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.