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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 = P

^{α}*Z

^{β}*(a+b*v+c*v

^{2}) + 1/2*ρ*C

_{D}*A*v

^{2}. 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 = c

_{1}+c

_{2}*v+c

_{3}*v

^{2}(v in km/h, and time in s)

My objective then is to find the constants c

_{1}, c

_{2}, and c

_{3}, 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*v

^{2})/m → dv/(c1+c2*b+c3*v

^{2}) = -dt/m →

∫dv/(c1+c2*b+c3*v

^{2}) = -∫dt / m

the ∫dv/(c1+c2*b+c3*v

^{2}) would be between v

_{0}= 115 and

**v**

the ∫dt would be between t

_{0}= 0 and

**t**

The integration of ∫dv/(c

_{1}+c

_{2}*v+c

_{3}*v

^{2}) is a very annoying one :

if 4ac-b

^{2}>0 I obtain the solution

-t/m = 2/√(4*c

_{3}*c

_{1}-c

_{2}

^{2}) * arctg [(2*c

_{3}*v + c

_{2}) /√(4*c

_{3}*c

_{1}-c2

^{2})] - 2/√(4*c

_{3}*c

_{1}-c

_{2}

^{2}) * arctgh [(2*c

_{3}*115 + c

_{2}) /√(4*c

_{3}*c

_{1}-c

_{2}

^{2})]

if 4ac-b

^{2}<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.