Scotch Yoke

How a Scotch yoke works

Imagine a crank arm of length R spinning around a fixed axis. The crank carries a pin at its tip; the pin sweeps a circle of radius R. Now thread that pin through a slot in a flat plate (the yoke), oriented perpendicular to the direction the yoke is allowed to slide. The yoke can only move back and forth along one axis — call it the x-axis — so any vertical position the pin takes simply slides up and down inside the slot without affecting the yoke's x-position.

The yoke's x-position is therefore exactly the x-coordinate of the pin: x = R·cos(θ), where θ is the crank angle. If the crank rotates at constant angular velocity ω, then θ = ωt and the yoke moves as:

x(t) = R · cos(ωt)
v(t) = −R · ω · sin(ωt)
a(t) = −R · ω² · cos(ωt)

This is the textbook definition of simple harmonic motion. The Scotch yoke is the only common mechanism that produces it without approximation. A slider-crank — the same crank but with a connecting rod replacing the slot — produces motion with a second-harmonic term that grows with the crank-to-rod ratio.

Worked example: 3,000 RPM at 50 mm crank radius

Take a Scotch yoke with crank radius R = 50 mm running at 3,000 RPM:

• Angular velocity: ω = 2π × 3000 / 60 = 314 rad/s
• Stroke: 2R = 100 mm
• Peak velocity: v_max = R·ω = 0.05 × 314 = 15.7 m/s
• Peak acceleration: a_max = R·ω² = 0.05 × 314² = 4,930 m/s² ≈ 503 G
• If the yoke + load mass is 0.5 kg, peak inertial force is F_max = m·a_max = 0.5 × 4,930 = 2,465 N

That 2.5 kN force flips direction every half-revolution. At top dead center the slot's outer face is loaded; at bottom dead center, the inner face. The crank pin experiences the same magnitude of force perpendicular to the slot. This is why Scotch yokes wear their slots — the loading is high, the contact patch is small, and the relative sliding velocity at the slot face equals R·ω·sin(ωt) (peaks at center, zero at TDC and BDC).

Scotch yoke vs slider-crank vs cardioid & other rotation-to-linear converters

	Scotch yoke	Slider-crank (rod L)	Cardioid cam	Rack & pinion	Ball-screw	Whitworth quick-return
Output motion	Pure sine x = R cos ωt	Sine plus 2nd harmonic R²/L	Constant-velocity rise	Linear with rotation	Linear with rotation	Asymmetric oscillation
Side load on slot/rod	High (slot face)	Lower (rod axial)	Cam normal force	Tooth pressure	Ball thrust	Sliding bushing
Mechanical efficiency	~85% (sliding friction)	~95% (rolling/journal)	~92%	~95%	~90% (high-end ball)	~85%
Speed limit	Slot wear-bound	Rod-bearing-bound	Cam wear-bound	Tooth-stress-bound	Ball-retainer-bound	Slider-bound
Best for	Pure sine motion	Engines, compressors	Custom motion law	Servo positioning	Precision positioning	Quick-return cycles
Compactness	Very compact	Length L + 2R	Cam diameter	Long rack required	Long screw	Compact
Maintenance	Slot lubrication critical	Standard piston-ring	Cam profile inspection	Tooth lubrication	Ball cleanliness	Sliders + crank

The Scotch yoke trades durability for kinematic perfection. Its constant-amplitude sine motion makes it the natural choice for vibration test fixtures, harmonic-balance shafts, and any rig where you need a known sinusoid as a forcing input.

Real-world specifications

• Bourke engine (1932). Russell Bourke designed a two-stroke piston engine using a Scotch yoke instead of a slider-crank, claiming reduced friction and better balance. The engine reached prototype stage but never mass production — slot wear was the killing factor.
• F1 balance shafts (1990s). Some Formula One V12 piston engines used Scotch-yoke balance shafts to cancel second-harmonic vibration produced by the inline cylinder layout. The yoke's pure-sine output was easier to phase-cancel than a slider-crank's distorted output.
• Industrial vibration test stands. Eccentric-driven shaker tables use Scotch yokes to deliver pure sinusoidal acceleration to a test article. Frequency 0.5 to 50 Hz, displacement up to 50 mm peak-to-peak.
• Oilfield reciprocating pumps. Some surface "horsehead" pumps and small downhole pumps use Scotch-yoke geometry for the simple slot-pin connection, where the deep stroke and slow speed match the slot-wear envelope.
• Quarter-turn valve actuators. Scotch-yoke pneumatic and hydraulic actuators convert linear cylinder motion into 90° valve rotation, with maximum torque at the end positions where it's needed most for ball- or butterfly-valve seating.
• Hand-pumped emergency steering pumps. A handle drives a crank; the crank's pin slides in a yoke connected to the pump piston. Pure sine motion is acceptable in low-speed manual operation.

Variants

• Standard (single-yoke) Scotch yoke. One slot, one pin, one yoke. Simple, but highly asymmetric loading: the same slot face takes load on each stroke. Standard for valve actuators.
• Double-acting Scotch yoke. Two cylinders or pistons on opposite sides of the same yoke. Loads partially cancel; widely used in valve actuators where each cylinder pressurizes alternately.
• Inverted Scotch yoke. The slot rotates with the crank, and a fixed pin engages it. Less common, used in some test fixtures where the rotating part needs to be the slot rather than the pin.
• Slotted (replaceable insert) Scotch yoke. The slot is machined into a hardened, replaceable insert rather than the yoke body. Worn inserts can be swapped out without scrapping the whole assembly — standard for industrial valve actuators rated for millions of cycles.
• Roller-pin Scotch yoke. The crank pin carries a needle bearing, replacing sliding contact with rolling. Reduces slot wear and friction at the cost of complexity. Found in higher-quality vibration shakers.
• Doubled (orthogonal) Scotch yoke. Two perpendicular yokes share one crank, producing sin(ωt) and cos(ωt) outputs simultaneously — useful in mechanical synthesizers, vintage analog computers, and circular-motion test rigs.

Common failure modes

• Slot face wear. The dominant life-limiter. The pin slides over the same slot region on every stroke, with the highest contact stress at the slot's center (where sliding velocity peaks). Worn slots produce play, knocking, and motion deviation from the ideal sine.
• Pin galling. Without adequate lubrication, the steel pin can gall against the slot face — local welding and tearing of metal. Usually catastrophic; the assembly seizes or the pin fractures.
• Crank pin shear. The pin sees full lateral load every cycle. Fatigue cracks initiate at the pin's root fillet; sudden shear failure during operation can damage the surrounding mechanism.
• Yoke distortion. Repeated lateral loading bends the yoke arms, shifting slot alignment relative to the crank. The result is increasing side load and accelerating wear.
• Bushing wear (yoke guide). The yoke slides on guide bushings or rails; wear here lets the yoke tilt, putting non-uniform pressure across the slot.
• Crank-bearing fatigue. The lateral force the slot puts on the pin is reacted at the crank's main bearing. High-speed designs can fatigue this bearing race even when the pin and slot remain in spec.
• Lubricant breakdown. Sliding contact at high pressure heats the lubricant; once viscosity drops, hydrodynamic film fails and metal-to-metal contact begins. Periodic regreasing or oil-mist lubrication is essential.

Common misconceptions

• Scotch yokes are obsolete. They're rare in IC engines but common in valve actuators, test fixtures, and balance shafts — where their kinematic purity earns its keep.
• The slot must be vertical. The slot must be perpendicular to the yoke's motion axis, but the assembly can be oriented any way.
• The yoke gives the same torque profile as a slider-crank. The yoke's torque profile is exactly cosinusoidal at the crank, while a slider-crank's is distorted by rod-angle effects.
• Two yokes at 90° produce circular motion. Yes — and this is why doubled Scotch yokes were used in some early mechanical computers to multiply by sin and cos simultaneously.
• Scotch yokes are inefficient. They lose a few percent more than slider-cranks in well-lubricated tests, but the loss is sliding-friction-bound and well-managed by hardened/coated slot inserts.