it said an involute curve is like unwinding a spool of thread …

so if unwinding thread, basically the length of your line is equal to the arc length it was unwound from.

we could describe that as a function of the angle of unwinding. for each point on the circumference, there’d be an associated line length.

- (point on circumference) + (vector of tangent) * arclength

function of angle

r = radius of spool

t = time/theta

x(t) = r * (cos(t) + t * sin(t))

y(t) = r * (sin(t) - t * cos(t))

p(t) = r * (exp(i t) - i t * exp(i t))

> parametric plot (cos(t) + t * sin(t)), (sin(t) - t * cos(t)), t=0..2pi/2

yayyy involute curve yayy

so now how is this related to gears?

after plotting an involute curve i felt more comfortable understanding material a little when a websearch led me back to 

https://khkgears.net/new/gear_knowledge/introduction_to_gears/involute_tooth_profile.html .

they draw a circle partway up the teeth of a gear and call it the base circle. if two gears move at the same velocity and are continuously touching teeth, the contact points follow a shared tangent of the two base circles as if they were connected via a strap with opposite wrapping on each.

so if a tooth contact point were at the radius of the base circle, at the endpoint of the shared tangent, then turning the gear toward the center would move it farther from its base circle, and toward the opposing gear. Meanwhile, the opposing gear has the opposite: its contact point progresses toward it.

looking a little more.

to make this contact line, an involute curve would be rotated as the thread is pulled. maybe like rotating a spool toward the thread pulled from it.

having some trouble engaging. the pitch point is the point of contact that is most distant from both gears and on a line between their centers. the pitch circle is a circle of this radius about a gear. the pressure angle is the angle of the tooth at this point — the angle of the tooth tangent from the circle normal or radius, or identically the angle of the tooth normal from the circle tangent or a line between the gears.

So a 0 degree pressure angle would mean the gears push in the direction of motion like stepping on somebody’s foot, and a high degree would mean the gears push against each other a lot, exerting pressure to spread apart, maybe.

1909

i’m at 

https://khkgears.net/new/gear_knowledge/gear_technical_reference/involute_gear_profile.html and it has this picture:

Here the involute curve can be seen unwrapping from a circle with labeled angles. the curve extends from a through b.

It looks roughly apparent that the angle of the curve’s tangent from the horizontal is equal to theta or t. (we guess the tangent of the circle is normal to the curve at every point :s)

I think the page says the pressure angle alpha = arccos(r_b/r) where r is the radius of the pitch circle. It also give formulas/formulae for the curve in terms of alpha.

alpha = arccos ( r_b / r )

inv alpha = tan alpha - alpha

x = r cos ( inv alpha )

y = r sin ( inv alpha )

It’s apparent sadly that r is a function of alpha or theta :/ . When plotting, they start by calculating r for each spot.

note: inv alpha means involute angle, not inverse.

This seems kind of roundabout to calculate r first without even a formula for it. I guess you could assume it starts at r_b and rises.

So your pressure angle is then defined solely by the distance between gears …?

Thinking on that idea a little. If the gears touch then the pressure angle is 0 degrees and the teeth have 0 height.

Then as the gears move apart higher teeth develop, but there would only be room for a theta of so many degrees.

So if I had say a 25% gap between gears, i could draw r from r_b to 1.125r_b and this might show the edge of a tooth. I wonder if it would work.

It seems like this would also relate with tan alpha - alpha: the involute angle, on the chart, appears to be how far around the circle the curve goes. So one can only fit on the curve as many teeth as twice the inv alpha fit on a circle.

There were 5 teeth on the gear bearing. I think. A commentor or troll also posted one with 6 teeth. Let’s assume 5. The small gears had five. The center bearing part had 10, and the outer bearing part had 20.

Let’s assume 5 teeth. What’s the maximum involute angle and pitch radius?

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