how to measure vibration
How do you measure a vibration? Put a mass and a spring inside a case, fasten the case to the thing you are measuring, and read the relative displacement of mass and case, z = x − y. The same device becomes a seismometer that reads displacement, or an accelerometer that reads acceleration, depending on where you place the natural frequency. That placement decides the kind of instrument.
The heart of the instrument is the mass and spring inside the case. When the case shakes with the object, what we read is the relative displacement between mass and case, z = x − y. Change r = ωωₙ. When it is slow the mass follows the case and z≈0. When it is fast the mass stays still in space and only the case moves, so z carries the object's motion as it is. What is read is not an absolute position. It is the difference of the two.
The relative-displacement curve ZY = r²√((1−r²)²+(2ζr)²) holds both instruments on one page. It has the same shape as the curve of A9. Move r. The left region, r≪1, is ZY ≈ r², proportional to acceleration, an accelerometer. The right region, r≫1, is ZY → 1, handing the displacement over as it is, a seismometer. The two ends of one curve, and the middle is neither.
A seismometer lives at the right end (r≫1). Set the natural frequency very low, a soft spring and a heavy mass, and in front of a fast shake the mass stays almost still in space by its inertia. That still mass becomes the reference, and the relative displacement z gives the object's displacement Y as it is. The natural frequency has to be low, so the device grows large and heavy. A classical seismometer is like that.
An accelerometer lives at the left end (r≪1). Set the natural frequency very high, a stiff spring and a light mass, and Z ≈ aωₙ², so the relative displacement is proportional to the object's acceleration. Raise ωₙ and the signal Z shrinks as 1ωₙ², so the sensitivity falls, but the frequency range you can measure widens and the device gets small. The MEMS accelerometer inside a phone got small in exactly this way.
Step through the examples. A seismometer, the displacement of the ground, a very low natural frequency. A phone or car MEMS accelerometer, tilt, steps, an airbag, a high natural frequency. Machine condition monitoring, the vibration of a bearing or a motor. An accelerometer is usually set at a damping ζ≈0.7 to widen the flat measuring range. Turning a shake into a number all comes out of this one device.