Spinning gear's speed based on number of links in bike chain in 1 spin

Antonin Ganner · 2012-02-13 03:28:09

Hey guys, this has been puzzling me for all of 5 minutes, but it's long enough to realize I don't stand a chance..

I'm an animator and I have a bike chain spinning around 2 gears in 3D. I have to simulate the rig with mathematics as calculated physics are temperamental and costly on my machine.

I calculated the number of links in my chain and the size of the gears and angles required for the links as so (Sorry if my way of explaining it is not correct, I haven't done maths for a long time.
When I put something in brackets it's annotations, not part of my equations as I haven't needed to use brackets that way.

L = Link (3Gs from gap to gap (spacing for gear's teeth))
G = Gridspace (reference integer for calculations)

Gear 1
Divisions: 80L

3G (1L) * 80 = 240 (Circumference)
/ Pi = 76.394372684109761169064206418807 (Diameter to account for circumference)
360 / 80 (L) = 4.5 (Angle for reference)

Gear 2
Divisions: 120L

3(1T)*120=360(Circumference)
/Pi=114.59155902616464175359630962821(Diameter)
360 / 120 (L) = 3 (Angle)

Now the problem is working out how fast to turn the gears relative to the chain turning..
I have an object stuck to the side of the chain for reference. By one full rotation of the chain, I mean that this object returns to it's starting position. That's 100% of my rotation of the chain.
What I need to know is how many degrees each gear would rotate in a full rotation.

Key data:
Number of links: 192
Number of gears: 2
Gear1: 80 teeth
Gear2: 120 teeth
I'm hoping just those statistics are enough to perfectly calculate the degrees each cog rotates in a full cycle of the chain, but if not I have provided all other details I could below.. some other aspects could not be achieved with precision.

Distance between centrepoints of gears: 106.687
(Distance between may be technically off by a few decimal places as placement of the cogs had to be by eye to line up the start and end points of the chain, a hazard of the only way to set up such a rig)

Other notes on possible variables:
I know based on my calculations that the chain links' centres will be constrained to the edges of my gears, I've shrunk the gears slightly inwards after creating them in preparation for the extrusion of the teeth to counteract that; if you imagine lines coming out from the centre of them spinning at a set speed, the size of the gear does not make a difference here (no physics are involved, the chain spins along a curve matching the gears' original circumferences, which is the most important thing to note on that).

I didn't bother with trigonometry to figure out the exact points of the gears that the chains would leave contact with the gears but I believe that shouldn't make a difference as a whole spin is a whole spin, right?

Of-course there are links not on the gears while between them and, as an average, half of each gear does not have contact with any links at all times.

I really hope I made sense, I have trouble explaining things, but I hate not having my animation mathematically accurate.

Thanks for your time,
Tony

Bob · 2012-02-14 02:20:00

hi Tony,

I did get a bit confused reading your explanation but I think my picture below is what you are trying to animate.

In this I make it 14 teeth on the small spocket and 48 on the large. It'll be easier to work with 16 rather than 14, so let's pretend that how many teeth it has.

On a real bike the chain has to be a bit slack as it goes round the sprockets or you get too much friction and chain wear, but too slack and it keeps falling off!

The actual number of links doesn't matter though; deraileur type gears rely on that to work. The idler wheel together with a spring takes up any slack.

For a chain wheel of 48 teeth and a rear sprocket of 16, the gear ratio is 3:1. What that means is this. Suppose you turn the chain wheel (where the pedals are attached) exactly one turn. That will wind on 48 pitches *

So the rear sprocket winds on 48 teeth; 16 per turn; so it turns three times for every one turn of the pedals.

Now let's choose an easy number for the chain itself. Let's make it 128 pitches.

When the chain does one full cycle ( so that a link arrives back at its start position ) 128 pitches have turned around the sprockets. For the small spocket that will be 128 / 16 = 8 turns of the sprocket.

For the large sprocket ( or chain wheel ) 128 / 48 = 2.66667 turns.

You can approximate the chain's shape by assuming it follows a circular path whilst on the sprocket and a straight line path between the sprockets. Neither is strictly true for the chain is made of unyielding metal. Its path around a sprocket is more like a polygon and there has to be a gradual deviation from a straight path to the polygonal path at the first point on contact.

Number of links: 192
Number of gears: 2
Gear1: 80 teeth
Gear2: 120 teeth

I'm going to assume you mean 192 pitches rather than links. See note below *

Gear1 will turn 192/80 and gear2, 192/120 .

So that will be 192/80 x 360 degrees and 192/120 x 360 degrees respectively.

That covers my initial thoughts on your post. You may certainly post again if that hasn't answered all your questions.

Bob

* The pitch is the distance between two roller pins on the chain. Because of the way a chain is made, a link is actually two pitches in length. There are some helpful diagrams at

http://www.gizmology.net/sprockets.htm

Last edited by Bob (2012-02-14 02:28:38)

Math Is Fun Forum

#1 2012-02-13 03:28:09

Spinning gear's speed based on number of links in bike chain in 1 spin

#2 2012-02-14 02:20:00

Re: Spinning gear's speed based on number of links in bike chain in 1 spin

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