Bourdon Tube Gauge

What is modelled

Pressure inside a flattened curved tube tries to make its section rounder, which uncurls the tube a little. The small tip movement is multiplied by a sector and pinion linkage into a pointer sweep of 270 degrees. The pointer, the linkage and the tube together behave like a damped spring, so the reading can overshoot, ring and show only part of a fast ripple.

Equations

• Opening of the tube Δα/α0 = κ (1 − ν2) p R2 / (E t h), with a wall thickness that grows with the dial range.
• Pointer u'' + 2 ζ ωn u' + ωn2 u = ωn2 r(p), with a cubic linkage error in r.
• Overshoot exp(−π ζ / √(1 − ζ2)) and gain 1/√((1 − r2)2 + (2 ζ r)2).
• Snubber: first-order lag 1/√(1 + (2π f τ)2). Sea water dp/dz = ρ(p) g with ρ = ρ0/(1 − p/K), solved by p = K (1 − √(1 − 2 ρ0 g z / K)).

Checks run in the page

• A Runge-Kutta step response against the closed-form overshoot.
• Zero linkage error at 0, 50 and 100 % and the cubic value in between.
• A numerical integration of compressible sea water pressure against the closed form.
• The pulsation gain from a long run against the transfer function.

Honest simplifications

The opening of the tube uses a lumped shape factor chosen so that a 40 bar gauge moves its tip by about 0.6 mm, hysteresis, temperature and friction in the linkage are ignored and the linkage error is a simple cubic. It is a teaching model, not a calibration tool.

• Press Pump pulsation, then Snubber fitted: the ripple on the pointer almost disappears.
• Press Resonance and watch the pointer swing wider than the ripple that drives it.
• Press Worn linkage and read the calibration error plot.