Kinematic Coupling Hertz Contact Calculator
Exact Constraint Mechanical Design: Determine Hertzian contact stresses ($\sigma_{max}$), elastic deformations, and sub-micron repositioning repeatability for Maxwell and Kelvin kinematic couplings.
Coupling Configuration & Preload
Contact Material Properties
Hertz Contact & Repeatability Output
Kinematic Coupling Mechanics & Hertzian Contact Stresses
Kinematic couplings provide exact 6-DOF deterministic location without overconstraint, delivering repeatable sub-micron tool changer and optical pallet docking.
1. Normal Contact Force Decomposition
F_normal = ( F_preload / 3 ) / [ 2 · sin(α_groove / 2) ]
2. Hertzian Spherical Contact Equations
a = [ ( 3 · F_normal · R ) / ( 4 · E* ) ]^(1/3) σ_max = 3 · F_normal / ( 2 · π · a² ) ≤ σ_yield
Frequently Asked Questions
What is the principle of exact constraint in kinematic couplings?
A rigid body in 3D space possesses exactly 6 degrees of freedom (3 translations, 3 rotations). A classic Maxwell kinematic coupling provides exactly 6 independent points of contact (3 spherical balls resting in three 90° V-grooves). Because the number of constraints equals the degrees of freedom ($N=6$), there is zero overconstraint, eliminating mechanical internal stress and ensuring sub-micron repositioning repeatability.
Why must Hertzian contact stress be checked in kinematic couplings?
Because a sphere contacts a planar groove over a microscopic elliptical area ($a \approx 30\sim 80\,\mu\text{m}$), localized contact stresses easily exceed $1000\sim 2500\,\text{MPa}$. If normal pressure exceeds the material yield limit, plastic deformation (Brinelling or fretting wear) creates small indentation flats, destroying the sub-micron repeatability.
What is the difference between Maxwell and Kelvin kinematic couplings?
In a Maxwell coupling, three identical V-grooves are arranged symmetrically at 120° pointing toward the center of the coupling. In a Kelvin coupling, constraint is asymmetric: one sphere rests in a 3-point tetrahedral cup (3 DOFs), the second rests in a V-groove (2 DOFs), and the third rests on a flat plate (1 DOF).