Mechanism design · A mechanical sine wave

Scotch Yoke

How does a slot turn a circle into a sine wave? The slot is vertical, so the pin slides freely up and down inside it, and only the pin's sideways position moves the yoke. A point going steadily round a circle has a sideways position that rises and falls as a sine wave, so the yoke moves in simple harmonic motion with a stroke twice the crank radius. Its acceleration is largest at the ends, where it stops and turns back.

Turn a crank whose pin runs in the open slot of a yoke. Follow the pin's horizontal projection, compare the phase of displacement and velocity, and see why the yoke has zero speed but maximum acceleration at the end of each stroke.

Yoke position $0$

Output velocity $0$

Output acceleration $0$

Total stroke $0$

Required input torque $0$

Signed load power $0$

Engaged slot residual $0$

Quasistatic power residual $0$

Turn & inspect

Follow the gold drive pin.

Cycle phase

psi

40.00\,{}^circ

Prescribed input speed

omega

60.00\,{}^circ/mathrm s

Crank radius

r

0.65\,mathrm m

Signed external output force

F_x

-5.00\,mathrm N

Blue is the input crank, gold the drive pin and coral the output. Motion happens in a plane with right and up positive; angles are positive counterclockwise, and the pin axes point along the depth. Drag the gold pin in the separate plan view, or use the cycle-phase slider with the keyboard.

The yoke moves exactly with the pin’s horizontal position, and its vertical slot takes up the pin’s up-and-down motion. Where the pin sits in the slot and where the yoke sits horizontally are two different quantities. The small gaps drawn on screen are schematic; backlash is not simulated.

At each end of the stroke the velocity is zero and the acceleration is largest in magnitude. So the load can need zero input torque there even though the acceleration is not zero. Inertial forces are not calculated.

The applied load keeps a fixed signed direction, and for part of each cycle it can supply power. With massless, frictionless parts, the required input torque comes from balancing virtual work. During a Geneva dwell the ideal brake carries the output torque at zero output power, and the input pin transfers no load.

Playback turns the input once and stops; replaying resets the cycle. Lengths are in metres, while widths, thicknesses, supports and the gaps on screen are schematic. Nothing here predicts impacts, wear, structural safety or dynamic drive capability. The CSV holds 721 phase samples, one every half degree, with angles and angular rates in radians.

The yoke is rigid and planar, with an ideal vertical slot and horizontal guides. The harmonic motion is exact, and a signed load passes through massless parts in static balance at each instant. Contact pressure, wear, inertia, compliance and motor sizing are not modeled.

Start with the essentials

Focus question: Why is acceleration largest where the yoke stops?
One-sentence intuition: The vertical slot lets the pin move up and down freely, so only the pin’s horizontal position reaches the yoke, and that position is a cosine of the crank angle.

Core mathematical model

Horizontal projection of the pin

x=r\cos\theta,\qquad y_{\rm pin}=r\sin\theta

The yoke slides horizontally, and the pin slides up and down inside its slot.

Harmonic motion

\dot x=-r\omega\sin\theta,\qquad\ddot x=-\omega^2x

The input angular speed is constant. Stroke is twice the crank radius.

Power balance with no inertia

\tau_{\rm in}=-F_x\frac{\mathrm dx}{\mathrm d\theta},\qquad\tau_{\rm in}\omega+F_x\dot x=0

Inertia and friction losses are left out.

Common difficulties

Smooth motion does not mean smooth forces

Typical misconceptionIf the animation looks smooth, the forces must be smooth too.

Better mental modelThe input speed is imposed, and inertia, impacts, friction and flexibility are left out. The Geneva wheel's acceleration jumps at pin entry and exit.

A constant load can help or resist

Typical misconceptionA constant external load always resists the motion.

Better mental modelA constant force or torque feeds power in when it points along the output velocity and takes power out when it points against it.

Run the experiment

01
Follow the pin
Play one turn, then scrub the cycle or drag the gold pin in the plan.
What to observe: The input pin, output and graph cursors move together.
02
Inspect a stroke end
Compare Mid-stroke and Stroke end. Read velocity and acceleration together.
What to observe: At the end of a stroke the yoke reverses, so its velocity is zero while its acceleration is not.
03
Look at the edge cases
Reverse the external force and export the cycle.
What to observe: The motion is prescribed, so reversing the force reverses the torque but leaves the path unchanged.

Theory & apparatus: Carnegie Mellon · Geneva Wheel · Purdue · Scotch Yoke