A sliding-and-rotating prism in cylinder is a lower kinematic pair where a non-circular prismatic shaft sits inside a matching cylindrical bore, free to translate along the axis and rotate about it — but constrained from any tilt or radial play. It's a foundational element in machine-tool spindle design and linear-rotary actuator construction. The prism's flats key against the bore so torque transmits while axial slide remains free. The result is a two-degree-of-freedom joint that does the work of a spline shaft and a linear bushing in one part.
Sliding-and-Rotating Prism in Cylinder Interactive Calculator
Vary prism size and engaged length to see the scaled clearance window, minimum engagement, and animated two-DOF sliding-rotating joint.
Equation Used
This calculator applies the article's two practical checks: a 25 mm hex prism in an H7/g6 sliding fit has about 13 to 41 um diametral clearance, and the engaged length should be at least 1.5 times the across-flats dimension.
FIRGELLI Automations - Interactive Mechanism Calculators.
- Clearance window is scaled from the article's 25 mm H7/g6 example for teaching.
- Minimum engaged length uses the article rule of at least 1.5 times the across-flats dimension.
- Final production fits should be checked against the relevant ISO tolerance table.
Operating Principle of the Sliding-and-rotating Prism in Cylinder
The pair is one of Reuleaux's lower element pairs — surfaces in continuous area contact, not point or line contact. You have a prismatic male element (think a hexagonal or square shaft, or a splined shaft) running inside a female cylinder bored and broached to match. The flats of the prism mate against corresponding flats inside the bore. That mating geometry is what kills 4 of the 6 rigid-body degrees of freedom: no X or Y translation, no pitch, no yaw. What's left is axial slide and axial rotation. Two DOF, locked together if you want, independent if the geometry allows.
Why build it this way instead of stacking a separate spline and a separate bushing? Part count, alignment error, and stack-up. A two-DOF joint in continuous area contact distributes load over the full engaged length, so contact pressure stays low and wear stays predictable. Get the clearance wrong and the whole thing falls apart fast — too tight and you get galling and stick-slip the moment any side load shows up; too loose and the prism rattles inside the bore, backlash climbs, and any torque reversal slams the flats into each other. For a typical 25 mm hex prism in a steel bore, you want a sliding fit around H7/g6 — roughly 13 to 41 µm diametral clearance. Not 5 µm. Not 100 µm.
Failure modes in the field are nearly always one of three things. Fretting on the flats from micro-oscillation under load. Brinelling at the bore ends from edge loading when the prism cantilevers out past full engagement. Or seizure from a contaminated lubricant film — these are linear-rotary bushings, and if abrasive ingress gets in, the flats score the bore and clearance opens up by 50 µm in days. Engagement length matters too: you want at least 1.5× the prism's across-flats dimension engaged at all times, otherwise the moment arm for any side load exceeds what the contact pressure can handle.
Key Components
- Prismatic Shaft (male element): The non-circular shaft — typically hex, square, splined, or D-profile — that carries torque through its flats while sliding axially. Surface hardness should sit at HRC 58-62 for steel-on-steel pairs, with flat-to-axis parallelism inside 0.02 mm over 100 mm of length.
- Matching Cylindrical Bore (female element): The host bore broached or wire-EDM cut to mirror the prism's profile. Bore-to-flat clearance lives in the H7/g6 sliding fit window — about 13 to 41 µm diametral on a 25 mm shaft. Bore straightness must hold inside 0.01 mm/100 mm or the prism binds at one end and rattles at the other.
- Edge-Relief Chamfers: Lead-in chamfers at the bore ends — typically 1 to 2 mm × 30° — that prevent edge loading when the prism enters or exits. Without them, the corner of a hex flat hits the bore mouth and creates a stress concentration that brinells the bore over a few thousand cycles.
- Lubrication Film: A grease or oil film on the flats that carries the boundary load. For sliding-rotating service, lithium-complex grease at NLGI 2 with EP additives works for moderate duty; higher-speed builds want oil mist or a polymer-coated bore (PTFE-impregnated bronze, for example) running dry.
- End Stops or Retaining Features: Mechanical stops — collars, snap rings, or keyed shoulders — that prevent the prism from sliding beyond the engaged length. Loss of minimum engagement (typically 1.5× across-flats) flips the joint from area contact to line contact and the failure clock starts.
Industries That Rely on the Sliding-and-rotating Prism in Cylinder
You see this pair anywhere a machine needs torque transmission with simultaneous axial slide. It's the geometric backbone of linear-rotary actuators, sliding-spline driveshafts, and quill spindles. The reason it persists across so many industries is simple — nothing else gives you two DOF in a single self-aligning area-contact joint with this kind of compactness.
- Automotive Driveline: The sliding yoke on a Spicer or Dana driveshaft uses a splined prism-in-cylinder pair to accommodate axle travel while transmitting engine torque to the differential.
- Machine Tools: The drilling quill on a Bridgeport Series 1 mill rides on a splined sleeve so the operator can feed the drill axially while the spindle rotates at full RPM.
- Industrial Automation: SMC and Festo linear-rotary actuators (the LZB and ERMH series) use a hardened hex or splined shaft inside a matched bushing to deliver pick-place-rotate motion in one cylinder body.
- Aerospace Actuation: Helicopter rotor pitch links and swashplate sliders use ball-spline prism-in-cylinder pairs (Tsubaki and NB ball-spline shafts) where load reversals demand zero backlash.
- Robotics and Pick-and-Place: The Z-theta axes on a Yamaha YK-X SCARA robot share a single ball-spline shaft so a servo handles vertical motion while a separate servo drives rotation through the same prism.
- Medical Devices: Surgical drill quills on Stryker and Medtronic powered handpieces use small-diameter splined prism pairs to allow burr depth adjustment while spinning at 75,000 RPM.
The Formula Behind the Sliding-and-rotating Prism in Cylinder
The number that drives the whole design is the diametral clearance between the prism flats and the bore — and how that clearance behaves as engagement length, applied torque, and side load change. At the low end of the typical clearance range you get bind and stick-slip; at the high end you get backlash and impact loading on torque reversal. The sweet spot sits in the middle of an H7/g6 sliding fit, and the formula below tells you how that clearance translates into angular backlash at the working end of the shaft.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| θbacklash | Angular backlash measured at the cantilevered end of the prism | rad (or °) | ° (degrees) |
| c | Diametral clearance between prism flats and matching bore | mm | in |
| W | Across-flats width of the prism (e.g. hex AF dimension) | mm | in |
| Lcantilever | Distance from the engaged contact zone centre to the loaded end | mm | in |
| Lengaged | Axial length of prism currently engaged inside the bore | mm | in |
Worked Example: Sliding-and-rotating Prism in Cylinder in a robotic Z-theta beverage capping head
You're designing the splined Z-theta drive shaft for a robotic beverage-cap torquing head on a Krones Modulfill bottling line. The shaft is a 25 mm across-flats hex prism running inside a hardened bushing. The cap chuck sits 80 mm below the bottom of the engaged bore zone. Engaged length varies between 40 mm at full-extension (cap pickup) and 120 mm at full-retract (idle). You want to know the angular backlash at the cap chuck across the operating range so you can verify the cap-torque feedback loop won't chase a phantom error.
Given
- c = 0.025 mm (mid-range H7/g6 fit)
- W = 25 mm
- Lcantilever = 80 mm
- Lengaged,low = 40 mm
- Lengaged,nom = 80 mm
- Lengaged,high = 120 mm
Solution
Step 1 — compute the angular tilt the clearance allows at the engagement zone alone, independent of cantilever:
Step 2 — at nominal 80 mm engagement, the cap chuck sits 80 mm beyond the engagement centre, so cantilever ratio is 80 / 80 = 1.0:
That's about 0.16 mm of arc at the cap chuck rim on a 75 mm cap — well below the cap-torque sensor's noise floor, so the feedback loop stays stable.
Step 3 — at the low end of the engagement range (40 mm engaged, fully extended for cap pickup), the cantilever ratio jumps to 80 / 40 = 2.0:
That doubles the arc swing at the chuck to roughly 0.30 mm — still tolerable, but you can feel the head wag if you push it sideways by hand. Push the engagement shorter than 40 mm and the relationship goes nonlinear because edge loading at the bore mouth opens the effective clearance further.
Step 4 — at full retract (120 mm engaged, idle position), cantilever ratio drops to 80 / 120 = 0.67:
Backlash falls to about 0.10 mm of arc at the chuck. The head feels rock-solid in this configuration, which is why cap-torque calibration cycles always run with the head retracted.
Result
Nominal angular backlash at the cap chuck is 0. 115°, or roughly 0.16 mm of arc on a 75 mm cap. The feedback loop won't see this — it sits below sensor noise. Across the operating range, backlash spans 0.077° at full retract (rock-solid) to 0.230° at full extension (just noticeable by hand), confirming that cap pickup at 40 mm engagement is the worst-case condition you design around. If you measure 0.5° or more of slop in the field, the usual culprits are: (1) prism flats fretted from micro-oscillation under cyclic torque, opening effective clearance by 30-80 µm, (2) bore mouth brinelled at the lead-in chamfer because the chamfer was machined at less than 1 mm × 30°, or (3) lubricant breakdown and abrasive ingress scoring the flats — pull the shaft and look for axial scoring lines, which confirm contamination over wear.
Choosing the Sliding-and-rotating Prism in Cylinder: Pros and Cons
The sliding-rotating prism in cylinder competes with two close cousins whenever a designer needs torque plus axial freedom. Compare on the dimensions that actually matter on a spec sheet: backlash, load capacity, speed limit, and cost.
| Property | Sliding-Rotating Prism in Cylinder (splined shaft) | Ball Spline Shaft | Separate Linear Bushing + Keyed Shaft |
|---|---|---|---|
| Typical backlash (angular) | 0.05-0.25° at working end | <0.02° (preloaded) | 0.1-0.5° (key + bushing stack-up) |
| Maximum surface speed | 1-3 m/s sliding | 3-5 m/s sliding | 0.5-2 m/s sliding |
| Static torque capacity (25 mm shaft) | ~400-600 N·m | ~250-400 N·m | ~300-500 N·m |
| Lifespan under cyclic load | 10-50 million strokes (lubricated) | 100+ million strokes (rolling contact) | 5-20 million strokes (key wears first) |
| Relative cost (25 mm bore, 300 mm stroke) | $40-120 (broached hex set) | $300-800 (NB or THK ball spline) | $60-150 (bushing + keyed shaft) |
| Tolerance to contamination | Moderate — sealed grease helps | Poor — balls jam on debris | Good — open geometry flushes |
| Best application fit | Moderate-speed actuators, driveshafts | High-precision robotics, CNC | Low-cost, low-cycle equipment |
Frequently Asked Questions About Sliding-and-rotating Prism in Cylinder
Under torque, the prism flats load asymmetrically — only two or three flats out of six carry the tangential force, so those flats press hard against the bore while the opposite flats unload. That asymmetric pressure tilts the prism inside the clearance envelope and effectively reduces axial clearance at the loaded flats to near zero. Now any imperfection in bore straightness or flat parallelism becomes a hard interference.
Diagnostic check: blue the flats with engineer's blue, run one stroke under load, and inspect. If only 2 of 6 flats show transfer, your prism geometry is fine but the bore is bowed. If the transfer is patchy along the length, you've got a bore straightness problem — typically over 0.02 mm/100 mm.
Multi-spline wins for high-cycle service. A 6-spline involute profile distributes torque across six contact zones simultaneously, while a hex only loads two or three flats at a time depending on torque direction. That means contact pressure on a spline drops by roughly half for the same torque, and fatigue life climbs proportionally.
Hex is fine for low-cycle, low-precision work — manual quills, hand tools, low-duty actuators. Square is the worst of the three for this use because it concentrates load on just two flats per torque direction. If you're running over a million cycles, spend the money on an involute spline or a ball spline.
H7/g6 is the bore-to-flat fit, but total joint backlash also includes angular clearance from any flat-to-flat parallelism error and from the chamfer geometry at the bore mouth. If your prism was hobbed or milled with 0.05 mm of flat parallelism error over 200 mm length, that adds angular play independent of the diametral fit.
Measure each flat with a height gauge against a surface plate. If any flat varies by more than 0.02 mm across the engagement length, the flat — not the fit — is your problem. Send the prism out for grinding, or accept the backlash.
Rule of thumb: minimum engaged length should be at least 1.5× the across-flats dimension, and 2× is safer for cantilevered loads. Below 1.5×, the contact zone shrinks toward the bore mouth and the prism starts pivoting on the bore edge — that's how brinelling starts, and it's irreversible.
If your stroke length forces you below this minimum at full extension, add a secondary support bushing at the cantilevered end, or extend the bore so engagement stays adequate at full stroke. Don't try to compensate with a tighter fit — tighter fit makes the edge-loading damage worse, not better.
Classic stick-slip. At low speed the lubricant film between the flats hasn't built up to hydrodynamic thickness, so the joint operates in boundary lubrication. Static friction releases in jerks as the drive force overcomes successive sticking points. Above some threshold velocity — typically 10-30 mm/s for a greased steel pair — the film transitions to mixed or hydrodynamic and the chatter stops.
Fixes in order of effort: switch to an EP grease with a lower base-oil viscosity, add a PTFE-loaded coating to the flats, or change to a polymer-lined bushing (oil-impregnated bronze or PEEK). If the actuator must hold position smoothly below 10 mm/s, a ball spline is probably the right answer instead.
Yes, but only with the right material pair. Steel-on-steel dry will gall within minutes under any meaningful load. Steel prism running in a self-lubricating polymer bushing — PEEK with 10% PTFE, or graphite-impregnated bronze (Oilite) — works dry up to about 0.5 m/s and moderate loads.
Watch the wear rate. Polymer bushings shed material as they run, and that debris has to go somewhere. In a food-contact application, design a wash-down path or a contamination shield so polymer dust doesn't migrate to product. Hardcoat-anodised aluminium bores running on PTFE-coated stainless prisms are another option used on Tetra Pak filler valves.
References & Further Reading
- Wikipedia contributors. Kinematic pair. Wikipedia
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