Page 384 - Mechatronics with Experiments

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is typically driven at a constant speed by a DC electric motor. The rotor is supported by
two bearings on the inner gimbal, which in turn is supported by two bearings on the outer
gimbal. The inner gimbal is further connected to the outer gimbal via a pair of springs. The
basic dynamic principle of operation of the gyro is that a rotational rate (angular velocity)
̇
about the z-axis, that is (t) results in a proportional angular displacement about the x-axis
z
in steady-state, that is
d (t)
z
(t) = K ⋅ (6.117)
x
dt
As a result, when this gyro is mounted on a vehicle by securing its outer gimbal to the
vehicle, the angular displacement about the x-axis is a measure of the rotational velocity
about the z-axis. This rotational speed about the z-axis can be integrated over time, given
the initial condition of the angle about the z-axis, to keep track of the orientation of the
vehicle about the z-axis. The dynamic relationship for this gyro is
2
d (t) d (t) d (t)
x
z
x
J x + C x + K (t) = J ⋅ w ⋅ ; transient (6.118)
y
y
x x
dt 2 dt dt
d (t)
z
K (t) = J ⋅ w ⋅ ; steady-state (6.119)
y
x x
y
dt
J ⋅ w y d (t)
y
z
(t) = ⋅ (6.120)
x
K x dt
d (t)
z
(t) = K (6.121)
x
dt
where J , C , K are the rotational moment of inertia of the gyro rotor and its inner gimbal
x
x
x
about the x-axis, the damping coefficient, and the spring constant. J is the rotary inertia
y
of the gyro rotor about the y-axis, w is the speed of the gyro rotor, which is typically kept
y
constant by a controlled electric motor.
6.6.2 Piezoelectric Accelerometers
Some materials (such as natural quartz crystal, silicon dioxide, barrium titanite, lead zir-
conate titanate (PZT)), called piezo crystals, produce a charge in response to a force (or
deformation) applied to them. This is called the direct piezoelectric effect. The same materi-
als also have the reverse phenomenon, that is, they produce force in response to an applied
charge. This is called the reverse piezoelectric effect. The Greek word “piezo” means
“to squeeze” or “pressure.” Quartz has excellent temperature stability and shows almost
no decay in its piezoelectric properties over time. PZTs are polarized by applying very
high DC voltages at high temperatures. PZTs show a natural decay in their piezoelectric
properties over time, hence may need to be periodically calibrated or re-polarized.
Piezoelectric accelerometers work on the principle that the acceleration times the
sensor inertia will apply a force on the sensor. Mechanically, the piezo element acts as a very
precise and stiff spring in the sensor design. As a result of the piezoelectric phenomena, the
charge output, hence the output voltage, from the sensor is proportional to the inertial force
̈ x → F → q → V out (6.122)
q = C ⋅ V out (6.123)
where q is the charge produced by the piezoelectric material, C is the effective capacitance,
and V out is the produced output voltage.
A calibrated piezoelectric acceleration sensor has the following input–output rela-
tionship,
V out = K ⋅ ̈ x (6.124)