What should be the geometries of two contacting solids that may have a relative rotation and translation along the same axis?

[apcosta]

TL;DR Summary: What should be the geometries of two contacting solids that may have a relative rotation and translation along the same axis?

a) Consider two rigid bodies that have a relative motion characterized by a rotation and a translation with respect to the same axis (like a bolt and a nut). The two solids may rotate around a certain axis and translate along the *same* axis (exactly as a bolt and a nut).

b) The two solids are separated by a surface so that the geometries of the two bodies match perfectly at all points of the surface.

c) What is the geometry of such a surface? In other words: what is the geometry of the screw head of a bolt so that it matches perfectly with the nut?

 

[Baluncore]

c) What is the geometry of such a surface?

Cylindrical.
Any constant pitch helical thread profile, with its complement, would work if the translation was due to rotation.

 

[apcosta]

Yes, the translation is due to rotation so that there is a permanent/persistent sliding at all points of the interface. So, you mean "a constant pitch helicoid surface"? Where could I find a proof?
Thank you very much for your fast reaction!

 

[Baluncore]

b) The two solids are separated by a surface so that the geometries of the two bodies match perfectly at all points of the surface.

That is not true of real helical threads. Most screw threads have truncated crests and troughs, so the opposed thread surfaces do not match exactly, but are easier to cut. Contact between the bodies is only made on one thread flank of the internal body, against the one opposed flank of the external body. There must always be a clearance to allow for tolerance and temperature changes. Without a lubricant film, the sliding surfaces would have high friction, or cold weld together.
Also consider a low friction ball-screw, where the recirculating balls are in the channel between the two bodies. There are gaps between the ball contact lines with the channel, the balls are ancillary bodies, so there is no single surface in contact at all points.

Where could I find a proof?

The proof will depend on what you are trying to do, and why you need a proof.

 

[Lnewqban]

c) What is the geometry of such a surface? In other words: what is the geometry of the screw head of a bolt so that it matches perfectly with the nut?

Welcome, @apcosta !

What the screw head of a bolt has to do with the nut?

What is guiding the relative movement of one surface respect to the other?

 

[FactChecker]

It seems like you must want something more than that. You could consider that nut and bolt as a single solid and then put a partitioning surface between them with practically any shape remaining in the interior of the combined solid.
CORRECTION: Sorry, I missed the significance of this part: "have a relative motion characterized by a rotation and a translation with respect to the same axis".

 

[Hill]

I think that we can take a segment of any flat curve and uniformly rotate and translate it around and along any axis in space to make such surface, as long as this surface does not intersect itself. Not only "nuts and bolts" with various grooves, but also "corkscrews" with various cross sections.

 

[apcosta]

It seems like you must want something more than that. You could consider that nut and bolt as a single solid and then put a partitioning surface between them with practically any shape remaining in the interior of the combined solid.
CORRECTION: Sorry, I missed the significance of this part: "have a relative motion characterized by a rotation and a translation with respect to the same axis".

Yes, that last part is fundamental. Thank you for showing interest in this problem!

 

[Baluncore]

Thank you for showing interest in this problem!

Is this a purely theoretical study, or is there a practical application?

The requirement that the geometry of the two bodies, match perfectly at all points on the contact surface, seems to be an impossible practical relationship, that defeats the purpose of the study.

Spline shafts are designed to allow free linear translation, while preventing rotation.
Hydraulic cylinders control translation, but allow freedom of rotation about the axis.
A cylindrical sleeve, running on a round bar, may meet your requirement.
A screw thread couples the translation to the rotation, but there are many special threads that do not meet the perfect-contact-everywhere requirement.

Can you tell us more about the intended application, or the reason for your interest, in such a mechanism?