Single Degree-of-Freedom Examples: Accelerometer and Seismometer
engineering fundamentals SDOF Example: Accelerometer
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Accelerometer

An accelerometer attached to a larger object can be modeled as a single degree-of-freedom vibration system excited by a moving base.

The above accelerometer model can be analyzed by the following free body diagram,

The equation of motion then becomes,

We assume that the object is under the harmonic excitation, to simplify the forcing function. The equation of motion becomes,

Accelerator Displacement Solution
The steady state solution for the accelerometer is found to be,

where the amplitude H, the phase f, the damping ratio z, and the natural frequency wn are given by,

Most accelerometers are constructed with a small mass and a short stiff spring, such that the natural frequency wn is much higher than the working frequency w. As a result, the denominator of the amplitude H is approximately 1,

This is important because the accelerometer can now track the acceleration of the target object directly, without the need for any amplitude corrections. To see this, compare the simplified displacement of the accelerometer with the acceleration amplitude of the target object, AObject,

Observe that the target object's acceleration amplitude is contained within the accelerometer's displacement directly,

For z = 0.707, the effective frequency range can be up to 0.4 wn with less than 1% error. In fact, the results are often acceptable up to 0.6 wn without adjustment.

For modern piezoelectric accelerometers, the damping ratio is close to zero. In addition, their mass is very small (approx. 10 grams; less than 1 oz) and they have a very high stiffness, resulting in natural frequencies of 30 kHz or more. Hence their working range can extend up to 5 kHz or higher.

Seismometer

In contrast to small accelerometers, bulky transducers with large masses and soft springs would have very small natural frequencies. If the natural frequency is much smaller than the working frequency, then the amplitude H from above simplifies even more,

The displacement of the object can then be obtained directly (adjusted for a phase delay),

This kind of transducer is used as a seismometer to detect earthquakes, or as vibrometers to measure vibration displacements. For z = 0.707, the effective frequency range can be as low as 3 wn.

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