Resonance of oil-filled diaphragms

Many industrial pressure sensors isolate the silicon die from the medium with a stainless-steel diaphragm and a silicone-oil fill. The oil also loads the silicon diaphragm and lowers its resonant frequency, which matters for dynamic pressure measurement.

Dry resonance

For a clamped plate the first mode is f₀ = λ² / (2π L²) · √(D / ρh), with λ² = 35.99 for a square of side L and 10.21 for a circle of radius L (Leissa, 1969).

Fluid loading

Fluid on one side moves with the plate and adds mass. Lamb (1920), revisited by Amabili and Kwak (1996), expressed this as a non-dimensional added virtual mass incremental factor:

β = Γ · ρ_f a / (ρ h),   Γ ≈ 0.6689 (clamped circular plate, first mode)
f_wet = f_dry / √(1 + β)

β grows with diaphragm size and falls with thickness, so large thin diaphragms lose the most frequency. DiaphragmCalc applies the same Γ to square plates using the radius of an equal-area circle, which is an approximation.

Example

1000 µm square, 20 µm thick silicon diaphragm in silicone oil (970 kg/m³): dry f₀ = 282 kHz, β = 7.9, oil-loaded ≈ 94.8 kHz.

Limits

The added-mass model assumes an unbounded, incompressible fluid on one side. A thin oil gap in a real package adds squeeze-film effects and cavity compliance, and the steel isolation diaphragm adds its own mode. Use this estimate to check whether you are near a problem, then model the package.

Results are analytical first-order estimates and should be independently validated before use in safety-critical or production designs.