Brake Simulation

On the workstation one of the main goals are to measure the performance of the drivetrain during load. The tractive torque is the torque that must be applied on the wheels for the car to move. It consists of several elements which largely depends on the current velocity and acceleration of the car. By knowing the tractive torque, one may simulate it by applying the translated braking torque at the given speed/rpm on the measuring motor.

The tractive torque is defined by the following equation:

𝑇_𝑤 = 𝐹_𝑡∗ 𝑟_𝑤 𝐹_𝑡 = 𝐹_𝑟+ 𝐹_𝑔+ 𝐹_𝑑− 𝐹_𝑖𝑒

Tw: Tractive torque at the wheel Ft: Tractive force

rw: Radius of the wheel

Maximum length of wheelbase: 2250 mm

Minimum length of wheelbase: 1500 mm

Side | 45 Fr = tyre rolling resistance

Fd = aerodynamics drag

Fie = equivalent inertial force (during acceleration) - (including linear and rotational inertias, due to vehicle mass and rotating component of gear train and wheels)

Fg = forces due to gradient

Since it is the roller that will be applying the tractive force, it is the torque used on the roller to achieve the tractive force on the wheel that must be clarified. The same formula may be used with minor tweaks:

𝑇_𝑅 = 𝐹_𝑡∗ 𝑟_𝑅

TR: Torque at the roller rR: Radius of the roller

After this has been clarified, all that must be done is to convert the torque necessary at the roller to the motor.

𝑇_𝑅= 𝐹_𝑡∗ 𝑟_𝑅∗ 𝑖₃ i3: Ratio in between the roller pulley and motor pulley

The tractive force

For the equation above to function, the individual components of the tractive force must be defined.

First the rolling resistance will be defined. In this case the rear tyres are still experiencing rolling resistance. Therefore, the only rolling resistance that must be added are from the front tyres. Considering there are two measuring motors, each motor will apply the brake for one front tyres. To calculate the rolling resistance of the wheels it is necessary to, either do extensive testing with the tyres in question, or use a provided data sheet from the tyre’s manufacturer. In this case Michelin 90-80 R16 tyres are being used, which the data sheet is provided for. This makes it possible to make an expression for the rolling resistance using the following equation:

𝐹_𝑅 = 𝐶_𝑅∗ 𝑁 𝑁 = 𝑚𝑔 = 𝑚𝑎𝑠𝑠 ∗ 9,81𝑚

𝑠²

Cr: constant defined by the manufacture through thorough testing. It is a constant specific to the speed.

N: normal force due to gravity

Side | 46

Figure 35 Rolling Resistance vs. Speed for Loads (Michelin, 2015)

Since the graph for the tyre is provided, one may find an estimated equation for the rolling resistance throughout the rpm range by using interpolation:

v@0,0015=35 km/h=9,72 m/s

CR@9,72m/s=0,0015 CR@0m/s=0,00075 m=143 Kg (Car=75 Kg, driver=78 Kg)

𝐹_𝑅(𝑣) = ((𝐶_𝑅@9,72^𝑚

𝑠 − C_R@0m

s) ∗ 𝑣

9,72𝑚 𝑠

+ C_R@0m

s) ∗ 𝑚𝑔 = 1,05 N ∗ 𝑣 9,72m

s

+ 1,05 𝑁

This is the rolling resistance for all four wheels. Each motor will only have to simulate the resistance caused by one front wheel. This results in the following equation:

𝐹_𝑅1(𝑣) =F_R(v)

4 = 1,05

4 N ∗ v

9,72m s

+1,05

4 𝑁 = 0,263 𝑁 ∗ 𝑣 9,72𝑚

𝑠

+ 0,263 𝑁

FR1: Rolling resistance for a single wheel

Side | 47 The next force to consider is the equivalent inertial force. Since the purpose of the dyno system is to measure the car’s performance, the equivalent inertial force caused by the rollers and pulleys must be removed. Both of which functions as a flywheel, braking the rotation of the car’s wheel.

To simplify the expression, only the barrel of the roller is considered. More in depth calculations may be done, but they are not included in this report.

The expression that will be used is defined accordingly:

𝑇 = 𝐼 ∗ 𝛼

I: Moment of inertia for solid cylinder T: Newtons second law

α: Angular acceleration m: mass of roller

The equivalent inertial force is largely dependent on the acceleration of the car. The

acceleration will have to be real time monitored so that the equivalent inertial force can be subtracted from the total tractive force.

Then the forces due to aerodynamic drag needs to be calculated. This force is variable, meaning that the motors will need to apply more braking power the faster the car drives.

The aerodynamic drag is defined by the following equation:

𝐹_𝐷 = 𝐶_𝑑∗𝜌 ∗ 𝑣² 2 ∗ 𝐴

Cd: Drag constant ρ: Air density

v: Speed measured in m/s

A: Reference area being affected by the drag

For the equation to function there are a couple of constants that must be known. The approximate drag coefficient and reference area of the current car is defined in the backbone of DNV GL Fuel Fighters. Which is based on simulations made in openFOAM.

Side | 48

Figure 36 (Trefall, 2019)

Depending on where SEM will be held, the density of air will vary. In this instance the

density at sea level and 15C will be utilized. This is a normal setting which also functions as a worst-case scenario.

Using the formula, it is clear that the only variable will be the speed. The speed variable can be scaled accordingly to a rpm sensor attached to the measuring motor. By using the rpm sensor and a gearing constant the speed of the car can be calculated.

The final force to consider are forces due to gradient. This will however be neglected during this phase of the subsystem. The reason for this is because the track simulator is not

finished yet, so there is no way of knowing the gradients the car will experience.

Additionally, the different gradients and turns would cause the angel of the car, relative to the flow of air, to change. This would ultimately cause the reference area to change, causing the need for more aerodynamic testing.

Result

Now that the individual tractive forces have been defined, it is important to consider that there are two motors delivering the braking power. Therefore, the final tractive force for each wheel must be defined like this:

𝐹_𝑡 =𝐹_𝑟 4 +𝐹_𝑑

2 − 𝐹_𝑖𝑒

Now that the basis for the tractive force is established, that can be used to make an expression for the torque needed on the roller to achieve this braking force:

Flau 3000=Flow 3000 (The type of grid refinement which gave the most accurate results), snappyHexMeshDict used

Side | 49 𝑇_𝑅 = (0,263 𝑁 ∗ 𝑣

9,72𝑚 𝑠

+ 0,263 𝑁 + 0,0792𝐾𝑔

𝑚 ∗ 𝑣²−1

2𝑚𝑎) ∗ 𝑟_𝑅

Finally, the gearing constant in between the roller pulley and motor pulley can be used to calculate the needed braking torque on the measuring motor at the given speed and acceleration:

𝑇_𝑀 = 𝑇_𝑅∗ 𝑖₃ = (0,525𝑁 ∗ 𝑣 9,72𝑚

𝑠

+ 0,263 + 0,0792𝐾𝑔

𝑚 ∗ 𝑣² −1

2𝑚𝑎) ∗ 𝑟_𝑅 ∗ 0,33 𝑆𝑒𝑒 𝐺𝑒𝑎𝑟𝑖𝑛𝑔 𝑖𝑛𝑝𝑢𝑡

This is the formula that must be inputted into a motor controller to achieve the correct amount of tractive torque on the wheels.

Side | 50

4 Result

Figure 37 Workstation