# Car transmission model

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Hi everyone,

I'm reviewing a car transmission model for a basic racing game, and I've noticed a slight discrepancy between what I would expect of the macroscopic properties of the model (when changing gear) and the numerical results.

Here is the basic model:

from math import pi
import numpy as np

class Car:

gearRatios = []
maxTorqueArray = []
rpmAtMaxTorque = []

differentialRatio = 0.0
Cd = 0.0
Crr = 0.0
transmissionEfficiency = 0.0
mass = 0

def __init__(self):
self.speed = 0.0
self.acceleration = 0.0
self.currentGear = 1

def update(self, dt):
throttle = 1.0

# Gear ratios
assert self.currentGear > 0
currentGearRatio = self.gearRatios[self.currentGear - 1]
effectiveGearRatio = self.differentialRatio * currentGearRatio
transmission = effectiveGearRatio * self.transmissionEfficiency

wheelCircumference = 2 * pi * self.wheelRadius

# Find RPM
wheelRPM = (self.speed / wheelCircumference) * 60
engineRPM = wheelRPM * transmission
engineRPM = min(max(engineRPM, self.rpmAtMaxTorque[0]), self.rpmAtMaxTorque[-1])

# Find torque at RPM
maxTorque = np.interp(engineRPM, self.rpmAtMaxTorque, self.maxTorqueArray)
engineTorque = maxTorque * throttle
wheelTorque = engineTorque * transmission
tractionForce = wheelTorque / self.wheelRadius

# Resolve forces
dragForce = -self.Cd * self.speed * abs(self.speed)
rollingResistanceForce = -self.Cd * 30 * self.speed

# Resultant
resultantForce = tractionForce + dragForce + rollingResistanceForce

self.acceleration = resultantForce / self.mass
self.speed += self.acceleration * dt

self.currentForce = resultantForce
self.currentRPM = engineRPM
self.currentSpeed = self.speed
self.currentTorque = wheelTorque

class BoxsterS2004(Car):

gearRatios = [3.82, 2.20, 1.52, 1.22, 1.02, 0.84]
maxTorqueArray = [220, 220, 310, 220]
rpmAtMaxTorque = [0, 1000, 4500, 6600]

differentialRatio = 3.42
Cd = 0.32
transmissionEfficiency = 0.7
mass = 1300

if __name__ == "__main__":
car = BoxsterS2004()
dt = 1 / 60

while True:
car.update(dt)


Now, in it's actual implementation, I'm running the update function in a fixed-step game loop, which allows me to inspect the car state and shift gear at runtime.

Now, the behaviour:

It is to be expected that the force of the car drops according to the speed for the same power.

It is also to be expected that the car cannot reach as high a speed in lower gears.

Yet, for these parameters, the top speed can only be reached in 1st gear, and decelerates according as the gears are shifted higher.

I'm not sure why this happens. It makes sense that the acceleration rate is lesser for higher speeds, because the drag force is proportional to v^2, and the force of the engine falls with higher gears, but my resultant forces are negative for higher gears when they're not for the lower gears. Shifting at any point reduces the max speed achievable.

Evidently, I don't quite understand the model well enough, could anyone point me in the right direction?

To my mind, given that the driving force of the car falls off at

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Ddoes your engine model have a torque curve?

This is really the reason we have multiple gears in a combustion engine in the first place.

As you increase the rpm in a low gear, The torque will rise up to a peak and then fall down again, forcing you to shift up to get back into the peak region.

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Indeed, the torque curve is defined as follows (see the BoxsterS2004 class)

maxTorqueArray = [220, 220, 310, 220]
rpmAtMaxTorque = [0, 1000, 4500, 6600]



Where maxTorqueArray are the Y values for the rpmAtMaxTorque X values.

I can see that the descending gear ratios will shift the engine back into the max torque region, but the applied force is reduced because the transmission factor (gear * differential * loss factor) is reduced by the gear reduction.

From this, I am led to believe that the model is fine, but good data is required.

Edited by Angus Hollands

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Did you experiment with the drag coefficient?

Your calculations seem to be correct

Edit: I played around with the drag in my simulation and get the same behaviour. If the drag is too big the car reaches top speed only in 1st gear

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Your rolling resistance force is all wrong

It should be in the region of 0.010 to 0.015 for a car tire on tarmac.

You have a value of around 0.32 * 30 = 9.6

A little bit of a difference

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