handbook of modern sensors

304 8 Velocity and Acceleration
Fig. 8.2. A frequency response of an accelerometer. f n is a natural frequency and f ref is the
reference frequency.
where a is the time-dependent impulse of the
√
accelerometer body and g(t) is the
inverse transform L −1 {G(s)}.Ifwesetω = ω 0 1 − ζ 2 , then Eq. (8.6) has two solutions.
One is for the underdamped mode (ζ < 1),
x(t)=
∫ t
whereas for the overdamped mode (ζ > 1),
x(t)=
0
∫ t
0
− 1 ω e−ζω 0(t−τ) sin ω(t − τ)a(t)dτ, (8.7)
− 1 ω e−ζω 0(t−τ) sinh ω(t − τ)a(t)dτ, (8.8)
√
where ω = ω 0 ζ 2 − 1. The above solutions can be evaluated for different acceleration
inputs applied to the accelerometer base [1].
A correctly designed, installed, and calibrated accelerometer should have one
clearly identifiable resonant (natural) frequency and a flat frequency response at which
the most accurate measurement can be made (Fig. 8.2). Within this flat region, as the
vibrating frequency changes, the output of the sensor will correctly reflect the change
without multiplying the signal by any variations in the frequency characteristic of
the accelerometer. Viscous damping is used in many accelerometers to improve the
useful frequency range by limiting the effects of the resonant. As a damping medium,
silicone oil is used quite often.
When calibrated, several characteristics of an accelerometer should be determined:
1. Sensitivity is the ratio of an electrical output to the mechanical input. It is usually
expressed in terms of volts per unit of acceleration under the specified conditions.
For instance, the sensitivity may be specified as 1 V/g (unit of acceleration:
g = 9.80665 m/s 2 at sea level, 45 ◦ latitude). The sensitivity is typically measured