Mechanism design · Rotary to reciprocating

Slider-Crank Workshop

Drive a guided piston with a crank and connecting rod. Change the rod length and guide offset, compare unequal strokes in time, and see why piston force produces no crank torque at dead center.

Slider position $0$

Slider velocity $0$

Slider acceleration $0$

Acute transmission angle $0$

Required input torque $0$

Signed load power $0$

Length-closure residual $0$

Quasistatic power residual $0$

Build & turn

Change the geometry. Follow the motion.

Input angle

\theta

40.00\,{}^\circ

Prescribed input angular speed

\dot\theta

45.00\,{}^\circ/\mathrm s

Crank length

a

0.40\,\mathrm m

Connecting link length

b

1.20\,\mathrm m

Guide vertical offset

e

0.00\,\mathrm m

Tracer fraction along coupler

u

0.50\,

Tracer normal offset

h

0.30\,\mathrm m

Signed slider force

F_x

-5.00\,\mathrm N

Positive angles turn counterclockwise from the rightward horizontal axis; positive guide offset is upward. Blue is the input crank, mint the coupler, coral the output, and gold the coupler tracer. Drag the gold crank-pin handle in the separate plan view; the angle slider supports keyboard input.

Playback prescribes constant input angular speed. It stops at the next geometric limit and never jumps across a missing assembly. A full turn is not guaranteed by the Grashof inequality alone: the grounded and driven links also matter. Coincident-circle configurations have no unique assembly and are reported explicitly.

The connecting rod uses the right-hand circle–guide intersection. Slider dead center has zero transmission velocity and zero required input torque under a finite slider force; it is distinct from a vertical-rod singularity where derivatives are undefined. An offset can make the two stroke durations unequal.

The load is a constant signed external force or torque, not always a resisting load: it can supply power on part of the cycle. Input torque balances its virtual work in the massless, frictionless quasistatic limit. This does not compute joint reactions, inertia forces, bearing safety or motor sizing.

Pin-center lengths and tracer coordinates use metres. Bar width, pin diameter, bearing blocks and separated depth layers are schematic; the physical motion is planar, with pin axes along the depth direction. Link overlap is allowed. The gold curve is a sampled calculated locus, not a physical rail. CSV includes 721 half-degree samples; missing assemblies or singular derivatives stay blank.

Right-hand slider assembly, rigid rod and ideal guide. Exact geometry and quasistatic constant signed slider force. No gas cycle, pressure, flywheel dynamics, bearing loads or combustion model.

Start with the essentials

Focus question: Why does the slider pause at an end while the crank keeps turning?
One-sentence intuition: The connecting-rod constraint makes slider displacement nonlinear in input angle. A finite slider force cannot back-drive the crank at dead center.

Core mathematical model

Right-hand slider assembly

x=a\cos\theta+\sqrt{b^2-(e-a\sin\theta)^2}

The square root must be real. A zero root is a singular vertical connecting rod, not a fabricated clipped position.

Velocity and acceleration

\dot x=x'(\theta)\dot\theta,\qquad\ddot x=x''(\theta)\dot\theta^2

Input angular speed is constant during playback. The displayed acceleration is a kinematic result, not a computed force.

Signed virtual power

\tau_{\rm in}=-F_x x'(\theta),\qquad\tau_{\rm in}\dot\theta+F_x\dot x=0

At a regular dead center the displacement derivative and required torque vanish. The sign of a constant external force does not reverse automatically.

Common difficulties

Zero is not undefined

Typical misconceptionEvery stationary output is a singular solution.

Better mental modelAt an ordinary output reversal the output velocity is zero and the solution can remain regular. At a closure singularity the differential equations cannot uniquely determine the motion.

Prescribed motion

Typical misconceptionThe animation proves this motor can drive the real mechanism.

Better mental modelInput speed is prescribed. Link inertia, friction, compliance and collision dynamics are omitted; power balance is quasistatic.

Run the experiment

01
Turn it
Play, then pause and drag the gold crank pin in the plan view.
What to observe: The 3D links and all plots share the same input angle.
02
Change a length
Compare Inline guide, Short connecting rod and Offset guide.
What to observe: A shorter rod changes the speed profile; an offset changes the angular duration of each stroke.
03
Inspect the limiting pose
Select Dead center and inspect input torque. Then export the full sweep.
What to observe: Missing and singular values remain distinct from valid zero values.

Theory & apparatus: Carnegie Mellon · Planar Linkages · Universidad de Almería · Four-bar mechanisms