A cylindrical pair in Reuleaux's classification — specifically the revolute form — is a lower kinematic pair where two rigid bodies share a cylindrical surface contact that permits one degree of freedom: pure rotation about the common axis. It solves the problem of constraining 5 of the 6 spatial degrees of freedom while keeping rotation frictionless and repeatable. The shaft turns inside the bore, the matched surfaces share area contact, and load spreads across that contact patch. You see it everywhere — door hinges, robot elbow joints, the trunnions on a 5-axis CNC like the Haas UMC-750.
Cylindrical Pair Revolute Interactive Calculator
Vary radial load, pin diameter, and bearing length to see projected bearing pressure and bronze-bushing margin.
Equation Used
Projected bearing pressure divides the radial journal load by the rectangular projected contact area, shaft diameter times effective bearing length. Lower pressure improves lubricant-film life; higher pressure pushes the bushing toward boundary lubrication and wear.
- Radial load is carried uniformly over the projected shaft area.
- Pressure uses SI units: N divided by mm^2 gives MPa.
- SAE 660 bronze static reference limit is 14 MPa.
How the Cylindrical Pair (revolute) Actually Works
The revolute cylindrical pair works because two matched cylindrical surfaces — a shaft and a bore — share full surface contact along their length. That area contact is what makes it a lower pair in Reuleaux's classification, as opposed to higher pairs like cams or gears that touch on a line or a point. The single degree of freedom comes from the geometry itself: any motion other than rotation about the common axis would require the surfaces to interpenetrate or separate, and neither is allowed if you want the joint to function.
The fit between shaft and bore is everything. Too tight and you get galling, seized journals, or thermal lock-up when the shaft heats and grows faster than the housing. Too loose and you get radial play, off-axis loading, and the kind of wobble that destroys precision in a robot wrist or a machine tool spindle. For a typical 25 mm steel shaft running in a bronze bushing, you want a running clearance around 25-50 µm — H7/g6 in ISO fits. Drop below 15 µm and a 30°C temperature rise will lock the joint. Open it past 80 µm and a payload arm shows visible deflection at the tool point.
Failure modes are predictable. Dry running scores the bore in minutes. Misaligned bores load the shaft ends, not the middle, and you'll see two bright wear bands at the journal extremes. Contamination — grit pulled in past a worn seal — turns the journal bearing fit into a lapping operation that opens clearance until the rotational pair behaves like a sloppy slot. If you notice a rhythmic clunk once per revolution, that's almost always a flat worn into the shaft from a static load cycling through the same angle.
Key Components
- Shaft (journal): The rotating cylindrical member. Surface finish matters — Ra below 0.4 µm is the working target for hydrodynamic lubrication. Above 0.8 µm you'll chew bushings. Hardness typically 55-60 HRC for hardened steel journals running against bronze.
- Bore (housing or bushing): The stationary cylindrical surface that mates to the shaft. Bronze, oil-impregnated sintered material, or a needle-bearing race depending on load and speed. Bore concentricity to the housing reference must be within 10-20 µm for precision robotics — anything looser shows up as TIR at the tool tip.
- Axial retainer: A snap ring, shoulder, or thrust face that prevents the shaft from walking along its axis. Without it the revolute pair effectively becomes a cylindrical pair (2-DOF) and you lose positional repeatability in one direction.
- Lubricant film: Grease, oil, or a self-lubricating PTFE/bronze matrix. Film thickness 1-5 µm under load. Lose the film and metal-to-metal contact starts within seconds at normal sliding velocities above 0.1 m/s.
- Seal or wiper: Keeps contamination out and lubricant in. A failed seal is the root cause of most premature revolute joint failures in field equipment — Bobcat skid steer pivot pins are the classic example.
Who Uses the Cylindrical Pair (revolute)
The revolute pair is the most common kinematic pair in machinery — if a part rotates relative to another part about a fixed axis, you're looking at one. The choice of bushing material, fit class, and lubrication strategy is what separates a 2-million-cycle industrial joint from a hobby-grade hinge that wears out in a season.
- Industrial Robotics: Every joint on a FANUC LR Mate 200iD 6-axis arm — each axis is a revolute pair built around a harmonic-drive output shaft running in cross-roller bearings.
- Construction Equipment: The boom and bucket pivots on a Caterpillar 320 excavator — hardened steel pins in steel-backed bronze bushings, greased every 10 operating hours.
- CNC Machine Tools: The trunnion axis on a Haas UMC-750 5-axis mill — a large-diameter revolute pair carrying the tilting rotary table.
- Aerospace: Control surface hinge points on a Boeing 737 flap track — self-aligning spherical bearings that approximate revolute behavior under wing flex.
- Consumer Products: The hinge on a MacBook Pro lid — a precision revolute pair using a friction-clutch design to hold any open angle without a detent.
- Bicycle Manufacturing: Bottom bracket and headset bearings on a Trek Madone — sealed cartridge bearings forming the revolute pair between frame and crank or fork.
The Formula Behind the Cylindrical Pair (revolute)
The single number that decides whether your revolute pair lives a long life or eats itself is the projected bearing pressure — the radial load divided by the projected contact area. At the low end of typical operating range (under 2 MPa for bronze) the joint runs cool and the lubricant film stays intact for tens of thousands of hours. At the high end (8-10 MPa for the same bronze) you're pushing into boundary lubrication and the joint runs hot. The sweet spot for most industrial revolute pairs sits between 3 and 6 MPa — high enough to use the bushing efficiently, low enough that PV (pressure × velocity) stays inside the material's safe working envelope.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| P | Projected bearing pressure | MPa (N/mm²) | psi |
| F | Radial load on the journal | N | lbf |
| D | Shaft diameter (bore inner diameter) | mm | in |
| L | Effective bearing contact length | mm | in |
Worked Example: Cylindrical Pair (revolute) in an articulated theatre rigging hoist pivot
You're sizing the main pivot pin for a 2-tonne-rated theatre rigging hoist arm — the kind used at venues like the Royal Albert Hall to fly speaker arrays. The pivot is a revolute pair: a hardened steel pin running in a bronze-lined bore, carrying the static and dynamic load of the swinging arm. Pin diameter 40 mm, bearing length 60 mm, peak radial load at full load is 18 kN. You need to know whether the projected pressure sits inside the safe working window for SAE 660 bronze (typical limit 14 MPa static, 7 MPa dynamic).
Given
- F = 18000 N
- D = 40 mm
- L = 60 mm
Solution
Step 1 — compute projected contact area at the nominal pin geometry:
Step 2 — compute projected bearing pressure at the nominal 18 kN load:
That sits right at the dynamic limit for SAE 660 bronze. Acceptable for a static-hold rigging point, marginal for a frequently-cycled arm. Now check the low end of the operating range — at 50% rated load (9 kN, typical when flying a single line array cabinet):
That's the sweet spot — comfortable middle of the bronze working envelope, lubricant film stays intact, you'd expect 20,000+ load cycles before measurable wear. At the high end, picture an overload condition where shock loading hits 1.5× rated capacity (27 kN, momentary on a sudden stop):
Past the dynamic limit. The bushing won't fail in one cycle, but repeated overloads will smear the bronze surface — you'll see the bore go oval within a few hundred shock events, and the pin will start showing a polished flat where the load concentrates.
Result
Nominal projected bearing pressure is 7. 5 MPa. In practice that means the pivot is sized correctly for static holds and occasional motion but is running hotter than ideal for a frequently-cycled application — you'd feel warmth in the pin housing after an hour of repeated load swings. Across the operating range, 3.75 MPa at half load is the comfort zone, while 11.25 MPa under shock pushes you into surface damage territory. If you measure visible play at the pivot after fewer cycles than predicted, the usual culprits are: (1) inadequate grease interval letting the lubricant film fail under cycling load, (2) bore misalignment between the two pivot lugs concentrating load on the inboard end of the bushing, or (3) a pin surface finish above 0.8 µm Ra acting as a lapping tool against the bronze.
Choosing the Cylindrical Pair (revolute): Pros and Cons
The revolute pair competes with a few alternatives when you need rotational motion. Each carries its own load capacity, precision, and cost profile. Pick on the dimensions that matter for your build.
| Property | Cylindrical Pair (Revolute) | Rolling Element Bearing | Flexure Pivot |
|---|---|---|---|
| Load capacity (radial) | High — limited by bushing pressure (3-14 MPa typical) | Very high — concentrated point/line contact rated in kN per bearing | Low — limited by flexure material yield, typically <100 N |
| Precision (radial runout) | 20-80 µm depending on fit class | 2-10 µm for ABEC 5+ | <1 µm — no sliding contact |
| Maintenance interval | Re-grease every 10-500 hours depending on duty | Sealed-for-life to 5,000+ hours | Zero — no wear surfaces |
| Lifespan (cycles) | 1-10 million depending on PV | 10-100 million at rated load | Limited by fatigue — 10⁵ to 10⁷ at rated deflection |
| Cost per joint (industrial 25 mm bore) | $2-15 (bushing + pin) | $10-80 (cartridge bearing) | $50-300 (custom flexure) |
| Speed (continuous RPM) | Up to ~3000 RPM with oil lubrication | Up to 20,000+ RPM | Not applicable — small angular travel only |
| Best application fit | High-load low-to-medium speed pivots, hinges, linkages | High-speed shafts, wheels, spindles | Precision instrumentation, optical mounts |
Frequently Asked Questions About Cylindrical Pair (revolute)
The clearance is too tight at operating temperature. Cold, the H7/g6 fit gives 25-50 µm of running clearance and the lubricant film holds. As the shaft heats faster than the housing — common when the shaft sits inside a motor or close to a heat source — it grows radially and squeezes the film out. Metal contact starts and you get stick-slip squeak.
Quick check: stop the machine, let it cool, and measure clearance with a dial indicator on the shaft. If cold clearance is under 20 µm on a 25 mm shaft, open the bore by 15-20 µm or switch to a fit class with more running clearance.
Speed and duty cycle. Below 0.5 m/s sliding velocity and intermittent duty, bronze wins on cost, shock tolerance, and tolerance for contamination. Above 1 m/s continuous, the PV product on bronze pushes you into hot-running territory and a needle bearing becomes the right choice.
The other deciding factor is shock loading. A bronze bushing absorbs impact loads that would brinell the races of a needle bearing — that's exactly why excavator pivots use bronze and not rolling elements despite the maintenance burden.
Almost always because you're measuring total joint compliance, not just bushing clearance. The dial indicator picks up housing flex, mounting bolt stretch, shaft bending, and any slop in adjacent joints. On a robot arm, lever-arm amplification at the tool tip can make 30 µm of bushing clearance read as 200 µm of tip play.
Isolate the bushing by indicating directly on the shaft at the bore exit, with the shaft loaded radially by a known force. Subtract the unloaded reading from the loaded reading and you'll get the actual clearance — usually within 10% of the catalogue number.
The formula computes the same number, but the allowable pressure changes drastically. Polymer bushings like Igus iglide G300 have static pressure limits around 80 MPa but dynamic PV limits much lower than bronze — the velocity term punishes you fast. A polymer bushing happily holds 20 MPa static and 5 MPa at 0.1 m/s, but drops to 2 MPa at 0.5 m/s.
Always check the manufacturer's PV chart, not just their pressure rating. The chart is what tells you whether your operating point sits in the green zone or the failure zone.
Edge loading. The projected pressure formula assumes uniform load distribution along the bearing length. If the two lugs holding the pin aren't perfectly coaxial — and they rarely are without line-boring as an assembly — the pin contacts only at the lug edges. Real pressure at those edges can be 3-5× the calculated average.
Fix by line-boring the lugs together after welding or by using a self-aligning spherical bushing that compensates for misalignment up to 1-2°. The wear pattern tells you which problem you have: edge bands at both ends mean lug misalignment, a single band at one end means the lugs are skewed.
If any axial load exists at all — even just the weight of the rotating member when the axis isn't perfectly vertical — add a thrust washer. A plain journal bushing has almost no axial load capacity. The end face of the bushing wasn't ground for sliding contact and it'll dig into the shaft shoulder within a few hundred cycles, smearing material and creating axial slop.
Bronze or PTFE thrust washers cost almost nothing and add 50+ N of axial capacity per mm² of washer area. Skipping them is a false economy you'll pay for at the first service interval.
References & Further Reading
- Wikipedia contributors. Kinematic pair. Wikipedia
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