Tractive Force Graphs

This is a logic path for understanding straight line acceleration performance.

Thirty second explanation of power, torque, and gearing:

Imagine a Civic and a Corvette traveling at the same speed.
30mph.
The Civic is in 2nd gear, turning 3000 rpm.
At that RPM, at full throttle, the Honda engine can produce 105 lb-ft and 60 hp at the crankshaft.
The Corvette is in 6th gear, turning 1000 rpm.
At that RPM, at full throttle, the small block can produce 315 lb-ft and 60 hp at the crankshaft.

Imagine both cars have the same weight, the same traction, the same aerodynamic drag.
When you give both full throttle, they accelerate at the same rate as long as they have the same power.
Given fairly flat torque curves, they will be neck and neck until 60mph when the Honda has to shift into 3rd.

How do you make the Corvette faster?
That's what the other five gears are for.

Put the Corvette in 1st gear, turning 3000 rpm.
Same as the Civic.
At that RPM at full throttle, the small block can produce 315 lb-ft (flat torque curve) and 180 hp.
3x the mechanical advantage in 1st gear vs 6th, 3x the force at the tires.
And the Civic won't see which way it went.

This is all a lot easier to do in metric units.
Especially EV teams: Save yourselves a lot of trouble.
Do all your calculations in metric.

As engineers, we settle disputes with math.
All of your questions need to be answered with math.
I'm not going to give you the equations.
Just the concepts.
But trust me, you can do all of this easily.
Whether you use a spreadsheet, MATLAB, or another language, iterate the tool and the parameters.
Sweep different factors and keep all those experiments.
Make sure it's well labeled and bring it, along with anything interesting you learn, to design judging.

When you put your analysis together, here's what you will confirm:
The car with more power at a given vehicle speed has more positive driving force at the wheels.
It doesn't matter:

• What peak power or torque is.

• Whether it's a big slow revving engine or a small high revving engine.

• Whether the wheels are 10's or 13's or any other size.

• What the final drive ratio is.

The car with more power at a given vehicle speed has more positive driving force at the wheels.
The more power there is, the more mechanical advantage can be applied.
Torque is nothing without rotational speed.

The graph shows the power curves intersecting at exactly the same speed as the wheel torque curves.
This is guaranteed for any power curve by the mathematical relationship between power and torque.
It is not a coincidence.
At a given speed, the gear with more power is the gear with more wheel torque.
The optimum shift point is a little different in each gear.

Let's talk about flat torque curves and flat power curves.

I use them as simplifications to make the concepts easier.

The only difference between motor power and wheel power is drivetrain efficiency (losses).
Drivetrain efficiency is generally approximated as constant, but not necessarily constant.
At 90% efficiency, power will be 90% as high, at 70% efficiency, power will be 70% as high, etc.

I guessed low, but the example flat power curve is reasonably well matched against the example flat torque curve.
The flat power example has an advantage after the flat torque example shifts into 2nd, 3rd, 4th, and 5th.
The flat torque example has an advantage over an increasingly large top rpm range in each gear.
This flat torque example will pull away in most of 5th gear where it has more power.
(Assuming the cars are not drag limited before that point.)

The graph shows that with a perfectly flat torque curve:
1. Hold each gear to maximum rpm.
2. Use as many gears as practical.

The graph shows that with a perfectly flat power curve:
3. It literally doesn't matter what gear is used.
4. Might as well use a single speed, as long as the flat power rpm range is large enough.

Make sure to prove that in your analysis tool if you're not convinced.

Information needed for a tractive force graph:

• Chassis dyno curve, or an engine dyno curve and estimate of drivetrain loss

• Total reduction ratio from motor to wheel in each gear

Useful extras:

• Estimate of tire longitudinal mu

• Estimate of Center of Gravity (CoG) location (longitudinal and height), and wheelbase

• Estimate of Area Coefficient of Drag (CdA)

• Estimate of Center of Pressure (CoP) height for drag

• Estimate of Area Coefficient of Lift (ClA) and CoP, if applicable

Traction is the result of combined normal forces multiplied by tire mu.
This example assumes rear wheel drive (RWD.)
All cars experience drag rear load transfer.
(Race Car Vehicle Dynamics, Milliken, Fig15.10 p504, 14th printing)
See Critical Specifications For Balance for a detailed discussion.
Acceleration load transfer can be modeled from flat power, stepwise, or recursive.
All models are wrong.
Make the model useful.

The relative scale of these factors will depend on your coefficients.

The tractive force graph shows how the flat power curve is tangent to each tractive force curve.
Peak power has the highest potential for acclerative force.
It could also be the curve of a CVT with equal mechanical efficiency to the standard gearbox.

The difference between tractive force and drag is the acceleration force available.
I find it easy to understand as shown, but another approach is to subtract drag from each in-gear curve.
The straight line performance envelope of your car is bounded by the traction limit, the area under the tractive force curves for each gear, and drag.

You can use tractive force graphs to determine:

• Overall gearing

• How many speeds your team wants to use

• RPM drops between gears and the optimum speed to run the engine

  • i.e. RPM for upshift and downshift lights

• What shape your team would like the powerband to have in that region

Maybe one option or another is not clearly superior.
Or the gains are marginal.
Especially compared to the amount of work.
Those can be given a lower priority than other decisions.
I always recommend going after the big advantages that are easy to do.
Followed by the big advantages that are hard, but justifiable.
Those require judgement calls.
Don't miss the small advantages that are easy.
Save the small ones that are hard for later.
Maybe triage them out of the plan.

Data is essential.
If you have a complex acquisition system with dozens of sensors, put them to work.
A stopwatch in a parking lot is also excellent data.
So are zip ties on dampers to show maximum travel.
Set up your tests to eliminate variables and do multiple measurements.

Show us how you can apply new techniques to last year's car, optimize its performance.
Then show us how much better this year's car is than last years.
Do this in a straight line.
Do this around a skidpad.
Do this through a slalom.
Do this through a braking event.

DO

• Figure out the equations that generated these graphs.

• Calculate, measure, improve, and optimize traction.

• Think about maps and setups for each different event.

• Maximize acceleration, possibly only compromising during endurance for fuel economy.

  • Train your team’s drivers to do all of the above.

• Take data, check calculations against data, revise, repeat.

  • Guess if no better estimate is available for calculations, test later for an empirical estimate.

DON'T

• Do not use RPM histograms to determine the most important rpm.

  • Do this instead: Look at speed histograms to determine the most important corner and peak speeds.

  • Do not forget to account for gearing and throttle position to determine the power being used.

• Do not assume shifting at maximum rpm is guaranteed to maximize acceleration.

• Do not make a lot of power without working on traction as well.

• Do not rely on explanations without checking the math yourself.

• Do not forget to update your calculations using the most current planned or measured changes in performance.

Critical Specifications For FSAE Electric Powertrain

Maximum acceleration is defined by traction mu (tire dependent, at all four wheels) and the 80kW limit.
The accumulator, inverter/controller, and motor each have a peak power limit, available for a short burst.
They each have a continuous power limit, which they can maintain indefinitely given sufficient cooling.
The continuous limit is best expressed as root-mean-square (RMS) power, to include regeneration.
The components have efficiency that varies with load.
Any difference between the component efficiency and 100% is heat.
Heat must be conducted to the cooling system and out of the car.
Heat transfer and heat buildup define the continuous power limit and peak time limits of the components.

Assuming the accumulator and inverter/controller can supply sufficient voltage and current:
A simplified EV motor model is peak torque from 0 until the start of peak power at the crossover rpm.
Then peak power from the crossover rpm until maximum rpm.
Continuous torque is often approximately 50% of peak torque.
Continuous power is often approximately 50% of peak power.
The continuous torque curve may or may not have the same crossover rpm.

Gearing will determine how close peak and continuous torque can get to the traction limit at the wheels.
Gearing may also limit the top speed below the drag limit.
If the crossover RPM is near the maximum rpm, the motor is defined by the flat torque curve.
It will respond to more gear speeds to stay near peak power until wheel force exceeds the traction limit.
If there is a large range between the crossover RPM and the maximum rpm, different gearing will not make a difference as long as the power is the same.