One Device, Two Instruments — Measuring Vibration
How do you measure vibration? Put a mass-spring inside a case, attach the case to the object, and read the relative displacement z = x − y between mass and case. Remarkably, the same device — depending on where you put its natural frequency — becomes either a seismometer that measures displacement or an accelerometer that measures acceleration.
The heart of the instrument is the mass-spring inside the case. As the case shakes with the object, what we read is the relative displacement z = x − y between mass and case. Change r = ωωₙ: slow, and the mass follows the case so z≈0; fast, and the mass holds still in space while only the case moves, so z captures the object's motion itself.
The relative-displacement curve ZY = r²√((1−r²)²+(2ζr)²) holds both instruments in one picture (the same shape as A9's curve!). Drag r: the left, r≪1, has ZY ≈ r², proportional to acceleration — the accelerometer; the right, r≫1, has ZY → 1, the displacement itself — the seismometer. Two ends of one curve.
A seismometer lives at the right end (r≫1). Put the natural frequency very low (soft spring, heavy mass), and against fast shaking the mass holds nearly still in space by its inertia. That still mass becomes the reference, so the relative displacement z gives the object's displacement Y directly. The price is a low ωₙ, which makes the device big and heavy — the classic seismometer.
An accelerometer lives at the left end (r≪1). Put the natural frequency very high (stiff spring, light mass) and Z ≈ aωₙ² — the relative displacement is proportional to the object's acceleration. Raise ωₙ: the signal Z shrinks as 1ωₙ² (less sensitive), but the usable frequency range widens and the device gets smaller. That is how the MEMS accelerometer in your phone got so tiny.
Flip through the examples — a seismometer (ground displacement, very low ωₙ), a phone or car MEMS accelerometer (tilt, steps, airbags, high ωₙ), and machine condition monitoring (tracking bearing and motor vibration). Accelerometers are usually damped to ζ≈0.7 to widen their flat measuring range. Turning shaking into numbers all comes down to this one device.