Continuous rotary via two paired link cranks is a mechanism where two cranks on parallel shafts share the load through a connecting link, with their crank pins phased 90° apart so neither crank ever sits at dead centre simultaneously. George Stephenson and his contemporaries refined this layout for steam locomotives in the 1820s. The 90° offset guarantees that one crank can always pull the other through its dead spot, producing uninterrupted rotary output from reciprocating input. You see it today on every preserved steam locomotive and in many twin-piston compressors and pumps.
Continuous Rotary via Two Paired Link Cranks Interactive Calculator
Vary crank phase and rotation angle to see the individual and combined normalized torque that keeps the paired cranks moving through dead centre.
Equation Used
This calculator expresses torque as a percentage of one crank's maximum torque, F*r. With phi = 90 deg and theta = 0 deg, crank A is at dead centre with zero torque while crank B is at maximum torque, so the pair still has a nonzero driving torque.
- Both cranks have equal radius and equal available driving force.
- Torque is normalized to one crank's maximum torque F*r.
- Absolute torque envelope represents useful driving torque from double-acting or correctly timed reciprocating input.
- Side rod, bearings, and shafts are treated as rigid and lossless.
Inside the Continuous Rotary via Two Paired Link Cranks
The problem this mechanism solves is brutally simple. A single crank driven by a reciprocating force — a piston, a beam, a pedal �� has two positions per revolution where the input force produces zero torque. These are the dead centres, top and bottom, where the connecting rod points straight along the crank axis. Push as hard as you like at dead centre and the crank does not turn. Pair two cranks on separate shafts, link them with a side rod, and offset their crank pins by 90°. Now when crank A sits at top dead centre, crank B sits at quadrature — the position where input force produces maximum torque. The pair pulls itself through every dead spot the other one hits.
The phasing tolerance is tighter than people expect. On a coupled-wheel steam locomotive the crank pins must be set within roughly ±0.5° of true 90° quadrature. Get it wrong by 2° or 3° and the locomotive develops a noticeable hunt — a periodic surge and slack in the side rods that loads the crank pin bushings unevenly and chews them out in months instead of years. The side rod itself must be machined so the centre-to-centre distance between its big-end bores matches the wheel-centre spacing within about 0.05 mm at operating temperature, accounting for thermal growth between cold and hot rod.
Failure modes cluster around three things — phasing drift, bearing wear in the crank pin bushings, and side rod fatigue at the bolt holes. If you notice a clicking that comes and goes once per revolution, you are almost always looking at a worn crank pin bushing letting the rod knock through dead centre. Quadrature cranks tolerate small errors gracefully right up until they don't, and then the side rod throws.
Key Components
- Crank A (primary crank): The first crank receives reciprocating drive directly from the prime mover — piston rod, beam, or pedal. Throw radius is typically 50-300 mm in industrial use; the throw determines stroke and must match crank B within 0.1 mm or the side rod fights itself.
- Crank B (paired crank): The second crank sits on a parallel shaft offset from crank A by the wheelbase distance. Its crank pin is phased exactly 90° from crank A's pin — the quarter-turn offset that eliminates simultaneous dead centres.
- Connecting side rod: Rigid steel link with bronze-bushed big ends at each crank pin. Centre-to-centre length must match shaft spacing within 0.05 mm at operating temperature. Carries the full coupling load whenever one crank is pulling the other through dead centre.
- Crank pin bushings: Bronze or white-metal bushings inside the side rod big ends. They take a reversing radial load every revolution and are the primary wear part — service life typically 5,000-15,000 hours depending on lubrication and load.
- Crank shafts and bearings: Parallel shafts on rigid frame mountings. Shaft parallelism must hold within 0.1 mm/m or the side rod runs in a skewed plane and loads the big-end bushings axially, which they are not designed for.
Industries That Rely on the Continuous Rotary via Two Paired Link Cranks
Where you see this mechanism most often is anywhere a long shaft of rotation needs to be driven through multiple points without losing motion at dead centre. The classic case is the coupled driving wheels of a steam locomotive, but the same arrangement runs in twin-piston compressors, oilfield pump jacks paired in tandem, and industrial reciprocating machinery where one crank alone would stall. The reason it survives in modern industry is that no other layout offers the same combination of mechanical simplicity and guaranteed uninterrupted torque from a reciprocating prime mover.
- Rail heritage and preservation: Coupled driving wheels on every preserved mainline steam locomotive — the LNER A3 Flying Scotsman, UP Big Boy 4014, and the SR Merchant Navy class all run paired link cranks across 3-4 driving axles.
- Compressed air: Twin-cylinder reciprocating air compressors such as the Ingersoll Rand Type 30 series use paired cranks at 90° phasing to smooth output and eliminate dead-centre stall on hand-cranked startup variants.
- Oil and gas extraction: Tandem pump jacks on shared-prime-mover well sites use paired link cranks between two beam pumping units to balance loads and prevent simultaneous stroke peaks.
- Marine propulsion (heritage): Triple-expansion marine steam engines like those preserved on SS Shieldhall use three cranks at 120° phasing — a direct extension of the paired-crank principle to three coupled shafts.
- Industrial pumping: Duplex piston pumps from manufacturers such as Gardner-Denver use 90°-phased paired cranks to deliver near-constant flow rate from two reciprocating pistons.
- Educational and demonstration: Working-model Stuart Turner twin-cylinder steam engines built by hobbyists worldwide demonstrate the mechanism in 1:16 scale with 25 mm crank throws.
The Formula Behind the Continuous Rotary via Two Paired Link Cranks
The number that matters most for this mechanism is the instantaneous combined torque output across the rotation cycle — specifically the minimum value, because that is what determines whether the system can self-start from any angular position. A single crank's torque varies as sin(θ) and goes to zero at θ=0° and θ=180°. With two cranks at 90° quadrature, the combined torque becomes the sum of sin(θ) and sin(θ+90°) = cos(θ). At the low end of the operating range — startup at the worst-case angle — combined torque never falls below a predictable minimum. At the high end of the cycle, peak combined torque is √2 times the single-crank peak. The sweet spot is anywhere in between, and that is exactly the point — the mechanism is designed so there is no bad spot.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Tcombined | Combined output torque from both cranks at angle θ | N·m | lbf·ft |
| F | Reciprocating input force per crank (assumed equal on both) | N | lbf |
| r | Crank throw radius (must be identical on both cranks) | m | ft |
| θ | Angular position of crank A measured from top dead centre | rad or ° | rad or ° |
| Tmin | Minimum combined torque across one full revolution — the self-start floor | N·m | lbf·ft |
Worked Example: Continuous Rotary via Two Paired Link Cranks in a heritage twin-cylinder traction engine restoration
You are commissioning the paired link cranks on a restored 1908 Burrell traction engine for a UK steam rally circuit. Each cylinder produces a peak piston force of 12,000 N at full boiler pressure. Crank throw on both wheels is 0.150 m. You need to know the combined torque at three angular positions — startup at the worst-case 45° offset, mid-stroke nominal, and peak — to confirm the engine will self-start from any wheel position without barring over.
Given
- F = 12,000 N
- r = 0.150 m
- Crank phasing = 90 °
- θ range of interest = 0 to 90 °
Solution
Step 1 — at the worst-case startup angle, θ = 45°, both cranks contribute equally because sin(45°) = sin(135°) = 0.707. This is the minimum combined torque across the cycle:
This 2,546 N·m floor is what guarantees self-start. The engine will turn over from any wheel position without a crew member having to bar the wheels round to a favourable angle. On a single-crank engine the same calculation at θ=0° would give zero — the locomotive would sit dead unless someone moved it.
Step 2 — at the nominal mid-stroke position θ = 0°, crank A is at dead centre and crank B is at quadrature carrying the full load alone:
Wait — that is lower than the 45° figure. Correct, and this is the counterintuitive bit. The minimum of the combined function actually sits at θ=0° and θ=90°, not at 45°. Recompute: at θ=0°, T = F·r·(0 + 1) = 1,800 N·m. At θ=45°, T = F·r·(0.707 + 0.707) = 2,546 N·m. So 1,800 N·m is the actual self-start floor.
Step 3 — at the peak combined position θ = 45°, both cranks pull together at 0.707 of their peak. This is the high end of the operating range:
So the torque output of the pair varies between 1,800 N·m and 2,546 N·m across one revolution — a ripple of about 30%. Compare this to a single crank, which varies from 0 to 1,800 N·m — a ripple of 100% with hard stalls twice per revolution.
Result
Nominal minimum combined torque is 1,800 N·m at θ=0° and θ=90°, with a peak of 2,546 N·m at θ=45°. In practice the engine pulls away from any wheel position with the regulator cracked open — no barring required, no hesitation at start. The 30% ripple between min and max is small enough that a flywheel of modest size (which the traction engine wheels themselves provide) smooths it to imperceptible. If you measure significantly less torque at startup, three failure modes account for almost all real cases: (1) crank pin phasing drifted from true 90° because a keyway has fretted oversize on one of the eccentric sheaves, easily checked with a protractor on the crank pins; (2) side rod big-end bushings worn beyond 0.3 mm radial clearance, which lets the rod knock and lose effective coupling at dead centre; (3) one cylinder producing reduced piston force from a leaking valve, which unbalances the pair and effectively reverts the engine to single-crank behaviour at the weak side.
When to Use a Continuous Rotary via Two Paired Link Cranks and When Not To
Paired link cranks compete with two main alternatives in modern reciprocating-to-rotary conversion — a single crank with a heavy flywheel, and a Scotch yoke with no dead centres at all. The choice comes down to how much torque ripple you can tolerate, how much shaft length you need to drive, and whether self-starting from any position is a hard requirement.
| Property | Two paired link cranks (90° phased) | Single crank with flywheel | Scotch yoke mechanism |
|---|---|---|---|
| Self-start from any angular position | Yes — guaranteed by 90° phasing | No — requires barring or starter motor at dead centre | Yes — no dead centres exist |
| Torque ripple per revolution | ~30% (sinusoidal sum) | 100% (zero at dead centre, peak at quadrature) | ~5% (near-constant) |
| Maximum practical shaft speed | Up to 600 RPM in industrial use, 80 RPM typical for locomotives | Up to 3,000 RPM with balanced flywheel | Limited to ~300 RPM by yoke wear |
| Side rod / coupling part lifespan | 5,000-15,000 hours bushing life | N/A — no coupling rod | 2,000-5,000 hours yoke slot wear |
| Tolerance to phasing or alignment error | ±0.5° phasing, ±0.05 mm length match | Not applicable | ±0.1 mm slot parallelism critical |
| Capital cost (relative) | Medium — two cranks plus rod and bushings | Low — single crank plus flywheel mass | High — precision-ground yoke and slot |
| Best application fit | Long shafts driven at multiple points, locomotives, twin pumps | Single-shaft engines with ample flywheel space | Compact constant-velocity reciprocating drives |
Frequently Asked Questions About Continuous Rotary via Two Paired Link Cranks
Once you go to three cranks the geometry changes. With three equal cranks spaced at 120° around the shaft, you get six power impulses per revolution instead of four, and the combined torque ripple drops to under 10%. The 90° phasing is specific to two-crank pairs because it places the second crank at peak torque exactly when the first is at zero. Add a third crank and you would have to choose: keep two at 90° and stick the third somewhere awkward, or go to 120° equal spacing and accept that no single crank is ever at peak when another is at zero — but the average is smoother. SR Bulleid Pacifics and the Gresley A4s both use 120° three-cylinder layouts for exactly this reason.
1° error is within the acceptable ±0.5° to ±1° band for most applications and you will not feel it on a flywheel-equipped engine. Where it bites is high-RPM continuous-duty applications like twin compressors running above 400 RPM — there the small phase error creates a periodic vibration at shaft frequency that the bearings will eventually pick up. For a heritage steam engine running at 80 RPM, leave it. For an Ingersoll Rand twin running at 600 RPM, recut the keyway or fit a tapered bush.
Length of driven shaft is the deciding factor. Paired cranks shine when you need rotary motion delivered to multiple points along a long axis — coupled wheels, multi-stage pumps, anything where one rotating output has to pull other rotating outputs in lockstep. A Scotch yoke gives you smoother torque but only at one shaft. If your compressor is single-shaft and you want minimum vibration, go yoke. If it has two cranks driving two cylinders that share a common output, paired cranks are simpler and longer-lived because the bushings wear gracefully whereas yoke slots wear into noticeable backlash.
Two-per-revolution knock points to one of the crank pins being slightly oversize relative to its bushing on one side only. A worn bushing on a single big end gives one knock per rev as that end passes through the load reversal. A two-per-rev pattern usually means the side rod itself is slightly bent — it knocks once when the bend is loaded in compression and once when loaded in tension. Pull the rod and check it on a surface plate. Anything over 0.2 mm bow across the length and it needs straightening or replacement.
No — the throws must be identical to within 0.1 mm. The side rod is a rigid link, so if one crank throws further than the other, the rod is forced to flex through every revolution. It will fail in fatigue at the big-end bolt holes within hundreds of hours. If you have two different cylinder sizes producing different forces but you need them coupled, equalise the throws and adjust the cylinder bore or stroke to balance the work, or use a separate gear coupling rather than a side rod.
Crank phasing controls torque smoothness, not flow smoothness. Flow output from each piston is sinusoidal in time, and even with 90° phasing the combined flow has a residual ripple of around 15-20% at twice shaft frequency. If the surge is bigger than that, look at valve timing — a slow-acting reed valve on one cylinder delays its flow contribution and turns the symmetric 90° phasing into an effective 70° or 110° as far as flow is concerned. Check both inlet and outlet reed valves for cracked or sticking petals.
5,000-15,000 hours for bronze bushings on a properly aligned and lubricated pair, with the spread driven mostly by load reversal frequency and lubrication quality. The bushings see a full load reversal twice per revolution because each crank alternately pulls and pushes the rod. At 600 RPM that is 72,000 reversals per hour. If you start seeing radial clearance grow past 0.3 mm or hear knock developing, you are typically at 70-80% of bushing life and replacement should be planned within the next 1,000 hours of running.
References & Further Reading
- Wikipedia contributors. Coupling rod. Wikipedia
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