Dial Indicator Reading

What is modelled

A shaft turning with eccentricity A moves the surface by A sin(phi). The probe touches at a small angle alpha to the measuring direction, so the plunger only moves by the cosine of that angle. A rack on the plunger drives a pinion and a gear train that gives one pointer revolution per millimetre, and the dial has 100 divisions of 0.01 mm. Play between the gears makes the pointer lag after each reversal.

Equations

• Plunger travel x = A sinφ cosα; total indicator reading over one turn 2 A cosα.
• Pointer angle θ = 360° x / (1 mm); the train ratio for a pinion of radius r is G = 2π r / (1 mm).
• Spring force F = F0 + k x, work F0 x + k x²/2.
• Backlash is a play operator: the pointer moves only when the plunger is more than half the play away from it, so the error never exceeds b/2.

Checks run in the page

• Gear train angle against a stepwise sum.
• Cosine error against 2 A cosα from sampled data.
• Spring work against the closed form.
• The backlash error bound over three turns.

Honest simplifications

Rigid gears and plunger, a linear spring, no friction, no thermal effects and no tip-radius effect. The 3D page shows the gauge mechanism; the numbers here are illustrative of the general instrument, not of one product.

• Press Tilted probe and compare the total reading with the true 2A: the pointer under-reads.
• Press Sloppy gears and watch the pointer loop in the right-hand graph.
• Raise the spring rate and see the contact force at full travel climb.