6 '16 at 16:47
• OK! Let's see if I have tanh aviable in blender:D Oh, and what if I use Pacejka Magical Formula taking in V where usually would slip be? Would it be more realistic than?:) – Adrians Netlis Mar 6 '16 at 17:06
• The Pacejka formula has a shape that would probably work pretty well for your application. Unless you're using it as intended (with slip ratio instead of velocity), the fact that it was developed for tires won't make it more physically meaningful, but that doesn't mean that you won't be able to find coefficients that produce the behavior that you want. – Dan Mar 6 '16 at 21:01
• On second though: If you're coding up the equation already, you might go ahead and use slip ratio etc. That would be very realistic and would set you up for traction-limited acceleration, cornering and braking! It would be more complicated because you'd be dealing with the fact that the wheel speed ($\Omega R$) won't match the ground speed most of the time, so there'l be some back and fourth coupling between the ground and the drivetrain. – Dan Mar 6 '16 at 21:06

to calculate fulcrum based rears wheels my formula : f²(w²divide by unit decimal digits) derive : V = mass gravitational force x psf vol : f²(w² x w¾) as value to minimal point to sling aerodinamic idea

time by ff (force G break) tq