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(#38) Involute Curves to AGMA Standards · Mechanical Engineering Made Simple

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  Welcome back, design junkies, to Mechanical Engineering Made
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  Simple. Today we are undertaking a a
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  deep dive into the fundamentals of gear design.
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  And we're not just talking definitions here.
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  Our mission is really to get into the calculations, the
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  tradeoffs, you know, what it takes to design A reliable gear
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  train. We'll be leaning pretty heavily
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  on the AGMA standards, the American Gear Manufacturers
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  Association ones. Yeah, it's a great topic.
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  I mean, gears are incredibly old tech, right?
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  Over 3000 years. They're everywhere from ancient
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  clocks right up to modern rockets.
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  And their main job is always control, controlling speed, RPM
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  and force, which is torque. Right.
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  And the precision that comes down to the tooth shape, doesn't
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  it? We always hear about the
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  involute curve. Why is that shape so important?
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  Ah, the involute. Well, fundamentally, it's the
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  path you'd trace if you unwound a string from a cylinder.
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  Think about the end of that string.
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  The reason it's essential is that it's the only shape that
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  guarantees a constant velocity ratio as the teeth mesh.
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  And maybe more importantly, it really maximizes that smooth,
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  pure rolling motion between the teeth and minimizes sliding.
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  OK, so maximizing rolling, that's the key to the high
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  efficiency. We see the secret sauce.
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  That's it exactly. For typical spur gears the
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  efficiencies are remarkably high often up there with good anti
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  friction bearings, we're talking high 90% range usually.
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  OK, let's. Get some basic terms down when
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  you have two gears meshing. The smaller one's the pinion,
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  the larger one's the gear. Always.
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  Always. Doesn't matter which one is
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  driving the other. Opinion.
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  Smaller gear is bigger. Got it.
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  And the ratio is just teeth on the gear divided by teeth on the
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  pinion. Yep.
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  Simple division. And it all seems to relate back
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  to the pitch diameter. Absolutely.
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  The pitch diameter is the sort of fundamental reference circle
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  for all the geometry. And just like the tooth count,
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  if you divide the gear's pitch diameter by the pinions pitch
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  diameter, you get the ratio again.
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  OK. So diameters give the ratio, but
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  how do we define the actual size, the teeth themselves?
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  This is where we hit that metric versus imperial thing again,
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  right? That's right, it's a classic
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  split. In the the English or US
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  customary system, we use diametral pitch.
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  We write it P sub D and it's defined as the number of teeth
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  divided by the pitch diameter in inches.
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  So P sub D and it's kind of counterintuitive.
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  Lower number means bigger teeth. Exactly.
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  A low P sub D, like maybe 1/2 means really large teeth.
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  A high P sub D, say 200 means tiny tiny teeth.
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  OK. And the metric approach.
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  Metric uses the module just written as M and module is the
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  pitch diameter, this time in millimeters divided by the
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  number of teeth. So module M is more direct,
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  bigger number, bigger teeth. Yeah, much more straightforward.
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  A small module like .2 is a small tooth.
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  A large module, maybe up to 50, is a very large tooth.
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  Let's zoom in on a single tooth. There's the addendum.
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  That's the height above the pip circle.
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  And the addendum. That's the depth below the pitch
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  circle. And if you add them together.
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  That gives you the whole depth of the tooth space.
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  For standard teeth, a quick way to estimate it is just 2.250
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  divided by the diametral pitch P sub D That little bit extra
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  depth beyond where the mating tooth reaches is the clearance.
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  It's critical. OK.
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  And the angle of the toothpace, the pressure angle Phi usually
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  like 14.5 to 25°. Right.
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  The pressure angle Phi is basically the slope of the tooth
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  right at that pitch circle. What's the advantage of going
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  with a higher pressure angle, say 25° instead of 14.5?
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  Well, it's a significant trade off in design.
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  A higher pressure angle gives you a tooth that's wider at the
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  base, it's just physically stronger so it can carry more
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  load. And crucially, it's much less
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  likely to suffer from undercutting when it's
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  manufactured. Undercutting.
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  Remind us what that is and why it's bad news.
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  Sure. Undercutting is when the cutting
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  tool like the hob actually cuts away material below the intended
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  envelope profile, right near the root of the tooth.
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  It essentially weakens the tooth right where the bending stress
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  is highest. Not good for fatigue life.
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  OK, so higher pressure angle means stronger root, less
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  undercutting risk. What's the downside then?
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  The potential downside is often in the operation.
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  Higher pressure angles can sometimes mean a lower contact
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  ratio. Fewer teeth sharing the load at
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  any given instant. Exactly, and that can lead to
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  running that's a bit rougher or noisier compared to gears with a
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  lower pressure angle, all else being equal.
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  Right, okay, let's broaden out from geometry to the different
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  types of gears, starting with the most common one, the spur
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  gear. Spur gears, Yeah, they're the
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  foundation. Simplest design, easiest to
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  manufacture. Their teeth are straight
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  parallel to the axis of rotation.
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  And importantly, the loads they put on the.
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  Shafts purely radial loads just perpendicular to the shaft.
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  That makes choosing bearings much simpler.
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  But if we need more strength or quieter running, we look at
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  helical gears. Correct.
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  The helical gears have teeth cut at an angle, the Helix angle
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  usually somewhere between say 10 and 35°.
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  Because of that angle, the teeth engage more gradually.
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  It's not an instant full width contact like a spur gear.
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  And that gradual engagement gives them the edge in strength
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  and noise. Yes, they are stronger and run
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  much more quietly than a spur gear of similar size.
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  But engineering is always about trade-offs.
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  What's the catch with helicals? 2 main things.
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  You lose a tiny bit of efficiency compared to spurs
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  though. It's usually small, but the big
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  one is that the angled teeth create an axial load, a thrust
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  load along the shaft axis. So now you need bearings that
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  can handle both radial and thrust loads.
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  Precisely, it complicates the bearing selection and the
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  overall support structure design quite a bit.
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  Is there a way around that thrust load?
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  There is for really high torque you can use double helical
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  gears, often called herring bone gears.
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  They look like two helical gears back-to-back.
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  The opposing Helix angles cancel out the thrust loads.
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  Very effective, but more complex and expensive to make.
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  OK, what about when shafts aren't parallel?
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  If they need to intersect maybe at 90°.
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  Then you need bevel gears. They're shaped like sections of
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  cones instead of cylinders. They're very efficient too,
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  usually operating in that same high 90% range, letting you
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  transmit power around corners essentially.
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  And there are variations here too, like spiral bevels.
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  Yes, spiral bevels have curved teeth, similar to how helical
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  teeth are angled on a cylinder. They run quieter than straight
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  bevels, but are slightly less efficient.
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  And then there are hypoid gears. These are a special type of
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  spiral bevel where the shaft axis don't actually intersect,
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  they're offset. Offset.
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  Why would you want that? The most common reason is in
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  automotive rear axles. It lets you lower the drive
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  shaft, giving more space in the vehicle cabin.
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  Hypoids are standard in most rear wheel drive car
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  differentials. Fascinating.
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  Now all these designs depend critically on the material.
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  You mentioned AGMA grades. Right, AGMA classifieds gear
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  steels generally into grades 1-2 and three based on quality and
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  things like material cleanliness.
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  And beyond the base material, heat treatment is huge.
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  Oh, absolutely critical. You have two main approaches
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  through hardening where the entire tooth is hardened to a
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  consistent level. Maybe like Rockwell C42 or you
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  have case hardening. This gives you a very hard outer
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  surface, maybe Rockwell C60 or even higher, but leaves the
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  inner core of the tooth softer and tougher.
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  And why leave the core softer? What's the advantage there?
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  That softer, tougher core gives the gear better resistance to
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  shock loads. It can absorb impact without
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  fracturing. Meanwhile, the Super hard case
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  provides the wear resistance and the high surface strength
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  needed, where the teeth actually match and carry the load.
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  It's often the best of both worlds for demanding
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  applications. OK, let's put on our engineering
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  hats. We've got a design challenge.
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  Laid it on me. We need a spur gear set.
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  It has a handle 100 horsepower, the pinions running at 1000 RPM.
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  We need roughly an 8 to 1 speed reduction and we want to use 25°
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  pressure angle teeth. OK, 100 horse left 1000 RPM
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  pinion 8.1 ratio 25°. Five first step teeth numbers
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  with a 25° pressure angle. AGMA says we need at least 14
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  teeth on the pinion to avoid that undercutting problem we
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  talked about. S 14 minimum.
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  Let's be safe. Maybe start with 17?
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  Good call. Starting with 17 teeth on the
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  pinion gives us a safety margin. OK, 17 pinion teeth and eight to
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  one ratio would mean 17 * 8. That's 136 teeth on the gear.
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  Mathematically, yes, but we need to consider where life we need
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  what's called a hunting ratio. The hunting ratio.
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  Explain that again. See, 136 is exactly divisible by
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  17. That means the same pinion tooth
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  would mesh with the same gear tooth every 17 revolutions of
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  the pinion. That concentrates where in
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  specific spots. Not ideal.
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  So you want the numbers to be relatively prime, not sharing
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  common factors other than one. Exactly.
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  So instead of 136, let's try 135 teeth on the gear. 135 and 17
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  don't share any common factors. Now every tooth on the pinion
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  will eventually mesh with every tooth on the gear, spreading the
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  wear evenly. OK, so our actual ratio is a 1
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  and 35 / 17, which is about 7.94 to one.
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  It's also enough to 8.1. Yep.
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  Usually acceptable. So 17 and 135 teeth.
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  It is now the rating. How do we know if it can handle
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  100 horsepower? AGMA says check two things.
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  Bending strength. Right, which is basically asking
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  if the tooth will physically snap off at the root.
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  And pitting resistance. And pitting is about surface
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  fatigue, tiny cracks forming on the toothpaste due to repeated
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  contact stress. Which one usually governs the
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  design in practice? For most industrial gears,
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  especially at decent speeds, pitting resistance is almost
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  always the limiting factor. The surface gives out long
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  before the tooth breaks, so we. Really need to focus on the
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  pitting resistance horsepower calculation.
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  The piece of AC that AGMA formula looks pretty intense.
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  It is complex, yeah. Lots of factors, but
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  conceptually what really drives a piece of AC rating?
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  It boils down to a few key things.
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  First, the allowable contact stress S of AC.
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  That's purely down to the material strength and heat
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  treatment. Higher S of AC, higher power.
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  Second, the pinion pitch diameter D Bigger gears can
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  handle more load. Power scales roughly with the
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  diameter squared actually. Diameter squares.
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  The size matters a lot. Hugely.
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  And 3rd there are factors related to the geometry and
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  material elasticity, like a term called the elastic coefficient C
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  sub P and the geometry factor I OK C sub P accounts for how
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  stiff the materials are. Yeah, basically how much the
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  teeth deform under load when they mesh steel and steel has a
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  certain C sub P value. So back to our 100 horsepower
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  example, let's try some materials, say AGMA Grade 1
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  steel. We could use a basic through
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  hardened steel, maybe Rockwell C42 or a better case hardened
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  steel Rockwell C60 on the surface.
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  Right. The case hardened is
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  significantly stronger, higher allowable stress, higher S sub
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  AC. Let's pick a starting two sides,
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  say a diametral pitch P sub D70 or 00.
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  What horsepower can we get? OK, running the numbers with P
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  sub D = 7 the through hardened RC42 steel.
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  Nope, it only gets us about 71 horsepower.
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  Not enough. And the stronger case hardened
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  RC60 at the same P sub D of seven.
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  Better. That gets us up to about 92
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  horsepower. Closer, but still short of the
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  100 horsepower target. So neither material works at
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  that tooth size, right? We need more power capacity.
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  What's the lever to pull? Remember, power scales with
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  diameter squared. The easiest way to boost the
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  rating is to make the teeth physically bigger.
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  Bigger teeth means a lower dimensional pitch number.
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  Exactly. We need to decrease P sub D
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  Let's take the cheaper through harden RC42 steel.
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  How low does P sub D need to go to hit 100 horse cranking
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  through the calculation? If we drop P sub D down to 600,
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  that makes the pinion diameter significantly larger, and that
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  design gives us about 112 horsepower safely over the 100
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  horsepower requirement. OK.
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  So even though the RC60 case hardened steel is stronger, we
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  could meet the spec with a lower cost RC42 steel simply by making
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  the gear physical larger. P sub D = 6.0.
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  That's often the case. Size can compensate for material
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  strength up to a point, and pitting, not bending, was the
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  limiting factor in all those calcs.
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  So the final choice might come down to cost versus size and
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  weight. The P sub D = 6.0 design with
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  RC42 works and might be cheaper. Precisely.
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  That's the engineering trade off.
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  Now, once we have a baseline design like that, under 12
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  horsepower one, engineers rarely stop there, right?
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  Optimization is key. Always looking for improvements,
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  yeah, especially if, say, the customer later needs a bit more
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  power but can't change the overall gearbox size or center
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  distance. So, stuck with the same basic
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  envelope, what tricks can we use?
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  You mentioned something called a long addendum design, ah?
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  Yes, that's a clever geometry tweak.
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  You basically modify the tooth proportion slightly.
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  You make the pinion tooth addendum a bit taller and the
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  gear tooth addendum a bit shorter, while keeping the
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  overall working depth the same. So you're just shifting the
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  tooth heights slightly between the pinion and gear.
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  What does that do? It does 2 very useful things.
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  It tends to make the pinion tooth stronger in bending, and
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  it often improves the surface conditions, reducing pitting
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  stress. It changes is that geometry
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  factor I we mentioned. How much difference going to
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  make? Let's say we had a baseline 115
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  horsepower spur gear design. Applying a standard long
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  addendum modification to that 115 horsepower design, purely
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  through changing the geometry factor, I could easily boost the
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  rating to something like 131 horsepower, a significant jump
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  with no material change. Wow, just from tweaking the
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  tooth heights. OK, what about materials?
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  If we stuck with the baseline geometry but upgraded the steel,
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  say from a grade 1-RC60 to a top tier grade 3 RC 60.
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  Now you're talking pure material strength improvement.
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  Grade 3 steel allows a much higher contact stress.
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  S sub AC. For grade 1-RC60S sub AC might
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  be around 180,000 PSI. For grade 3 it could jump to
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  maybe 275,000 PSI. And the impact on power rating,
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  that 115 horse design. With that jump in S of AC, the
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  rating would shoot up dramatically.
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  You could potentially get up to maybe 268 horsepower from the
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  same size gear just by using the higher quality steel.
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  Huge difference. OK, one more option, change the
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  type of gear. What if we replaced that 115 HP
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  spur gear with a helical gear, say with a 30° Helix angle?
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  Going helical now, the improvement comes mainly from
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  better load sharing due to the angle teeth and gradual
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  engagement. Again, it improves the geometry
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  factor. I that 115 HP spur gear replaced
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  with a similar size 30° helical might jump up to around 222
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  horsepower. So helical gives a big boost
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  too. But now you have that axial
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  freslo to deal with which the spur gear didn't have.
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  Again, trade-offs. It's clear these calculations
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  are precise and they rely on achieving that perfect involute
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  curve or those complex helical or bevel shapes.
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  How is that actually done in manufacturing?
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  How do we guarantee the shape? Well, it's less about drawing
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  the shape perfectly and more about generating it through a
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  controlled kinematic process. It's really quite elegant.
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  Mathematically. We model the manufacturing
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  process itself. Imagine the cutting tool.
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  Maybe a rack shaped cutter for hopping spur gears, or a conical
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  cutter for bevels rolling in perfect mesh with the gear
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  blank. The actual tooth profile created
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  on the blank is the boundary or the envelope traced by the
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  surface of that cutting tool as it moves.
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  So the final shape isn't static, it's the result of this dynamic
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  interaction between the tool and the blank.
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  Exactly right. The fancy term for it is the
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  envelope of A1 parameter family of curves.
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  Engineers use this math to simulate the cutting.
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  They saw the equations representing the cutter's
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  surface simultaneously with a constraint equation related to
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  the gear blank's rotation. The solution to that system of
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  equations defines the precise complex tooth profile that will
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  be generated. So we've really seen that
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  designing gears that last. Is this this delicate balancing
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  act? It really is.
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  Small tweets in geometry like the long addendum or changes in
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  material great can have massive impacts on power capacity, and
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  often it's not about preventing the tooth snapping off, but
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  preventing that microscopic surface damage to the pitting.
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  That's the key take away for most industrial gears.
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  Yes, pitting is usually the life limiter and especially when you
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  get into high performance gears like spiral bevels and hypoids,
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  the complexity goes up another notch.
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  The exact tooth surface you get, and therefore it's performance
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  in life, isn't just defined by the blueprint dimensions, it's
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  critically dependent on the precise kinematic settings used
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  on the specialized gear cutting machine.
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  OK, so here's a final thought for you to chew on based on
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  that. If the actual functional
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  geometry of A high performance gear is determined less by the
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  CAD drawing and more by the specific complex motion settings
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  of the manufacturing machine, where does the design truly
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  happen? Is it in the engineer software
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  or is it on the shop floor when those machine settings are
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  dialed in? That's a fantastic question.
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  Maybe the future really lies in CAD systems that don't just draw
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  static shapes, but can fully simulate that complex 3D
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  envelope generation process, predicting and preventing
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  failures based on the actual manufactured geometry before a
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  single chip of metal is even cut.
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  I think you've hit the nail on the head.
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  That's where the cutting edge of gear design and analysis is
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  heading.