This page now uses the documented normalized Chebyshev geometry and exact rigid-link closure. The former page interchanged the side-link and coupler dimensions and converted pivot play into an unsupported wander band.
Chebyshev 4:5:2 Geometry and Line-Deviation Screen
Scale the documented normalized linkage. The animation solves each four-bar position by circle intersection and reports the midpoint tracer's geometric departure from y=4m over the declared 45-to-90-degree sweep.
Equation Used
- The linkage is planar, rigid and assembled on the displayed open branch.
- Pins are ideal revolute joints with no clearance, deformation or friction.
- The fixed pivots are separated by 4m, both side rods are 5m and the coupler is 2m.
- P is exactly the midpoint of the coupler CD.
- The reported line-deviation screen covers only the declared 45-to-90-degree input-angle sweep and target line y=4m.
- Loads, dynamics, tolerances, wear, speed, life, ratings, selection and safe operation are excluded.
Corrects the inverted Chebyshev link proportions, replaces the fabricated clearance-wander band with exact rigid-link closure and a directly sampled geometric line-deviation screen, and removes unsupported performance and field claims.
Two circles close the documented four-bar
MIT's normalized Chebyshev mechanism fixes A=(-2,0) and B=(2,0), uses equal side rods AC=BD=5 and a coupler CD=2, and places tracer P at the coupler midpoint.
The animation drives C through 45 to 90 degrees and finds D from the intersection of the radius-2 circle about C and radius-5 circle about B. That preserves every link length.
Use this page only as an ideal geometry screen
The worksheet scales the normalized linkage and evaluates its midpoint path over one declared branch and angular interval.
It does not predict a manufactured mechanism's accuracy, load capacity, speed, life or suitability.
Rigid-link constraints and midpoint tracer
The maximum shown is sampled every 0.05 degree from 45 through 90 degrees using the same exact circle-intersection closure.
A transparent scale example
At m=50 mm, the ground is 200 mm, each side link is 250 mm and the coupler is 100 mm. The sampled maximum departure from y=200 mm over the declared sweep is about 0.488 mm.
The line is approximate
A four-bar coupler point can trace a nearly straight segment without a sliding pair, but the path is not an exact line. The University of Illinois and Cornell references distinguish approximate four-bar paths from exact straight-line linkage classes.
The former swapped geometry, pivot-clearance wander rule, tolerance, speed, life, deployment, selection and troubleshooting claims are removed.
Chebyshev geometry questions
Is the tracer path exactly straight?
No.
Does the calculator include pivot play?
No.
Are the side links twice the coupler length?
No; in this normalized source geometry each side link is 5m and the coupler is 2m.
Does scaling change the dimensionless path shape?
No; it scales every coordinate and absolute deviation linearly.
Chebyshev and four-bar references
- MIT OpenCourseWare 18.900, Lecture 20: normalized fixed pivots, length-5 side rods, length-2 coupler, midpoint tracer and approximate-line distinction.
- University of Illinois Mechanics Reference, Four-Bar Linkages: four-bar closure, coupler-point paths and approximate versus exact straight-line mechanisms.
- Cornell Reuleaux Collection, Approximate Four-Bar Straight-Line Linkage: historical university-collection description of approximate four-bar straight-line motion.
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