Grinding Contact Length Calculator

Equivalent diameter and arc-of-contact for surface, external OD and internal ID grinding: De from the wheel-work curvatures, lc = √(ae·De) — why internal work runs softer wheels.

Machining 4 inputs 3 results

Inputs

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Configuration

Wheel diameter (d_s)

Work diameter (d_w)

Depth of cut (infeed) (a_e)

Results

Default result

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Equivalent diameter(D_e)

1.714in

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The single curvature the contact zone behaves as — bigger De = longer contact = harder-acting wheel.

Also computed

Contact length(l_c)0.0367in

Geometric arc of contact; real contact runs somewhat longer under deflection.

Contact ÷ depth(l_c/a_e)46.7

Method notes 4 notes

Curvature sum: 1/De = 1/ds ± 1/dw (+ external, − internal); a flat is the wheel alone. lc = √(ae·De) is the geometric value — elastic deflection stretches real contact somewhat longer.
Per-grit chip thickness falls as contact grows: the same wheel acts HARDER on internal work and SOFTER on small external diameters — the reason ID grinding picks softer grades than OD grinding of the same steel.
Burn risk tracks contact length at a given removal rate; when an OD recipe moves to a bore, drop the infeed or the wheel grade, not just the speed.
Kinematics only — grit, bond, dressing and coolant data come from the wheel maker; speeds live in the grinding wheel speed calculator.

Wheel behavior is mostly contact geometry: the curvatures sum to an equivalent diameter (1/De = 1/ds + 1/dw external, − internal, wheel-only on a flat) and the arc of contact is lc = √(ae·De). A 12 in wheel over a 2 in shaft acts like a Ø43 mm wheel (0.93 mm contact at 0.02 mm infeed), while a 40 mm quill in a 50 mm bore reaches De = 200 mm and twice the contact — thinner grit chips, a harder-acting wheel, and the geometric reason internal grinding runs softer grades. This calculator covers all three configurations and flags near-conformal bore work.

How to use this calculator

Pick the configuration. External OD, internal ID, or surface — the curvature signs differ.
Read De and the contact length. lc = √(ae·De) at your infeed; the ratio to depth shows how stretched the contact zone is.
Compare configurations. Moving a recipe from OD to ID multiplies contact — drop the infeed or wheel grade accordingly.
Watch conformity on bores. A wheel filling >90% of the bore is near-conformal: very long contact, loading and burn territory.

How it works

Wheel behavior is mostly contact geometry. Adding the wheel and work curvatures gives the equivalent diameter, and the contact arc follows from the infeed:

1/De = 1/ds ± 1/dw (+ OD, − ID) · lc = √(ae·De)

Longer contact divides the same infeed among more grits for longer — thinner chips, more rubbing, a harder-acting wheel. That single relation explains why bores grind differently from shafts with the identical wheel. Wheel and work speeds for the same setup live in the grinding wheel speed calculator, and the finish consequences in the surface finish calculator.

Worked example

Verified against the live calculator

A 12 in (305 mm) wheel at a 0.02 mm infeed, across the three configurations:

OD 2 in shaft: De = 43.5 mm, lc = 0.93 mm · flat: De = 305 mm, lc = 2.5 mm

And a 40 mm quill wheel in a 50 mm bore: De = 40×50/(50−40) = 200 mm, lc = 2.0 mm — over twice the contact of the big wheel on the shaft, from a wheel an eighth its size. Same steel, same infeed, three different wheels' worth of behavior: the OD contact self-sharpens, the bore contact loads and burns unless the grade softens.

Frequently asked questions

What is the equivalent diameter in grinding?

The single curvature the wheel-work contact behaves as: 1/De = 1/ds + 1/dw for external work, 1/ds − 1/dw for internal, and just the wheel diameter on a flat. A 12 in wheel over a 2 in shaft acts like a Ø43 mm wheel on a flat — far "smaller" than it looks.

How do you calculate the arc of contact in grinding?

lc = √(ae × De): the geometric contact length is the square root of depth of cut times equivalent diameter. At a 0.02 mm infeed, that 12-in-wheel-on-2-in-shaft contact is just 0.93 mm long; the same wheel on a flat contacts 2.5 mm.

Why does internal grinding use softer wheels?

Conformal contact. A 40 mm wheel in a 50 mm bore has De = 200 mm — five times the external equivalent — so each grit takes a thinner, longer chip, the wheel acts harder, and it loads and burns unless the grade drops. The geometry forces the wheel choice.

Does a bigger wheel act harder or softer?

Harder, at the same settings: larger De lengthens contact, thins each grit chip, and reduces self-sharpening. Conversely a worn-small wheel acts softer — one reason behavior drifts over wheel life even at constant surface speed.

Method & assumptions

Standard grinding-theory kinematics (Malkin; Rowe): rigid geometric contact, lc = √(ae·De). Elastic deflection lengthens real contact beyond the geometric value.
Equivalent diameter covers plunge-type contact; traverse dressing leads, crowns and form wheels add geometry not modeled here.
No grit, bond, hardness or dressing data is embedded — the wheel maker's grade recommendations govern; this screen explains the direction geometry pushes them.
Internal configuration requires the wheel smaller than the bore; conformity above 90% flags the near-conformal regime.