Mechanism design · Closed loops

Four-Bar Linkage

Turn a three-dimensional four-bar mechanism. Change link lengths, flip its assembly and move a tracer on the coupler. Find which designs rotate freely and which reach a geometric limit.

Rocker angle $0$

Rocker angular velocity $0$

Rocker angular 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

1.10\,\mathrm m

Output link length

c

0.85\,\mathrm m

Fixed pivot spacing

d

1.20\,\mathrm m

Oriented assembly: minus or plus

b_{+}

1.00\,

Tracer fraction along coupler

u

0.50\,

Tracer normal offset

h

0.30\,\mathrm m

Signed output torque

\tau_o

-1.00\,\mathrm{N\,m}

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 two assembly branches are defined by the sign of the output pin relative to the directed input-pin-to-fixed-pivot line. The acute transmission angle is zero when coupler and output align. At singular closure, rates, acceleration and load transfer are undefined rather than silently capped.

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.

Rigid planar links and ideal pins. Exact position geometry, analytic velocity and acceleration, and quasistatic signed load transfer. Prescribed input speed; no inertia, friction, collisions or structural strength model.

Start with the essentials

Focus question: Which link lengths allow this input to complete a turn?
One-sentence intuition: Assembly is the intersection of two circles. Where circles separate, no rigid assembly exists; where they touch, the velocity constraint becomes singular.

Core mathematical model

Vector loop closure

a\begin{bmatrix}\cos\theta\\\sin\theta\end{bmatrix}+b\begin{bmatrix}\cos\beta\\\sin\beta\end{bmatrix}=\begin{bmatrix}d\\0\end{bmatrix}+c\begin{bmatrix}\cos\phi\\\sin\phi\end{bmatrix}

The fixed pivots lie on the horizontal axis. The two oriented circle intersections define the assembly branches.

Continuous input rotation

d+a<b+c,\qquad |d-a|>|b-c|

These strict inequalities keep this chosen input away from circle tangency for an entire turn. Equality is a change or limiting configuration and is treated separately.

Signed virtual power

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

The output torque is an applied external torque, positive counterclockwise. This relation requires a regular closure and negligible link inertia.

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 the two assemblies, then select Geometric limit.
What to observe: Only feasible positions are drawn. A warning replaces missing links instead of stretching them.
03
Inspect the limiting pose
Select At the toggle and inspect the undefined derivatives. Then export the 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