A two-degree cylindrical pair is a lower kinematic pair where a cylindrical shaft sits inside a matching cylindrical bore and is free to both rotate about and slide along the common axis — two independent degrees of freedom on one axis. Unlike a screw pair, which couples rotation and translation through a helical thread, the cylindrical pair leaves them uncoupled. This lets a single joint serve as both a bearing and a slide, which is why you find it on lathe quills, hydraulic ram glands, and indexing turret shafts where you need linear and angular motion through one feature.
Two-degree Cylindrical Pair Interactive Calculator
Vary friction, engagement, arm length, shaft diameter, and axial force to estimate cocking-jam side-load threshold and L/D guidance.
Equation Used
The cocking-moment equation estimates the side load Fjam at which a sliding-rotating cylindrical pair begins to bind. Friction mu and engagement length L increase cocking tendency, while a larger moment arm a reduces the muL/a index. Keep a greater than muL for the formula to remain physically valid.
- Inputs use SI units; L and a must use the same length unit.
- Formula is valid for a > mu * L; otherwise the joint is treated as self-locking.
- Shaft diameter is used for L/D guidance and is not part of the Fjam equation.
- Only the 25 mm shaft diameter is explicitly provided in the worked-example excerpt; remaining defaults are practical adjustable design values.
The Two-degree Cylindrical Pair in Action
The geometry is the simplest possible — a round shaft in a round hole — but the way it constrains motion is what classifies it as a Reuleaux lower pair. A revolute pair locks translation and only allows rotation. A prismatic pair locks rotation and only allows translation. The cylindrical pair allows both, simultaneously and independently, which is why it counts as 2 degrees of freedom out of the 6 a free body has in space. The contact is surface-on-surface (that's what 'lower pair' means — area contact, not line or point contact), so load capacity is high and wear distributes over the full engagement length.
What makes or breaks a cylindrical pair in practice is the diametral clearance and the surface finish. Run the bore too tight and the shaft binds the moment temperature rises or the housing distorts under load — you'll feel it as a sudden stall on the slide stroke or as torque spikes during rotation. Run it too loose and you get radial slop, which on a quill or a turret post shows up as poor concentricity and chatter. A typical sliding-rotary fit on a 50 mm shaft sits at H7/g6 — about 9 to 50 µm of clearance — with shaft surface finish at Ra 0.4 µm or better. Push the finish above Ra 0.8 µm and you'll see scoring within a few hundred cycles, especially under boundary lubrication.
Failure modes split into three buckets. Galling and pickup — usually from inadequate lubrication or from running steel-on-steel without a bronze or PTFE liner. Brinelling of the bore — from shock loads concentrating contact at one angular position when the shaft is stationary. And axial scoring from contamination — a single chip of swarf trapped in the gland will plough a groove down the entire stroke length within one cycle.
Key Components
- Cylindrical Shaft (Pin): The male element. Hardened and ground to the working diameter with surface finish at Ra 0.4 µm or finer for sliding service. Hardness typically HRC 58–62 for steel-on-bronze pairs; the shaft is the harder partner so wear concentrates on the replaceable bushing.
- Cylindrical Bore (Bushing or Housing): The female element. Often a pressed-in bronze, sintered, or PTFE-lined bushing rather than the parent housing material, so wear is contained to a serviceable part. Diametral clearance follows ISO H7/g6 or H8/f7 depending on whether sliding speed or rotational accuracy dominates.
- Lubrication Path: Grease grooves, oil holes, or impregnated bushing matrix. Without continuous film between the surfaces the pair degenerates from full hydrodynamic to boundary lubrication and friction coefficient jumps from 0.05 to 0.15+. On vertically mounted pairs you also need a wiper or scraper at the exit to keep contamination out of the gland.
- Axial Retention (Optional): Snap rings, shoulder collars, or thrust faces that limit the stroke without removing the rotational freedom. The retention must constrain only the translational extreme — not pinch the shaft and kill the spin DOF. This is where new designers often accidentally convert a C-pair into a revolute pair.
Real-World Applications of the Two-degree Cylindrical Pair
You see two-degree cylindrical pairs everywhere a single joint must carry both a slide and a spin without the cost or complexity of two separate bearings stacked together. The reason designers reach for it over a screw pair is exactly because rotation and translation must remain uncoupled — the user, the actuator, or a second mechanism controls each DOF independently. When the bore is the wrong fit, the symptom is always the same: one DOF starts dragging the other along with it, and the machine loses repeatability.
- Machine Tools: Tailstock quill on a Hardinge HLV-H toolroom lathe — the quill rotates with the work but also translates under handwheel control to advance a drill or live centre.
- Hydraulics: Piston rod gland on a Parker 2H-series hydraulic cylinder — the rod translates under pressure and is free to rotate against the seal pack so a misaligned clevis pin doesn't impose a side load.
- Indexing Machinery: Turret post on a Hauser jig borer — the head must rotate to index between tools and translate vertically for tool changes through one cylindrical journal.
- Aerospace: Landing gear oleo strut sliding member, like the inner cylinder of a Messier-Bugatti-Dowty gear leg — translates under shock absorption and rotates to allow nose-wheel steering through the same joint.
- Robotics: Z-θ end-effector shaft on a SCARA robot like the Epson G6 series — a single shaft drives both the rotational wrist and the vertical Z-stroke, mechanically a textbook cylindrical pair.
- Firearms & Defence: Bolt body in a rotating-bolt rifle action — the bolt translates to chamber a round and rotates to lock the lugs, both motions running through the same cylindrical bore in the receiver.
The Formula Behind the Two-degree Cylindrical Pair
There's no single performance equation for a cylindrical pair the way there is for a gear or a cam — what governs whether the joint actually behaves as 2 DOF without binding is the relationship between diametral clearance, length-to-diameter ratio, and applied moment. The cocking-moment formula below tells you the side load at which the shaft jams in the bore. At the low end of typical designs (L/D ≈ 1, light load) the joint slides freely with almost no concern. At the nominal sweet spot (L/D between 1.5 and 2.5) you get good concentricity and free motion. Push L/D above 3 or load the shaft with a large overhung moment and the pair self-locks — same physics that makes a drawer jam when you pull on one corner.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Fjam | Side load at which the shaft binds in the bore (jamming threshold) | N | lbf |
| μ | Coefficient of friction between shaft and bore | dimensionless | dimensionless |
| L | Engagement length of shaft inside bore | mm | in |
| a | Moment arm from bore centre to applied load | mm | in |
| W | Axial sliding force applied to shaft | N | lbf |
Worked Example: Two-degree Cylindrical Pair in a precision optics microscope focus column
You are designing the cylindrical pair that supports the rotating-and-translating focus column on a metallurgical microscope similar to a Nikon Eclipse MA200 — a 25 mm diameter hardened steel column running in a bronze bushing, carrying a 4 kg objective turret offset 80 mm from the bore axis. The column must rotate freely to swap objectives and translate vertically for coarse focus. You need to know whether the design will jam under the offset load and what engagement length keeps it running clean.
Given
- D = 25 mm
- μ (steel on bronze, oil-lubricated) = 0.10 dimensionless
- a (overhang) = 80 mm
- W (turret weight) = 39.2 N (4 kg × 9.81)
- L (nominal engagement) = 50 mm
Solution
Step 1 — at the nominal engagement length of L = 50 mm (L/D = 2.0, the sweet spot), check whether the geometry can support the offset load without self-locking. The jamming criterion requires a − μ × L > 0:
Positive, so the pair will not self-lock at this geometry. Now compute the actual side reaction needed to overcome friction during a focus stroke:
The user has to push only 2.6 N harder than gravity to slide the column — a feather touch on the focus knob. This is exactly what a microscope feels like when it works.
Step 2 — at the low end of typical engagement, L = 30 mm (L/D = 1.2, short bushing):
Even lower drag, but you pay for it in concentricity — a 1.2 L/D ratio lets the column tilt visibly under the turret weight, and the image walks across the field as you rotate the objective. Useful only for very light loads.
Step 3 — at the high end, L = 100 mm (L/D = 4.0, deep bushing):
Still safely positive, but climbing fast. Push L to 800 mm against this 80 mm overhang and the denominator goes to zero — the column self-locks and no amount of force on the knob will move it. That's the classic drawer-jam failure, and it is why deep bores with offset loads need either an anti-rotation key or a recirculating ball bushing instead of a plain cylindrical pair.
Result
The nominal design needs about 2. 6 N of additional axial force to slide the focus column under the offset turret load — well within the comfortable feel range for a focus knob. Across the operating envelope, drag scales from 1.5 N at L = 30 mm up to 5.6 N at L = 100 mm, with the sweet spot landing near L/D = 2 where you get clean rotation, repeatable concentricity, and a focus action that doesn't fight the user. If you build this and measure 8 N or more of stroke force instead of the predicted 2.6 N, the most common causes are: (1) bushing bore distortion from over-press fit during assembly — check ovality with a bore gauge, anything over 8 µm out of round will spike friction, (2) lubricant breakdown or contamination in the bronze pores leading to boundary-lubrication friction at μ ≈ 0.18 instead of 0.10, or (3) the turret bracket clamping screw torquing the bushing housing oval — back it off and re-check before blaming the fit.
Choosing the Two-degree Cylindrical Pair: Pros and Cons
The cylindrical pair competes against three close cousins depending on how many degrees of freedom you actually need and how decoupled they have to stay. Pick wrong and you either over-constrain the motion or pay for hardware you don't need.
| Property | Cylindrical Pair (2 DOF) | Revolute Pair (1 DOF rotation) | Linear Ball Bushing + Separate Bearing |
|---|---|---|---|
| Degrees of freedom | 2 (rotation + translation, decoupled) | 1 (rotation only) | 2 (rotation + translation, decoupled) |
| Typical diametral clearance | 9–50 µm (H7/g6 on 50 mm) | 0–5 µm (interference or precision running) | Zero clearance (preloaded balls) |
| Concentricity under offset load | Moderate, depends on L/D ≥ 1.5 | Excellent | Excellent |
| Side-load tolerance | Limited by jamming when a < μ × L | High (full radial bearing) | High |
| Cost per joint (50 mm class) | $10–$40 (bushing + ground shaft) | $25–$150 (sealed bearing assembly) | $80–$400 (recirculating ball bushing + bearing) |
| Lifespan under continuous duty | 10⁶–10⁷ cycles with clean lube | 10⁸+ cycles | 10⁸+ cycles |
| Best application fit | Quill spindles, oleo struts, hydraulic glands | Pivots, hinges, rotating shafts | High-speed Z-θ robotic axes, CMM probes |
| Failure mode | Galling, axial scoring, brinelling | Race fatigue, seal wear | Ball pitting, raceway brinelling |
Frequently Asked Questions About Two-degree Cylindrical Pair
This is almost always asymmetric clearance — the bore has gone slightly oval. Rotation averages over the full circumference so it feels fine, but translation has to pass through the tight diameter on every stroke and stalls there.
Common causes are over-torqued housing clamps, a press fit done with too much interference (anything beyond 0.0005 × D for a bronze bushing in steel), or thermal expansion mismatch when the assembly heats up. Pop the bushing out and check it with a bore gauge at 0°, 60°, and 120° — more than 8 µm of variation and you've found the problem.
Three triggers. First, when sliding speed exceeds about 0.5 m/s continuous — plain bushings struggle to maintain hydrodynamic film above that without forced lubrication. Second, when the L/D ratio you need for stiffness pushes you near the jamming boundary (a − μL approaching zero). Third, when both DOF need to move simultaneously at high speed, like a SCARA Z-θ axis at 1 m/s linear and 500 RPM — friction-coupled motion in a plain pair causes the rotation to drag the translation and vice versa.
Below those thresholds the plain pair is cheaper, more compact, and easier to seal.
Once diametral clearance drops below roughly 5 µm on a 25 mm shaft, surface roughness asperities start interlocking and the joint behaves intermittently as a friction-locked single body. You see this on cold mornings when a tightly fitted quill won't budge until the user warms it with a hand.
The practical floor is H7/g6 for sliding-rotary service. If you need tighter than that for accuracy, you've actually outgrown the cylindrical pair concept and want a hydrostatic or aerostatic bushing where the film itself sets the clearance.
The cylindrical pair classification of a rod gland assumes both DOF are free, but the seal pack only tolerates rotation up to a certain PV (pressure × velocity) limit. Spinning a rod under axial pressure heats the elastomer locally, hardens it, and breaks down the dynamic lip.
If your application genuinely requires rotation under pressure — a clevis that swivels under load — specify a rotary-rated rod seal like a PTFE-energised lip seal, not a standard U-cup. Standard polyurethane rod seals are designed for the rotational freedom to be used only intermittently to relieve cocking, not continuously.
Compute the tilt angle the bushing allows: θ ≈ clearance / L. For a 50 mm engagement at 25 µm clearance, θ ≈ 0.0005 rad, which at an 80 mm overhang puts the load point 40 µm off-axis. If your application can't tolerate that — optical, metrology, or fine indexing — you need L/D ≥ 2 minimum.
The rule of thumb in machine-tool design is L/D = 1.5 for rough service, 2.0 for general precision, and 2.5–3.0 for instrument-grade work. Beyond 3.0 you risk creeping toward the jamming boundary and gain nothing in stiffness.
Reuleaux's lower/higher distinction is about contact geometry, not DOF count. Lower pairs make surface contact between the two elements; higher pairs make line or point contact. The cylindrical pair has surface contact (a cylinder mating with a cylinder), so it's lower — same family as the revolute, prismatic, screw, planar, and spherical pairs.
Higher pairs like a cam-follower or a gear tooth contact only along a line, which is why they wear faster under the same load and why a cylindrical pair can carry far more force per unit footprint than any cam joint of similar size.
References & Further Reading
- Wikipedia contributors. Kinematic pair. Wikipedia
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