Bricard’s exact straight-line mechanism combines a parallelogram, a rhomboid linkage and a crossed linkage. Seven moving links connect to a frame at three fixed pivots. The ends Q and D of one rigid output bar trace perpendicular straight lines.
This is the planar straight-line mechanism documented by Artobolevsky. It is distinct from the spatial Bricard 6R linkage. The calculator below derives link lengths and output travel from the documented proportions and shows the actual pin connections.
Bricard Exact Straight-line Interactive Calculator
Scale the documented linkage and choose a rocking sweep. Calculate its link dimensions and the perpendicular travels of Q and D.
Equation Used
- OA=AB=MN=PQ=a.
- AM=ON=NP=MQ=MD=√2a; BP=2a.
- Q, M and D are collinear; M is the midpoint of DQ.
- Continuous crossed MNPQ branch, 5°≤θ≤175°.
- Ideal planar rigid links and pins; physical layers and clearance are not modeled.
Planar exact geometry, not the spatial Bricard 6R mechanism. Dashed paths are not sliders. The selected branch avoids collinear change points.
Three fixed pivots and seven moving links
O, A and B lie on the same fixed line, with OA=AB=a. The driven crank AM and the parallel crank ON both have length √2a; MN has length a. These form the parallelogram ONMA.
The link NP has length √2a and the rocker BP has length 2a. The crossed four-link group MNPQ has MN=PQ=a and NP=MQ=√2a. M is the midpoint of the straight output bar DQ, so MD=MQ.
The crank AM rocks about A. Q moves vertically through A, while D moves horizontally along OAB. Dashed colored lines show their paths; they are not rails or sliding joints. The only pin connections are the marked circles. A crossing between bars does not add a joint.
The drawing separates link layers visually, as a physical pin-jointed model would need to do. It does not determine spacer thickness, pin clearance or collision-free construction details.
Explore exact paths and an elliptical coupler
The model is useful for studying how rigid bars can generate a straight path without a sliding guide. Adjust the pivot spacing to scale the mechanism, and adjust the crank sweep to see how horizontal and vertical travel change differently.
Although Q and D follow straight lines, the bar between them rotates. A point partway along DQ generally traces an ellipse. M, its midpoint, follows the circle centered at A because M is also the crank pin.
The animation follows a specified rocking interval above the fixed-pivot line. It avoids the fully collinear configurations at crank angles 0° and 180°, where the linkage can change assembly branch. A real mechanism needs appropriate layer spacing and joint construction; a mathematical trace alone is not a precision or load rating.
Exact geometry and travel calculation
Set A=(0,0), O=(−a,0), B=(a,0), and l=√2a. At crank angle θ, M=(u,v)=(l cosθ,l sinθ), N=(u−a,v), Q=(0,2v) and D=(2u,0).
The continuous crossed-linkage branch is P=(2v²/(3a−2u)−a, 2v(2a−u)/(3a−2u)). These coordinates satisfy BP=2a, NP=l and PQ=a while maintaining MN=a and MQ=MD=l.
The selected interval is θ=90°−β to 90°+β. D spans from +2l sinβ to −2l sinβ, giving horizontal travel 4l sinβ. Q rises from 2l cosβ to 2l and returns, giving vertical travel 2l(1−cosβ). Q’s travel is its maximum minus minimum height, not the distance traveled up and back.
For animation, θ=π/2−β cos(2πt/T), with β in radians and T=60/N seconds. Thus the crank slows smoothly at each end of its rocking motion. One complete rocking cycle makes D travel left and back, while Q makes two up-and-down excursions.
Example with 50 mm pivot spacing
With a=50 mm, AM=ON=NP=MQ=MD=70.71 mm, BP=100 mm and DQ=141.42 mm. Choosing β=60° rocks the crank from 30° to 150°.
D’s horizontal travel is 244.95 mm. Q’s height varies from 70.71 mm at either end of the crank sweep to 141.42 mm at vertical, so its vertical travel is 70.71 mm.
At six complete rocking cycles per minute, the displayed cycle lasts ten seconds. Doubling the spacing doubles every link dimension, coordinate and travel value; it does not change the cycle time.
Exact geometry is not a manufacturing accuracy forecast
The ideal coordinates put Q exactly on x=0 and D exactly on y=0. Manufacturing error, clearance, elastic deformation and assembly details can alter the physical paths. This calculator does not infer micrometre accuracy or a percentage probability of binding from an arbitrary sensitivity constant.
The documented link ratios must be kept together. Independently changing one bar would require solving a different linkage; it would no longer be this exact construction. The scale slider therefore changes all related lengths consistently.
The view is a planar kinematic reconstruction, not a fabrication drawing, force analysis or bearing specification. It does not claim use in specific commercial measuring instruments or balances without evidence.
Bricard straight-line questions
Is this a six-revolute spatial linkage?
No. This page shows the planar exact straight-line construction with seven moving links and a fixed frame. Bricard’s spatial 6R mechanisms are a different subject.
Are Q and D constrained by hidden sliders?
No. Their paths follow from the rigid-link geometry. The dashed path lines are visual references.
Why does Q return to the same height at both ends?
The selected crank interval is symmetric about vertical. Sine has the same value at 90°−β and 90°+β, so Q is lowest at both endpoints and highest midway.
Why does the output bar tilt?
Q and D move along perpendicular lines while their separation remains constant. That makes DQ rotate as it translates.
Can I estimate a build’s error from this animation?
The animation verifies nominal geometry. A tolerance analysis needs actual dimensional errors, pin clearances and assembly information; this page does not fabricate an accuracy or binding-risk score.
Why is the sweep limited to 85° each side of vertical?
It leaves the driven crank clear of the collinear change-point positions at 0° and 180°. That makes the chosen assembly branch unambiguous throughout the shown interval.
Geometry references
DMG-Lib / Technische Universität Ilmenau: Bricard exact straight-line mechanism, digitized from Artobolevsky’s Lever mechanisms. The record specifies the length ratios and the three interconnected linkage groups.
University of Crete mathematical demonstration: Bricard exact straight-line mechanism. Its labeled diagram and fixed-length construction independently confirm the connections and the perpendicular paths of Q and D. The animation here is newly drawn and calculated from those geometric constraints.
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