Mechanism design · Closed loops

Four-Bar Linkage

Which link lengths let a four-bar linkage's input crank turn all the way around? Lengths that let the loop close at every crank angle. As the crank turns, its pin's distance from the output pivot swings between the pivot spacing plus the crank length and the difference of the two. That whole range must stay strictly between the difference and the sum of the other two moving links. Otherwise the crank reaches a point where those two links line up, and it can go no further.

Turn the crank of a three-dimensional four-bar mechanism. Change the link lengths, flip the linkage into its other assembly and move a tracer point along the coupler. Find out which designs turn all the way round and which run into 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 and 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}

Angles are measured counterclockwise from the horizontal axis pointing right, and a positive guide offset is upward. Blue is the input crank, mint the coupler, coral the output and gold the tracer point on the coupler. Drag the gold crank-pin handle in the separate plan view, or use the angle slider with the keyboard.

Playback turns the input at a constant angular speed. It stops at the next geometric limit and never jumps across a range where the linkage cannot be assembled. The Grashof inequality alone does not guarantee a full turn; which links are grounded and driven also matters. When the two circles coincide, the assembly is not unique, and the Lab says so.

The two assembly branches are told apart by which side of the line from the input pin to the fixed pivot the output pin lies on. The acute transmission angle is zero when the coupler and output line up. At a singular closure, rates, acceleration and load transfer are left undefined; they are not quietly capped.

The load is a constant external force or torque with a sign. It does not always resist; on part of the cycle it can supply power. In the massless, frictionless, quasistatic limit, the input torque balances the load's virtual work. Joint reactions, inertia forces, bearing safety and motor sizing are not computed.

Lengths between pin centres and tracer coordinates are in metres. Bar widths, pin diameters, bearing blocks and the separate depth layers are schematic; the real motion is planar, with pin axes along the depth. Links may overlap. The gold curve is the computed path of the tracer at sampled angles, not a physical rail. The CSV holds 721 samples, one every half degree; entries stay blank where the linkage cannot be assembled or the derivatives are singular.

The links are rigid, move in one plane and turn on ideal pins. Positions come from exact geometry, and velocity and acceleration from analytic derivatives. An output load of either sign is carried back to the input quasistatically. The input speed is prescribed. Inertia, friction, collisions and the strength of the parts are not modeled.

Start with the essentials

Focus question: Which link lengths let the input crank turn all the way round?
One-sentence intuition: The joint between coupler and output link sits where two circles cross. If the circles separate, the rigid links cannot be assembled; where they just touch, the velocity equations become singular.

Core mathematical model

Closing the vector loop

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}

Both fixed pivots lie on the horizontal axis. The two circles cross at two points, and each crossing, told apart by its orientation, gives one assembly branch.

When the input can turn a full circle

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

When both strict inequalities hold, the two circles never touch during a full turn of the chosen input. If either becomes an equality, the linkage has a change point or a limiting position, which is handled separately.

Input torque from 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 external torque applied to the output link, positive counterclockwise. The relation holds only away from closure singularities and when link inertia is negligible.

Common difficulties

A zero velocity is not a singularity

Typical misconceptionWhenever the output stops moving, the mechanism must be at a singular position.

Better mental modelWhen the output simply reverses, its velocity is zero but the solution stays regular. Only at a closure singularity do the differential equations fail to determine the motion uniquely.

The input speed is set, not computed

Typical misconceptionIf the animation runs, this motor could drive the real mechanism.

Better mental modelThe input speed is simply prescribed. Link inertia, friction, flexing of the links and collisions are left out, and the power balance is quasistatic.

Run the experiment

01
Turn the crank
Press play, then pause and drag the gold crank pin in the plan view.
What to observe: The 3D links and every plot move together because they all use the same input angle.
02
Change a length
Compare the two assemblies, then select Geometric limit.
What to observe: Only positions the links can actually reach are drawn. When the links cannot be assembled, a warning appears; the bars are never stretched to fit.
03
Look at the limiting position
Select At the toggle and look at the derivatives that become undefined. Then export the sweep.
What to observe: Missing and singular values are flagged on their own and never mixed up with genuine zeros.

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