Bennett’s linkage is a spatial four-bar chain with four revolute joints. Its hinge axes are skew, and motion depends on a specific relationship between alternating link lengths and twists. The calculator checks that geometry and shows the actual three-dimensional centreline construction.
Bennett's Linkage Interactive Calculator
Set both alternating link lengths and twists. Compatible dimensions animate as a closed spatial linkage. Incompatible dimensions remain visible as an open-chain preview with the unclosed joint marked in red.
Equation Used
- Opposite links have equal lengths and equal twists, with zero joint offsets.
- The animation uses the four-revolute closure equations and rigid homogeneous transforms.
- The numerical compatibility threshold is not a permitted manufacturing error.
- Bearing bodies are illustrative; collision, clearance, stiffness and force are not calculated.
Compare alternating link lengths and twists, including supplementary-angle solutions. Required twist, ratio error, twist error and geometry condition describe ideal compatibility; they do not set a manufacturing tolerance.
A spatial four-revolute mechanism
Unlike a planar four-bar, this linkage has skew hinge axes. Adjacent links turn about the same shaft at each joint, but the axes at the ends of a link have a fixed twist relative to each other. Opposite link lengths and twists repeat around the loop. Their sine-to-length ratios must match for the ideal Bennett construction.
The view shows steel hinge shafts, three moving links and a grey fixed link DA. The input drives joint A through an illustrative back-and-forth sweep. All endpoints and axes come from three-dimensional rigid transforms; perspective alone does not create the apparent spatial motion.
When the input geometry is incompatible, the final connection is deliberately left open. Its red dashed gap reveals the mismatch. That preview must not be read as a real rigid closed mechanism moving despite an error.
Deployable structures
Bennett loops are studied as building blocks for deployable structures. Taylor and You’s Oxford report develops and tests a shelter assembled from such linkages. The present calculator concerns one ideal loop, not the structural performance of a network.
Geometry and closure
For alternating lengths a, b and twists α, β, the compatibility condition is sin(α)/a = sin(β)/b, with opposite pairs equal and all joint offsets zero. The joint variables obey θA + θC = 2π, θB + θD = 2π and tan(θA/2) tan(θB/2) = sin((β+α)/2)/sin((β−α)/2).
The program composes four rigid transforms to evaluate position and orientation closure. The animated sweep is prescribed for inspection and does not predict a motor speed. Required twist is derived from sin(β) = b sin(α)/a. Both β and 180°−β solve that sine relation; the output chooses the nearer solution.
If the sine target exceeds one, displaying 90° would hide an impossible request. The calculator instead reports no real solution. Ratio error is the relative difference between the two sine-to-length ratios. The geometry condition identifies compatible, mismatched and special-case inputs.
Compatible and incompatible examples
With a = b = 100 mm, α = 60° and β = 120°, both sine-to-length ratios are equal. The centreline linkage closes throughout the displayed sweep. Changing b to 200 mm while retaining α = 60° requires sin(β) = 1.732, so no real second twist can meet the condition.
Equal a and b with α = β is a special congruent case. It has discontinuous motion branches rather than the ordinary nondegenerate motion. The preview labels the coincident-link branch instead of treating it as a general working example.
Ideal geometry versus hardware
The comparison checks geometric compatibility only. Small manufacturing errors, joint play and elastic deformation require a separate analysis. A centreline model also cannot verify whether thick links, bearings or supports collide. The red gap is a positional diagnostic; it is not a suggested clearance.
Bennett-linkage questions
Why are the hinge shafts not parallel?
The fixed twist between adjacent revolute axes is part of the spatial mechanism. The coloured links connect their common-normal locations.
Why is the required second twist sometimes greater than 90°?
An angle and its supplement have the same sine. The calculator includes both solutions and chooses the one nearest the entered angle.
Can a mismatched linkage move as shown?
Only as the displayed open chain. The broken connection marks where it fails to meet the fixed joint. A rigid closed loop cannot follow that diagnostic path.
Does compatible mean ready to manufacture?
No. It means the ideal centreline geometry satisfies the numerical check. Clearances, collisions, material strength and tolerances are outside this model.
References
Jonathan Taylor and Zhong You, Innovative Design for a Rapidly Deployable Shelter, equations 1–8 and figures 1–2, provides the spatial linkage convention and closure equations.
Jian Qi, Yuan Gao and Fufu Yang, Synthesis of clearance for a kinematic pair to prevent an overconstrained linkage from becoming stuck, discusses the distinction between ideal Bennett geometry and joint-clearance behavior.
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