A kinematic coupling locates two bodies using a deliberately arranged set of contacts. For spatial exact constraint, the contact geometry must restrain six independent motions while the contacts remain engaged. Six contact points alone do not prove this. The calculator below counts assumed contacts; it does not analyze their directions, stiffness or repeatability.
Kinematic Coupling Interactive Calculator
Compare an assumed contact count with a selected number of freedoms. Equality is a count comparison, not proof that a physical coupling is exactly constrained.
Equation Used
The calculator multiplies balls by entered contacts per ball, then reports the difference and ratio relative to the selected freedom count. It also reports equal angular spacing. Contact normals, locations, engagement and constraint independence are not evaluated.
FIRGELLI Automations - Interactive Mechanism Calculators.
- Each engaged frictionless point contact contributes a normal constraint; independence depends on the arrangement and is not checked.
- A free spatial rigid body has six freedoms. A smaller selected value assumes other freedoms have already been constrained elsewhere.
- Grooves are evenly spaced around the coupling center.
Geometry matters as well as count
A common ball-and-groove coupling uses three balls seated in three grooves, with two nominal contacts per ball. Exact constraint depends on where the contacts act and on their normal directions, not only the total of six.
Preload helps maintain engagement. Real contacts deform and can exhibit friction and hysteresis. Their stiffness and load capacity require contact mechanics and material data beyond this calculator.
Repeatable mounting interfaces
Kinematic couplings are studied and used for locating removable parts in instruments and precision fixtures. Their performance is specific to the geometry, surfaces, loading and environment. A contact count cannot guarantee a particular repeatability or eliminate the need for calibration.
Contact-count arithmetic
For n balls, c assumed contacts per ball and f selected freedoms, C=nc, count difference=C−f, count ratio=C/f and equal angular spacing=360°/n.
A zero difference means only that the two numbers match. A constraint analysis must establish which motions are actually restricted. The animation shows a fixed three-ball, three-groove example separating and reseating. The upper plate is transparent to reveal the balls attached beneath it. Each ball meets two faces of a radial 90-degree V-groove. Gold markers identify the six nominal contact locations when seated. Preload is omitted. The count inputs affect the numerical comparison, not the illustrated mount geometry or a predicted stiffness.
Three-ball count example
At n=3, c=2 and f=6, C=6, the count difference is zero, the ratio is 1 and equal spacing is 120°.
At n=2, c=3 and f=6 the count also equals six, but that does not establish a usable spatial mount. Contact positions and directions, along with engagement, still have to be evaluated.
What the count leaves out
The inputs specify no contact radii, modulus, preload, external load, surface finish or temperature. They therefore cannot calculate contact stress, elastic deflection, life or positioning error. The illustrated mount always has three balls and two contacts per ball. Other count settings remain arithmetic comparisons; they do not generate another seat geometry.
Frequently asked questions
Do six contacts always constrain six freedoms?
No. Their geometry and independent constraint directions must be checked, and the contacts must remain engaged.
Is the difference a safety margin?
No. It is only assumed contact count minus selected freedoms.
Do equal counts establish stability and repeatability?
No. Equal counts alone do not establish independent constraint directions, continued engagement or positioning performance.
Can this calculate preload or Hertzian stress?
No. The necessary load, material and contact-geometry inputs are absent.
Why can the body freedom input be less than six?
That represents an assumption that other freedoms are already constrained. The calculator does not model those additional constraints.
Further reading
Alexander Slocum, Kinematic couplings: A review of design principles and applications, 2010. The paper addresses coupling design and structural performance beyond contact counting.
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