Machin's Performance prediction calculations

[machin]

On another thread I got into a debate about performance curves I had created for a fictional F1-type car. Rather than discussing my calculations on that thread I thought I would start another thread.

Here is an example of the 0-100mph acceleration curve that is one of the outputs:-



The full program predicts speeds around a racetrack. Race track data is derived from data-logger information from my track car. The original purpose of the program was for driver training purposes and as a tool for performance-benefit analysis to be conducted for future modifcations to the vehicle.

The summary of the acceleration calculations is as follows:-

At 1 mph increments across the speed range 0 to 250mph I have calculated acceleration in the following way:-

1. Convert miles per hour to metres/second

2. Determine Engine Speed, rpm, in each gear for the speed calculated in step 1.

3. Calculate the Driving Force at the wheels, using the engine’s output curve, gear ratio, final drive ratio and rolling diameter. Note, I add an additional calculation to take into account the effect of the driver’s ability to slip the clutch. I only apply this in 1st gear.

4. Select the appropriate gear: -i.e. if engine rpm in gear”X” > “Upshift rpm” use gear”X+1”, etc

5. Determine maximum driving force at the wheels using the figure from step 3 that corresponds to the gear chosen in step 4. Multiply this by the “transmission efficiency coefficient” to obtain the driving force available at the driven tyres.

6. Determine the maximum accelerative force that can be transmitted by the front and rear wheels (separately) using static mass distribution, downforce generation, and the maximum load transfer due to CG height, wheelbase length, mass and tyre grip coefficient. Since the tyre’s grip coefficient is dependant on the vertical load placed on it you’ll need to iterate this step a couple of times at least. Apply a “grip factor” to the result of the above based on the road bumpiness and the suspension’s compliancy and damping.

7. Find the maximum accelerative force that can be transmitted by the car, based on step 7, where:- RWD = Rear grip only, FWD = Front grip only, 4WD = Front and rear grip.

8. Find the actual maximum driving force which is the lower value of step 5 or step 7.

9. Determine the Total Resistive Forces, where Total resistance = Air drag + Rolling resistance + Gradient “resistance”

10. Determine the acceleration at the chosen road speed using the accelerative force, total resistive force, the car’s mass and the inertia of the car’s rotating parts.

11. Determine the time taken to accelerate from the last mph to this mph using the result of step 10. If there has been a gear change between the last mph and this one, add in the time required to change gear.

12. Determine the distance covered whilst accelerating from the last mph to this mph based on the average speed and the time calculated in step 11.


The above calculations are performed at all speeds from 0 -250 mph. Time taken to accelerate between any two speeds can be determined by summing all the individual times from one mph to the next, and the distance travelled in the same way.

The force data can be assembled into a graph, which shows the total resistive forces (Dark Blue), Maximum force that can be transmitted by the driven wheels (Light Blue), and the accelerative forces available in each gear (green). The top speed of the vehicle is determined when there is no difference in the green and dark blue lines (i.e. where they cross).

The graph below is for an F1 type car with a theoretical constant torque engine between 1000 and 12000rpm, with peak power of 600bhp and 7 roughly equally spaced gears.



The next graph is for the same car model, but fitted with a theoretical constant power engine making 600bhp at all engine speeds from 0 to 12000rpm, and only one gear with the same ratio as gear 7 in the example above.



The third graph shows the two graphs above superimposed showing that the accelerative force of the constant power engine joins all the points equivalent to 600bhp of the constant torque engine/7 gear example.



(In reality it is unlikely that the 2013 engine will be constant torque or constant power)

[Jersey Tom]

Not bad. Looks decent.

[747heavy]

nice job machin, looks correct to me
A other way to look at it.
If we integrate the graph in your graphic 1(constant torque) and graphic 2(constant power) to get the area under the line,
and then subtract the area graphic2 - graphic1 we see the power which we don´t use
for acceleration with a constant torque engine.

Ergo more gears will make this area difference smaller, and we can utilize more of the power our constant torque engine produces.

The goal is it to bring it closer to the constant power line (more often).
This is where CVT was intended to go (the "1000 gear" gearbox).

But if we could build a constant power engine for the whole range of speed (rpm), non of this would be necessary.

The practical aim would be to build an engine which can produce constant power in a wide(r) range of speed (rpm).
If you can do this, you can reduce the number of gears.

A practical example would be the R10 (TDI Audi, LeMans car) which had a 5 speed gearbox.

[machin]

Spot on 747... I suspect with the 2013 engines they will have a range of Rpm at the top end where they make constant power due to the fuel flow regs... This should get them up to the 'ideal' line more of the time, even if it doesn't decrease the number of gears...

[machin]

Does anyone have any real world efficiency data for a CVT gearbox? (Hard numbers, not words!)

[747heavy]

from this paper are the following numbers.

Transmission type Efficiency
Manual 97%
Automatic 86%
CVT belt 88%
CVT toroidal 93%
Table 1. Efficiencies of various transmissions over a
drive cycle.

From this paper (year 2000) are the following claims.

CVT Mechanism Efficiency Range
Rubber Belts 90-95%
Steel Belts 90-97%
Toroidal Traction 70-94%
Nutating Traction 75-96%
Variable Geometry 85-93%

Dunno, how accurate or correct the claimed figures are.

A thesis on a belt CVT, for automotive use, can be found here

[xpensive]

I doubt if the CVT has an efficiency as low as 88%, when considering the 800 Hp Williams Renault would have had a 70 kW loss, thats some 140 toasters.

[ringo]

What kind of downforce do you have at 100mph? not the drag, or resistance. 2000N of resistance at 100mph is some serious forces.

This graph is not bad, but it doesn't say anything about the acceleration, since the graph is not based on time or distance.

Some information is missing as well, that car will spin the wheels in first gear, and in second, it can't make full use of the available grip.

With that constant power gear ratio, the car will never move.
The wheels will slip anywhere bellow 53mph. For it to move you have to be above that.
But you can't get to 53mph if the car can't roll there to begin with.
You may need a tow truck to free roll the car to 53mph, so that gear can be used. This can be avoided with much more gears, much higher gears, then gear down back to the lower one after 53mph.

The issue with constant power is the torque is higher down low. This is counter intuitive, so a very anemic gear ratio has to be used to get it moving.
It's best a car only have constant power after a certain rpm.

I am going to try make one of these graphs.

[machin]

@ 747: Good find! Thanks!

[machin]

The wheels will slip anywhere bellow 53mph. For it to move you have to be above that

Oh Ringo! You don't have to drive everywhere at full throttle you know?!

You simply use less throttle at speeds where the grip is limited!

[machin]

and in second, it can't make full use of the available grip

These aren't fully optimised gears: I've just put them equally spaced as I said in my first post.

[ReubenG]

I had to put together a similar model as part of my final year mechanical engineering project. It was some time ago but I do remember the following:

1. For normal road vehicles, the early part of the acceleration is very dependent on your clutch model. This is not a problem if your focus is racing vehicles which are traction limited for this phase.

2. There is a (small) parasitic loss of torque due to the rotary inertia of the drive train - you have to account for the relative rotational speeds of the different stages of the transmission to get an equivalent rotary inertia, but this is constant for all speeds.

[machin]

Yeah, I take the rotational inertia of the transmission and engine into account. In lower gears the inertia of the engine has a greater effect since it must accelerate faster for a given increase in road speed. I take that into account too...

Good heads-up about the clutch... but you're right, I only ever look at high performance (traction limited at low speed) cars....

[autogyro]

Yeah, I take the rotational inertia of the transmission and engine into account. In lower gears the inertia of the engine has a greater effect since it must accelerate faster for a given increase in road speed. I take that into account too...

Would be nice if you could accelerate the vehicle initialy without even needing to have the engine running would it not?

[machin]

Auto; possibly yes... Especially if the throttle is a bit sensitive!

Ringo; some more clarification; the green force lines represent the max force available at full throttle... At part throttle the force lines in each gear would be lower... Although I agree the gear ratios aren't optimised... Might as well make 1st gear on the first example longer, but no point on the constant power engine as it makes constant power -changing the gear ratio makes no difference!

2000N at 100mph? That's about 120 bhp... About right considering the downforce an f1 car generates...