Mechanism design · A mechanical sine wave

Scotch Yoke

Turn a crank through a real open yoke slot. Follow its horizontal projection, inspect the phase between displacement and velocity, and see why the end of a stroke has zero speed but maximum acceleration.

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 is in the horizontal-right, vertical-up plane, with counterclockwise positive angles and pin axes along depth. Drag the gold pin in the separate plan, or use the cycle-phase slider with a keyboard.

The yoke follows the horizontal pin coordinate exactly. Its vertical slot absorbs the pin’s vertical motion. The pin’s position inside the slot and the yoke’s horizontal position are different quantities. The finite rendered clearances are schematic, not a backlash simulation.

At a stroke end, velocity is zero and acceleration has its largest magnitude. The force can therefore give zero input torque without implying zero acceleration. No inertial force is calculated.

The applied load keeps its signed direction and can supply power during part of a cycle. Required input torque balances virtual work with massless, frictionless parts. During a Geneva dwell the ideal brake carries the applied output torque at zero output power; the input pin transfers none.

Playback performs one input revolution and stops. Replay explicitly resets the cycle. Lengths use metres; widths, thicknesses, supports and display clearance are schematic. No impacts, wear, structural safety or dynamic drive capability are inferred. CSV contains 721 half-degree phase samples and uses radians for angles and angular rates.

Rigid planar yoke, ideal vertical slot and horizontal guides. Exact harmonic kinematics and massless quasistatic signed load transfer. No contact pressure, wear, inertia, compliance or motor-sizing model.

Start with the essentials

Focus question: Why is acceleration largest where the yoke stops?
One-sentence intuition: The vertical slot removes the pin’s vertical coordinate from the output motion. Its horizontal coordinate is a cosine.

Core mathematical model

Projection constraint

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

The yoke slides horizontally while the pin slides vertically inside it.

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.

Quasistatic virtual power

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

No inertia or frictional power is included.

Common difficulties

Prescribed motion

Typical misconceptionA smooth-looking animation proves smooth forces.

Better mental modelInput speed is imposed. Dynamic inertia, impacts, friction and flexibility are omitted. Geneva boundary acceleration is discontinuous.

Signed load

Typical misconceptionA constant external load must always resist motion.

Better mental modelA constant signed force or torque can supply or absorb power depending on output velocity.

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 an end the yoke changes direction; zero velocity does not mean zero acceleration.
03
Check the boundary
Reverse the external force and export the cycle.
What to observe: The motion is prescribed; reversing force reverses torque, not the trajectory.

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