For the past few years I have been working on a very similar concept of kinetic energy transfer via a variable pitch cylindrical cam, so I was pleasantly surprised to have come across your project. I also derived my transmission ratio curve in a similar way by enforcing conservation of energy across discrete steps of the solution then minimizing my objective function, so it was quite satisfying and validating to see that you followed a similar route.
What I'm struggling with, which you hinted at in your previous post, is the exact shape of the rolling cam followers. Since my implementation will be on a much larger scale, I am trying to figure out how to optimize the efficiency by choosing a rolling follower shape and conjugate cam surface geometry which will operate across the varying pitch range with no or minimal slipping (like a car wheel steering at a slip angle) and without "scrubbing" (like trying to roll a cone in a straight line across a flat plane).
So far, I've managed to work out that a hyperboloid can roll without slipping on a constant-pitch helicoid, but its axis must be offset and this offset distance changes along with pitch, which will not work since we need pitch to vary. In the end I may have to set up another generalized free-form shape and somehow optimize it in a similar way by minimizing the relative velocities along the contact line.
Have you given any thought to this aspect? I imagine at the scale of your mechanism such details aren't worth the effort, but I'm nonetheless curious what ideas you might have for approaching it.
Props on the impressive engineering, I'm really looking forward to seeing your prototype in action!
I've thought about it a little bit, but haven't done any detailed analysis. Having a cylindrical cam follower (like I've shown) is about as bad as it gets, although as the ratio of cam diameter to cam follower length goes up, it gets better - rolling a cylinder along a flat surface in an arc, there's less scrubbing as the arc diameter goes up or the cylinder length goes down.
The "easy" improvement I see is making the cam follower tapered. If the point of the cone that defines the tapered cam follower intersects the screw axis, then at least for shallow helix angles there should be much less scrubbing - at zero helix angle, it's a tapered roller bearing so there's no scrubbing. I haven't worked through what happens as helix angle increases, though.
Sorry for the possibly stupid question; I'm in high school, so still an engineering novice. Could you elaborate on what exactly the inefficiencies are with a circular cam follower? I understand the concept of the follower scrubbing - the walls on either side of the cam aren't perfectly parallel- and why a noncircular follower might fix this. But I don't get what the issue is with slipping (if all you want is the follower to travel with minimal friction, why do you care if it slips?), and how a non cylindrical cam follower remedies it.
What I'm calling "scrubbing" doesn't have to do with the cam walls not being parallel. It has to do with the radius increasing as you go radially outward on the cam surface. Since the radius increases, the linear velocity of the points on the cam surface also increase. With a cylindrical cam follower, all the points contacting the cam move at the same velocity as each other, since the cam follower radius is constant. So there must be a velocity difference between some points on the cam follower and some points on the cam. This relative velocity is what I'm calling "scrubbing", and it results in a friction torque on the cam. It's equivalent to forcing a cylinder to roll around an arc on a flat surface - naturally it "wants" to roll in a line - you have to apply a torque to the cylinder to get it to roll in an arc.
A cone, on the other hand, naturally rolls in an arc on a flat surface (or on another cone of the right angle). This is how tapered roller bearings work without the rollers scrubbing. I'm not sure if this works perfectly for the cam/cam follower, however, the varying helix angle probably complicates things.
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ReplyDeleteHi Ben,
ReplyDeleteFor the past few years I have been working on a very similar concept of kinetic energy transfer via a variable pitch cylindrical cam, so I was pleasantly surprised to have come across your project. I also derived my transmission ratio curve in a similar way by enforcing conservation of energy across discrete steps of the solution then minimizing my objective function, so it was quite satisfying and validating to see that you followed a similar route.
What I'm struggling with, which you hinted at in your previous post, is the exact shape of the rolling cam followers. Since my implementation will be on a much larger scale, I am trying to figure out how to optimize the efficiency by choosing a rolling follower shape and conjugate cam surface geometry which will operate across the varying pitch range with no or minimal slipping (like a car wheel steering at a slip angle) and without "scrubbing" (like trying to roll a cone in a straight line across a flat plane).
So far, I've managed to work out that a hyperboloid can roll without slipping on a constant-pitch helicoid, but its axis must be offset and this offset distance changes along with pitch, which will not work since we need pitch to vary. In the end I may have to set up another generalized free-form shape and somehow optimize it in a similar way by minimizing the relative velocities along the contact line.
Have you given any thought to this aspect? I imagine at the scale of your mechanism such details aren't worth the effort, but I'm nonetheless curious what ideas you might have for approaching it.
Props on the impressive engineering, I'm really looking forward to seeing your prototype in action!
Eric
Hi Eric,
DeleteI've thought about it a little bit, but haven't done any detailed analysis. Having a cylindrical cam follower (like I've shown) is about as bad as it gets, although as the ratio of cam diameter to cam follower length goes up, it gets better - rolling a cylinder along a flat surface in an arc, there's less scrubbing as the arc diameter goes up or the cylinder length goes down.
The "easy" improvement I see is making the cam follower tapered. If the point of the cone that defines the tapered cam follower intersects the screw axis, then at least for shallow helix angles there should be much less scrubbing - at zero helix angle, it's a tapered roller bearing so there's no scrubbing. I haven't worked through what happens as helix angle increases, though.
Hi Ben,
DeleteSorry for the possibly stupid question; I'm in high school, so still an engineering novice. Could you elaborate on what exactly the inefficiencies are with a circular cam follower? I understand the concept of the follower scrubbing - the walls on either side of the cam aren't perfectly parallel- and why a noncircular follower might fix this. But I don't get what the issue is with slipping (if all you want is the follower to travel with minimal friction, why do you care if it slips?), and how a non cylindrical cam follower remedies it.
Hi Jeffrey,
DeleteWhat I'm calling "scrubbing" doesn't have to do with the cam walls not being parallel. It has to do with the radius increasing as you go radially outward on the cam surface. Since the radius increases, the linear velocity of the points on the cam surface also increase. With a cylindrical cam follower, all the points contacting the cam move at the same velocity as each other, since the cam follower radius is constant. So there must be a velocity difference between some points on the cam follower and some points on the cam. This relative velocity is what I'm calling "scrubbing", and it results in a friction torque on the cam. It's equivalent to forcing a cylinder to roll around an arc on a flat surface - naturally it "wants" to roll in a line - you have to apply a torque to the cylinder to get it to roll in an arc.
A cone, on the other hand, naturally rolls in an arc on a flat surface (or on another cone of the right angle). This is how tapered roller bearings work without the rollers scrubbing. I'm not sure if this works perfectly for the cam/cam follower, however, the varying helix angle probably complicates things.
Hopefully that explanation makes some sense...
o.O lets gooooo
ReplyDelete