19.2 Velocity
Translational velocity cannot be
measured directly and therefore must be calculated indirectly by other means as
set out below.
19.2.1 Differentiation of
displacement measurements
Differentiation of position
measurements obtained from any of the translational displacement transducers
described in section 19.1 can be used to produce a translational velocity
signal. Unfortunately, the process of differentiation always amplifies noise in
a measurement system. Therefore, if this method has to be used, a low-noise
instrument such as a d.c. excited carbon film potentiometer or laser
interferometer should be chosen. In the case of potentiometers, a.c. excitation
must be avoided because of the problem that harmonics in the power supply would
cause.
19.2.3 Conversion to rotational
velocity
Conversion from translational to
rotational velocity is the final measurement tech[1]nique
open to the system designer and is the one most commonly used. This enables any
of the rotational velocity measuring instruments described in Chapter 20 to be
applied.
19.3 Acceleration
The only class of device available
for measuring acceleration is the accelerometer. These are available in a wide
variety of types and ranges designed to meet particular measurement
requirements. They have a frequency response between zero and a high value, and
have a form of output that can be readily integrated to give displacement and
velocity measurements. The frequency response of accelerometers can be improved
by altering the level of damping in the instrument. Such adjustment must be
done carefully, however, because frequency response improvements are only
achieved at the expense of degrading the measurement sensitivity. Besides their
use for general[1]purpose motion
measurement, accelerometers are widely used to measure mechanical shocks and
vibrations.
Most forms of accelerometer consist
of a mass suspended by a spring and damper inside a housing, as shown in Figure
19.15. The accelerometer is rigidly fastened to the body undergoing
acceleration. Any acceleration of the body causes a force, Fa, on the mass, M,
given by:
This is the equation of motion of a
second order system, and, in the absence of damping, the output of the
accelerometer would consist of non-decaying oscillations. A damper is therefore
included within the instrument, which produces a damping force, Fd,
proportional to the velocity of the mass M given by:
One important characteristic of
accelerometers is their sensitivity to accelerations at right angles to the
sensing axis (the direction along which the instrument is designed to measure
acceleration). This is defined as the cross-sensitivity and is specified in
terms of the output, expressed as a percentage of the full-scale output, when an
acceleration of some specified magnitude (e.g. 30g) is applied at 90° to the
sensing axis.
The acceleration reading is obtained
from the instrument by measurement of the displacement of the mass within the
accelerometer. Many different displacement[1]measuring
techniques are used in the various types of accelerometer that are commercially
available. Different types of accelerometer also vary in terms of the type of
spring element and form of damping used.
Resistive potentiometers are one such
displacement-measuring instrument used in accelerometers. These are used mainly
for measuring slowly varying accelerations and low-frequency vibrations in the
range 0–50g. The measurement resolution obtainable is about 1 in 400 and
typical values of cross-sensitivity are ±1%. Inaccuracy is about ±1% and life
expectancy is quoted at two million reversals. A typical size and weight are
125 cm3 and 500 grams.
Strain gauges and piezoresistive
sensors are also used in accelerometers for measuring accelerations up to 200g.
These serve as the spring element as well as measuring mass displacement, thus
simplifying the instrument’s construction. Their typical characteristics are a
resolution of 1 in 1000, inaccuracy of š1% and cross-sensitivity of 2%. They have
a major advantage over potentiometer-based accelerometers in terms of their
much smaller size and weight (3 cm3 and 25 grams).
Another displacement transducer found
in accelerometers is the LVDT. This device can measure accelerations up to 700g
with a typical inaccuracy of ±1% of full scale. They are of a similar physical
size to potentiometer-based instruments but are lighter in weight (100 grams).
Accelerometers based on variable
inductance displacement measuring devices have extremely good characteristics
and are suitable for measuring accelerations up to 40g. Typical specifications
of such instruments are inaccuracy ±0.25% of full scale, resolution 1 in 10 000
and cross-sensitivity 0.5%. Their physical size and weight are similar to
potentiometer-based devices. Instruments with an output in the form of a
varying capacitance also have similar characteristics.
The other common displacement
transducer used in accelerometers is the piezoelectric type. The major
advantage of using piezoelectric crystals is that they also act as the spring
and damper within the instrument. In consequence, the device is quite small (15
cm3) and very low mass (50 grams), but because of the nature of piezoelectric
crystal operation, such instruments are not suitable for measuring constant or
slowly time-varying accelerations. As the electrical impedance of a
piezoelectric crystal is itself high, the output voltage must be measured with
a very high-impedance instrument to avoid loading effects. Many recent
piezoelectric crystal-based accelerometers incorporate a high impedance charge
amplifier within the body of the instrument. This simplifies the signal
conditioning requirements external to the accelerometer but can lead to
problems in certain operational environments because these internal electronics
are exposed to the same environmental hazards as the rest of the accelerometer.
Typical measurement resolution of this class of accelerometer is 0.1% of full
scale with an inaccuracy of ±1%. Individual instruments are available to cover
a wide range of measurements from 0.03g full scale up to 1000g full scale.
Intelligent accelerometers are also now available that give even better
performance through inclusion of processing power to compensate for
environmentally induced errors.
Recently, very small microsensors
have become available for measuring acceleration. These consist of a small mass
subject to acceleration that is mounted on a thin silicon membrane.
Displacements are measured either by piezoresistors deposited on the membrane
or by etching a variable capacitor plate into the membrane.
Two forms of fibre-optic-based
accelerometer also exist. One form measures the effect on light transmission
intensity caused by a mass subject to acceleration resting on a multimode
fibre. The other form measures the change in phase of light trans[1]mitted through a
monomode fibre that has a mass subject to acceleration resting on it.
19.3.1 Selection of accelerometers
In choosing between the different
types of accelerometer for a particular application, the mass of the instrument
is particularly important. This should be very much less than that of the body
whose motion is being measured, in order to avoid loading effects that affect
the accuracy of the readings obtained. In this respect, instruments based on
strain gauges are best.
19.4 Vibration
19.4.1 Nature of vibration
Vibrations are very commonly
encountered in machinery operation, and therefore measurement of the
accelerations associated with such vibrations is extremely important in
industrial environments. The peak accelerations involved in such vibrations can
be of 100g or greater in magnitude, whilst both the frequency of oscillation
and the magnitude of displacements from the equilibrium position in vibrations
have a tendency to vary randomly. Vibrations normally consist of linear
harmonic motion that can be expressed mathematically as:
X = X0
sin (ωt) (19.6)
where X is the displacement from the
equilibrium position at any general point in time, X0 is the peak
displacement from the equilibrium position, and ω is the angular frequency of
the oscillations. By differentiating equation (19.6) with respect to time, an
expression for the velocity v of the vibrating body at any general point in
time is obtained as:
v = -ωX0
cos(ωt) (19.7)
Differentiating equation (19.7) again
with respect to time, we obtain an expression for the acceleration, α, of the
body at any general point in time as:
α = -ω2
X0 sin (ωt) (19.8)
Inspection of equation (19.8) shows
that the peak acceleration is given by:
αpeak
= ω2 X0 (19.9)
This square law relationship between
peak acceleration and oscillation frequency is the reason why high values of
acceleration occur during relatively low-frequency oscillations. For example,
an oscillation at 10 Hz produces peak accelerations of 2g.
Example
A pipe carrying a fluid vibrates at a
frequency of 50 Hz with displacements of 8 mm from the equilibrium position.
Calculate the peak acceleration.
Solution
From equation (19.9),
αpeak = ω2 X0
= (2π50)2 × (0.008) = 789.6 m/s2
Using the fact that the acceleration
due to gravity, g, is 9.81 m/s2 , this answer can be expressed
alternatively as:
αpeak
= 789.6/9.81 = 80.5g
19.4.2 Vibration measurement
It is apparent that the intensity of
vibration can be measured in terms of either displacement, velocity or
acceleration. Acceleration is clearly the best parameter to measure at high
frequencies. However, because displacements are large at low frequencies
according to equation (19.9), it would seem that measuring either displacement
or velocity would be best at low frequencies. The amplitude of vibrations can
be measured by various forms of displacement transducer. Fibre-optic-based
devices are particularly attractive and can give measurement resolution as high
as 1 µm. Unfortunately, there are considerable practical difficulties in
mounting and calibrating displacement and velocity transducers and therefore
they are rarely used. Thus, vibration is usually measured by accelerometers at
all frequencies. The most common type of transducer used is the
piezoaccelerometer, which has typical inaccuracy levels of ±2%.
The frequency response of
accelerometers is particularly important in vibration measurement in view of
the inherently high-frequency characteristics of the measurement situation. The
bandwidth of both potentiometer-based accelerometers and accelerometers using
variable-inductance type displacement transducers goes up to 25 Hz only.
Accelerometers including either the LVDT or strain gauges can measure
frequencies up to 150 Hz and the latest instruments using piezoresistive strain
gauges have bandwidths up to 2 kHz. Finally, inclusion of piezoelectric crystal
displacement transducers yields an instrument with a bandwidth that can be as
high as 7 kHz.
When measuring vibration,
consideration must be given to the fact that attaching an accelerometer to the
vibrating body will significantly affect the vibration characteristics if the
body has a small mass. The effect of such ‘loading’ of the measured system can
be quantified by the following equation:
a1
= ab ( mb/mb + ma)
where a1 is the acceleration
of the body with accelerometer attached, ab is the acceleration of
the body without the accelerometer, ma is the mass of the accelerometer and mb
is the mass of the body. Such considerations emphasize the advantage of piezoaccelerometers,
as these have a lower mass than other forms of accelerometer and so contribute
least to this system-loading effect.
As well as an accelerometer, a
vibration measurement system requires other elements, as shown in Figure 19.16,
to translate the accelerometer output into a recorded signal. The three other
necessary elements are a signal-conditioning element, a signal analyser and a
signal recorder. The signal-conditioning element amplifies the relatively weak
output signal from the accelerometer and also transforms the high output
impedance of the accelerometer to a lower impedance value. The signal analyser
then converts the signal into the form required for output. The output
parameter may be either displace[1]ment, velocity or
acceleration and this may be expressed as either the peak value, r.m.s. value
or average absolute value. The final element of the measurement system is the
signal recorder. All elements of the measurement system, and especially the
signal recorder, must be chosen very carefully to avoid distortion of the
vibration waveform. The bandwidth should be such that it is at least a factor
of ten better than the band[1]width of the
vibration frequency components at both ends. Thus its lowest frequency limit
should be less than or equal to 0.1 times the fundamental frequency of
vibration and its upper frequency limit should be greater than or equal to ten
times the highest significant vibration frequency component.
If the frequency of vibration has to
be known, the stroboscope is a suitable instrument to measure this. If the
stroboscope is made to direct light pulses at the body at the same frequency as
the vibration, the body will apparently stop vibrating.
19.5 Shock
Shock describes a type of motion
where a moving body is brought suddenly to rest, often because of a collision.
This is very common in industrial situations and usually involves a body being
dropped and hitting the floor. Shocks characteristically involve large-magnitude
deceleration (e.g. 500g) that last for a very short time (e.g. 5 ms). An
instrument having a very high-frequency response is required for shock
measurement, and for this reason, piezoelectric crystal-based accelerometers
are commonly used. Again, other elements for analysing and recording the signal
are required as shown in Figure 19.16 and described in the last section. A
storage oscilloscope is a suitable instrument for recording the output signal,
as this allows the time duration as well as the acceleration levels in the
shock to be measured. Alternatively, if a permanent record is required, the
screen of a standard oscilloscope can be photographed. A further option is to
record the output on magnetic tape, which facilitates computerized signal
analysis.
Example
A body is dropped from a height of 10
m and suffers a shock when it hits the ground. If the duration of the shock is
5 ms, calculate the magnitude of the shock in terms of g.
Solution
The equation of motion for a body
falling under gravity gives the following expression for the terminal velocity,
v:
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