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    Excellent Discussion of Moment of Inertia

    Excellent discussion of moment of inertia by Rennworx GT3 on 6speedonline that you might want to save for future reference and ARGUMENT as to why you spend $5000 to replace a perfectly fine set of stock wheels. Smiley Smiley While the discussion is about Dymag, the weight saving advantage of course applies to all after-market light weight wheels.

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    Moment of inertia facts



    Determination of moment of inertia for a wheel is very difficult because the “normal” calculations are for a thin wall cylinder or solid rod, the calculation being a function of the mass multiplied by the square of the radius. With the Dymag carbon / magnesium wheel being of different materials and different masses as you radiate out from the hub, then the calculation needs to take on some assumptions.

    Therefore if we take the whole wheel, then the basic formula is:

    MOI = ½ mr²

    If the carbon barrel alone is measured versus a metal rim it would be:

    MOI = mr²

    So because of the radius being squared in each calculation, the greater the diameter of any wheel then the greater the effect of reducing the MOI.

    As you can see if we use the generally accepted first formula then we are very conservative in our result, especially if we compare the Dymag wheel with an aluminium centred wheel where we could be as much as 50% greater on MOI than in reality! Hence the importance of measuring MOI of 2 wheels accurately on the same machine to make any comparison.

    Porsche 997 rear wheel comparison

    Comparing a 19x11.5” Porsche 997 Carrera S wheel with a Dymag carbon/magnesium 19x12” wheel.

    Porsche: 30lb x19” M=30lb, R=9.5”

    Therefore: ½ x 30 x 9.5² = 1,353.75 lb/in² MOI

    Dymag: 19lb x 19” M=19lb, R=9.5”

    Therefore: ½ x 19 x 9.5² = 857.375 lb/in² MOI which is 63% of the MOI of the Porsche wheel.


    Why is this important?

    Consider 2 factors of the wheel in use, rotating and steering.

    To rotate the wheel, the work energy required is calculated as force of net angular position change = ½ MOI x angular velocity², strictly speaking it is Force net θ = Δ( ½ MOI ω²)

    Which in English means that the energy consumption goes up as a function of the moment of inertia x the square of the speed, or as you go quicker it takes much more energy! This equation also shows that both acceleration and braking are both effected significantly by a reduction in MOI

    Steering changes are even easier to understand. The change in direction is governed by the momentum of the wheel which is calculated as the MOI x angular velocity, so in the above example, the Porsche on the Dymag wheel will use 47% less energy input to steer the car, either driver or power steering input, this is why the car feels “lighter” to drive and more responsive to steering input.

    Rule of thumb calculations

    This is a minefield of assumptions!! 2 of the old tuning rules of thumb were that 6lb weight saved on a car was equal to 1 bhp and that 1lb of rotating weight was worth 10lb of static weight, so in the Porsche example above, we are saving 11lb per wheel, x 10 = 110lb x 4 = 440lb ÷6 = 73.33 bhp we think this is probably excessive as the 10 factor does not take into account the diameter of the rotating part. A carbon propshaft would not have the same effect for example.

    We have been stating that the rotating/static weight factor is about 6:1, this would give a result as above of 44bhp, which is roughly the gain effect that Parr Porsche said about the original tests of the carbon car wheels on the 996 GT3 RS!!

    Power to weight calculation

    Now if we take the Porsche wheels, the 997 Carrera S weighs 3131lbs and has 355 bhp, which gives it a power to weight ratio of 8.82:1 or, if you lose 8.82 lbs it is equivalent to 1bhp equivalent gained.

    So if we take the Dymag 6x factor for the carbon wheels then saving11lb per wheel x 6, x 4 wheels ÷ 8.82 lb/bhp = 29.93 bhp an important 8.43% power gain simply by bolting on different wheels.

    But if we take the sometimes accepted10x rotating to static weight factor this figure comes to 49.88 bhp

    Obviously using this type of formula, the more power to weight ratio, the bigger the wheel effect on performance – not too surprising really!

    Fuel saving and emissions reduction

    The best estimates available for fuel and emissions savings by using carbon wheels are between 3.0% and 8.0% fuel savings. The increased % would be with stop – start driving for the obvious reasons of overcoming inertia and momentum. Also air drag factors mask the wheels performance at higher & more consistent speeds.

    Emissions are much more difficult to calculate, especially when taking into account warm up and urban type cycles of car use. However as gasoline produces an average of 2.35kg per litre of fuel at the tailpipe, then it is safe to assume an initial saving of emissions to match fuel consumption. Again urban cycles will show a greater effect from the carbon wheel. So if the average car produces 165g of CO2 per km (at 7litres per 100km) the carbon wheels may reduce this to nearly 151g, a 9% improvement.

    The fuel and emissions savings/reductions caused by carbon wheels need much more work. It is not possible to calculate these effects due to the other rotating components of the car which contribute to the overall moment of inertia, plus there are effects on non-rotating parts such as suspension where lighter wheels would give softer settings which require less energy to use the car in normal road conditions due to the reduction in energy losses with tyre deflections.

    Conclusion

    Carbon rims have a noticeable effect on a car’s performance and handling, the variations in moment of inertia of the complete system are not well enough researched to give absolute figures of the improvements that the wheels give. That the improvement is significant is beyond doubt, the independent testing and standards agencies now need to make these evaluations, equally condemning many of the very heavy aftermarket products.


    --
    Cargraphic Exhaust/PSS10 http://www.rennteam.com/forum/thread/449462/Bilste...

    Re: Excellent Discussion of Moment of Inertia

    Can, to add to this alot of people overlook the weight of the tires as well. Since the tire mass is even further away from the axis of rotation it contributes even more to the MOI. From the stock Bridgestones to Pilot Cups there is about a 3-5# difference per corner.
    --

    track vid

    0-300kph

    chasing a 997GT2




    Re: Excellent Discussion of Moment of Inertia

    We deal with these kinds of problem everyday in integrated circuit electronics design, electrical instead of mechnical but a similar solution path exists.

    Break the wheel up into seperate pieces alone the logical lines of tire, wheel center and wheel outter section. Further break each up into smaller increments, tire into sidewall section and outter tread, do the same with the wheel spokes vs. the rim. The finer you go in detail the more accurate the result. If you think of the wheel that has a radius and depth (width of wheel), then think of where the material changes (wheel/tire) and where the depth changes (tire sidewall/tread, wheel spokes/rim). These are good places create seperate analysis, kind of like cutting the wheel into incremental cylinders of analysis where the material within the thin cylinder wall is homogenius in both composition and depth. After you have analyzed each section then add up for the result. You can do this because this is called a "linear system" and the method of superposition (analyze each section independantly then add the results) applies.

    Those with science/engineering backgrounds will recognize this as incremental integration ( we call it Red Neck Calculus here in South Florida!).  This would be a great problem for a undergrad student in engineering or physics. They could estabilish a density vs. radius relationship for each composition, include this in the MOI formula, integrate over the incremental radius and sum the results for the final.

     

    Mike


    --
    993C2 & 997TT

     
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