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to the magnitude of acceleration ( ̈ X) of the object provided that the frequency of
acceleration is well within the natural frequency (bandwidth) of the sensor.
̈ X → x → V (6.98)
out
2. Piezoelectric based accelerometers provide a charge (q) proportional to the iner-
tial force as a result of the acceleration. Piezoelectric materials provide a charge
proportional to the strain which is proportional to the inertial force.
̈ X → F → q → V (6.99)
out
3. Strain-gauges can be used for acceleration measurement if the sensor can transduct a
strain ( ) proportional to the acceleration. Once that is accomplished, the change in
the strain is measured by the change in the strain-gauge resistance and hence as an
output voltage from a Wheatstone bridge and op-amp circuit.
̈ X → F → → R → V (6.100)
out

6.6.1 Inertial Accelerometers
An inertial accelerometer is basically a small mass-spring-damper system with high natural
frequency. Consider the figure shown in Figure 6.37, which is the concept of an inertial
accelerometer connected to a body whose acceleration is to be measured. Figure 6.38
shows pictures of a number of accelerometers. The dynamic relations between the relative
displacement of the sensor inertia with respect to its enclosure, x, and the acceleration of
the body, ̈ x base ,are
m ⋅ (̈ x(t) + ̈ x (t)) + c ⋅ ̇ x(t) + k ⋅ x(t) = 0 (6.101)
base
̈ x(t) + (c∕m) ⋅ ̇ x(t) + (k∕m) ⋅ x(t) =−̈ x (t) (6.102)
base
Notice that if the accelerometer parameters m, c, k are chosen such that the motion of the
accelerometer is critically damped, then the displacement of the accelerometer relative to
its enclosure, x(t), is proportional to the acceleration of the base in steady-state. The speed
of response is determined by the c∕m and k∕m ratios. Let

c∕m = 2 w (6.103)
n
2
k∕m = w (6.104)
n
Mass

Output
transducer
m
Spring
Damper

Input
motion

Object FIGURE 6.37: Operating principle
in motion of an inertial accelerometer.