A cylinder-on-cylinder lower pair is a kinematic joint where one cylindrical surface mates inside or against another cylindrical surface, allowing two independent motions — rotation about the common axis and translation along that axis. Surface-on-surface contact lets it carry radial loads of several kN with running clearances as tight as 5–25 µm on a 25 mm bore. The pair exists to combine spin and slide in one compact joint without needing two separate bearings, and you see it in hydraulic cylinder rod glands, lathe tailstock quills, and pneumatic rotary actuators across factory automation.
Cylinder-on-Cylinder Lower Pair Interactive Calculator
Vary bore and shaft diameters to see the diametral clearance, radial running gap, and margins to the tight/loose fit limits.
Equation Used
Diametral clearance is the bore diameter minus the shaft diameter. The worked fit in the article uses a 25 mm shaft in a 25.020 mm bore, giving 20 µm diametral clearance and 10 µm radial running gap.
- Bore and shaft are coaxial cylindrical surfaces.
- Diameters are entered in mm and clearance is reported in micrometres.
- For the article's 25 mm class fit, below about 8 µm risks seizure and above about 40 µm risks cocking or chatter.
Operating Principle of the Cylinder-on-cylinder Lower Pair
The cylinder-on-cylinder lower pair is what Reuleaux called a cylindrical pair — two mating cylindrical surfaces that share a single axis. Because the contact is surface-to-surface (not point or line, like a higher pair), the joint counts as a lower pair and gives you exactly 2 degrees of freedom: rotation θ about the axis and translation z along it. Every other motion — radial drift, tilt, side-to-side shimmy — is constrained by the cylinder walls themselves.
The geometry is designed this way so a single machined bore can replace what would otherwise be a rotary bearing stacked on top of a linear bushing. You get rotation and translation along axis from one feature, which saves length, parts count, and alignment headaches. The catch is that the two surfaces have to share the axis to within tight tolerances, or the joint binds. On a 25 mm shaft running in a 25.020 mm bore, you have 20 µm of diametral clearance to work with — drop below 8 µm and thermal growth alone will seize the pair on a hot day, push above 40 µm and the shaft starts to cock under side load and chatter.
If tolerances are wrong you see predictable failures. Too tight and the pair galls — aluminium-on-steel is the worst offender because the soft surface picks up and tears. Too loose and you lose positional accuracy in both axes simultaneously, which is why a sloppy lathe quill makes both bad bores AND bad facing cuts. Surface finish matters as much as fit: Ra above 0.8 µm on either surface destroys the lubricant film at low speeds, and the joint stops behaving like a surface contact bearing and starts behaving like a chattering line contact.
Key Components
- Outer cylinder (bore or housing): The fixed or guiding member with an internal cylindrical surface. Bore tolerance is typically H7 (e.g. 25 +0.021/0 mm on a 25 mm nominal), and roundness must hold within 5 µm or the shaft will bind at one rotational position and run free at another.
- Inner cylinder (shaft or quill): The moving member that rotates and slides inside the bore. Shaft tolerance is normally g6 or h6 to give a running fit. Surface finish needs Ra ≤ 0.4 µm — anything rougher chews up the lubricant and accelerates wear.
- Lubricant film: A 2–10 µm oil or grease film keeps the two surfaces from metal-to-metal contact. Without it the pair galls within seconds of rotation under load. ISO VG 32 hydraulic oil is the standard pick for unsealed industrial pairs.
- End seals or wipers: On hydraulic and pneumatic versions, U-cup or rod seals stop fluid escaping while still allowing rotation and translation along axis. Seal drag adds 5–15 N of friction force that you must size into the actuator.
- Anti-rotation feature (optional): When you want a true cylindrical pair you leave it free in both DOF. When you want a prismatic pair only, a key or pin removes the rotation. Removing translation instead — leaving only rotation — turns it into a turning pair.
Industries That Rely on the Cylinder-on-cylinder Lower Pair
The cylinder-on-cylinder lower pair shows up anywhere you need rotation and linear motion sharing one axis. The reason it dominates over splined or two-bearing alternatives is cost and length — one bored hole, one ground shaft, no stack-up. The trade you accept is that both motions share the same friction surface, so you cannot optimise for low-friction sliding without also affecting rotation behaviour, and vice versa.
- Machine tools: The tailstock quill on a Hardinge HLV-H toolroom lathe — the operator hand-rotates the wheel to translate the quill forward, and the quill itself rotates freely as the centred workpiece spins.
- Hydraulics: Rod-end glands on Parker 2H series hydraulic cylinders, where the rod must translate under load while the seal carrier accommodates small angular rod rotation without leaking.
- Pneumatic automation: Festo DSL series rotary-linear pneumatic actuators used on SMT pick-and-place stations to lift and rotate a vacuum nozzle in one compact stroke.
- Firearms: The bolt-in-receiver fit on a Remington 700 — the bolt rotates to lock the lugs and translates to chamber the round, both motions through one cylindrical pair.
- Robotics: SCARA Z-axis spindles such as those on Epson G-series robots, where the output shaft both spins and translates on a 150 mm stroke without a separate ballspline.
- Medical devices: Syringe-pump plunger guides where a stainless plunger rotates to engage anti-rollback threads while translating through the barrel.
The Formula Behind the Cylinder-on-cylinder Lower Pair
The useful design number for a cylinder-on-cylinder pair isn't a motion equation — both motions are independent — it's the diametral clearance and the resulting bearing pressure when a side load shows up. Below the low end of the typical clearance range (under about 8 µm on a 25 mm shaft) the pair seizes from thermal growth. Above the high end (over about 40 µm) the shaft cocks and pressure spikes locally. The sweet spot is the clearance and projected-area combination that keeps mean bearing pressure below the material limit while leaving room for thermal expansion and a working oil film.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| pmean | Mean projected bearing pressure between the two cylindrical surfaces | MPa (N/mm²) | psi |
| Fside | Side load applied perpendicular to the cylinder axis | N | lbf |
| D | Nominal diameter of the cylindrical pair | mm | in |
| L | Engaged contact length along the axis | mm | in |
| cd | Diametral clearance (D<sub>bore</sub> − D<sub>shaft</sub>) — sets film thickness and binding margin | µm | thou (0.001 in) |
Worked Example: Cylinder-on-cylinder Lower Pair in a tailstock quill on a toolroom lathe
You're spec'ing the cylinder-on-cylinder pair for a 50 mm diameter tailstock quill on a Hardinge-style toolroom lathe. The quill engages the bore over 120 mm of length and carries a side load from a slightly off-axis workpiece pushing 800 N against the centre. You need to confirm the bearing pressure is safe for a cast-iron-on-hardened-steel pair and check the clearance window across the operating range.
Given
- D = 50 mm
- L = 120 mm
- Fside = 800 N
- Allowable p for cast iron on steel = 7 MPa
Solution
Step 1 — at the nominal 800 N side load, compute the mean projected bearing pressure:
That's 0.133 MPa against a 7 MPa allowable for a cast-iron quill housing on a hardened steel shaft — sitting at less than 2% of the limit. The pair has plenty of headroom. The quill will run smooth and the operator won't feel any binding when winding the handwheel.
Step 2 — at the low end of the typical operating range, a light centring load of 200 N (small workpiece, well aligned):
At this pressure the oil film floats the shaft cleanly — feed force is dominated by seal drag and the operator barely feels resistance on the handwheel. This is where the pair behaves most like a true surface contact bearing.
Step 3 — at the high end, a heavy 3,000 N side load from a poorly supported long workpiece deflecting against the centre:
Still well under the 7 MPa allowable, but now you're squeezing the oil film. If the diametral clearance cd is on the tight end of the H7/g6 fit (around 9 µm on a 50 mm shaft), thermal growth from a hot afternoon in the shop can drop you to 4–5 µm of running clearance — close enough to the boundary lubrication regime that you'll feel the handwheel stiffen and hear faint stick-slip.
Result
Nominal bearing pressure is 0. 133 MPa, comfortably under the 7 MPa cast-iron-on-steel limit. Across the operating range the pair runs from 0.033 MPa (light work, oil film floating) through 0.133 MPa (typical) up to 0.500 MPa (heavy off-axis load) — the sweet spot is the middle band where the film stays intact and the handwheel feels consistent. If you measure handwheel torque higher than the predicted seal-drag value of about 4–6 Nm, the most likely causes are: (1) bore roundness out of spec — over 8 µm of out-of-round on a 50 mm bore makes the quill bind once per revolution, (2) shaft surface finish above Ra 0.4 µm chewing through the lubricant film and shifting you into mixed lubrication, or (3) chips and grit pulled past worn end wipers, scoring the bore and creating localised high-pressure spots that read as a gritty handwheel feel.
When to Use a Cylinder-on-cylinder Lower Pair and When Not To
The cylinder-on-cylinder lower pair competes mostly against splined-shaft assemblies and stacked rotary-plus-linear bearing arrangements. Each option trades parts count, friction, accuracy, and load capacity differently. Pick by what matters most for your application — compactness, precision, or independent control over each axis.
| Property | Cylinder-on-cylinder lower pair | Splined shaft + bearing | Stacked linear bushing + rotary bearing |
|---|---|---|---|
| Degrees of freedom | 2 (rotation + translation, coupled friction) | 2 (rotation + translation, partly decoupled) | 2 (fully decoupled) |
| Typical positional accuracy | ±10–25 µm on a 50 mm pair | ±15–40 µm (spline backlash adds error) | ±5–10 µm (each bearing optimised separately) |
| Load capacity (radial) | High — full surface contact, several kN per cm² | Medium — load on spline teeth | Low to medium — limited by bushing rating |
| Friction at rotation | Medium — sliding contact, oil film | Low — rolls on bearing | Low — rolls on rotary bearing |
| Cost | Low — one bore, one shaft | Medium — splined shaft costs 3–5× | High — two bearing systems plus housing |
| Length / packaging | Most compact — single feature | Medium — spline plus bearing stack | Longest — bearings stacked in series |
| Maintenance interval | Re-oil every 200–500 operating hours | Re-grease spline every 500–1000 hours | Bearing service every 2000+ hours |
| Best application fit | Lathe quills, hydraulic rod glands, pneumatic rotary-linear actuators | SCARA Z-axes needing low rotation friction with stroke | Lab automation requiring micron-level independent positioning |
Frequently Asked Questions About Cylinder-on-cylinder Lower Pair
That's almost always a roundness or straightness defect that ovality measurement missed. A bore can be in tolerance on diameter at every measurement point but still have a 5–10 µm cam lobe along its axis from boring-bar deflection. As the shaft rotates, it picks up the high spot once per revolution and binds.
Check it by sweeping a test indicator inside the bore axially, not just radially. If you see a periodic rise of more than 5 µm on a 50 mm bore, hone the bore — don't try to compensate by opening up the shaft fit, because that just makes the quill sloppy in every other rotational position.
H7/g6 gives you 9–34 µm of diametral clearance on a 50 mm shaft — enough running clearance for a continuous oil film under combined motion. H7/h6 is a location fit with 0–25 µm clearance, fine for occasional sliding but it'll squeeze the film and gall under continuous rotation.
Rule of thumb: if the joint sees both motions simultaneously for more than 10% of duty cycle, use g6. If translation and rotation are sequential (lock the rotation, then slide), h6 is acceptable and gives you better positional accuracy.
Differential thermal expansion. The shaft and the housing are usually different materials (steel shaft, aluminium body is common on Festo DSL-class units), and aluminium grows roughly twice as fast as steel per °C. Over a 30 °C rise across an 8-hour shift, a 50 mm aluminium bore grows about 35 µm while the steel shaft grows only 17 µm — net clearance increases by 18 µm.
That's enough to add measurable angular slop and translational drift. Fix is either matched-material construction, or designing the cold clearance at the tight end of the fit so the warm clearance lands in the middle of the spec.
U-cup and lip seals are designed for translation. They handle rotation, but rotation drags the seal lip in a helical pattern that pumps fluid past the lip if the surface finish or runout is wrong. Translation alone keeps the seal lip in a clean axial wipe.
If you measure rod runout above 25 µm TIR or surface finish above Ra 0.4 µm, rotation will leak even though translation seals fine. Either tighten the rod runout spec or switch to a rotary-rated seal like a PTFE energised lip seal designed for combined motion.
Add the key when angular position carries information you don't want lost — a sensor target, a keyed tool interface, an anti-roll feature on a clamp. Don't add it just to feel safer about the design, because the key itself becomes the new primary wear point and it limits the joint to one rotational orientation forever.
If the application only needs anti-rotation occasionally (quill locked during drilling, free during retract), use a clamp or detent rather than a permanent key. You preserve the cylindrical pair's flexibility and only constrain it when needed.
That's break-in roughness transfer, not wear. The two surfaces start at their as-machined finish — say Ra 0.6 µm on the shaft and Ra 0.8 µm on the bore. Under load and motion, the asperities flatten and material transfers in micro-particles, temporarily increasing effective contact area before the film stabilises.
Friction torque should plateau and then drop slightly after 100–200 hours as the surfaces polish to a steady-state finish around Ra 0.2 µm. If torque keeps climbing past 200 hours, you've moved from break-in into actual wear — pull the shaft and check for bright scoring lines, which mean grit ingress past the wipers.
Not on its own. The pair has nothing inherently self-locking — both motions are free, and a vertical load will drop the shaft as soon as you release torque or pressure. Static friction from seals and oil film holds maybe 5–15 N on a 50 mm pair, nowhere near enough for a typical Z-axis tool weight.
You need either a separate brake (pneumatic spring-loaded brake collar, electric solenoid brake), a counterweight or gas spring sized to the tool mass, or a mechanically self-locking element like a leadscrew with under-5° lead angle in series with the pair. Don't trust friction alone for hold-up duty.
References & Further Reading
- Wikipedia contributors. Kinematic pair. Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
