20.2 Rotational velocity
The main application of rotational
velocity transducers is in speed control systems. They also provide the usual
means of measuring translational velocities, which are transformed into
rotational motions for measurement purposes by suitable gearing. Many different
instruments and techniques are available for measuring rotational velocity as
presented below.
20.2.1 Digital tachometers
Digital tachometers, or to give them
their proper title, digital tachometric generators, are usually non-contact
instruments that sense the passage of equally spaced marks on the surface of a
rotating disc or shaft. Measurement resolution is governed by the number of
marks around the circumference. Various types of sensor are used, such as
optical, inductive and magnetic ones. As each mark is sensed, a pulse is
generated and input to an electronic pulse counter. Usually, velocity is
calculated in terms of the pulse count in unit time, which of course only
yields information about the mean velocity. If the velocity is changing,
instantaneous velocity can be calculated at each instant of time that an output
pulse occurs, using the scheme shown in Figure 20.16. In this circuit, the
pulses from the transducer gate the train of pulses from a 1 MHz clock into a
counter. Control logic resets the counter and updates the digital output value
after receipt of each pulse from the transducer. The measurement resolution of
this system is highest when the speed of rotation is low.
Optical sensing
Digital tachometers with optical
sensors are often known as optical tachometers. Optical pulses can be generated
by one of the two alternative photoelectric techniques illustrated in Figure
20.17. In Figure 20.17(a), the pulses are produced as the windows in a slotted
disc pass in sequence between a light source and a detector. The alternative
form, Figure 20.17(b), has both light source and detector mounted on the same side
of a reflective disc which has black sectors painted onto it at regular angular
intervals. Light sources are normally either lasers or LEDs, with photodiodes
and phototransistors being used as detectors. Optical tachometers yield better
accuracy than other forms of digital tachometer but are not as reliable because
dust and dirt can block light paths.
Inductive sensing
Variable reluctance velocity
transducers, also known as induction tachometers, are a form of digital
tachometer that use inductive sensing. They are widely used in the automotive
industry within anti-skid devices, anti-lock braking systems (ABS) and traction
control. One relatively simple and cheap form of this type of device was
described earlier in section 13.2
(Figure 13.2). A more sophisticated version shown in Figure 20.18 has a
rotating disc that is constructed from a bonded-fibre material into which soft
iron poles are inserted at regular intervals around its periphery. The sensor
consists of a permanent magnet with a shaped pole piece, which carries a wound
coil. The distance between the pick-up and the outer perimeter of the disc is
around 0.5 mm. As the disc rotates, the soft iron inserts on the disc move in
turn past the pick-up unit. As each iron insert moves towards the pole piece,
the reluctance of the magnetic circuit increases and hence the flux in the pole
piece also increases. Similarly, the flux in the pole piece decreases as each
iron insert moves away from the sensor. The changing magnetic flux inside the
pick-up coil causes a voltage to be induced in the coil whose magnitude is
proportional to the rate of change of flux. This voltage is positive whilst the
flux is increasing and negative whilst it is decreasing. Thus, the output is a
sequence of positive and negative pulses whose frequency is proportional to the
rotational velocity of the disc. The maximum angular velocity that the
instrument can measure is limited to about 10 000 rpm because of the finite
width of the induced pulses. As the velocity increases, the distance between
the pulses is
reduced, and at a certain velocity,
the pulses start to overlap. At this point, the pulse counter ceases to be able
to distinguish the separate pulses. The optical tachometer has significant
advantages in this respect, since the pulse width is much narrower, allowing
measurement of higher velocities.
A simpler and cheaper form of
variable reluctance transducer also exists that uses a ferromagnetic gear wheel
in place of a fibre disc. The motion of the tip of each gear tooth towards and
away from the pick-up unit causes a similar variation in the flux pattern to
that produced by the iron inserts in the fibre disc. The pulses produced by
these means are less sharp, however, and consequently the maximum angular
velocity measurable is lower.
Magnetic (Hall-effect) sensing
The rotating element in Hall-effect
or magnetostrictive tachometers has a very simple design in the form of a
toothed metal gearwheel. The sensor is a solid-state, Hall-effect device that
is placed between the gear wheel and a permanent magnet. When an intertooth gap
on the gear wheel is adjacent to the sensor, the full magnetic field from the
magnet passes through it. Later, as a tooth approaches the sensor, the tooth
diverts some of the magnetic field, and so the field through the sensor is
reduced. This causes the sensor to produce an output voltage that is
proportional to the rotational speed of the gear wheel.
20.2.2 Stroboscopic methods
The stroboscopic technique of
rotational velocity measurement operates on a similar physical principle to
digital tachometers except that the pulses involved consist of flashes of light
generated electronically and whose frequency is adjustable so that it can be
matched with the frequency of occurrence of some feature on the rotating body
being measured. This feature can either be some naturally occurring one such as
gear teeth or the spokes of a wheel, or it can be an artificially created
pattern of black and white stripes. In either case, the rotating body appears
stationary when the frequencies of the light pulses and body features are in
synchronism. Flashing rates available in commercial stroboscopes vary from 110
up to 150 000 per minute according to the range of velocity measurement
required, and typical measurement inaccuracy is ±1% of the reading. The
instrument is usually in the form of a hand-held device that is pointed towards
the rotating body.
It must be noted that measurement of
the flashing rate at which the rotating body appears stationary does not
automatically indicate the rotational velocity, because synchronism also occurs
when the flashing rate is some integral sub-multiple of the rotational speed.
The practical procedure followed is therefore to adjust the flashing rate until
synchronism is obtained at the largest flashing rate possible, R1.
The flashing rate is then carefully decreased until synchronism is again
achieved at the next lower flashing rate, R2. The rotational
velocity is then given by:
V = R1R2/R1 - R2
20.2.3 Analogue tachometers
Analogue tachometers are less
accurate than digital tachometers but are nevertheless still used successfully
in many applications. Various forms exist.
The d.c. tachometer has an output
that is approximately proportional to its speed of rotation. Its basic
structure is identical to that found in a standard d.c. generator used for
producing power, and is shown in Figure 20.19. Both permanent-magnet types and
separately excited field types are used. However, certain aspects of the design
are optimized to improve its accuracy as a speed-measuring instrument. One
significant design modification is to reduce the weight of the rotor by
constructing the windings on a hollow fibreglass shell. The effect of this is
to minimize any loading effect of the instrument on the system being measured.
The d.c. output voltage from the instrument is of a relatively high magnitude,
giving a high measurement sensitivity that is typically 5 volts per 1000 rpm.
The direction of rotation is determined by the polarity of the output voltage.
A common range of measurement is 0–6000 rpm. Maximum non-linearity is usually
about ±1% of the full-scale reading. One problem with these devices that can
cause difficulties under some circumstances is the presence of an a.c. ripple
in the output signal. The magnitude of this can be up to 2% of the output d.c.
level.
The a.c. tachometer has an output
approximately proportional to rotational speed like the d.c. tachogenerator.
Its mechanical structure takes the form of a two-phase induction motor, with
two stator windings and (usually) a drag-cup rotor, as shown in Figure 20.20.
One of the stator windings is excited with an a.c. voltage and the measurement
signal is taken from the output voltage induced in the second winding. The
magnitude of this output voltage is zero when the rotor is stationary, and
otherwise proportional to the angular velocity of the rotor. The direction of
rotation is determined by the phase of the output voltage, which switches by
180° as the direction reverses. Therefore, both the phase and magnitude of the
output voltage have to be measured. A typical range of measurement is 0–4000
rpm, with an inaccuracy of ±0.05% of full[1]scale
reading. Cheaper versions with a squirrel-cage rotor also exist, but measurement
inaccuracy in these is typically ±0.25%.
The drag-cup tachometer, also known
as an eddy-current tachometer, has a central spindle carrying a permanent
magnet that rotates inside a non-magnetic drag-cup consisting of a cylindrical
sleeve of electrically conductive material, as shown in Figure 20.21. As the
spindle and magnet rotate, a voltage is induced which causes circulating eddy
currents in the cup. These currents interact with the magnetic field from the
permanent magnet and produce a torque. In response, the drag-cup turns until
the induced torque is balanced by the torque due to the restraining springs
connected to the cup. When equilibrium is reached, the angular displacement of
the cup is proportional to the rotational velocity of the central spindle. The
instrument has a typical measurement inaccuracy of š0.5% and is commonly used
in the speedometers of motor vehicles and as a speed indicator for
aero-engines. It is capable of measuring velocities up to 15 000 rpm.
Analogue-output forms of the variable
reluctance velocity transducer (see section 20.2.1) also exist in which the
output voltage pulses are converted into an analogue, varying-amplitude, d.c.
voltage by means of a frequency-to-voltage converter circuit. However, the
measurement accuracy is inferior to digital output forms.
20.2.4 Mechanical flyball
The mechanical flyball (alternatively
known as a centrifugal tachometer) is a velocity[1]measuring
instrument that was invented in 1817 and so might now be regarded as being
old-fashioned. However, because it can act as a control actuator as well as a
measuring instrument, it still finds substantial use in speed-governing systems
for engines and turbines in which the measurement output is connected via a
system of mechanical links to the throttle. The output is linear, typical
measurement inaccuracy is š1%, and velocities up to 40 000 rpm can be measured.
As shown in Figure 20.22, the device consists of a pair of spherical balls
pivoted on the rotating shaft. These balls move outwards under the influence of
centrifugal forces as the rotational velocity of the shaft increases and lift a
pointer against the resistance of a spring. The pointer can be arranged to give
a visual indication of speed by causing it to move in front of a calibrated scale,
or its motion can be converted by a translational displacement transducer into
an electrical signal.
In equilibrium, the centrifugal
force, Fc, is balanced by the spring force, Fs, where:
Fc = Kcω2 ; Fs
= Ksx
and Kc and Ks
are constants, ω is the rotational velocity and x is the displacement of the
pointer.
Thus:
20.2.5 The rate gyroscope
The rate gyro, illustrated in Figure
20.23, has an almost identical construction to the rate integrating gyro
(Figure 20.14), and differs only by including a spring system which acts as an
additional restraint on the rotational motion of the frame. The instrument
measures the absolute angular velocity of a body, and is widely used in generating
stabilizing signals within vehicle navigation systems. The typical measurement
resolution given by the instrument is 0.01°/s and rotation rates up to 50°/s
can be measured. The angular velocity, α, of the body is related to the angular
deflection of the gyroscope, θ, by the equation:
where H is the angular momentum of
the spinning wheel, M is the moment of inertia of the system, β is the viscous
damping coefficient, K is the spring constant, and D is the D-operator.
This relationship (20.2) is a second
order differential equation and therefore we must expect the device to have a
response typical of second order instruments, as discussed in Chapter 2. The
instrument must therefore be designed carefully so that the output response is
neither oscillatory nor too slow in reaching a final reading. To assist in the
design process, it is useful to re-express equation (20.2) in the following
form:
The static sensitivity of the
instrument, K’, is made as large as possible by using a high-speed motor to
spin the wheel and so make H high. Reducing the spring constant K further
improves the sensitivity but this cannot be reduced too far as it makes the
resonant frequency ω of the instrument too small. The value of β is chosen such
that the damping ratio ξ is close to 0.7.
20.2.6 Fibre-optic gyroscope
This is a relatively new instrument
that makes use of fibre-optic technology. Incident light from a source is
separated by a beam splitter into a pair of beams a and b, as shown in Figure
20.24. These travel in opposite directions around an optic-fibre coil (which
may be several hundred metres long) and emerge from the coil as the beams
marked a’ and b’ . The beams a’ and b’ are directed by the beam splitter into
an interferometer. Any motion of the coil causes a phase shift between a’ and b’
which is detected by the interferometer. Further details can be found in
Nuttall (1987).
20.2.7 Differentiation of angular
displacement measurements
Angular velocity measurements can be
obtained by differentiating the output signal from angular displacement
transducers. Unfortunately, the process of differentiation amplifies any noise
in the measurement signal, and therefore this technique has only rarely been
used in the past. The technique has become more feasible with the advent of
intelligent instruments, and one such instrument which processes the output of
a resolver claims a maximum velocity measurement inaccuracy of š1% (Analogue
Devices, 1988).
20.2.8 Integration of the output from
an accelerometer
In measurement systems that already
contain an angular acceleration transducer, it is possible to obtain a velocity
measurement by integrating the acceleration measurement signal. This produces a
signal of acceptable quality, as the process of integration attenuates any
measurement noise. However, the method is of limited value in many measurement
situations because the measurement obtained is the average velocity over a
period of time, rather than a profile of the instantaneous velocities as motion
takes place along a particular path.
20.2.9 Choice between rotational
velocity transducers
Choice between different rotational
velocity transducers is influenced strongly by whether an analogue or digital
form of output is required. Digital output instruments are now widely used and
a choice has to be made between the variable reluctance transducer, devices
using electronic light pulse counting methods, and the stroboscope. The first
two of these are used to measure angular speeds up to about 10 000 rpm and the
last one can measure speeds up to 25 000 rpm.
Probably the most common form of
analogue output device used is the d.c. tachometer. This is a relatively simple
device that measures speeds up to about 5000 rpm with a maximum inaccuracy of ±1%.
Where better accuracy is required within a similar range of speed measurement,
a.c. tachometers are used. The squirrel-cage rotor type has an inaccuracy of
only ±0.25% and drag-cup rotor types can have inaccuracies as low as ±0.05%.
Other devices with an analogue output
that are also sometimes used are the dragcup tachometer and the mechanical
flyball. The drag-cup tachometer has a typical inaccuracy of ±5% but it is
cheap and therefore very suitable for use in vehicle speedometers. The
Mechanical flyball has a better accuracy of ±1% and is widely used in speed
governors, as noted earlier.
20.3 Measurement of rotational
acceleration
Rotational accelerometers work on
very similar principles to translational motion accelerometers. They consist of
a rotatable mass mounted inside a housing that is attached to the accelerating,
rotating body. Rotation of the mass is opposed by a torsional spring and damping.
Any acceleration of the housing causes a torque JR on the mass. This torque is
opposed by a backward torque due to the torsional spring and in equilibrium:
A damper is usually included in the
system to avoid undying oscillations in the instrument. This adds an additional
backward torque B P to the system and the equation of motion becomes:
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