The Chebyshev straight-line linkage uses two grounded rocking arms and a short coupler. With the classic proportions, the coupler midpoint follows an approximately straight segment during part of its motion. The arms cross without a joint at their crossing.
Chebyshev Linkage Interactive Calculator
Adjust the four pin-to-pin lengths. The animation solves the linkage and traces the coupler midpoint while the left arm rocks from 37° to 90° and back. Path measurements update from the solved geometry.
Equation Used
- Rigid planar links and ideal pin joints.
- The fixed-pivot line is horizontal.
- The selected crossed assembly is used continuously wherever it exists.
- Working window is fixed at 37°–90°; this is not the complete allowable motion range.
- An unclosed window produces no path-size result. Proportion deviation is not a straightness tolerance.
Four-bar geometry solved from circle intersections. The displayed path uses 213 samples over a fixed 37°–90° left-rocker window. Horizontal span and vertical variation are measured over that window; they are not a guaranteed straight-line travel or tolerance.
A coupler midpoint produces the tracing motion
Two arms turn about separate fixed pivots O and D. Their moving ends A and B connect to a short coupler. P lies halfway along that coupler. The long arms can cross in projection; their crossing is not another pin joint.
The classic proportions are ground:left arm:coupler:right arm = 2:2.5:1:2.5, as illustrated by the UC Davis reference. This double-rocker arrangement moves its grounded input arm through a limited working arc.
The green curve traces the calculated midpoint positions. Vertical path variation measures its departure from a horizontal line over the selected working arc.
Exploring approximate straight-line motion
This linkage is a useful teaching example of generating a path with rotary joints. The midpoint has an approximately straight portion without a prismatic guide.
A practical design still needs an appropriate driving arrangement, joint clearance, load analysis and space for the links to pass. The planar drawing separates intersecting arms visually; it does not design their physical layers or fasteners.
Closed-link position calculation
For each input angle, the left arm sets A from its length and angle. The right moving pin B must be both one coupler length from A and one right-arm length from D. The calculator finds B at the corresponding intersection of two circles.
If the circles cannot intersect, that pose cannot assemble. Otherwise, their selected intersection defines the coupler and its midpoint. Every valid frame preserves all entered pin-to-pin lengths.
The calculator samples 213 positions from 37° to 90°. It reports the midpoint’s horizontal coordinate range and vertical coordinate range. These are sampled measurements in the displayed reference frame, not a best-fit straightness assessment. The separate proportion-deviation output compares each moving-link-to-ground ratio with the classic ratios.
The classic proportions
The defaults are 100 mm between fixed pivots, two 125 mm rocking arms and a 50 mm coupler. These give the classic 2:2.5:1:2.5 proportions. P is at the middle of the 50 mm coupler.
The selected working arc starts just beyond a limiting pose and follows the nearly horizontal part of the midpoint curve. Changing a length can increase the vertical variation or make part of that window impossible to close. The animation continues to show the geometry and identifies an unclosed pose with a dashed gap.
Scaling all four lengths by the same factor scales both path measurements by that factor. The proportion deviation and dimensionless rocker/ground ratio stay unchanged.
Approximate straightness and assembly limits
A small vertical variation over this chosen window does not imply exact straight-line motion over the whole mechanism cycle. A different angle range or different proportions will produce a different path.
The model includes neither manufacturing clearance nor elastic deflection. It also does not calculate torque, bearing load or motion through a singular limiting pose. The animation reverses within the stated working window and uses an illustrative timing law.
Calculator questions
Why does the input arm rock instead of spin?
The classic arrangement shown here is a double rocker. Its grounded input arm cannot be treated as a continuously rotating crank.
Is there a joint where the long arms cross?
No. The only joints are the four endpoint pins. The crossing arms need separate physical planes in a real model.
Does the proportion-deviation output measure straightness?
No. It compares length ratios. The vertical path variation is calculated separately from midpoint positions.
What does Not closed mean?
At least one sampled angle in the specified working window cannot satisfy all four entered lengths. The drawing shows the geometry, but a path-span result for the complete window is not reported.
References & Further Reading
- UC Davis Integration Engineering Laboratory: Chebyshev Straight Line Linkage — link proportions, coupler midpoint and double-rocker classification.
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