Mechanism Linkage Lab
Place fixed pivots A and D, connect crank A–B, coupler B–C and rocker C–D, then drive the crank through an angle sweep. Each closure solves the intersection of circles centered on B and D. Choose the left or right assembly by the sign of cross(D−B,C−B); the tool keeps that sign across regular poses and breaks its path at toggles or impossible angles. The rigid coupler marker P moves with B–C. A toggle has coincident branches, so actual physical continuation is not uniquely determined. Distances are consistent millimeters in a planar, perfectly rigid model; forces, collisions and real-machine safety are outside its scope.
Key features
- Exact circle–circle closure for a rigid planar four-bar, with user-specified fixed pivots and three moving lengths
- Two explicit elbow branches defined by cross-product orientation, without silent branch switching
- Analytic insertion of feasibility-boundary angles even when a coarse sweep would skip a narrow blocked interval
- Separate regular, toggle, unreachable and coincident-center indeterminate states, plus closure residual
- Movable crank preview, segmented coupler-point trajectory and color-coded feasible/infeasible angular strip
- Local sweep CSV, standalone SVG, result JSON and strict restorable project JSON
How to use
- Enter fixed A and D coordinates plus crank, coupler and rocker lengths in consistent millimeters; use the full-rotation or blocked-angle example to start.
- Select the left or right assembly branch and a driven crank angle. Optionally position a marker by its fraction along B–C and its signed perpendicular offset.
- Choose sweep start/end and angle step, then analyze. The sweep inserts exact circle-tangency candidates in addition to your grid angles.
- Scrub the angle slider and inspect A/B/C/D/P coordinates, output rocker angle, transmission angle and closure error. Unreachable and coincident-center poses do not invent a moving joint.
- Review the segmented trace and status table; save SVG/CSV/report or a project JSON, and restore a project from a local file or pasted JSON.
Use cases
- Compare two assembly branches of the same four-bar dimensions
- Locate crank angles with no geometric closure before discussing a design
- Visualize a coupler marker trajectory without interpolating through inaccessible poses
- Teach why a toggle is a singular branch decision and why a physical mechanism needs more than geometry
Frequently asked questions
What do left and right branch mean?
They are the positive and negative signs of cross(D−B,C−B), where B is the driven crank endpoint. The selected sign is held at regular positions. At a toggle both circle intersections meet and the physical continuation cannot be inferred uniquely from geometry alone.
How are impossible crank angles identified?
The distance from B to fixed D must lie between |coupler−rocker| and coupler+rocker. Outside that interval the two circles do not intersect. The sweep also inserts analytic boundary angles, so a coarse grid cannot hide a narrow inaccessible range.
Why is a pose called indeterminate?
If B and D coincide and the coupler and rocker have equal lengths, the two closure circles are identical and infinitely many C points satisfy lengths. No unique moving-joint location or path is reported there.
Is the trajectory continuous across a toggle?
Regular samples on one selected branch form a continuous segment. A toggle terminates that segment because both branches meet and next motion is ambiguous without dynamics, stops or assembly constraints. The SVG does not draw a false connecting line across the singularity.
What units and accuracy apply?
All coordinates, lengths and marker offsets are millimeters; angles are degrees. Circle intersections are floating-point calculations with a relative tolerance of 1e−9 and a displayed length-closure residual. This is numerical kinematics, not exact symbolic or certified dimensional analysis.
Can this certify a machine or predict forces?
No. The model has perfectly rigid, planar, zero-clearance joints. It does not check collisions, bearings, elastic deflection, torque, speed limits, fabrication tolerance, guards or operator safety.
Are project files uploaded?
No. The solver and exports run in this browser tab. JSON/CSV/SVG are downloaded only on request; the project is not submitted to an application API.
Privacy
Link lengths, pivot coordinates and results remain in this browser tab. Local JSON, CSV and SVG are created only after you request a download; no linkage data is submitted to an application API.
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