Page 381 - Mechatronics with Experiments

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JWST499-c06
JWST499-Cetinkunt
SENSORS 367 254mm×178mm

General purpose Low frequency High frequency

General purpose 3-axes High temperature

FIGURE 6.38: Pictures of various accelerometers.

The values of m, c, k are chosen such that = 0.7to1.0 range, and w can be chosen up to
n
a few hundred Hz. The smaller the mass and the stiffer the spring constant of the sensor,
the higher will be its bandwidth.
Let us consider a sinusoidal base displacement and acceleration as a function of time,
x base (t) = A sin(wt) (6.105)
The resulting acceleration of the base as a result of constant magnitude sinusoidal displace-
ment of the base is,
2
̈ x base (t) =−Aw sin(wt) (6.106)
The steady-state response of the displacement of the sensor inertia with respect to its
housing, x(t), is the steady-state solution for
̈ x(t) + (c∕m) ⋅ ̇ x(t) + (k∕m) ⋅ x(t) =−̈ x base (t) (6.107)
2
̈ x(t) + (c∕m) ⋅ ̇ x(t) + (k∕m) ⋅ x(t) = Aw sin(wt) (6.108)
In steady-state,
2
A (w∕w ) sin(wt − )
n
x (t) = (6.109)
ss
2 2
2 1∕2
{[1 − (w∕w ) ] + [2 (w∕w )] }
n
n
where
2 (w∕w )
n
−1
=tan (6.110)
1 − (w∕w ) 2
n
The steady-state displacement of the sensor has the following properties:
1. the sensor displacement has the same frequency as the acceleration of the base,
2. there is a phase shift between the sensor displacement and base acceleration and
the phase angle is a function of acceleration frequency (w) as well as the sensor
parameters ( , w ),
n
3. the magnitude of the sensor displacement is proportional to the acceleration magni-
tude. However, the proportionality constant is a function of acceleration frequency
(w) as well as the sensor parameters ( , w ).
n