5/24 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 >> <https://khkgears.net/new/gear_knowledge/introduction_to_gears/involute_tooth_profile.html#:~:text=In%20effect%2C%20the%20involute%20shape,ideal%20shape%20for%20gear%20teeth.> >> . >> 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 >> > 2050 > > 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? > > 2113 >
I think I am missing something because later down the page it says you can shift the teeth to account for near or far neighboring gears, but let’s try this naive approach. 360 deg / 5 = 72 deg 72 deg / 2 = 36 deg = pi/5 radians So inv alpha might rise from 0 to pi/5 or 36 deg … I’m thinking you could start at > 0 deg to get shallower angles at the start. This could relate to clearance too. humm … naive approach still it’s nice to form a relation between r/r_b and inv alpha alpha = arccos ( r_b / r ) inv alpha = tan alpha - alpha i guess i’m realizing that you could actually draw any portion of this if you strted at r>r_b and maybe solving for alpha(inv alpha) is not needed … found this unsent at 1027 following day. was thinking of considering r rather than alpha and making naive gear unsure. post might need trim
