Classical smooth elastic non-conforming contact
Documented Hertz Circular & Parallel-Line Contact Calculator
Calculate effective modulus, signed relative curvature, contact size and maximum pressure for either circular sphere contact or the central region of parallel-cylinder line contact.
Documented Hertz result
Shared compliance and signed curvature
κ = 1/R1 + s/R2 ; R* = 1/κ
s=+1 external convex, s=0 flat, s=−1 internal concave
For the internal case, R2 is a positive magnitude but its curvature is subtracted; R2 must exceed R1 so κ remains positive. Treating a concave groove as another positive convex radius is a different geometry.
Circular point contact
p0 = 3F/(2πa²)
Ac = πa²
δ = a²/R*
The displayed δ is the total normal elastic approach for this circular Hertz model. It is not used for the parallel-cylinder result.
Parallel-cylinder line contact
b = [4wR*/(πE*)]^(1/2)
p0 = 2w/(πb)
projected footprint = 2bL
The line model represents the central region of straight parallel cylinders. The footprint is a nominal rectangle; end pressure and edge effects are not calculated.
Primary published references
The equations are classical engineering Hertz relations, not “ISO formulas.” The NIST Engineering Metrology Toolbox documentation states the smooth, elastic, homogeneous and negligible-friction assumptions and links the published Puttock–Thwaite technical paper. NIST separately implements two external spheres, a sphere in an internal spherical surface, and parallel cylinders. Its published 4.45 N, two 25.4 mm diameter steel-sphere case gives 0.519 µm compression; the displayed equations reproduce 0.5193248 µm before NIST rounding. For a 25.4 mm ball diameter inside a 50.8 mm spherical diameter with the same properties/load, NIST gives 0.327 µm; subtracting curvature reproduces that result.
Why the former page was changed
The former “ball in groove” preset added both positive radii, although an internal groove requires subtracting curvature. It also advertised approach but did not calculate it, accepted broad gear/bearing/wheel-rail applications that commonly require general elliptical contact, embedded unsourced material/geometry presets, and used partial-number parsing.
Not a strength, fatigue or bearing-life decision: independently verify elastic limits, subsurface stress state, residual stress, hardness/yield relation, material anisotropy, coatings, roughness, lubrication/traction, thermal effects, misalignment, edge loading, plasticity, repeated loading and the applicable component standard or validated numerical model.
Nikolai Shelkovenko
Nikolai Shelkovenko is a vibration analysis engineer and the founder and CEO of Vibromera. For more than 15 years he has balanced rotating equipment in the field rather than on a test bench: mulchers, industrial fans, crushers, centrifuges, shafts and spindles. That work is what the Balanset instruments grew out of — they were designed as a tool a specialist can carry to the machine and use alone, on site, not as laboratory equipment. Vibromera was founded in 2017 and has been based in Porto, Portugal, since 2023. Development, assembly and support of the Balanset line all happen here. The flagship instrument is the Balanset-1A, a portable analyser for single- and two-plane balancing and for vibration diagnostics. Nikolai is personally involved in customer support, in working through difficult balancing cases and in the development of the software. He works with customers worldwide, in any language.
