Closed Four-Link Chain: Planar Mobility Calculator | FIRGELLI

Closing a loop with four links

The ground link fixes two pivot locations. An input link connects to the coupler, and the coupler connects to the output link at a fourth pivot. Closing the loop constrains the relative positions of the moving links.

For an ordinary nonsingular four-bar with independent constraints, specifying one suitable joint coordinate determines the local configuration once the assembly branch is selected. Not every link necessarily rotates through a full turn. That requires the link lengths, fixed-link assignment and operating branch to be considered.

The illustration uses a fixed set of four-bar dimensions for every input combination. Adding links or joints in the controls does not add them to the picture. The picture also keeps moving when the entered mobility count is zero or negative, so its motion is not evidence that the entered counts describe a movable assembly. The white marker follows one fixed point on the coupler, and the pale curve shows that point’s path through a complete crank revolution.

Where a four-link loop is useful

Four-link loops can transfer motion between pivots, guide a coupler or produce rocking motion from a suitable rotating input. Related loops appear within larger lifting, wiping and positioning mechanisms.

A real machine can include additional links, sliders, joints and constraints. Count its actual connectivity before treating a visible group of four members as an isolated four-bar. The mobility calculator does not select bearings, actuators or structural dimensions.

What the counting equation measures

Each unconstrained moving rigid body in a plane has three coordinates. With L total links and one fixed link, the starting count is 3(L − 1). Each ordinary binary pin joint supplies two constraints; each higher pair counted here supplies one. Subtracting gives M = 3(L − 1) − 2J − H.

The Constraints output is the algebraic count 2J + H, not a verified rank of independent geometric constraints. A joint shared by three links must be counted as equivalent binary joints, not simply as one visible pin. Special geometry, redundant constraints and passive freedoms can make the simple count differ from actual mobility.

The non-ground-link count is L − 1. The loop estimate is max(0, J + H − L + 1). Without the clamp, E − L + 1 is the independent cycle count for a connected graph with E binary edges. The tool does not know which links the edges join. If there are too few edges to connect all links, its zero loop output does not establish a connected mechanism.

Mobility is different from Grashof classification. Grashof compares four link lengths; this calculator has no length inputs and makes no full-rotation or transmission-angle calculation. A positive mobility count does not prove that the geometry can be assembled.

Example: the standard four-bar count

Enter L = 4, J = 4 and H = 0. The mobility count is 3(4 − 1) − 2(4) = 1. There are three non-ground links, eight counted constraints and one independent graph loop under the connected binary-joint assumption.

Keeping four links but entering three pin joints gives M = 3 and a loop estimate of zero. A connected three-joint chain of that kind is open, despite the four-bar picture remaining on screen.

At L = 2, J = 12, H = 6, the algebra returns M = −27. That is a sign that the independent-constraint count cannot be interpreted literally as physical freedom; it is not “negative motion.” At L = 10, J = 1, H = 0, the result is 25, but one binary joint cannot connect ten links. The numerical formula alone cannot verify the assembly.

Counting before solving geometry

A mobility count is a useful first description of a mechanism's topology. A position analysis then needs lengths, angles and connectivity. A force analysis additionally needs loads and operating conditions.

Four-bars, slider-cranks and cam mechanisms impose different constraints. A slider-crank guides one member along a line, while a cam and follower depend on their contact geometry. Counts do not determine cost, speed, precision or service life for any of them.

Frequently asked questions

Does Links include the frame?

Yes. L counts all links, including the one fixed ground link.

Does M = 1 guarantee a working four-bar?

No. It is the usual count under independent-constraint assumptions. Dimensions, assembly and singular positions still need checking.

What does a negative mobility count mean?

The algebra has subtracted more constraints than the unconstrained coordinate count. It does not represent physical negative freedom; the actual constraints and geometry require examination.

Why does the picture always contain four links?

It is a fixed four-bar example. Changing the count controls updates the readouts, not the topology of the drawing.

Can I use this tool to check Grashof or rocker swing?

No. Those require link lengths and geometric analysis. This calculator only counts links and joint constraints.

How are multiple links on one pin counted?

Use equivalent binary joints. A common pin joining three links corresponds to two binary revolute pairs for this count.

Does a zero loop estimate prove the chain is open and connected?

No. The graph formula assumes connectivity. The controls do not specify which links are joined and clamp a negative loop expression to zero.

Do the colors or motion establish safety?

No. The drawing is a four-bar reference, independent of the entered counts. Loads, interference, stiffness, clearance and speed limits are outside the model.