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Model checks
The science
What is modelled
A tube is driven at resonance in its symmetric bending mode. Mass flowing through the vibrating tube feels a Coriolis force that excites the antisymmetric mode in quadrature, so the two sensors near the tube ends see signals that are no longer in phase. The phase difference is proportional to the mass flow.
Equations
Two coupled oscillators: ẍ + 2ζω_s ẋ + ω_s² x = F_drive + g·ẏ, ÿ + 2ζω_a ẏ + ω_a² y = −g·ẋ, with gyroscopic coupling g = c·ṁ (skew-symmetric, so it adds no energy). Sensors see x·sin(πu) ± y·sin(2πu). Quadrature lock-in amplitudes give the amplitude ratio, which yields ṁ.
Honest simplifications
A two-mode demonstrator, not an instrument-grade finite-element model: single tube, linear dynamics, idealised sensors, no temperature or pressure effects. The animation is slowed about 7 times and the twist is magnified relative to the drive motion.
Built-in checks
Energy conservation of the undriven, undamped coupling, the recovered flow against the set flow after a 4 s lock-in, and the proportionality of the phase shift to flow.