A planar four-bar linkage joins four rigid links with four pin joints, with one link fixed as the ground. Its geometry constrains the motion of the other three links. Depending on lengths and which link is fixed, the input and output may rotate or rock. The calculator compares Grashof’s length condition and sampled positions; it does not predict joint forces or a safe operating speed.
Four-bar Linkage Interactive Calculator
Compare the four lengths, Grashof margin and sampled angles. Results describe an ideal geometric sampling rule, with branch and angle-span limitations explained below.
Equation Used
The margin compares the two intermediate lengths with the shortest plus longest. The selected grounded link still matters. The solver samples 721 input angles, retaining positions that close and selecting the higher circle intersection. Its angle span and illustrative index are not force or binding predictions.
- Planar rigid links with ideal pin joints.
- Ground pivots are horizontal and separated by the ground length.
- At each angle the higher circle intersection is selected. This rule does not guarantee a continuous assembly branch for every geometry.
- Included angles are sampled where closure is possible. The minimum is not the acute transmission-angle minimum when values exceed 90°. No closure produces zero angle outputs, not a physical zero-angle motion.
Closing a four-link loop
The fixed pivots are separated by ground length g. An input link r locates its moving pin B. The coupler c and output link k then place their shared pin C at an intersection of two circles: one centered at B with radius c, the other at the fixed output pivot with radius k.
There may be two intersections, one tangent intersection or no intersection. Those cases correspond to possible assemblies, a limiting position or an input position the lengths cannot reach.
For a strict Grashof chain with a unique shortest link, grounding the shortest gives a double crank. Grounding a link adjacent to the shortest gives a crank-rocker; grounding the opposite link gives a double rocker. The length inequality alone does not certify full rotation of whichever control is named “Crank.” Equality requires care at change points.
Constrained motion from linked pivots
Four-bar geometry is useful in pivoted mechanisms such as hinges, motion-transfer linkages and suspension arrangements. A coupler point can follow a curved path or an approximately straight segment, depending on the lengths and the point’s location.
A complete machine may combine several loops and additional constraints. The simple four-link model should not be taken as the full mechanism of every walking machine, aircraft flap or suspension that uses related ideas.
Length condition and sampled angles
Sort the four lengths into shortest s, longest l and intermediate p and q. Grashof margin is M = p + q − s − l. Positive M satisfies the strict length inequality; zero is a change-point boundary. Grounded-link assignment and physical assembly still determine the motion.
For each input angle, the solver tests circle closure. Its included angle μ at C lies from 0° to 180°. The acute angle often used to assess transmission would be min(μ, 180° − μ); the displayed minimum is instead the smallest included μ directly sampled.
Sampled output span is the largest minus smallest atan2 output angle. It is not angle-unwrapped, and selection of the higher intersection can switch assembly branches. Treat it as a sample statistic, not a guaranteed continuous rocker excursion for every geometry.
The Illustrative index combines missing samples, angle thresholds and absolute millimetre margins, then clamps the result to 0–100. Its percent sign is only a display scale: it is not a probability, safety rating or validated binding criterion. Uniform scaling can change the index even when link ratios remain identical. A zero result does not certify the mechanism.
The model omits friction, clearance, interference, elasticity, loads and inertia. A geometric closure or Grashof margin does not establish force capacity, lubrication requirements or allowable speed.
Example: the default length set
Enter ground 120 mm, input 30 mm, coupler 100 mm and output 90 mm. The shortest and longest sum to 150 mm; the intermediate lengths sum to 190 mm. Grashof margin is therefore 40 mm.
The input is the unique shortest link and is adjacent to the fixed ground. For the sampled assembly, the calculator reports minimum included angle about 56.3° and sampled output span about 39.5°. The illustrative index is 0 on its scale; that is not a tested binding risk.
For ground 300 mm, input 10 mm, coupler 30 mm and output 30 mm, no input position closes. The angle readouts fall back to zero. Those zeros mean unavailable geometric results, not a working mechanism with no swing.
Geometry is the first check
A four-bar can provide a compact constrained path with revolute joints. Its path is fixed by geometry; changing the desired path generally requires changing link lengths, pivot locations or the coupler point.
A slider-crank constrains a slider along a guide, while a cam provides motion through its profile and follower arrangement. Choosing among them also requires loads, contact forces, clearances, lubrication and manufacturing requirements.
There is no universal transmission-angle cutoff that proves binding, nor a universal RPM or service-life limit. Investigate the actual branch and operating range before using angle results in a loaded mechanism design.
Frequently asked questions
Does positive Grashof margin guarantee that the chosen input rotates fully?
No. It classifies the length set. Which link is fixed and which link is driven still determine the mechanism type.
Is Illustrative index a binding probability?
No. It is an unvalidated numerical combination of sample closure, angle thresholds and length margins. Its percent display is not a risk percentage or safety rating.
Why can scaling all lengths change the index?
The index uses absolute millimetre thresholds as well as angles. It is therefore not scale-invariant even though geometric link ratios are unchanged.
What does Min included angle measure?
It is the smallest sampled 0–180° angle between coupler and output link. It is not necessarily the smallest acute transmission angle.
Is Sampled output span always the true rocker swing?
No. The output angles are not unwrapped, and the higher-intersection rule can switch branches. A continuous physical branch must be checked separately.
What do zero angle outputs mean when the mechanism cannot close?
They are fallback values when no valid sample is found. They should not be interpreted as physical zero-angle motion.
Does Grashof tell me how much torque is needed?
No. Torque depends on geometry together with loads and operating conditions. This tool has no force or inertia model.
Does the blue coupler trace always form one physical path?
No. It joins sampled valid points and can bridge gaps or branch changes. Inspect continuity before treating it as a realizable path.
Linkage references
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