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  1. #1
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    Engineer/Mathematician needed

    I am trying to calculate moment of inertia for my rear wheel.

    i=m*r sqd.
    i=m*333.6mm sqrd.

    m*9.81m/s sqrd.=1645.88g (16.14 newtons).

    I am math stupid. I need to do this on a couple of wheel builds. If someone could explain I be appreciative.

    THanks

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    In SI unit, you need to use kg for m (mass of hub, rim, spokes, freehub, cassette, tire, and inner tube), and meter for r (radius from center of hub to the outer most edge of the rim). Why at the edge of the rim? Because the tire also has mass, so we can cheat by approximating the center of mass located at the outer edge of the rim. The resulting moment of inertia I has a unit of radian per sec. Note that I=mrr is valid for a thin cylinder with mass concentrated at the outermost edge.

    The actual rear wheel also has a sizable amount of mass from the spokes, cassette, and freehub...somewhere between a disc with uniform thickness (I for disc=0.5mrr) and a thin cylinder with mass at the outer edge (I for cylinder=mrr). I would calculate I of the wheel using 0.90 or 0.95mrr to compensate for the spokes and the cassette/freehub.

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    Quote Originally Posted by furballi View Post
    In SI unit, you need to use kg for m (mass of hub, rim, spokes, freehub, cassette, tire, and inner tube), and meter for r (radius from center of hub to the outer most edge of the rim). Why at the edge of the rim? Because the tire also has mass, so we can cheat by approximating the center of mass located at the outer edge of the rim. The resulting moment of inertia I has a unit of radian per sec. Note that I=mrr is valid for a thin cylinder with mass concentrated at the outermost edge.

    The actual rear wheel also has a sizable amount of mass from the spokes, cassette, and freehub...somewhere between a disc with uniform thickness (I for disc=0.5mrr) and a thin cylinder with mass at the outer edge (I for cylinder=mrr). I would calculate I of the wheel using 0.90 or 0.95mrr to compensate for the spokes and the cassette/freehub.


    i=.95*m*rr??

    333.6mm is the radius of a 700c wheel with a 23mm tire. Which is what I have.

    I convert 1645.88 g to kg, then convert 333.6mm to m

    I=.95*0.167775739 kg s / m*.3336m
    I=0.0531714872 kg s???

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    r is the distance in meter from the center of the hub to the outermost edge of the rim...0.334m
    m is the mass of the whole wheel including tire, tube, rim spokes, freehub, and cassette...1.65kg (1645gr or 1.645kg)
    I = 0.95 x 1.65 x 0.334 x 0.334 = 0.1749 radian per second

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    I don't know what level of accuracy you need, but to really do the calculation right you'd need to know the masses of each part of the wheel and the radius from the center of that part. So you'd take the mass of the tire times its radius squared, add it to the mass of the rim times its radius squared, etc. The hub has a significant mass but since it has such a small radius it contributes very little to the moment of inertia.

    Furballi's 0.9 or 0.95 factor is a guess as to the weight distribution but it would be better to actually have some values for the various parts. And, BTW, the units for moment of inertia should be kg-m^2, not radians/sec.

    Do you have separate masses for each part of the wheel?
    Last edited by prathmann; 03-24-11 at 06:19 PM.

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    Senior Member Flying Merkel's Avatar
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    Red bikes are faster.

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    True...kg m squared (m x r x r). My bad for thinking about angular velocity. You can weight each spoke, multiply by the total number of spokes to get m, and use an r value from the center of the hub to the mid-point of the spokes. Post back with your result. Curious to see the actual value vs. the thin cylinder approximation.

  8. #8
    Insane Bicycle Mechanic Jeff Wills's Avatar
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    Quote Originally Posted by I_like_cereal View Post
    I am trying to calculate moment of inertia for my rear wheel.
    Since you're in Portland, it'll creep lower as the rim wears away.

    Why do you "need" to do this? Are you building the ultimate criterium wheelset?
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    To answer all the questions:
    175g for the tire
    220g Tube - Bontrager slime type - I commute and this saves my bacon.
    241g cassette
    10.88g nipples
    24g spokes
    525g rim
    450g hub

    There was post here about an article on Sheldon's website about aerodynamic wheels and the myth that they are better fundamentally. I was curious to see how my DT Swiss Straight gauge 32 spoke, RX-100, CXP21 rim held up in the grand scheme of things.

    If I=0.1749 then my rear wheel falls in between the Mavic Cosmic Expert and Cane Creek Crono. Yes the article is old as are the wheels. Since I am in the market for a better set of wheels I thought I'd start here and see where I stand.

    I think if I shed between 300 and 600 grams off the back end it would improve my climbing and overall speed. I can stick with a box profile on the back because the seat post and bottle cage will deform the air enough. I could use drive side aero, but I don't think that would be reasonable.

    I can do the same with the front and possibly use a deeper section rim with an insanely light hub with radial laced aero spokes, thus turning my front into a semi-disc. Basically I am trying to maximize my money and squeeze some blood from a turnip. I know it can be done.

    @Jeff - That is problem I am having now. My rear wheel brake track has a dip and dive in it and will need to be replaced soon.

  10. #10
    Senior Member meanwhile's Avatar
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    Losing weight will help with climbing but not speed on the flat. 600g won't help *much* with except by tdf standards.

    Read

    http://en.wikipedia.org/wiki/Bicycle...rotating_wheel

    http://www.training4cyclists.com/how...on-alpe-dhuez/

    Frankly, telling you the answer to your question wouldn't help you. Using the answer correctly requires more knowledge of physics than obtaining it. (However, a good approximation for a spoked wheel is too consider it as two point masses - one at the distance of the rim with mass of the rim + tyre + tube, the other at half the radius of the spokes and having the total mass of the spokes.)

    But: aero resistance is a speed cubed law, which means it matters proportionately more as you go faster, and that you only get the cube root of power reductions as increased speed. Which is a way of saying that what you are doing is utterly pointless outside of a race.

    Maths: Your rear wheel might be 10% of your total drag, cutting its drag by amazing 50% gives you an overall reduction in power use of 5%. You take the cube root of 1.05 to find out how much faster you're going - and its a might 1.6% speed boost. In the real world I'd expect you to get only a fraction of that. You'd almost certainly do better to fit latex inner tubes and reduce hysteresis energy - that would give you around a 10% reduction in rolling resistance and cost hardly anything. (Hmm - you're using slime and are probably crappy tyres: you should be able to get rr down by MUCH more than 10%!)

    And, yes, I have a physics degree.
    Last edited by meanwhile; 03-25-11 at 10:52 AM.

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    @ Meanwhile. Slime tubes and Specialized Armadillos Elites. I commute over a lot of glass and metal and nasty things that were giving me several flats per week from punctures in my old Vittoria Rubinos. I cut the casing through on a piece of metal in my Rubinos.

    So then to decrease my drag coefficient by 5% I would need, better tubes, lower rolling resistance in my tires, and a lighter all around wheel?

    Not really worried about raw speed because I am not a crit racer or TT person. I just want to get where I am going faster and with less effort. Which I know means lighter wheels, but how much lighter?

    Right now my rear wheel weighs in at 1645.88g. That's huge considering that the Williams 30x rear is 890 without a cassette. If you add 200g for a cassette you are still at 1090g. This would reduce the overall weight by 555.88g. A 30mm deep aero section is useless unless you are at speed that I know from basic physics of flight.

    Switching tubes, and perhaps a better tire would reduce rolling resistance. Would a standard butyl tube reduce the rolling resistance enough. Just worried about latex and commuting.

  12. #12
    Senior Member meanwhile's Avatar
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    Quote Originally Posted by I_like_cereal View Post
    @ Meanwhile. Slime tubes and Specialized Armadillos Elites. I commute over a lot of glass and metal and nasty things that were giving me several flats per week from punctures in my old Vittoria Rubinos. I cut the casing through on a piece of metal in my Rubinos.

    So then to decrease my drag coefficient by 5% I would need, better tubes, lower rolling resistance in my tires, and a lighter all around wheel?
    What I'm saying is that a lighter wheel is pointless, unless you are riding hill trials or crits. Read the articles I linked and you should understand why. As a commuter, on a reasonably flat road most of the work you do is against the air resistance of your own body. Most of what is left you do against rolling resistance. Spending money on wheel aero can only yield minute returns.

    And if you're commuting on hills? Then about 90% of the work you do will be lifting your own weight.

    As for reducing rolling resistance - with a route like that it is probably better to concentrate on puncture resistance. But if you're riding a road racer or close relative (something with drops and narrow tyres - otherwise none of this makes sense!) then your main options for speed are

    1. Riding in the drops, on the lowest stem you can (bad for riding in traffic)

    2. Aerobars (terrible in traffic!)

    3. Lowering rolling resistance. Maybe try an alternative tyre? Go to the road forum and ask what the most puncture proof and fastest sanely priced tyre is. Definitely keep your tyre pressure topped up - do it every week.

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    If you're commuting, then one extra red light can kill your commute time...even if you're riding a 15 pounder. The lighter wheel may net you 0.1mph going up a grade. That's noise in my book. Day to day variation in wind speed has a much greater influence on your commute time than a lighter bike.

    In general, you want to avoid a tire with high TPI rating. A low TPI number translates to a larger thread, and a more puncture resistant tire. I ride Michelin City with 33 TPI. Only one flat in almost 3K miles. The culprit was a tiny wire from a delaminated auto tire. The Michelin Pilot comes with a thick 3mm breaker, instead of the 1mm breaker found on basic City.

    I had a 21lbs Cannondale. Currently ride a 24lbs Bridgestone. My 19.6 mile commute is still 65 to 80 minutes...head wind in one direction, tail wind in the other.

  14. #14
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    I read the articles and they sorta made sense.

    In essence reducing weight is meaningless in return compared with lessening rolling resistance. Which now that I think about it makes sense because if you can't over come the friction then you stand still. Lessing the grip of friction reduces the energy needed to accelerate the object. Highschool physics is slowly coming back. Lessening rolling resistance comes in the form of a better rear hub. Better than my 1997 RX-100. Perhaps better tires. I am not afraid to drop money on those as the Armadillos are $60 per tire. The Armadillos were recommended by a friend that does a lot of Cycle Oregon Tours with goat head thorns and other nasty road crap.

    My roadie is a Reynolds 853 steel frame. Probably 19 pounds?

    @Furballi I wish I could ride that fast. My 10 mile commute one way is 50 - 55 min with no wind. However it is 90% hill climb. I start a 200 feet and climb to around 600'. This does not include the small rollers. Descend to about 20'. The return is a 6.5% grade from 25' to around 600' and then bomb down to 200' then rollers home. If it was flat it would be no problem. It ain't and that's Portland's west side.

    Armadillo TPI 120 which is lower than the Krylion by Michelin 127tpi. I do not have the clearance for the City or City Pilot.

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    I agree with furballi and meanwhile that weight of the wheels isn't going to play any significant factor in your commute time - and the moment of inertia will be even less significant. The importance of 'rotating weight' has been blown way out of proportion. Yes, accelerating a rim requires twice the energy of accelerating the same weight on the frame, but a cyclist spends only a tiny amount of his overall energy expenditure in actually accelerating. Almost all of the energy goes into overcoming air resistance and gravity (assuming some hills).

    If you still want the moment of inertia calculated based on the component weights, let use know. But first could you check over the weights you gave - e.g. the tire weight of 175g seems very low for an Armadillo and the spoke weight of 24g is too high for a single spoke but too low for the set of 32. Are you using aluminum nipples? Not what I'd recommend for a commuting bike in Portland.

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    Sitting here at lunch.
    So to make sure I have my ducks in a row.
    A decrease in drag is an increase in something, just don't know what.
    Example air is fluid so to not disturb the fluid and keep it laminar you would need a 23mm wide rim for a 23mm wide tire with 32 bladed spokes to keep the rotating surface mostly disc shaped. Thus keeping the fluid mostly undisturbed. Resulting in a decrease of drag. This only comes into play at speed greater than 15mph.

    Reducing the friction co-efficient of the rubber and contact surfaces of the wheel results in a "faster" ride. Reducing the friction in the bearings and hub along with a lower RR in the tire means that less power is needed to rotate the wheel. Thus producing a quicker rotation of the tire.

    Even if a lighter wheel was used the contact oval shaped patch of tire to road would not change. Assuming no fluctuations in inflation. The oval would not change because the weight of the bike and rider remain constant which creates a "standard" oval. Lightening the rider and the bike would result in minimal gains of decreasing the contact oval.

    Glad to know that I was looking in the right area of the world when I was looking at replacement wheels.

  17. #17
    CRIKEY!!!!!!! Cyclaholic's Avatar
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    you're not serious about faster commute times until you're commuting on this...

    There are 10 types of people in the world - the ones that can count in base 2, the ones that can't count in base 2, and the ones that didn't expect this to be in base 3.

  18. #18
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    That right there. Now that's funny. I hate recumbent bikes. Plus look at that bog air pocket between his helmet and waist. All that drag.

  19. #19
    Insane Bicycle Mechanic Jeff Wills's Avatar
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    Quote Originally Posted by I_like_cereal View Post
    That right there. Now that's funny. I hate recumbent bikes. Plus look at that bog air pocket between his helmet and waist. All that drag.

    You ought to see my commuter/tourer/TT bike: http://home.comcast.net/~jeff_wills/jeff-big.jpg
    FWIW: I use wide-ish tires with Mr. Tuffy's to avoid flats. I hate flats, and I'll sacrifice some rolling resistance and weight if it keeps me moving.

    I hope you get your wheel rebuilt before this happens: http://home.comcast.net/~jeff_wills/...s/rites042.htm . Loud bang- my ears hurt.

    You be hatin'... we be rollin': http://www.ohpv.org/HPC/index.html
    Jeff Wills

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  20. #20
    Senior Member meanwhile's Avatar
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    Quote Originally Posted by I_like_cereal View Post
    I read the articles and they sorta made sense.

    In essence reducing weight is meaningless in return compared with lessening rolling resistance. Which now that I think about it makes sense because if you can't over come the friction then you stand still. Lessing the grip of friction reduces the energy needed to accelerate the object. Highschool physics is slowly coming back. Lessening rolling resistance comes in the form of a better rear hub.
    Rolling resistance isn't friction but hysteresis energy. And even cheap hubs are amazingly efficient.

    Better than my 1997 RX-100. Perhaps better tires. I am not afraid to drop money on those as the Armadillos are $60 per tire. The Armadillos were recommended by a friend that does a lot of Cycle Oregon Tours with goat head thorns and other nasty road crap.

    My roadie is a Reynolds 853 steel frame. Probably 19 pounds?

    @Furballi I wish I could ride that fast. My 10 mile commute one way is 50 - 55 min with no wind. However it is 90% hill climb. I start a 200 feet and climb to around 600'.
    Ok... Your problem is a lack of power to weight. Trying to cut aero or RR to speed up a 10mph commute is, well, like trying to understand why people go to Star Trek conventions - futile. And it isn't bike weight that matters but vehicle weight - i.e. bike + you.

    If you're fat, you need to lose weight. If you're not, then you need stronger muscles - unless you're getting out of breath, in which case you need better cardio. Do a search on "interval training" and another on "hill repeats."

  21. #21
    Senior Member meanwhile's Avatar
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    Quote Originally Posted by I_like_cereal View Post
    Sitting here at lunch.
    So to make sure I have my ducks in a row.
    A decrease in drag is an increase in something, just don't know what.
    It's this simple: at 10mph aero drag is tiny - it consumes 1/8 the power it does at 20mph. Rolling resistance is minute too. You need to weigh less - much less, lopping a few hundred grams off bike weight is pointless - or to pedal harder. Because it is gravity that is slowing you down.

    Reducing the friction co-efficient of the rubber and contact surfaces of the wheel results in a "faster" ride. Reducing the friction in the bearings and hub along with a lower RR in the tire means that less power is needed to rotate the wheel. Thus producing a quicker rotation of the tire.

    Even if a lighter wheel was used the contact oval shaped patch of tire to road would not change. Assuming no fluctuations in inflation. The oval would not change because the weight of the bike and rider remain constant which creates a "standard" oval. Lightening the rider and the bike would result in minimal gains of decreasing the contact oval.

    Glad to know that I was looking in the right area of the world when I was looking at replacement wheels.
    No!
    Last edited by meanwhile; 03-26-11 at 03:38 AM.

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    @ OP. Lighter and more aero components will probably net you an extra 0.1 to 0.3mph. A world class cyclist can put out about 0.5 continuous horsepower (~400W). Some of us are good for 0.3 continuous horsepower on a great day. Dropping a few lbs isn't going to make a huge difference in power to weight ratio. A cyclist worst enemies are head wind and hill. You have no control over these variables, so you need to ride smart. Position the saddle and stem so that you'll be comfortable way down on the drop, and up to 45 degree lean angle. This will allow you to ride in many positions to give various muscle groups a break. Less fatigue will allow you to output more power at the crank.

    Rise smoothly. Don't rock the bike when you climb. Try to apply power throughout the whole pedal stroke, not just on the down-stroke. Keep the crank rpm north of 60, especially on a climb. Switch to a lower gear if needed to maintain the rpm to deliver the power to the road. A skinny 23C at high pressure can cause your bike to bounce on the road, reducing traction needed to propel the bike. I prefer 28 or 32 for city riding.

    Get a speedometer. Go out on a calm day and record your average speed. Practice proper riding technique. Note the change in speed. It can take months to add 2mph to your cruising speed.

  23. #23
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    Just to clear some things up in this thread:
    - A lighter bike will not give you a faster top speed, but will allow you to accelerate faster (F = ma) (Note: the exception is on an incline. However, assuming you start and end at the same altitude, you're doing pretty much the same amount of work either way.)
    - Chopping off weight in the wheels/tires will give you the greatest benefit in this area as you need to both move and rotate that mass (i.e. you must provide the energy for both linear and rotational motion)
    - However, lighter wheels also means that the work done by drag forces will be more effective, and thus you will lose speed more quickly when you are coasting.
    - Your greatest enemy, as indicated in this thread, are wind and inclination
    - The evil thing about wind is that your ground velocity doesn't matter; it's the relative velocity. A 10 mph headwind will give you 20 mph air resistance if you travel at 10 mph.
    - As cliche as it is to say, you will stand to gain the most speed by building muscle and improving form. It's when you already have the muscle and form that you try to tune the little bits to gain the edge against other guys in top shape/form.

  24. #24
    Senior Member meanwhile's Avatar
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    Quote Originally Posted by itsthewoo View Post
    Just to clear some things up in this thread:
    - A lighter bike will not give you a faster top speed, but will allow you to accelerate faster (F = ma)
    Yes. Marginally, because bike weight is low as a % of vehicle weight.

    (Note: the exception is on an incline. However, assuming you start and end at the same altitude, you're doing pretty much the same amount of work either way.)
    The second part is untrue because air resistance dissipates kinetic energy: energy for the total bike + rider + air system is conserved, bike energy is not.

    - Chopping off weight in the wheels/tires will give you the greatest benefit in this area as you need to both move and rotate that mass (i.e. you must provide the energy for both linear and rotational motion)
    Yes, but it will be minute. Because cuttable wheel weight is so small compared to rider mass and work against air and road resistance.

    - However, lighter wheels also means that the work done by drag forces will be more effective, and thus you will lose speed more quickly when you are coasting.
    Not in anyway that matters. Energy stored in your wheels is tiny. Spin them with the wheels off the ground and then apply the brake. Both responses speed up and slow down are instant.

  25. #25
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    So as a 150 pound rider on a 19# bike. Technique, form, muscle, and endurance will make more difference than a lighter setup.

    So more squats, more hill repeats, and form practice will make more difference that lighter setups. On a flat road I know the bike can fly.

    Thanks guys I learned a lot.

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