Linked torus

Two identical tori that are mutually and symmetrically linked.

These things have no immediate practical uses (that I know of), but I am interested in their geometry and kinematics. I used them as an exercise to figure out Tinkercad basics (navigation, sizing, alignment), to experiment with minimum practical clearances and printer/surface quality/errors, and as a shape element that can be used in future projects (for example, as the core element of a simple universal joint).

Clearances:

The three models have a difference of minimum clearance between the two tori. If printed at the original scale (each torus radius-to-center-of-tube = 6mm) the centered nominal clearance between tori is:

• no clearance: 0 mm (rigid, the two tori are fused at near-tangential contact)
• low clearance: .1 mm
• high clearance: .4 mm

Scaling the STL model will increase or decrease the clearance, proportional to the scaling factor.

For the low clearance (.1 mm, nominal) models that I printed, the two component tori seperated from the supports and each other with only finger force. Sliding the tori through each other tends to smooth the high points, and the final actual clearance is close to the nominal clearance. The surface radii variations due to faceting are small compared to the low clearance value.

Geometry:

The torus radius (center to center-of-tube) of all models is 6 mm. The tube radius for the 'no clearance' model is 3 mm (half the torus radius). The maximum dimension of the full model is 24 mm; 4 x diameter of the tube.

With positive clearance the tori naturally tilt into contact. Any slight relative torsion about the axis between torus centers results in contact on a full circle on both tori. This circle of contact is a Villarceau circle, at the intersection of a plane and a torus that passes through the center of the torus and touches it tangentially at two antipodal points.

When in contact at a mutual Villarceau circle each torus can be rotated about its symmetry axis such that it will slide but stay in full contact at a Villarceau circle. Both can be rotated in this fashion independently, one at a time or simultaneously. There are two internal degrees of freedom, a rotation of each torus about its rotational symmetry axis. The bounding shape is unchanging after any of these internal rotations.

If the size and clearance is right the tori can be slid back and forth between a thumb and first finger, mutually rotating and sliding both tori through the other's center.

The zero tolerance model has tori holes that are the same diameter as the tori tube. The tori will not be able to slide through each other -- there are no internal degrees of freedom. The shape rolls on a slightly inclined plane in a smooth but periodically wobbling way. Its center of mass both rises-and-falls and moves side-to-side during one full rolling turn, and the two contact points follow a curved path with both a small and large curvature phase, with each side a half-turn out of phase with the other. The rise and fall of the center of mass is small compared to the tori dimensions which results in nearly smooth rolling motion on a very gentle incline (low slope).

Printing instructions:

Technical details and recommended print settings:

The STL files are compatible with all 3D printers.

I used PLA filament, using a Prusa Mini+. The print time on a Prusa Mini+ is about 16 min. for one of the models.

PrusaSlicer non-default values:

• 0.2 mm (Quality) layer height for clean and precise results.
• Layers and perimeters:
  • Solid layers: Top: 0 Bottom: 0
• Support materials:

  • Raft layers: 4
    

  • Style: snug
  • XY separation between an object and its support : 30%

Warnings:

Small parts are a choking hazard, do not make these available to young children.

Acknowledgements:

I modeled this using Tinkercad and used Prusa Mini+ printers at the free maker space, JDLC, at the Southeast Campus of Portland Community College.

Thanks goes to the helpful staff, Hacienda CDC and all supporters of JDLC.