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when properly employed. Here we consider techniques for the measurement of linear and
angular speed.
Displacement, velocity, and acceleration measurements are made with respect to some frame
of reference. Consider the case of a game of billiards in a moving railway car. Observers on
the platform and on the train would assign different velocity vectors to the balls during play. The
velocity vectors would differ by the relative velocity of the two observers. Simply differentiating the
vector velocity equation, however, shows that the accelerations of the balls are the same in all
reference frames moving relative to one another with constant relative velocity.
Linear Velocity Measurements
As previously discussed, measurements of velocity require a frame of reference. For example, the
velocity of a conveyor belt might be measured relative to the floor of the building where it is housed.
On the other hand, advantages may be realized in a control system if the velocity of a robotic arm,
which ‘‘picks’’ parts from this same conveyor belt, is measured relative to the moving conveyor. In
the present discussion, it is assumed that velocity is measured relative to a ground state, which is
generally defined by the mounting point of a transducer.
Consider the measurement of velocity relative to a fixed frame of reference. Typically, if the
measurement of linear velocity is to be made on a continuous basis, an equivalent angular rotational
speed is measured, and the data is analyzed in such a way so as to produce a measured linear velocity.
For example, a speedometer on an automobile provides a continuous record of the speed of the car, but
the output is derived from measuring the rotational speed of the drive shaft or transmission.
Velocity from Displacement or Acceleration
Velocity, in general, can be directly measured by mechanical means only over very short times or
small displacements, due to limitations in transducers. However, if the displacement of a rigid body
is measured at identifiable time intervals, the velocity can be determined through differentiation of
the time-dependent displacement. Alternatively, if acceleration is measured, the velocity may be
determined from integration of the acceleration signal. The following example demonstrates the
effect of integration and differentiation on the uncertainty of velocities computed from acceleration
or displacement.
Example 12.2
Suppose our goal is to assess the merits of measuring velocity through the integration of an
acceleration signal as compared to differentiating a displacement signal. The following conditions
are assumed to apply.
For both y t
ð Þand y
00
t
ð Þ the data acquisition system and transducers have the characteristics
described in Table 12.2. Note that the uncertainties are derived directly from the analog-to-digital
(A/D) resolution error and accuracy statements. Assume that the signals for both acceleration and
displacement are sampled at 10 Hz and numerical techniques are used to differentiate or integrate
the resulting signals.
SOLUTION Consider first determining the velocity through differentiation of the displacement signal. Displacement is measured digitally by the data acquisition system, with a digitized
12.2 Sensors 517
13:54:10 Page 517
when properly employed. Here we consider techniques for the measurement of linear and
angular speed.
Displacement, velocity, and acceleration measurements are made with respect to some frame
of reference. Consider the case of a game of billiards in a moving railway car. Observers on
the platform and on the train would assign different velocity vectors to the balls during play. The
velocity vectors would differ by the relative velocity of the two observers. Simply differentiating the
vector velocity equation, however, shows that the accelerations of the balls are the same in all
reference frames moving relative to one another with constant relative velocity.
Linear Velocity Measurements
As previously discussed, measurements of velocity require a frame of reference. For example, the
velocity of a conveyor belt might be measured relative to the floor of the building where it is housed.
On the other hand, advantages may be realized in a control system if the velocity of a robotic arm,
which ‘‘picks’’ parts from this same conveyor belt, is measured relative to the moving conveyor. In
the present discussion, it is assumed that velocity is measured relative to a ground state, which is
generally defined by the mounting point of a transducer.
Consider the measurement of velocity relative to a fixed frame of reference. Typically, if the
measurement of linear velocity is to be made on a continuous basis, an equivalent angular rotational
speed is measured, and the data is analyzed in such a way so as to produce a measured linear velocity.
For example, a speedometer on an automobile provides a continuous record of the speed of the car, but
the output is derived from measuring the rotational speed of the drive shaft or transmission.
Velocity from Displacement or Acceleration
Velocity, in general, can be directly measured by mechanical means only over very short times or
small displacements, due to limitations in transducers. However, if the displacement of a rigid body
is measured at identifiable time intervals, the velocity can be determined through differentiation of
the time-dependent displacement. Alternatively, if acceleration is measured, the velocity may be
determined from integration of the acceleration signal. The following example demonstrates the
effect of integration and differentiation on the uncertainty of velocities computed from acceleration
or displacement.
Example 12.2
Suppose our goal is to assess the merits of measuring velocity through the integration of an
acceleration signal as compared to differentiating a displacement signal. The following conditions
are assumed to apply.
For both y t
ð Þand y
00
t
ð Þ the data acquisition system and transducers have the characteristics
described in Table 12.2. Note that the uncertainties are derived directly from the analog-to-digital
(A/D) resolution error and accuracy statements. Assume that the signals for both acceleration and
displacement are sampled at 10 Hz and numerical techniques are used to differentiate or integrate
the resulting signals.
SOLUTION Consider first determining the velocity through differentiation of the displacement signal. Displacement is measured digitally by the data acquisition system, with a digitized
12.2 Sensors 517