Week 6: Finalizing the Linear Motion Slide

Topics

Design of a Linear Motion Slide, Part 2

The goal this week was to finalise the linear motion module design and evolve the analysis that predicts its stiffness, accuracy and repeatability.

1. Cutting forces

Following the analysis in Kalpakjan and Schmid’s Manufacturing Engineering and Technology (chapters 21 and 23), I built a spreadsheet predicting the cutting forces the slide will see.

Cutting force summary table for facing an aluminium part: 1600 rpm spindle speed, 0.5 mm depth of cut, 100 mm/min feed rate, predicted cutting force Fy of 62.5 N and thrust and normal forces Fx and Fz of 43.76 N each

Those forces feed the axis error apportionment. For the lathe to hold 100 µm of total allowable error, the load-induced deflection from bearings and structure on one axis gets 19.9 µm. Working back from the expected forces and that allowance gives a net system stiffness requirement of 3834 N/mm — a reasonable target.

2. Carriage geometry and stiffness

Having settled on a boxway and T design, I built a spreadsheet that predicts carriage stiffness about and along each axis directly from the geometry, so dimensions can be changed in CAD and re-evaluated quickly.

Hand-drawn carriage cross-section annotated with dimension letters A through T and the constraint relations F minus 2t greater than A plus 2B, with a second sketch below defining the effective plate lengths S, U and V used in the stiffness calculation

Bearing pad material took real thought. I found some 3M Scotch foam double-sided tape that adheres well to aluminium and whose backing is quite slippery. Two pieces of MDF and a tilt test gave the friction coefficient from the tangent of the sliding angle — 0.12. Stiffness came from sticking two pieces to an aluminium block and measuring deflection under an extra 2.6 kg block: 204.7 N/mm, nowhere near stiff enough for this slide.

Three photographs of the friction and stiffness testing: tilting two taped MDF strips by hand to find the sliding angle, an aluminium cylinder standing on a taped block on the mill table, and a second aluminium mass added on top under the mill quill

I then found UHMW polyethylene in my lab. Using the material properties from MatWeb, a thin square of UHMW calculates out at 138 N/µm — which is the number the design needed.

Screenshot of the carriage geometry sheet: rail and carriage properties in millimetres and metres, tool offsets, bearing pad width, thickness, length, Young's modulus and a resulting bearing pad stiffness of 138 N per micron, alongside material moduli for wood, aluminium and steel and the computed moments of inertia

Load-induced errors come from dividing the forces at the carriage’s centre of stiffness by the carriage stiffness about that axis. Geometric errors come from the manufacturing and assembly tolerances — since the carriage is preloaded, they are dominated by how the slide is mounted and by parallelism error from flatness. I estimated the geometric error about and along each axis, then used homogeneous transformation matrices to carry each one through to the tool tip in the reference frame, computing both the linear stack-up and the RMS and taking the expected error as the average of the two.

Screenshot of the geometric error sheet: manufacturing tolerances for rail flatness, mounting misalignment and surface finish at the top, rail geometric errors, an error gain matrix, HTMs for each of the x, y, z and rotational error terms, and the expected error at the tool tip highlighted in red

So the expected accuracy of the slide is the sum of the geometric and load-induced errors, and its repeatability is the load-induced deflection alone.

Summary table: load-induced errors of 0.92, 6.66 and 4.67 microns in x, y and z; geometric expected errors of 30.56, 21.27 and 67.72 microns; total slide accuracy of 31.48, 27.92 and 72.40 microns; and repeatability equal to the load-induced figures

Geometric error dominates, and will eat nearly the whole error budget. Going forward I’ll add reference features to guarantee the two linear systems end up perpendicular.

3. Design and manufacture

Several features are worth calling out. The Tivar squares double as bearing pad and gib — by sizing the set screw almost as wide as the pad, the pad preloads against the rail without uneven pressure. Three spheres act as kinematic coupling components to locate the spindle and tool; here the spheres and grooves only locate, with bolts providing the attachment stiffness. And two keeper plates bound the pads, which sit in slots milled into the inside faces of the carriage, so they can’t slide out during assembly.

The LMS bill of materials drawing listing the T-shaped rail slide, boxway carriage, twelve UHMW polyethylene gibs, set screws, 3D-printed keeper plates, three quarter-inch ball bearings and an 8 mm leadnut, with three component drawings below for the rail, carriage and keeper plate

I had intended to machine the carriage from a single piece of aluminium, but with the schedule pressing I couldn’t find stock large enough to start immediately. The choice was to redesign the carriage around available stock, or keep the dimensions and modify it to be made from two pieces. I took the second option so machining could start straight away.

Beginning manufacture of the first carriage: milling the T slot into an aluminium plate clamped in the mill vise, and the two plates bolted together with the material still to be removed highlighted in purple

Two identical 25 mm × 90 mm aluminium plates got four corner clearance holes on the CNC mill so they can be bolted together and machined as one piece, and I milled the “T” into one of them. Next is finishing the carriage outline and adding locating pins so the halves can be separated, machined individually and reassembled repeatably.

4. Conclusions and thinking ahead

The past two weeks have been stressful. I felt genuinely behind on the error analysis, and I’m not the fastest machinist. The linear motion module feels better now, and I’m hoping to use spring break to gather my thoughts.

I’m torn about how to proceed next week: either manufacture two complete slide assemblies and test them roughly, or thoroughly build and test one and take the time to reflect and improve the design before making the second. If my classmates feel the same, perhaps we can build and test a single assembly properly and make the second over spring break.

Supplemental material

LMS_Analysis_RMH.xlsx — cutting forces, error apportionment, carriage geometry and T-slide analysis — 142.1 MB, too large to host here