Off-Topic wheels - the maths

Discussion in 'Off-Topic Chat' started by jimjams, Saturday 3rd Oct, 2015.

  1. jimjams Expert Advisor ★ ★ ★ ★ ★

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    I promised to show, mathematically, what I was saying in another thread, and I decided to bite the bullet.

    In that thread, there was the statement "I'd also be interested to see the impact of the wider profile tyres now fitted as standard, along with heavier alloy versus steel wheels on emissions as rotating mass has one of the highest impact on emissions".

    My reply was "the rotating mass of a wheel is negligible compared with the mass of the car itself i.e. the linear energy required to accelerate a car body on a mag-lev would be hundreds of times greater in magnitude than the torque energy required to overcome the moment of inertia of a heavy car wheel to get it up to the equivalent rotational speed at the circumference." I did also add "Basically, 'Rotational mass' in terms of acceleration or cruising is no different to its static mass, the energy required to move its mass in a straight line is the same", by which I meant that when accelerating, the linear energy is still required, and that the energy to accelerate rotationally is comparable. Also, when cruising, the cruising energy is only in proportion to its static mass, and rotational inertia has nothing to do with energy required to maintain cruising.

    I'm not looking for a "who was right" (indeed the truth may be considered somewhere between the two). all I am doing is presenting the mathematical proof that I promised.

    I intend to use SI units (kg, metres), apart from the speed in kph.

    ----------------

    On page 459 of my "Accord Tourer" owner's manual (00X32-SED-6030) the kerb weight of my car is 1582 kg.
    Yesterday evening I removed one wheel and weighed it, finding it to be exactly 20 kg. I have also looked on the internet at the weight of the Michelin Pilot Sport 3 (225/45 ZR 17 91Y) and found it is about 10 kg, which means that the alloy wheel (without tyre) weighs about 10 kg.

    With the car on all 4 wheels on the ground, the radius to the ground is 0.3m. The radius to the inner rim is about 0.2m

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    The linear energy (work done) required to accelerate a body of mass m from rest up to a speed, is Force x distance moved during the acceleration, which is written W_lin = F.d

    From Newton's laws F = m.a

    The average acceleration to reach a speed v from rest is v/t (t is the time taken for the body to reach velocty v)

    So F = m.(v/t)

    The distance travelled to reach the speed v from rest is t.v/2

    So W_lin = m.(v/t).(t.v/2)

    The "t" cancels out i.e. we do not need to know the time or the acceleration or the distance to calculate the energy to accelerate a body mass m from rest up to speed v

    W_lin = m.v.v/2 = m.v²/2

    [incidentally, when accelerating from speed v1 to speed v2, the equation is m.(v2²-v1²)/2, so "from rest" is a "special case" where v1 = 0]

    With v in kph, the equation is

    W_lin = m.(kph.1000/3600)²/2 = m.(kph.10/36)²/2
    W_lin = m.50.kph²/1296

    ----------------

    For the rotational energy, similar mathematical methods apply giving

    W_rot = I.ω²/2 (for the energy to rotationally accelerate from rest)

    where I is the moment of inertia and ω is the angular velocity

    The equation for the moment of inertia for a "rim" with radius r to the rim is

    I = m.r²

    Also the angular velocity can be calculated from the RPM by

    ω = RPM.2π/60
    ω² = RPM².π²/900

    The RPM of the wheel depends on the linear velocity of the car and the circumference of the outer edge of the tyre i.e. speed in metres per minute / circumference in metres

    The circumference of the outer edge of the tyre = 2π.r (r in metres)

    The car speed in metres per minute = speed in kph.1000/60

    So wheel RPM = (kph.1000/60) / (2π.r) = kph.1000/(120.π.r)

    so ω² = [kph.1000/(120.π.r)]².π²/900 = 100.kph²/(1296.r²)

    Giving
    W_rot = m.r².[100.kph²/(1296.r²)]/2
    W_rot= m.50.kph²/1296

    ----------------

    So thus we see that, for a wheel, the linear and rotational energies required to accelerate are the same (note that you include both in the total acceleration energy calculation).

    BUT THEY ARE ONLY THE SAME IF ALL THE WHEEL MASS CAN BE CONSIDERED TO BE AT THE EDGE OF ROTATION.

    As the "radial centre of mass" moves inwards from the edge of rotation, you have to multiply the mass in W_rot by the ratio of "radius to the radial centre of mass" / "radius to the edge of rotation".

    So if the "radial centre of mass" is one third of the way inwards from the edge of rotation, then the ratio for the mass in W_rot is 2/3

    The comparison for 60 mph (96.6 kph), with the total wheel mass 20 kg, and the ratio 2/3

    W_lin = 7200 joules
    W_rot = 4800 joules

    For 4 wheels, the totals are
    W_lin = 28800 joules
    W_rot = 19200 joules

    But for the car itself, which weighs 1582 kg (including wheel mass), plus say 120 kg for fuel and driver = 1700 kg
    W_lin = 612024 joules

    As a percent of the energy to accelerate the car
    Actual wheel mass = 4.7%
    Rotational wheel mass = 3.1%
    Total = 7.8%

    Finally, it would be interesting to know the weight of steel wheels with slightly narrower (and taller) profile.
    Assuming that it weighs 20% less, but does not shift the "radial centre of mass" then obviously those percent values are reduced by 20% giving contributions of 3.7% and 2.5%

    Note also that at a constant speed, W_rot = 0 (because v is not changing)
    But the rolling resistance (which is always present) is proportional to the total mass of the car, and has nothing to do with rotational energy of the wheel, so shedding ANY weight impacts rolling resistance, as does the choice of tyre and the tyre pressure

    edit: 04:50 slight change to % calcs
    ----------------

    btw it took me about 20 minutes to do this (quite differently) in a spreadsheet, but over 2 hours to write this
    and no I don't have a degree in Physics, I just enjoyed Pure Maths and Applied Maths at school
    and no, I'm not a teacher
     
    Last edited: Saturday 3rd Oct, 2015
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  2. HondaHeritage Expert Advisor ★ ★ ★ ★ ★

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    so what are you ??:Grin:
     
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  3. jimjams Expert Advisor ★ ★ ★ ★ ★

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    If I told you I'd have to kill you :lol:
     
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  4. HondaHeritage Expert Advisor ★ ★ ★ ★ ★

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    so i take it your some kind of night predator???
     
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  5. DeviateDefiant Co-Founder Staff Team

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    That's a good old chunk you've written there, and I certainly haven't spend the time to digest all of it - though I appreciate how long it must have taken.

    But just making this an awful lot simpler. From my understanding, you're now saying that wheel weight does affect acceleration? Was the math not to prove that it didn't affect acceleration? Or was this in reply to what @John Dickson wrote instead?

    Big letters, short words and lots of pictures and big words will help the rest of us get on the same page :toothy:
     
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  6. jimjams Expert Advisor ★ ★ ★ ★ ★

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    yeah, often in trees in Val Verde, I'm using a sat phone to write this stuff
     
    Last edited: Saturday 3rd Oct, 2015
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  7. jimjams Expert Advisor ★ ★ ★ ★ ★

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    The reason is in the intro, but to answer explicitly, and so as not to rake over any coals, let's just say it was about the effect of rotational inertia, and there was a point where I said I would do the maths behind what we both had said.

    Using the mass of the "alloy" wheels & tyres on my CM2 (2.4 Accord Tourer) and the total mass of my Accord itself, the extra "energy drain" during acceleration, caused by the rotational inertia of all 4 "alloy" wheels & tyres, is about 3.1%, but it depends on where you consider the "radial centre of mass" of the wheel & tyre to be (I used two-thirds of the tyre's outer-edge-radius). But also note that rotational inertia has zero effect on emissions or MPG during cruising.
     
    Last edited: Saturday 3rd Oct, 2015
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  8. DeviateDefiant Co-Founder Staff Team

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    Okay, so wheel weight does affect acceleration. MPG will suffer between different sized wheels anyway let alone weight due to the rolling resistance changes (even between the same tyres on different widths, due to the tyre aspect ratio) - of course there's an affect on emissions from that alone, as more resistance means more work for the engine, which means increased fuelling and thus greater emissions - but these are all such minute points I don't get why we're here debating it ...there wouldn't be such a huge lightweight alloy market for motorsport if wheels didn't mean diddly squat :Smile:

    Either way, props on the new paper Prof. - I'm sure someone could learn something from it :Thumbup:
     
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  9. jimjams Expert Advisor ★ ★ ★ ★ ★

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    Well it was about the "moment of inertia" of the wheel, not its gravitational mass

    A wheel's gravitational mass obviously contributes to the overall gravitational mass of the car, which has an effect on the rolling resistance, which will affect the emissions & MPG

    On the other hand, the "moment of inertia" of the wheels does also contribute to the energy used during acceleration (speed increase), but not to the rolling resistance (because it is not a gravitational mass). The same is true for anything that is rotating in the car during acceleration, so the engine's flywheel will also contribute a bit (I'm not going to bother with that too LOL)

    But note that everything that I have written is very well known, and it was very quick to do the calcs in a spreadsheet. What did take the time was trying to write it in a way that drops out to two seemingly identical equations based only on the vehicle speed and the wheel mass, which I think one will struggle to find on the internet (as I say, I didn't do it this way in the spreadsheet, it was only when I started to group the formulae that I found that the wheel radius and the wheel RPM, and π, all became irrelevant).
     
    Last edited: Saturday 3rd Oct, 2015
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  10. DeviateDefiant Co-Founder Staff Team

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    I guess what I was saying is that you seemed to have been arguing a point that wasn't raised instead of the one that was.

    Flywheel, alternator, power steering pump, A/C compressor, oil pump - all cause parasitic drain on the engine's output and far more important than the weight of the wheels. But getting to a point, @John Dickson's comment that bought all this about:

    Is still correct, in the sense that wider tyres, heavier alloys and all the rest will have a knock-on affect on emissions - albeit not only for the reason he raised.

    It's great seeing someone get into their equations to evidence their argument, but it wasn't really what was being raised in the first place - this has gone just a little bit down the garden path my friend.
     
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  11. jimjams Expert Advisor ★ ★ ★ ★ ★

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    well no, I am arguing the point that was raised, I don't see where the confusion is

    He's talking explicitly about the rotating mass of the wheels, saying their "rotating mass has one of the highest impact on emissions", which I disputed by saying "the rotating mass of a wheel is negligible compared with the mass of the car itself" .

    The maths shows that it only contributes about 3.1%, and it only contributes during upward speed change, not on cruising or on downward speed change.. So overall, I dispute that it has "one of the highest impact on emissions".

    What has a wider impact, is the gravitational mass, because it contributes to 4.7% of the upward speed change, and to cruising i.e. to 4.7% of the rolling resistance, which is there all the time that power is required (and I never said that the gravitational mass was negligible, which in #5, you seemed to think that I'd said)

    So although you say it's gone a little bit down the garden path, it most definitely has not.
     
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  12. DeviateDefiant Co-Founder Staff Team

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    You've proven that wheels have an effect on emissions which is what John was originally saying, the fact that he chose to highlight the rotating mass and not the gravitation mass is kinda a mute point - and pretty silly thing to go so far as to prove when you're quite capable of making the same deductions about the statement.
     
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  13. Zebster Expert Advisor ★ ★ ★ ★ ★

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    Hi Jim - I apologise for not going through the equations, but I have no reason to doubt you (if this was a 'work thing' then I'd make more of an effort and devote the time needed to fully understand the maths).

    But it instinctively occurs to me that the rotation energy loss from the wheels is negligible... energy that was 'tied up' when accelerating a rotational mass is released when that mass decelerates, so energy isn't lost in this process (but instead transferred from when it MAY be needed to a time when it MAY not be).
     
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  14. jimjams Expert Advisor ★ ★ ★ ★ ★

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    John said, in regard to wheels alone, "rotating mass has one of the highest impact on emissions", it's not up to the reader to deduce that he also meant gravitational mass.

    And in #5 you said "From my understanding, you're now saying that wheel weight does affect acceleration?" .
    "Weight" = gravitational mass, and the fact that you used "now" implies that I had been denying that the gravitational mass of wheels was not a significant contributor.

    I say again

    The maths shows that rotational mass only contributes about 3.1%, and it only contributes during upward speed change, not on cruising or on downward speed change.. So overall, I dispute that it has "one of the highest impact on emissions".

    What has a wider impact, is the gravitational mass, because it contributes to 4.7% of the upward speed change, and to cruising i.e. to 4.7% of the rolling resistance, which is there all the time that power is required (and I never said that the gravitational mass was negligible).

    - - - Updated - - -
    If car brakes had energy-recovery then that would be true, but all of the braking energy is lost as heat. The same is true of deceleration without using the brakes, where engine braking does the "work". Note that petrol engines go into fuel cut-off, so there are no emissions during engine braking (unless you have a car with a carb). I can't comment on diesels, if diesels do not have fuel cut-off, then John has more ammo.
    - - - Updated - - -
    Anyway folks, I have to go, it was not my intention to argue, dispute, Troll, etc, etc

    I was intrigued, I took the wheel off, weighed it, went onto a spreadsheet, did some quick calcs, and felt that I had done what I said I would do, and I have thus reported it.

    I guess I will never know whether anyone was going to ask if I had got anything to report re: my "promise" to show the maths. Assuming that that was gong to happen, I am damned if I do and damned if I don't.

    But IMO I'm damned right :lol:
     
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  15. Zebster Expert Advisor ★ ★ ★ ★ ★

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    I don't think anyone was going to ask! Nevertheless, it IS an interesting point, but was so far off-topic at the time that it was bound to be lost in the noise.
     
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  16. DeviateDefiant Co-Founder Staff Team

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    Quoting out of context doesn't remove the context Jim.

    You offered to do the math on the "rotational inertia" in the other thread, to which I replied heavier "Heavier wheels take more effort to turn my dear friend, you don't even need a degree in physics for that one." and linked up relevant content going through rotational inertia and its effect on acceleration, cruising, braking and all the rest. You then claimed it was irrelevant, said you'd show what you meant, and recycled and presented many of the same concepts from the page in a different way here anyway.

    It was hard to gauge what you meant other than that it seemed like you were disagreeing about my original statement that heavier wheels take more effort to turn, because I didn't agree nor disagree with what you said in the first instance.

    I've been trying to stop an argument here Jim, I don't think anyone was trying to dispute things further, that's the only reason I ever commented.
     
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  17. Duc de Pommfrit Moderator Staff Team

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    This is why pro-plus should be banned from sale. :shock:
     
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  18. HondaHeritage Expert Advisor ★ ★ ★ ★ ★

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  19. FirstHonda Premium Member Club Supporter

    ^^Am I the only person who thinks that sounds like a line from the original Star Trek - Spock to Kirk?

    :Laughing:
     
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  20. DeviateDefiant Co-Founder Staff Team

    United Kingdom Leo Northants
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    Yeah! Well, a Bones to Kirk line.

    Dammit Jim, I'm an enthusiast not a motorsports engineer.
     
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