A Scotch Yoke is a reciprocating motion linkage that converts rotary motion into linear motion using a crank pin sliding inside a slotted member called the yoke. Compared to a conventional slider-crank with a connecting rod, it eliminates the pitman entirely, producing pure sinusoidal output instead of the asymmetric motion a connecting rod creates. Engineers reach for it when they need clean simple harmonic motion, compact packaging, or high force at the stroke ends — which is why it dominates quarter-turn valve actuators on pipelines moving thousands of cubic metres of gas per hour.
Scotch Yoke Interactive Calculator
Vary crank radius, crank angle, and speed to calculate sinusoidal yoke position, stroke, velocity, and acceleration.
Equation Used
The Scotch yoke converts crank rotation into simple harmonic linear motion. The crank radius r sets the total stroke as 2r, while the instantaneous yoke position is x = r cos(theta). With constant angular speed, velocity and acceleration follow by differentiating the same sinusoidal displacement.
- Ideal Scotch yoke with no backlash, friction, or slot clearance error.
- Output displacement is measured from the mid-stroke centerline.
- Crank speed is constant and theta is the instantaneous crank angle.
- Acceleration output uses meters per second squared.
Inside the Scotch Yoke
The Scotch Yoke, also called the Slotted Yoke in valve actuator catalogues, works by trapping a rotating crank pin inside a transverse slot machined through a sliding carriage. As the crank rotates, the pin travels in a circle but is forced to drag the yoke back and forth along a single linear axis — the slot lets the pin slide vertically while pushing the yoke horizontally. Mathematically this is a Double-slider chain (Scotch yoke) in the Reuleaux classification: two prismatic joints and one revolute, no connecting rod. The output position follows x = r × cos(θ) exactly, which is why engineers describe the motion as pure sinusoidal — no second-harmonic distortion the way a slider-crank produces.
The design lives or dies on slot fit. The crank pin must run in the slot with about 0.02 to 0.05 mm diametral clearance on a hardened pin and a bronze or needle-bearing follower. Tighter than 0.02 mm and the pin binds as the yoke flexes under load. Looser than 0.08 mm and you get audible knock at each stroke reversal — and the slot edges peen over within a few thousand cycles. The most common failure mode in field service is exactly that: a worn slot that has gone from a clean rectangle to a barrel-shaped opening, letting the yoke shift a millimetre or two off-axis on every stroke. Lubrication matters too — a dry slot will gall a stainless pin in under an hour at 60 RPM.
Why build it this way at all? Because the dwell at each end of the stroke is generous and symmetric — the velocity goes through zero smoothly, so the inertial loads on whatever the yoke drives are predictable. That matters in valve actuators where you want maximum torque exactly when the valve is closing onto its seat, and it matters in test rigs where you want clean SHM input for vibration studies.
Key Components
- Crank (driving wheel): The rotating input member that carries the crank pin at a fixed radius r from the shaft centreline. Crank radius sets stroke directly — stroke = 2r — so a 50 mm crank radius gives a 100 mm stroke. Concentricity of the pin to the shaft must hold within 0.05 mm or the yoke develops a periodic side load that wears the guide rails.
- Crank pin with follower bearing: The cylindrical pin that engages the yoke slot. On industrial valve actuators the pin runs in a hardened steel sleeve or needle bearing — sliding contact wears too fast above 30 RPM. Pin hardness should be 58-62 HRC against a bronze slot insert, or use a roller follower for duty cycles above 10,000 cycles/day.
- Yoke (slotted carriage): The sliding member with the transverse slot. Slot length must equal at least 2r plus 4-6 mm clearance on each end so the pin never bottoms. The slot faces are typically induction-hardened and ground to Ra 0.4 µm — rougher than that and the pin chatters at low speed.
- Linear guide rails: Constrain the yoke to pure linear motion. Two parallel rails or a single dovetail are both common. The rail-to-slot perpendicularity must hold within 0.1° or the pin pulls against one slot face for the whole stroke and wears it asymmetrically.
- Output rod or rack: Transmits the yoke's linear motion to the load — a piston, a valve stem, a rack-and-pinion for quarter-turn rotation. Coupling alignment to the yoke matters: any angular offset shows up as a side load on the linear guides.
Where the Scotch Yoke Is Used
The Scotch Yoke shows up wherever sinusoidal motion or symmetric stroke-end torque matters more than mid-stroke speed efficiency. It is the dominant drive in pipeline valve actuators, common in piston pumps and compressors, and a staple of mechanical demonstration equipment. The Crank with slotted yoke (no pitman) configuration is what gives it the compactness that competing slider-crank designs cannot match in tight valve-mounting envelopes.
- Oil & Gas Pipeline: Bettis G-series and Rotork GP quarter-turn pneumatic valve actuators use a symmetric Scotch Yoke driving a rack-and-pinion to deliver peak torque at the closed and open positions where a ball or butterfly valve needs the most seating force.
- Steam & Internal Combustion Engines: The Bourke engine, designed by Russell Bourke in the 1930s, uses a Scotch Yoke to drive twin opposed pistons with reduced side loads on the cylinder walls compared to a connecting-rod engine.
- Test & Measurement: Vibration shaker tables for ASTM D999 package testing use a Scotch Yoke driver to produce true sinusoidal displacement input across 1-5 Hz.
- Reciprocating Pumps: Metering pumps from suppliers like Milton Roy use a Scotch Yoke to convert motor rotation into precise sinusoidal piston travel for chemical dosing at flow rates of 1-100 litres per hour.
- Education & Demonstration: University mechanism kits — including the Hydrosym and TecQuipment TM21 mechanism boards — feature a Scotch Yoke as the canonical example of sinusoidal motion conversion.
- Steam Locomotive Power Take-Offs: Some early gas-engine air compressors used a Scotch Yoke instead of a connecting rod to keep overall length short for installation on locomotive frames.
The Formula Behind the Scotch Yoke
The position, velocity, and acceleration of the yoke as functions of crank angle tell you everything about how the mechanism feels in operation. At low crank speeds — say 10-20 RPM on a manual test rig — peak velocity and peak acceleration are both small enough that the only thing that matters is stroke length. At a nominal industrial speed of 60 RPM the acceleration at the stroke ends starts dominating the bearing load on the crank pin. Push above 200 RPM and peak acceleration scales with the square of speed, so doubling RPM quadruples the inertial load — this is where Scotch Yokes start eating crank pin bearings if you didn't size them for it. The sweet spot for most pneumatic valve actuators sits in the 30-90 RPM band where motion is fast enough to actuate within a 2-3 second stroke spec but slow enough that pin loads stay manageable.
v(θ) = −r × ω × sin(θ)
a(θ) = −r × ω2 × cos(θ)
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| x(θ) | Yoke position from centre at crank angle θ | m | in |
| r | Crank radius (half the stroke) | m | in |
| θ | Crank angle from top-dead-centre | rad | rad |
| ω | Crank angular velocity | rad/s | rad/s |
| v(θ) | Yoke linear velocity | m/s | in/s |
| a(θ) | Yoke linear acceleration | m/s<sup>2</sup> | in/s<sup>2</sup> |
Worked Example: Scotch Yoke in a pipeline ball valve actuator
You are sizing the Scotch Yoke drive inside a Bettis-style double-acting pneumatic actuator that has to stroke a 12-inch ANSI 600 ball valve from open to closed in 2.5 seconds on a natural gas trunk line near Fort St. John. The crank radius is 35 mm (giving a 70 mm yoke stroke that drives a rack-and-pinion through 90°), and the typical operating speed band runs from 18 RPM (slow stroke spec) through 36 RPM nominal up to 72 RPM (emergency shutdown). You need peak yoke acceleration at each operating point to size the crank pin bearing.
Given
- r = 0.035 m
- Nnom = 36 RPM
- Nlow = 18 RPM
- Nhigh = 72 RPM
Solution
Step 1 — convert the nominal 36 RPM to angular velocity in rad/s:
Step 2 — peak acceleration at the stroke ends (where cos(θ) = ±1) at nominal speed:
That is roughly 0.05 g — the yoke decelerates and reverses smoothly enough that you can feel it as a gentle thump on the actuator housing. Peak yoke velocity at mid-stroke is r × ω = 0.132 m/s, which gives the 2.5 second stroke time the spec calls for.
Step 3 — at the low end of the operating band, 18 RPM:
alow = 0.035 × 1.882 = 0.124 m/s2
At 18 RPM the motion is so smooth you cannot feel the stroke reversal by hand on the housing. Crank pin load is dominated by the valve seat reaction, not by inertia. This is the comfortable operating point for routine open-close cycles.
Step 4 — at the high end, 72 RPM emergency shutdown speed:
ahigh = 0.035 × 7.542 = 1.99 m/s2
That is 4× the nominal acceleration because acceleration scales with ω2. The crank pin bearing now sees a peak inertial load you have to add on top of the valve torque reaction — and this is the load case that sizes the bearing, not the nominal 36 RPM duty cycle.
Result
Nominal peak yoke acceleration is 0. 498 m/s<sup>2</sup>, with peak velocity of 0.132 m/s — the actuator strokes the valve in the 2.5 second target without any objectionable end-of-stroke shock. At 18 RPM the peak acceleration drops to 0.124 m/s<sup>2</sup> (barely noticeable), while at 72 RPM emergency speed it climbs to 1.99 m/s<sup>2</sup> — meaning the bearing must be sized for the high-speed case even though the actuator spends 99% of its life at nominal. If your measured stroke time is 20% longer than the predicted 2.5 seconds, the most common causes are: (1) air supply pressure dropping below 80 psi at the actuator port and starving the piston, (2) crank pin follower bearing seizure from contamination — the pin starts skidding instead of rolling and friction climbs sharply, or (3) yoke slot wear past 0.10 mm clearance, which lets the yoke cock in its guides and bind on the rails.
When to Use a Scotch Yoke and When Not To
The Scotch Yoke competes mostly with the conventional slider-crank and the rack-and-pinion when an engineer needs to convert rotary to linear motion. Each has a sweet spot, and getting this choice wrong is one of the more expensive mistakes in actuator design — a slider-crank in a tight valve enclosure adds 30-40% length you do not have, while a Scotch Yoke driving a constant-velocity conveyor wears its slot in months. Pick on the actual motion profile and duty cycle, not on familiarity.
| Property | Scotch Yoke | Slider-Crank with Connecting Rod | Rack-and-Pinion |
|---|---|---|---|
| Motion profile | Pure sinusoidal (SHM) | Asymmetric — second-harmonic distortion | Constant velocity proportional to crank speed |
| Typical operating speed | 10-200 RPM continuous | 100-6000 RPM continuous | Limited by pinion tooth strength, often 50-500 RPM |
| Stroke length per package size | Compact — stroke = 2r, no rod length added | Long — needs rod length ≥ 3r for low side-load | Limited by rack length |
| Side load on output rod | Zero (rod is collinear with motion) | Significant — angles up to 15-20° | Zero |
| Wear point and maintenance interval | Yoke slot — re-grease every 50k cycles, slot replace at 500k-1M cycles | Big-end bearing — service every 5000 hours | Pinion teeth — inspect every 100k cycles |
| Cost (industrial actuator scale) | Moderate — slot machining is the cost driver | Low — mass-produced for IC engines | Low to moderate |
| Best application fit | Quarter-turn valve actuators, metering pumps, vibration test rigs | Reciprocating engines, compressors, high-speed pumps | Linear conveyors, CNC axes, steering systems |
Frequently Asked Questions About Scotch Yoke
Catalogue torque curves assume the crank pin is exactly at top-dead-centre when the valve hits its seat. If the rack-and-pinion engagement was assembled with the pinion 5-10° away from TDC at the closed position, you lose torque fast — output torque scales with sin(θ) of the crank angle past TDC, so even a 10° offset costs you about 17% of peak torque.
Diagnostic: pull the actuator cover and check the timing mark on the pinion against the yoke centreline with the valve fully closed. If they are not aligned within 2°, re-shim the pinion-to-stem coupling. This is the single most common cause of underperforming Scotch Yoke actuators in the field.
Three conditions push you toward the Scotch Yoke: (1) you need pure sinusoidal output for a vibration test rig or a controlled-flow metering pump, (2) package length is constrained — the absence of a connecting rod saves typically 2-3× the stroke length in axial space, or (3) you want maximum mechanical advantage at both stroke ends, as in a quarter-turn valve actuator where seat torque is the design load.
Three conditions push you the other way: continuous operation above 500 RPM (slot wear becomes the limit), constant-velocity output requirements (the slider-crank is closer to constant velocity over the middle 60° of crank rotation), or extreme cost sensitivity in high-volume products where the slider-crank has a 50-year manufacturing tooling lead.
Peak acceleration occurs at the stroke ends where cos(θ) = ±1, and it is the load case that sizes the bearing — not the nominal mid-stroke condition. Multiply r × ω2 by the moving mass (yoke + output rod + half the load mass) to get the inertial force, then add the maximum static load (valve seat torque divided by crank radius for valve actuators).
Rule of thumb: for a needle-bearing crank pin follower running at 100 RPM or less, target a dynamic load rating of at least 3× the calculated peak load to give you an L10 life over 10 million cycles. Above 100 RPM bump that to 5×, because surface fatigue in the slot face becomes a parallel failure mode and bearing margin masks early slot wear.
Audible knock at reversal almost always traces to backlash between the crank pin and the slot, but not necessarily because the slot is worn. Check three things in this order: (1) the crank pin follower bearing — if a needle bearing has spalled even one roller you will hear a sharp tick once per revolution, (2) yoke-to-guide-rail clearance — over 0.05 mm of play in the linear guides lets the yoke shift sideways at reversal and slap, (3) output rod coupling — a worn clevis pin at the rod-to-load connection produces knock that sounds like it comes from the yoke but is actually downstream.
If none of those are the cause, then yes — the slot has worn beyond the 0.08 mm clearance limit and needs to be re-machined or the yoke replaced.
Peak velocity is r × ω, but average flow per stroke depends on the integral of velocity over the displacement portion of the cycle — and on the check valves keeping up. Two things commonly cause that 5-10% shortfall on metering pumps: check valve lag and slip-back during the suction stroke. The discharge check needs roughly 30-50 ms to fully open at typical spring rates, and during that delay some of the displaced fluid leaks back past the not-yet-seated suction check.
Diagnostic: run the pump at half speed. If flow per stroke recovers, your check valve dynamics are the problem and you need lighter springs or a different valve geometry. If it stays low, look for piston seal leakage or air entrainment in the suction line.
Technically yes, historically people have — the Bourke engine ran at over 4000 RPM — but the engineering trade is brutal. Slot face contact stress scales with ω2, so going from 100 RPM to 1500 RPM increases peak slot pressure 225×. You need a roller-follower crank pin (sliding contact will gall in minutes), induction-hardened and ground slot faces at HRC 58+, forced lubrication into the slot, and you should expect to replace the yoke every 500-2000 hours.
For 99% of high-speed reciprocating applications a slider-crank with a connecting rod is the right answer. The Scotch Yoke earns its keep below 200 RPM where its motion profile and packaging advantages outweigh the wear penalty.
References & Further Reading
- Wikipedia contributors. Scotch yoke. Wikipedia
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