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(#67) Calculus Cams and Global Quality Standards · Mechanical Engineering Made Simple

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  Welcome back, design junkies to Mechanical Engineering Made
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  simple. Today we're we're undertaking a
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  really critical deep dive into the very core of machine design.
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  We're talking about the components that control motion,
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  transmit power, and, you know, most importantly, manage all the
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  stress and longevity. That's right.
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  Gears, cams, bearings, linkages, the works.
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  A whole vocabulary. Our analysis is coming from some
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  seriously dense engineering handbooks.
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  I mean, they focus intensely on the geometry, the manufacturing,
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  and the sometimes brutal calculations you need to make
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  sure these things actually survive.
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  So our mission today is really to pull out the functional
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  essence of these parts. Yeah, and to show you why every
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  single moving part is a small hard won victory against the
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  physics of failure. What becomes immediately clear,
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  and you see it right away in the sources, is that it doesn't
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  matter if you're looking at a simple little detent or a, you
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  know, full on aerospace actuator.
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  The design is just this massive balancing act.
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  Every millimeter, every angle, every material choice has to be
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  absolutely precise. There's no room for error.
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  OK, so let's start with rotational power.
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  Let's start with gears. The fundamental language of
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  gears and 1st we have to define tooth size that really splits
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  into two major systems. Metric and US.
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  Right. In the metric world, we use
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  what's called the module, and it's actually pretty simple.
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  It's just the gears pitch diameter in millimeters divided
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  by the number of teeth, OK. Straightforward enough, but in
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  the US customary system we use the diameterial pitch.
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  And this is one that always, always trips people up.
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  OK, it's inverse. Totally inverse.
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  It's the ratio of the number of teeth per inch of pitch
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  diameter. So the physically larger the
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  gear tooth is, the smaller the diametral pitch number becomes.
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  It feels completely backwards. It does, but that's just how the
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  ratio works. And that demand for precision,
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  it flows right into the manufacturing process.
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  You can't just cut the stuff out of a block of metal, no.
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  Absolutely not. To create a gear face that can
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  withstand constant high load, you're talking about highly
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  specialized operations. You need milling then maybe
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  hobbing for spur gears, or end milling and shaping for internal
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  gears. And some things, like bevel
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  gears need their own specialized machines.
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  Exactly. And it's not just gears.
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  It's astonishing when you look at something like a standard
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  roller chain sprocket, right? They have to adhere strictly to
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  ANI standard geometry. The dimensions aren't just
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  picked, they are derived from these complex equations based on
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  the pitch, which we call P and the number of teeth N.
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  So every single curve is mathematically defined.
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  To the thousandth of an inch. I mean take 1/2 inch pix roller
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  chain. The source gives a minimum
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  seating curve value of .317 inches.
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  It's calculated down to that level because if that contact
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  point is even slightly off. You just destroy the chain.
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  You destroy the chain in the sprocket.
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  It's a rapid failure which, you know, leads us right into
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  material selection. Because the material has to
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  handle those forces. Precisely.
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  For something low speed, light load, sure.
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  You might get away with engineering plastics, acedal,
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  polycarbonate, things like that. But for any serious power
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  transmission. You immediately jump to heat
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  treated ferrous and non ferrous alloys, no question.
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  And this is where it gets really interesting because this
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  quality, both material manufacturing, it's all
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  quantified by the AGMA quality numbers, yes.
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  And this is where you get a just a profound insight into the
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  mechanical hierarchy of the world.
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  AGMA numbers define exactly how accurate your gear manufacturing
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  has to be, so. Give us an example.
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  What's the range look like? OK, think about a simple
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  agricultural bailing machine. It might be perfectly functional
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  with quality numbers between, say, 5:00 and 7:00.
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  OK. So that's pretty loose
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  tolerance. Very loose, but then you jump
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  into something sensitive like electronic instrument control
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  and guidance systems. That's a whole different ball
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  game. It is.
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  Suddenly you need quality numbers from 10 all the way up
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  to 14. The tolerance just tightens
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  massively. And what's at the very top end?
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  Well, if you were engineering something truly demanding like a
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  space navigation sextant and a Star Trekker, you need quality
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  #14 and up. The required accuracy for a
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  mechanism pointing to a star is just.
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  It's a different universe compared to 1 compacting A.
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  It shows that precision isn't just a cost factor.
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  No, it's a mission critical constraint, and that material
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  choice also dictates how you can even machine the part.
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  Machinability is hugely affected by heat treatment.
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  I've heard that some of the high carbon steels are just a
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  nightmare to work with. They are a 4150 grade.
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  Steel is notoriously difficult to machine.
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  You have to specify tempering for steels like 4140 just to
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  make them manageable on the shop floor.
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  And all this is before the system is even running.
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  Once it is, you have to deal with the operational forces.
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  Right, which the designers have to anticipate.
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  A simple spur gear will generate tangential and radial loads, but
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  systems like helical gears or even more so, worm gearing.
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  They introduce that third dimension of.
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  Force, a massive axial load rust and the calculation for the
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  forces in a worm gear system is where things get really complex.
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  You can't just use simple geometry.
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  Not at all. Yeah, you have to factor in the
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  coefficient of dynamic friction and the system's overall
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  efficiency. And that complicates everything,
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  especially selecting the right support bearings, because that
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  friction is what converts the rotational input into a crushing
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  axial load that the bearings have to absorb day in and day
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  out. OK, calculating those continuous
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  forces is hard enough, but what about controlling the timing and
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  the specific path of a movement? That's even harder.
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  It is, and that brings us to the systems that manage selective
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  controlled movements cams. Right, so cams are what convert
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  that simle rotation into something much more complex.
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  Exactly. They're defined as mechanical
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  components that convert continuous Rotary motion into a
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  specific, you know, predetermined translating or
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  oscillating action using a follower and.
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  The absolute dictator of which Cam profile you choose.
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  It's. Speed.
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  It's always speed, so. For low speed movement you.
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  Might use something called a uniform velocity motion, but
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  this is actually terrible because when the follower hits
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  the flat part of the Cam, it goes from zero acceleration to
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  maximum acceleration instantly. It's like hitting a wall.
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  The shock is incredible and. That shock is exactly what
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  you're trying to engineer out of the System 1. 100% uniform
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  acceleration and parabolic motions are better for moderate
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  speeds because they smooth out that transition, but for any
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  kind of high speed application you absolutely must use
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  cycloidal motion. Why is cycloidal motion the go
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  to for high speed? Because.
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  It's the only profile that keeps the acceleration curve smooth.
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  The shape is mathematically designed to ramp up and ramp
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  down both the velocity and the acceleration gradually.
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  It prevents those massive G forces from destroying the
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  follower and, you know, the mechanism itself.
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  We see. That design battle against shock
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  loads everywhere in Cam design. I was reading that the slope
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  lines defining the rise and fall have to be terminated with a
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  parabolic curve. Specifically to prevent shock
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  loads. If you skip that tiny design
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  detail, your follower mechanism will literally hammer itself
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  apart. It's amazing.
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  Now there are two main types of follower systems, right?
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  Open track and close track, correct?
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  You have open track cams which require the follower to be held
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  against the Cam surface by a spring.
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  Think of a standard engine valve train.
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  OK. So a spring is pushing it back
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  exactly. But then you have close track
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  cams where the follower actually travels inside a groove.
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  This gives you positive action meaning.
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  No spring is needed, no. Spring.
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  This is critical in manufacturing machinery because
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  if a spring breaks in an open track system, the whole machine
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  fails immediately. Closed track cams eliminate that
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  specific risk, but. The challenge with the spring
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  based systems, the open track ones, is what happens at high
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  speeds you get. Spring floating.
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  The spring just can't keep up with a Cam profile and the
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  follower loses contact. It literally floats off the
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  surface. How do you fix?
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  That. The fix is designing the spring
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  with a natural frequency below the cyclic rate of operation,
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  and often the solution is to put multiple springs in parallel.
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  That's exactly what you see in the complex valve springs of
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  high performance automotive engines and.
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  Then, on the total opposite end of complexity, you've got the
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  simplest Cam of all, the. Eccentric Cam the most
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  economical to make. It's often just a round disk
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  that's offset from its center axis.
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  You. See those used for simple rise
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  functions or actuating limit switches and timing devices.
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  Yeah. But you can't run them at high
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  speed. For ultra high speed we have to
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  look at some really advanced systems like the desmodromic
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  system. This.
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  Is the system that was famously used by a Mercedes-Benz racing
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  engines back in the 50s it? Was it actually used a second
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  Cam to physically force the follower back down?
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  It completely eliminated the need for springs so.
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  If you ever hear a high revving 1950s Grand Prix engine that
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  really distinct mechanical clatter, that's the.
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  Sound of springs being completely engineer and out of
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  existence. It's the noise of pure positive
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  mechanical action. Amazing.
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  Now, when you're designing these Cam profiles, there are other
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  constraints too, right? You have to look at the contact
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  interface. Oh.
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  Absolutely, the geometry is critical.
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  You have to make sure the roller radius is smaller than the
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  minimum radius of curvature along the Cam profile.
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  Otherwise you'd actually cut into the Cam itself when you're
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  manufacturing it. You'd.
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  Undercut it and you also have to calculate the contact stress
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  which we call S sub S For a material like SAE 4340 steel he
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  treated to Rockwell C50, the Max allowable compressive stress is
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  220,000 PSI 2. £120,000 per square inch.
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  And if your calculation exceeds that stress limit, you have
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  designed a fatigue failure point.
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  It's guaranteed to break OK. Let's shift from cams to
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  linkages, the building blocks of movement.
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  We all remember the six basic, simple machines, but we're
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  focusing on mechanisms that translate motion in really
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  ingenious ways and. Maybe the most dramatic is the
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  toggle joint linkage. This thing is just incredible.
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  It's a mechanism designed specifically to multiply force
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  massively as the linkage angle approaches 90°.
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  The mechanical advantage and the velocity ratio both approach
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  Infinity. That's.
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  A trick of geometry it. Is the classic examples are the
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  bolt lock action of the Luger pistol or you know, giant rock
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  crushing machinery. Both use the toggle to generate
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  thousands of pounds of crushing force with a relatively small
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  input then. You have the four bar linkage
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  which is just everywhere. Absolutely.
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  Ubiquitous. Just 4 links connected by 4
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  pivots. You see it in everything from
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  car suspensions to construction equipment, but it.
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  Looks so deceptively simple it. Does, but mathematically its
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  action is extremely complicated. Before the 1950s, figuring out
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  the precise motion for all points was often just trial and
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  error. You'd build a prototype and see
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  what it did. It wasn't.
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  Until the mathematician Freudenstein developed his
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  geometric relationship, that designers had a really robust
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  mathematical tool that's. Right, but even then, the sheer
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  difficulty means that mathematical solutions are often
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  approximations at best. Which?
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  Is why even today high speed motion analysis is so critical
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  you have to actually film the prototype.
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  Exactly. You watch the complex actions
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  happening at 12,000 frames per second just to confirm the
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  mechanism is performing the way you thought it would and.
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  This analysis has its own defined vocabulary.
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  You see these letters RP and S? It's a kinematic vocabulary.
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  R is for a revolute joint, which is just simple rotation.
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  P stands for a prism joint, which allows sliding, and S is
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  the sphere joint, your classic ball joint and.
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  These designations let engineers define these incredibly complex
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  3 dimensional mechanisms. They.
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  Do. But then there are some linkages
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  like the RSSR linkage that just defied the standard rules of
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  kinematics. The.
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  Maverick space mechanisms, that's what.
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  They're called because they achieve this complex non
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  standard motion that you just can't analyze using basic planar
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  geometry. They're a class of their own
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  all. Right.
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  So all of these moving, complicated parts, the gears,
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  the cams, the linkages, they all have to be supported.
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  Which brings us to Part 3. Bearings and the science of
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  precision fits. Let's start.
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  With the simplest type, the plain bearing, often centered
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  bronze. And their selection comes down
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  to one critical calculation, the.
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  PV calculation The pressure velocity factor P is the
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  pressure on the projected area and V is the surface velocity
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  and designers have to balance these and this.
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  Balances everything. You can't just Max out all the
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  values. No, you.
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  Can't use the maximum for all three simultaneously.
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  If you run your machine at a super high velocity, you have to
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  lower the load. If you don't, the bearing
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  material will simply melt or disintegrate from the frictional
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  heat. It's a hard limit.
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  A very hard limit. For example, lead based Babbitt
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  has a maximum PV of 35,000, but PTFE fabric, a more advanced
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  synthetic material, only has a limit of 25,000.
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  These material limitations directly constrain how fast and
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  how heavy your machinery can possibly be then.
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  We move up to rolling element bearings, your ball, roller,
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  tapered and cylindrical types. Here we're.
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  Relying on authoritative bodies like the Anti Friction Bearing
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  Manufacturers Association, the ABMA for standards on load
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  ratings and fatigue life and. This is where the cost of
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  miscalculation just becomes brutally clear it.
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  Does there are two critical approximate rules for bearing
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  life that every single engineer has to know?
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  The first one is this. Doubling the bearing load
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  reduces bearing life to 110th 1. 10th not half. 110th That is a
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  monumental penalty for a small increase in load.
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  It's. Huge.
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  And the second rule is just as important.
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  Doubling the speed reduces bearing life to 1/2, so.
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  Controlling load and speed isn't just about performance, it's.
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  About controlling the maintenance schedule, the
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  downtime, the entire life cycle, cost of the machine itself, the.
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  Sensitivity brings us to maybe the ultimate constraint in all
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  of mechanical design. The precision fit, the fit of a
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  bearing into its housing, is often the last and most critical
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  step. It really.
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  Is EU customary classes of fit range from RC1 which are close
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  sliding fits just for accurate location all the.
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  Way U to FN 5 which? Are the heavy force fits use for
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  parts that are designed to be highly stressed and never ever
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  move again let's. Use an example, an RC3 precision
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  running fit on a 1 1/2 inch nominal shaft the.
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  Tolerance window required is just incredibly narrow.
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  The shaft and the hole have to be machined to clearances that
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  are often less than two thousandths of an inch across
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  the entire diameter. That's.
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  Tighter than a human hair. Tighter.
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  Than a human hair across the whole part.
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  If you miss that tolerance by even a fraction, the shaft will
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  either seize up from heat expansion or it will vibrate
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  itself to pieces. Immediate fatigue, failure and.
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  For the tightest connections the interference or force fits, you
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  have to calculate the stresses you.
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  Have to designers must treat the parts as thick walled cylinders
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  to calculate the internal stresses.
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  This determines how much torque can be transmitted and the axial
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  force. You need to physically press
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  them together and the. Stress analysis is absolutely
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  critical here. Because the maximum tangential
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  and radial stresses happen right in the contact surfaces,
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  designers have to use stress concentration factors, maybe 1.5
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  to 2.0, especially for thick hubs pressed onto a shaft.
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  If you neglect those factors, you.
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  Risk a sudden brittle fracture or a catastrophic fatigue
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  failure somewhere down the line? Absolutely.
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  So our deep dive today has moved us all the way from the
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  generalized motion of gears, through the controlled linear
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  action of cams and linkages, and finally into the critical
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  support systems defined by bearings.
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  These hyper precise fits the. Complexity forces designers to
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  master combined disciplines. You can't just be a mechanical
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  engineer anymore. You.
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  Can't you have to understand mechanical, electrical and
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  electronic systems and? That really is the ultimate
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  insight when you look at modern electromechanical products,
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  whether it's an industrial switch assembly or a high
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  precision titanium camera shutter.
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  It demands that engineers become general practitioners.
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  Yeah. You have to grasp all three
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  disciplines rather than sticking to a narrow specialty.
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  Precision in one area requires mastery in all the others, and.
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  This complexity brings us right back to the limitations of pure
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  analysis. You know, we discussed how
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  incredibly difficult it is to mathematically solve the precise
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  motion of something like a four bar linkage and.
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  That difficulty often forces modern engineers to rely on
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  computer aided design and critically, physical dynamic
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  analysis. So.
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  Here's the final thought when you look at any complex machine
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  operating smoothly today. An engine, a robotic arm, a
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  printing press, anything. Ask yourself this.
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  Are the specific movements you are watching the result of
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  elegant mathematics that perfectly modeled the entire
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  system from the start, or were? They finely tuned by watching a
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  prototype's slow motion video captured at 12,000 frames per
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  second, with engineers correcting the geometry based on
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  observed physical reality.