Involute Gears 2: Undercut and Profile Shift 

I did not know sliding is related to contact ratio. Is it really? I need to look into that, I'll think about it. On the technical side, that last animation uses a lot of control points to superimpose a wave on the gear surface curve, meaning it renders quite slowly, so it's a bit hard to work with.

thanks :)

I talk about it later at around 11 minutes - I'm still not entirely sure what's right. I use 1.25m in order to include the clearance, and simulate a perfect trapezoid-shaped cutting tool with 1.25m height. With those assumptions, the math works out that way with 1.25m. I know most sources use 1m for this, I think because real cutting tools are not perfect trapezoids, but rather have a straight edge up to 1m, and the rest of the 0.25m for the clearance is rounded off, and usually does not cause undercut. Or they just hand-wave and simplify away the clearance.

@@gergelybencsik8626 Ahh you anticipated my question >.< Thank you so much for taking the time to answer and create this terrific explainer on involute gears. Keep up the good work and I look forward to watching what you release in the future!

Eh no, I just used scipy's root finder to get that, and it's a bit finicky, sometimes screws up and you need to adjust the initial guess, but it gets the job done. And since it worked, I moved on, never really tried to get the closed form - but maybe one day I'll punch it into Wolfram and see if it can solve it.

Thanks for the comment. Sorry though, I won't make more gear videos, this is about as much as I know about them. So for example, I don't know how to measure the undercut after the gear is already made...

I haven't seen a any closed formula for it, sorry. I used scipy root finder to solve that problem numerically for these animations.

@@gergelybencsik8626 Yeah, that might be the only option. The real reason i want it is to make a continuous parametric in desmos which can be filled with a solid color :3

I don't really understand the question. Do you mean 'pinion' as the smaller gear out of a pair? Profile shift can be applied to any gear. It's just that larger ones don't necessarily need it.

@@gergelybencsik8626 Pinion - in general is a smaller gear of the two.

They're literally everywhere. Cars. Power drills. Clocks. Printers. Any gadget or machine that moves, I'd say has about 40% chance it has gears in it somewhere. They're not new and exciting, I'll give you that, but far from obsolete.

You don't want undercut gears, you want small gears with low tooth count to maximize reduction ratio. Undercutting is the trade-off you need to manage for that.

@@gergelybencsik8626 certainly but if we strive for greater reduction to the point where the teeth cant bear the load anyways, whats the point? surely theres a breakpoint where the optimal tooth thickness lies.

@@Mobosh yes, I guess there is an optimal thickness, but I never looked into engineering gears for strength. Profile shift can mitigate the thickness issue fairly well, but increases the alpha angle and puts more load on the bearings of the gears. Gear design is not actually my field, so I have no real world experience here... But I saw some colleagues working on them, and it ends up being a delicate balance. You need to design for high strength, small size, large tolerance or low precision, wear, lubrication, even sound. You'll end up with a somewhat optimal design for those... but it all depends on the use case.