# Go Kart resistance force

• Auto/Motor
Hello to all,

I am trying to design an electric Go kart as part of a personal project. I am now in the preliminary design phase, trying to find some literature regarding to the resistance force (Rolling resistance + aerodynamic drag).

I was able to find a document in which some guys did some testing, obtaining a graph speed/time (I understand that the controller was plotting a graph of values of speed and time). The graph shows how they disengaged the motor with a clutch and left the kart to decelerate by its "own" (in reality due to all the resistances = Rolling + aerodynamic...considering the bearing friction and rolling resistance negligible). I also consider the lifting down force negligible (measured to be around less than 1% the kart weight).

The steps I was following to find out the resistance force are the following:

I check the graph and wrote few +/- accurate points (speed in km/h, time s) contained in the graph. I got (115, 0) (80, 4) (60, 10) (40, 20).

I am going to consider the resistance force Fr= Frolling resistance +Drag = Pα*Zβ*(a+b*v+c*v2) + 1/2*ρ*CD*A*v2. I used the SAE J2452 for the rolling resistance (where v is in km/h). I am considering P, α, Z, β, a, b, c constant. Therefore, I ended up with a Fr = c1+c2*v+c3*v2 (v in km/h, and time in s)

My objective then is to find the constants c1, c2, and c3, using the graph points I showed before.

since the kart is decelerating only by the resistance force Fr, then I know that the deceleration -a = Fr/m

and a=dv/dt, therefore dv/dt = -(c1+c2*b+c3*v2)/m → dv/(c1+c2*b+c3*v2) = -dt/m →
∫dv/(c1+c2*b+c3*v2) = -∫dt / m

the ∫dv/(c1+c2*b+c3*v2) would be between v0 = 115 and v

the ∫dt would be between t0 = 0 and t

The integration of ∫dv/(c1+c2*v+c3*v2) is a very annoying one :

if 4ac-b2 >0 I obtain the solution

-t/m = 2/√(4*c3*c1-c22) * arctg [(2*c3*v + c2) /√(4*c3*c1-c22)] - 2/√(4*c3*c1-c22) * arctgh [(2*c3*115 + c2) /√(4*c3*c1-c22)]

if 4ac-b2 <0

Similar but with the Ln.

When I try to resolve the function v (t) using excel solver...it cannot converge into any solution.

Could somebody help me finding a way to solve the problem?

I have other ideas to calculate the Resistance torque/Force, but it would required a torque transducer that would complicate the design of the vehicle.

Excuse me if I had any mistake or if the formulas are not very well explained. it isn't easy to write those formulas in the forum.

## Answers and Replies

anorlunda
Staff Emeritus
It's a good question. This is not a full answer, but I think you made an error or a typo in

(c1+c2*b+c3*v2)
Shouldn't it be (c1+c2*v+c3*v2) ?

Hi Anorlunda,

Thank you, yes it was a typo but only here at the forum, the (c1+c2*v+c3*v2) is the correct polynomial I used in the problem.

Regards,

Pablo

jrmichler
Mentor
Here's how I measured the drag and rolling resistance of my truck: https://ecomodder.com/forum/showthread.php/coastdown-test-06-gmc-canyon-20405.html. I used a simpler model, where total drag = fixed rolling resistance plus a speed squared term for aerodynamic drag. It solved easily because I was fitting only two coefficients to a much larger number of data points, and the data points went down to zero speed.

When fitting an equation to data, several conditions must be met to get good results:
1) Enough data points. More is better.
2) Data must cover a wide enough speed range. If you are fitting a curve with a quadratic term, you should have speed data down to zero speed.
3) Noise level must be low enough.

berkeman and anorlunda
jack action
Science Advisor
Gold Member
Here's an empirical method from Bosch Automotive Handbook 4th ed. (p.331):

Empirical determination of coefficients for aerodynamic drag and rolling resistance

Allow vehicle to coast down in neutral under windless conditions on a level road surface. The time that elapses while the vehicle coasts down by a specific increment of speed is measured from two initial velocities, ##v_1## (high speed) and ##v_2## (low speed). This information is used to calculate the mean deceleration rates ##a_1## and ##a_2##. See the following example.

The example is based on a vehicle weighing ##m## = 1450 kg with a cross section ##A## = 2.2 m².

The method is suitable for application at vehicle speeds of less than 100 km/h.

1st trial (high speed)

Initial velocity: ##v_{a1}## = 60 km/h
Terminal velocity: ##v_{b1}## = 55 km/h
Interval between ##v_a## and ##v_b##: ##t_1## = 6.5 s

Mean velocity: ##v_1 = \frac{v_{a1} + v_{b1}}{2}## = 57.5 km/h

Mean deceleration: ##a_1 = \frac{v_{a1} - v_{b1}}{t_1}## = 0.77 km/h/s

2nd trial (low speed)

Initial velocity: ##v_{a2}## = 15 km/h
Terminal velocity: ##v_{b2}## = 10 km/h
Interval between ##v_a## and ##v_b##: ##t_2## = 10.5 s

Mean velocity: ##v_2 = \frac{v_{a2} + v_{b2}}{2}## = 12.5 km/h

Mean deceleration: ##a_2 = \frac{v_{a2} - v_{b2}}{t_2}## = 0.48 km/h/s

Drag Coefficient
$$C_d = \frac{6m(a_1-a_2)}{A(v_1^2 - v_2^2)} = 0.36$$

Coefficient of rolling resistance
$$f = \frac{28.2(a_2 v_1^2 - a_1 v_2^2)}{1000(v_1^2 - v_2^2)} = 0.013$$

berkeman and anorlunda
Thank you for the responses!

jrmichler, Not sure why I didn't think about a polynomial regression, you just solved my issue right away!

jack action, Thank you! I didn't know about this empirical solution. It will be handy to compare it with a polynomial regression. Maybe I should check the bosh handbook, since its quite inexpensive

jrmichler and berkeman