-
CHAPTER 15
MEASUREMENTS IN ROTATING MACHINERIES
In chapters 13 and 14, we studied some of the experimental
methods to estimate unbalances and bearing dynamic parameters,
respectively. These methods involve measurement of input (forces)
and output (vibration responses) in time domain. For subsequent
processing often these measurements are required in frequency
domain. In the present chapter, we would describe the overall
measurement and analysis systems. General terminologies associated
with measurement systems are presented. Sensitivity analyses of the
estimated parameters due to errors involved in the measurements are
presented. Various kinds of transducers, the conditioning and
analyzing instruments, and vibration exciters are described
especially those are suited for the measurement in rotating
machineries. Transducers include the displacement, velocity,
acceleration, force and acoustic transducers. In subsequent
chapter, the focus would be to describe the basic techniques of the
signal processing and associated error involved.
From experiments quantities that are desired may be the
velocity, acceleration, displacement, force, and its phase. These
quantities may be useful in predicting the fatigue failure of a
particular machine element of a machine or may play important role
in analyses, which are used to reduce the structure vibration or
noise level. It may be useful in estimating system parameters,
while the force is also measured which causes the vibration, by the
modal analysis or model updating. The central problem in any type
of motion or vibration measurement concerns a determination of the
appropriate quantities in reference to some specified state, i.e.,
the velocity, displacement, or acceleration with reference to the
ground. A vibration transducer is connected to the machine element
in motion, and it gives an output signal proportional to the
variational input. The transducer should be independent of its
application, i.e., it should function equally well whether it is
connected to a vibrating structure on the ground, in an aircraft or
in a space vehicle. Sound may be classified as a vibratory
phenomenon, and we shall discuss some of the important parameters
used for specification of sound level. The measurement and analysis
of sound levels (or signals) is very specialized subject, which are
becoming increasingly important in modern rotating machinery
design.
Machinery acoustics and vibrations are measured to monitor the
condition of the machine. It enables to detect the machine fault so
that it can be corrected as soon as possible. High levels of noise
and vibration are indicative of high levels of component stress,
high noise levels, and reduced machine fatigue life. Measurements
are usually taken of the system acoustics and vibration amplitude,
its phase and its frequency. The acoustics and vibration may be
composed of several sinusoidal signals all at
-
899
different frequencies and it is necessary to distinguish the
components signals from each other. These measurements can be
processed and displayed in such a way as to enable judgments to be
made about the condition of the machine. It will help in the
diagnosis of some fault conditions by estimation of dynamic
parameters of machine components and of faults.
When we investigate the causes of vibration, we first
investigate the relationship between frequencies and the rotational
speed. We can do such spectrum analysis using spectrum analyzer
equipments (i.e., by the Fast Fourier transformation). Spectrum
analysers have various convenient functions, such as the tracking
analysis, Campbell diagram, and waterfall diagram. In tracking
analysis, dynamic characteristics of a rotating machine are
investigated by changing the rotational speed. A Campbell diagram
is the variation of whirl frequency with respect to the rotor spin
speed. A waterfall diagram is a 3-dimensional plot of the spectra
at various speeds.
Vibration testing of rotors involves availability of various
hardware and software components such as one schematically shown in
Figure 15.1, which shows a typically layout for a simple
measurement system. Basically, there are three main measurement
mechanisms: (i) the exciter mechanism, (ii) the sensing mechanism
and (iii) the data acquisition, conditioning and processing
mechanism. In the present chapter all of these modules (except test
rigs) will be described in detailed.
Figure 15.1 A simple measurement system
15.1 Specifications of Measuring Instruments An important part
of the performance of rotating machinery depends on the efficiency
of the vibration (displacement, velocity, acceleration, and force)
sensors that are used. In order to measure the position of a
rotating rotor, contact-free sensors must be used which, moreover,
must be able to measure a rotating surface. Consequently, the
geometry of the rotor, i.e., the surface quality, and the
homogeneity of the material at the sensor location will also
influence the measuring results. A bad surface will thus produce
noise disturbances, and geometry errors may cause disturbances with
the rotational frequency or with multiples thereof. Some of the
terms which are used in specifications of measuring systems are
described below:
Readability: of an instrument indicates the closeness with which
the scale of the instrument may be read; an instrument with a 10 cm
(or 120o) scale would have a higher readability than an
instrument
Exciter mechanism
Test rig
Sensing mechanism
Data acquisition, conditioning
and processing mechanism
-
900
with a 5 cm (or 60o) scale for the same range of a measuring
parameter (e.g., 100 m or 10 m/s or 100 m/s2, or 1 kN). With modern
digital display (e.g., liquid crystal display: LCD) readability is
related with the display size and its brightness relative to the
varied ambient brightness (e.g., in the aircraft cockpit).
Least count: It is the smallest difference between two
indications that can be detected on the instrument scale. It
depends upon the scale length, spacing of graduations, size of
pointer, and parallax effects. For the digital display it is the
lowest decimal point of the measured physical quantity that can be
seen in the display.
Hysteresis: An instrument exhibit hysteresis when there is a
difference in readings depending on whether the value of the
measured quantity is approaches from above or below. It may be the
result of mechanical friction, inertia, magnetic effects, elastic
deformation, or thermal effects.
Accuracy: For an instrument it indicates the deviation of the
reading from a known input (true values). The deviation is called
the error. In many experimental situations we may not have a known
value with which one can compare instrument readings and yet we
feel fairly confident that the instrument is within the plus or
minus range of the true value. In such cases the plus or minus
range expresses the
uncertainty of the instrument. Accuracy is usually expressed as
a percentage of full-scale readings, for
example, for a 100 m displacement dial gauge with an accuracy of
1 percent would be accurate
within 1 m over the entire range of the gauge.
Precision: The precision of an instrument indicates its ability
to reproduce a certain reading with a given accuracy. The
difference between the instruments reported values during repeated
measurements of the same quantity. As an example of the distinction
between precision and accuracy, consider the measurement of a known
speed of 1000 rpm with a tachometer. Five readings are taken, and
the indicated values are 1040, 1030, 1050, 1030 and 1050 rpm, which
has maximum deviation from the actual value of 50 rpm, average
value of 1040 rpm, and the maximum deviation from the measured mean
value is 10 rpm. From these values, it is seen that the instrument
could not be
depended on for an accuracy of better than 5 percent
(50100/1000), while a precision of 1 percent (10100/1040 = 0.96 1%)
is indicated, since the maximum deviation from the mean reading of
1040 rpm is only 10 rpm. It may be noted that the instrument could
be used to dependably measure speed
within 10 rpm. Hence, the accuracy gives the measure of absolute
error, and the precision gives that
of the relative error. This simple example illustrates that the
accuracy can be improved up to but not beyond the precision of the
instrument by calibration.
-
901
Resolution: In addition to the useful signal, each sensor system
produces noise disturbances in the output signal. The smallest
increment of change in the measured value that can be determined
from the instruments readout scale is called the resolution.
Typically this value is often on the same order as the precision;
sometimes it is smaller. The minimum value of the useful signal,
which can be distinguished from the noise disturbance (mostly
peak-to-peak value of the noise disturbance), is called resolution.
The resolution is usually indicated in absolute values - for
instance in m for a displacement sensor. Noisy signal cannot be
improved by resolution, however, can often be improved by low-pass
filters at the expense of the frequency range. For spectrum
analyzer a resolution of 1 Hz is very common.
Sensitivity: The change of an instrument or transducers output
per unit change in the measured quantity. A more sensitive
instruments reading changes significantly in response to smaller
changes in the measured quantity. Typically, an instrument with the
higher sensitivity will also have finer resolution, better
precision, and higher accuracy. The sensitivity indicates the ratio
of the signal over
the quantity to be measured: for a displacement transducer, for
instance, it is indicated in mV/m. For example, a 7.8-mV proximity
voltage is equivalent to 1-mm displacement then its sensitivity
would be 7.8 mV/mm; here it is assumed that the measurement is
linear for the give displacement. Similarly, for the velometer,
accelerometer and force transducers the sensitivity are indicted in
mV/m-s-1,
mV/m-s-2 (or C/m-s-2) and mV/N, respectively. Generally, a
operating (linear) range of the measurement is specified with the
transducers. The sensitivity can be enhanced by the electronic
amplification of the output signal.
Calibration: The calibration of all instruments is important,
since it checks the instrument against a known standard (comparing
with another instrument of known accuracy, i.e., the accuracy of
the instrument must be specified by a reputable source) or known
input source (direct calibration with a primary or alternative
measurement procedure) and subsequently to reduce error in
accuracy. It is the calibration, which firmly establishes the
accuracy of the instruments. In principle, the calibration has to
be performed before taking important measurement or at least
periodically to ensure quality of the measured data. Since during
operation due to improper handling of instrument, there is
possibility of damage.
Measuring range: The output signal of a sensor changes according
to a physical effect as a function of the measured quality. The
range in which the output signal can be used often corresponds to
that range having an approximately linear correlation between
measured quality and output signal (i.e., the specified sensitivity
is valid in this range of operation). This linear measuring range
can be considered smaller than the physical one, where nonlinear
effects also will be there. For example, the proximity
-
902
sensor can have linear measuring range as 0 to 1 mm, and for
general purpose accelerometers the range would be up to 1000 g (1 g
= 9.81 m-s-2).
Linearity: The linearity is usually represented as a percentage
of the maximum measuring range. It shows to what extent the
measured quantity deviates from a linear relationship between
measured quantity and output signal.
Frequency range: A linear frequency response, i.e., a
sensitivity independent of the frequency, is necessary in some
applications, especially when working with the displacement and
accelerometer transducers. The frequency with a sensitivity reduced
by 3 dB is usually called cut-off frequency. One must consider here
that the output signal at the cut-off frequency, depending on the
transducer, may already show a significant phase lag. For general
purpose accelerometers the frequency range of operation is usually
up to 1.2 kHz with resonance frequency of the accelerometer of the
order of 40 kHz.
Impedance matching: In many experimental setups it is necessary
to connect various items of electrical equipment in order to
perform the overall measurement objective. When connections are
made between electrical devices, proper care must be taken to avoid
the impedance mismatching.
Figure 15.2 Two-terminal instrument with internal impedance Ri
and external load of R
The input impedance of a two-terminal device may be illustrated
as in Figure 15.2. The device behaves as if the internal resistance
Ri is connected in series with the internal voltage source E. The
connecting terminals for the instrument are designated as A and B,
and the open circuit voltage presented at these terminals is the
internal voltage E. Now, if an external load R is connected to the
device and the internal voltage E remains constant, the voltage
presented at output terminals A and B will be dependent on the
value of R. The potential presented at the output terminals is
11 ( / )AB i i
RE E ER R R R
= =
+ + (15.1)
-
903
The larger the value of R, the more closely the terminal voltage
approaches the internal voltage E.
Thus, if the device is used as a voltage source with some
internal impedance, the external impedance (or load) should be
large enough that the voltage is essentially preserved at the
terminals. If one wish to deliver power from the device to the
external load R. The power is given as
2ABEPR
= (15.2)
The value of the external load that will give the maximum power
for a constant internal voltage E and internal impedance Ri can be
obtained as follows. On substituting equation (15.1) into equation
(15.2), we get
( )22 2
2i i
E R E RPR R R R R
= =
+ + (15.3)
and the maximizing condition 0dPdR
= is applied. It results in
( ) ( ) ( ) ( )2 2 2
2 2 32 0; 2 0; i
i i i
dP d E R E E R R R RdR dR R R R R R R
= = + = + = + + +
iR R= (15.4)
That is, the maximum amount of power may be drawn from the
device when the impedance of the external load just matches the
internal impedance. This is the essential principle of impedance
matching in electric circuits. The internal impedance and external
load of a complicated electronics device may contain the inductive
and capacitive components that will be important in AC transmission
and dissipation. However, the basic idea is the same. The general
principle of impedance matching is that the external impedance
should match the internal impedance for maximum energy transmission
(minimum attenuation), and the external impedance should be large
compared with the internal impedance when a measurement of internal
voltage of the device is desired.
The impedance matching can be important in mechanical systems
also. Consider a simple spring-mass system as a mechanical
transmission system. From the frequency response function
describing the system behaviour, it is seen that frequencies below
the natural frequency are transmitted through the system, i.e., the
force is converted to displacement with a little attenuation. Near
the natural frequency undesirable amplification of the signal is
performed and above this frequency severe attenuation is
-
904
present. It is a case of a system that exhibit behaviour
characteristics of a variable impedance, which is the
frequency-dependent. When it is desirable to transmit mechanical
motion through a system, the natural frequency and damping
characteristics must be taken into account so that good matching is
present.
15.2 Uncertainty Analysis of Estimated Parameters Uncertainty of
the test data is a result of the individual uncertainties inherent
with each instrument. The method described by Holman (1978) is
briefly described here to estimate the uncertainty in rotor dynamic
parameters (RDPs). The method is briefly stated as follows. Let the
results R (e.g., RDPs) is given as the function of independent
variables 1 2, , , nx x x (e.g., the rotor speed, inlet pressure,
pressure drop, diameter, length, clearance, temperature, force,
excitation frequency, displacement, acceleration, etc.). Thus,
( )1 2, , , nR R x x x= (15.5)
Let Rw be the uncertainty in the result and 1 2, , , nw w w be
the uncertainties in the independent
variables. Then the uncertainty in the result is given as
1/ 222 2
1 21 2
R nn
R R Rw w w w
x x x
= + + +
(15.6)
with
( ) ( )1 1 11 1
R x x R xRx x
+ =
;
( ) ( )1 1 11 1
R x x R xRx x
+ =
; (15.7)
where 1 2, , , nx x x are the small perturbations of the
independent variables. It should be noted
that the uncertainty propagation in the results Rw predicted by
equation (15.7) depends on the squares of the uncertainty in the
independent variables ( )1,2, ,kw k n= . This means that if the
uncertainty in one variable is significantly larger than the
uncertainties in the other variables, then it is the largest
uncertainty that predominates and other may probably be negligible.
The relative magnitude of uncertainties is evident when one
considers the design of an experiment, procurement of instrument in
force, excitation frequency, displacement, and acceleration
measurements on the rotor dynamic parameters.
Example 15.1: A voltage is impressed on the resistor R and the
power dissipation is to be calculated in two different ways (i)
from P = E2/R and (2) from P = EI. In (1) only voltage measurement
will be
-
905
made, while both current and voltage will be measured in (2).
The register has a nominal stated value of 5 1 percent. Calculate
the uncertainty in the power determination in each case when
the
measured values of E and I are: E = 50 V 1% (for both cases) and
I = 5 A 1%.
Solution: The schematic of the power measurement across a
register R is shown in Figure 15.3. The
uncertainty of voltage and current would be 50 0.01 0.5Ew = = V
and 5 0.01 0.05Iw = = A.
Figure 15.3 The power measurement across a register
Case (1) For the first case P = E2/R, we have two independent
parameters to be measured that is E and R, which will have the
uncertainty. Hence, the uncertainty in the power measurement would
be
( ) ( ) 1/ 22 22 2, ,p E R
P E R P E Rw w w
E R
= +
(a)
with
( ), 2P E R EE R
=
and
2
2( , )P E R E
R R
=
(b)
On substituting equation (b) into equation (a), the uncertainty
in the power could be written as
1/ 222 22 2
22
p E RE E
w w wR R
= +
or
1/ 22 2
4p E Rw w w
P E R
= +
(c)
Inserting numerical values for the uncertainty, we get
( ) ( ) ( ) ( )1/ 2 1/ 22 2 2 2100 100 4 5/ 50 0.5 / 5 100 4
0.01 0.01 2.24%pwP
= + = + =
(d)
(2) For the second case P = EI, we have two independent
parameters to be measured that is E and I, which will have the
uncertainty. Hence, we have
( ),P E II
E
=
and ( , )P E I E
I
=
(e)
-
906
On using equation (e), the uncertainty in the power could be
written as
( ) ( ) 1/ 22 22 2, ,p E I
P E I P E Iw w w
E I
= +
or ( ) ( ) 1/ 22 22 2P E Iw I w E w = + or
1/ 22 2P E Iw w w
P E I
= +
(f)
On substituting numerical values for the uncertainty, we get
( ) ( ) 1/ 22 2100 100 0.01 0.01 1.414%PwP
= + =
(g)
Since, calculations are based on percentage that is why actual
values of various parameters have no effect on the final
uncertainty. However, the second method of power determination
provides considerably less uncertainty than the first method, even
though the primary uncertainties in each quantity are the same. In
this example, the uncertainty analysis is that it affords the
individual a basis for selection of a measurement method to produce
a result with less uncertainty. It should be noted that from
transducers generally we get these electrical parameters only and
with the sensitivity
subsequently it is converted to vibration parameters.
Example 15.2 In most of the practical voltmeter an internal
resistance Rm is always present. The power measurement in Example
15.1 is to be conducted by measuring the voltage and the current
across the resistor with circuit shown in Figure 15.4. Calculate
the nominal value of the power
dissipated in R and the uncertainty for the following
conditions: R 120 , Rm = 1200 5%, I = 5
A 1% and E = 500 V 1%.
Figure 15.4 Effect of the meter impedance on the measurement
Solution: The uncertainty in various parameters are: 500 0.01
5Ew = = V, 5 0.01 0.05Iw = = A, and
1200 0.05 60mR
w = = . Let I1 and I2 are currents flowing through registers R
and Rm, respectively. A
current balance on the circuit gives
-
907
1 2I I I+ = m
E E IR R
+ = 1
m
EI IR
= (a)
The power dissipated in the resistor R is give as
2
1m
EP EI EIR
= = (b)
so that
2m
P EIE R
=
,
P EI
=
,
2
2m m
P ER R
=
(c)
The nominal value of the power is thus calculated as
2500500 5 2292 W1200
P = = (d)
In terms of known quantities the power has the functional from (
, , )mP f E I R= and so the uncertainty for the power is now
written as
( )1/ 2 1/ 22 2 22 2 2
22 2 2 2 2 22
2m mp E I R E I R
m m m
P P P E Ew w w w I w E w w
E I R R R
= + + = + +
(e)
On substituting the given numerical values in equation (d), we
get
( ) ( ) ( )
[ ]
1/ 222 22 2 22
2
1/ 2
2 500 5005 5 500 0.05 601200 1200
434 625 108.5 34.2 W
pw = + +
= + + =
(f)
or
34.2100 100 1.49%2292
Pw
P= =
(g)
From equation (f), the order of influence on the final
uncertainty in the power are as follows: (i) the uncertainty of the
current determination, (ii) the uncertainty of the voltage
measurement, and (iii) the uncertainty of the knowledge of internal
resistance of the voltmeter. However, it should be noted from
-
908
equation (15.1) that this results from the fact that mR R ( 10
)mR R= . Moreover, if the uncertainty
in one variable is significantly larger than the uncertainties
in the other variables, say, by a factor of 5 or 10, then it is the
largest uncertainty that predominates and others may probably be
neglected. The relative magnitude of uncertainties is evident when
one considers the design of an experiment, procurement of
instrumentation, etc. Very little is gained to reduce the small
uncertainties. Because of the square propagation it is the large
ones that predominate, and any improvement in the overall
experimental technique connected with these relatively large
uncertainties.
A simple device for the measurement of the vibrational frequency
is shown in Figure 15.5(a). The small cantilever beam mounted on
the block is placed against the vibrating surface to provide a base
excitation (Figure 15.5(b)). Provision is made to varying the beam
length, which in term is expected to vary its natural frequency.
When the beam length is attuned so that its natural frequency is
equal to the frequency of the base excitation, the resonance
condition as shown in Figure 15.5(c) will result, which can be
visualize by naked eye also. The aim would be to measure the length
of the beam each time we attune the resonance condition to obtain
the natural frequency. However, due to uncertainty in measurement
of the beam length would lead to uncertainty in the measurement of
the frequency. It
should be noted that there could be so many other uncertainty
(e.g., uncertainty in attuning the resonance, etc.) that might
affect the uncertainty of frequency measurement, however, for
brevity only a single uncertainty have been considered.
(a) No excitation (b) Excitation other than n (c) Excitation at
n Figure 15.5 Cantilever beam used as frequency measurement
device
Considering the beam as a continuous system, the fundamental
natural frequency of the beam is given by
43.52nfEI
mL =
(15.8)
where n is the natural frequency in rad/s, E is the Youngs
modulus in N/m2, I is the moment of
inertia in m4, m is the beam mass per unit length in kg/m and L
is the beam length in m. We use
-
909
equation (15.8) to determine the allowable uncertainty in the
length measurement in terms of the uncertainty in the frequency
measurement. From equation (15.8), we have
37.04nf EI
L L m
=
(15.9)
The uncertainty in the natural frequency is given by
1/ 222
nfnf
Lw wL
= (15.10)
where Lw is the uncertainty in length measurement in m. On
substituting equation (15.9) into equation (15.10), and after
simplification we obtain
3
7.04 /nf
L
w Lw
EI m
= (15.11)
Now with an example the above method will be illustrated.
Example 15.3 A 0.6 mm diameter spring-steel rod to be used for a
vibration-frequency measurement as shown in Figure 11.5(a). The
length of the rod may be varied between 60 mm to 200 mm. The mass
density of this material is 7800 kg/m3 and the modulus of
elasticity is 112.1 10 N/m2. Calculate
the range of frequencies that may be measured with this device
and the allowable uncertainty in L at 200 mm in order that the
uncertainty in the frequency is not greater that 2 percent. Assume
the material properties are known exactly.
Solution: We have
112.1 10E = N/m2; 4 4 3(0.6) 6.362 1064 64
I dpi pi = = = mm
4 156.362 10= m4
( )2 62 37800 0.6 10 2.205 104 4
dm
pipi
= = = kg/m (a)
For L = 60 mm, from (15.8), we have
-
910
( ) ( ) 1/ 211 154 3 4
2.1 10 6.362 103.52 3.52 761.1
2.205 10 0.06nfEI
mL
= = =
rad/s (b)
Similarly, for L = 200 mm, we will have 68.50n = rad/s. Hence,
the range of the frequency is from
68.50 rad/s to 761.1 rad/s.
For L = 200 mm, we have 0.02 68.50 1.37nf
w = = . From equation (15.11), we have the allowable
uncertainty in the measurement of length
3 33
11 15 3
1.37 0.2 1.999 107.04 / 7.04 2.1 10 6.362 10 / 2.205 10
nfL
w Lw
EI m
= = =
m = 2.0 mm (d)
Hence, the uncertainty of 200 1% would be allowable.
15.3 Transducers: A large number of devices transform values of
physical variables into equivalent electric signals and such
devices are called transducers (e.g., LVDT (linear variable
differential transformer) gauges, eddy current, inductive,
capacitive, piezoelectric, photoelectric, photoconductive, pressure
transducers, nuclear radiation detectors, etc.). For measuring
motion, there are two basic types of transducers; the first being
the seismic that produces a signal proportional to the absolute
motion in space; and the second a signal proportional to the
relative motion between a reference point and the point of
interest. Most of displacement sensors are based on the relative
motion, and most of accelerometers are based on the absolute
motion.
15.3.1 Displacement Sensors a. Potentiometer. The simplest form
of displacement transducer is the potentiometer. Although they are
available for measurement of the linear and rotational
displacements, they tend to be noisy and are only suitable for
relatively low frequency and large displacement applications.
b. Linear Variable Differential Transformers (LVDT): This is
another form of displacement transducer that has been used
successfully for vibration measurements for many years. The
principle of operation of an LVDT is that a freely-moving magnetic
core is used to link the magnetic flux between a surrounding
primary coil and two secondary coil as shown in Figure 15.6. A
schematic cross-section of an LVDT is shown in Figure 15.7.
The primary coil is energised by an external AC source. The
alternating magnetic flux induces voltages at the null position are
equal in magnitude but opposite in phase. When these two coils
are
-
911
connected together, the net output of the transducer at the
central position is zero. As the magnetic core is moved away from
the central position the induced voltage in one of the secondary
coils increases. At the same time, the induced voltage in the other
coil decreases, resulting in a differential voltage output that
varies linearly with the position of the magnetic core. In moving
from one side of the central position to the other, the polarity of
the demodulated output changes instantaneously. The
core has a small rod and is separated from the coil structure by
a low friction lining that produces an almost frictionless device
that is insensitive to radial motion of the core. For vibration
measurements, the core is usually connected to the structure via a
push-rod (stinger). The push-rod has two functions: to decouple
lateral motion and to provide a convenient method of attachment to
the structure. To maintain the calibration of the device, the
push-rod should be nonmagnetic.
Figure 15.6 Schematic diagram of a differential transformer
Figure 15.7 A typical construction of a linear variable
differential transformer (LVDT)
c. Rotary Variable Differential Transformers (RVDTs): It is used
for measurement of the angular displacement. RVDT is an
electromechanical transducer that provides a variable alternating
current (AC) output voltage that is linearly proportional to the
angular displacement of its input shaft. When
-
912
energized with a fixed AC source, the output signal is linear
within a specified range over the angular displacement. RVDT
utilizes brushless and non-contacting features to ensure long-life
and reliable, repeatable position sensing with very high
resolution.
d. Proximity transducers (Relative motion transducers):
Proximity transduers use sensors that are able to detect the
presence of nearby objects without any physical contact. A
proximity transduers often emits an electromagnetic or
electrostatic field, or a beam of electromagnetic radiation
(infrared, for instance), and looks for changes in the field or
return signal. The object being sensed is often referred to as the
proximity sensor's target. Different proximity transduers targets
demand different sensors.
For example, a capacitive or photoelectric sensor might be
suitable for a plastic target; an inductive proximity sensor
requires a metal target. The relative-motion transducers are the
proximity probe type, which sense the gap (i.e., the displacement)
between the mounting point (usually the bearing housing) and the
point of interest (usually the rotating shaft). Proximity probes
are widely used on the turbo-machinery as the sensor for permanent
monitoring systems. They are particularly suitable for such
machineries, where there are small internal clearances.
Displacement sensors are necessary to detect the radial (and
sometimes axial) movement of the rotor. The requirements for a
displacement sensor are as follows: (i) wide frequency response,
(ii) low noise, (iii) low interference noise, (iv) low temperature
drift, (v) good linearity, (vi) compactness, and (vii) reliability.
Displacement sensors detect the linear position during the movement
of an object without a mechanical contact. When selecting the
displacement sensors, depending on the application, measuring
range, linearity, sensitivity, resolution and frequency range are
to be taken into account as well as: temperature range, temperature
drift of the zero point and sensitivity; noise immunity against
other sensors, magnetic alternating fields of the electromagnets,
electromagnetic disturbances from switching amplifiers, dust,
aggressive media, or vacuum.
There are basically three types of displacement transducers:
electromagnetic, capacitive, and optical.
Brief outline of each of these transducers will be discussed
now.
-
913
Figure 15.8 Principle of a displacement sensor Figure 15.9 An
equivalent circuit of the displacement sensor
(i) Electromagnetic displacement transducers: Electromagnetic
displacement transducers are of two types. First is the inductive
while the other is the eddy current type. Figure 15.8 shows the
structure and the principle of operation of an electromagnetic
displacement sensor. An E-shaped magnetic core has a winding with
two terminals. A target (i.e., the rotor shaft) is drawn as a
rectangular solid having air gap. The input impedance, Zin, at the
terminals varies with the air gap. When input terminals are
excited by a high frequency voltage then the coil impedance will
be dominated by the inductance (which is the variable part of the
impedance: An electric current i flowing around a circuit produces
a magnetic field and hence a magnetic flux through the circuit. The
ratio of the magnetic flux to the current is called the inductance,
or more accurately self-inductance of the circuit); and it is
obtained by detecting the terminal voltage and the current. Figure
15.9 shows the equivalent circuit of a sensor winding. The
inductance L0 is a constant while the inductance L1 is dependent on
the length of air-gap.
Inductive displacement transducers: An inductive transducer is
an electronic proximity sensor, which
detects metallic objects without touching them. An inductor coil
placed in a ferrite core is a part of an oscillating circuit
(Figure 15.10). The excitation frequency is in the range of 20-100
kHz and the inductance varies as a function of air gap length
(approximately inversely). If the air gap is small then there is a
high impedance. When a ferromagnetic object (of high permeability
such as laminated silicon steel, ferrite and carbon steel)
displacement to be measured approaches the coil the inductance
changes and the oscillating circuit is detuned. The signal is
demodulated and linearised and becomes proportional to the gap
between the sensor and the object of which the displacement to be
measured. Two sensors opposing each other are frequently arranged
on a rotor (Figure 15.11). They are operated differentially in a
bridge circuit with a constant bridge frequency, producing a nearly
linear signal.
Inductive sensors are operated with modulation frequencies from
approximately 5 kHz up to 100 kHz.
-
914
The cut-off frequency of the output signal lies in a range
between one tenth and one fifth of the modulation frequency.
Figure 15.10 Inductive displacement sensor Figure 15.11
Differentially measuring sensors
Eddy-current transducers: The transducer function by detecting
changes in the eddy current loss as the gap between the probe and
the target surface varies (Fig. 15.12). The high-frequency
alternating current runs through the air-coil cast in a housing.
The electromagnetic coil section induces eddy currents in the
conducting object (of low resistance such as copper, non-magnetic
stainless steel, aluminum, carbon steel and other metallic
material) to be measured, thus absorbing energy from the
oscillating circuit. Depending on the clearance, the amplitude of
oscillation varies. This amplitude variation will provide a voltage
variation proportional to the clearance, once it is
demodulated,
linearised and amplified. The usual modulation (excitation)
frequency lies in a range of 1-2 MHz and have measuring frequency
ranges of approximately 0 Hz up to 20 kHz.
Figure 15.12 An eddy current displacement sensor
-
915
Precautions and limitations: In-homogeneities in the material of
the moving rotor cause disturbances (noise) and reduce the
resolution accordingly. If the target is close to the sensor core
then eddy currents are induced into the target which reduces the
flux (almost as a short-circuit transformer) and produces a low
input impedance. As the target moves away, the coupling decreases
which increases the input impedance (which is opposite to the
inductive type). Manufacturers usually indicate the sensitivity
used on aluminum. When measuring steel, sensitivity is smaller.
Shielded sensors must be used for applications where high frequency
magnetic field occurs. Sensors may cause mutual interference.
Therefore, the minimum clearance between sensors is mostly defined
in the mounting
guide. Within the linear range, which typically extends from
250-2250 m gap, current standards
require either a 4 mV/m or 8 mV/m proportionality between gap
and voltage. Thus, a 250 m
change in gap should produce a voltage change of 1 volt at 4
mV/m or 2 volts at 8 mV/m (some times instead of mm the mils; is
used and the mils is one thousands of an inch). The standard
sensitivity for these transducers is 8 mV/m (or 8.0103 mV/mm) for
the normally used target materials (e.g., steel).
The extension cable and oscillator demodulator of the transducer
make up a turned resonant circuit. In order to establish and
maintain a constant ratio between gap and voltage, the transducer,
oscillator demodulator and extension cable must be properly matched
and calibrated. Most manufacturers will specify the type of probe,
generally the tip diameter, and the total electrical length of the
extension and probe cables, which must be used, with each
oscillator demodulator.
The slope of the curve, the linear range, and the DC output
corresponding to a given gap will vary with changes in a targets
conductivity and permeability. If a probe and oscillator
demodulator calibration for 4140 steel are used without
recalibration on a material as such stainless steel, the curve
shifts to the left, producing a higher-output voltage for a given
gap. Due to this shift and potential inaccuracies, a non-contact
probe system calibrated for one material should not be used with
another without recalibration.
Temperature may also affect the range limits of a non-contact
probe and the DC output at a given gap;
however, the shift is generally small across the temperature
range experienced within a bearing housing. Elevated pressures may
affect the sensitivity of a non-contact probe. If the probe is
installed in an area of high or fluctuating pressure, its response
should be tested in actual environment to determine what changes in
sensitivity or output will occur. With everything else equal, the
maximum linear range obtainable with a non-contact displacement
measurement system will increase with increasing probe tip diameter
and likewise increase with an increasing supply voltage. At a
sensitivity
of 8 mV/m , linear range of typical non-contact measuring
systems observing 4140 steel will vary
-
916
from approximately 1525 m with 5 mm tip diameter and -18 VDC
supply to 2160 m with a 8 mm tip diameter and 24 VDC supply.
When the target is moving surface such as the periphery of a
shaft, the displacement measuring system cannot distinguish between
shaft motion or vibration and defects such as scratches, dents and
variations in conductivity or permeability. As a result, the
output, rather than being pure vibration, is the sum of vibration
and all surface variations passing beneath the probe. Since the
magnetic field of the probe penetrates the surface of the observed
material, any repair which results in an interface between two
materials (when the shaft is plated or metal sprayed) will
introduce distortion in the output signal measured by an eddy
current displacement transducer.
Eliminating excessive runout is often a very difficult task. The
first and obvious step is during manufacture when every effort must
be taken to ensure the surface which will be observed by the shaft
probe is concentric with the journal, has a smooth finish, and is
protected from damage during handling and assembly. Produced when
the shaft is machined, ground or degaussed incompletely following a
magnetic particle inspection, electromagnetic runout can generally
be eliminated by degaussing the shaft surface observed by the
probe. If after degaussing, runout persists despite a smooth and
concentric shaft surface, it is likely due to a changing
permeability or conductivity around the circumference of the shaft.
Often a problem with high-alloy, precipitation-hardened shafts,
this type of runout has been successfully reduced by burnishing the
area- running it on balancing machine
rollers or producing similar effect in a lathe with a special
roller tool.
Should all these steps fail or be impossible to implement for
one reason or other, runout can be eliminated electronically with a
runout subtractor on-line or off-line. The runout subtractor
digitally memories a phase-referenced shaft motion at slow roll
when all motion is assumed to be runout then automatically
subtracts the slow-roll waveform from the raw waveform observed by
the probe to produce a corrected waveform representative of actual
shaft motion. The off-line procedure is explained in more details
in chapter related to balancing of rotors.
Inductive sensors are not as sensitive as the eddy current
sensors and it is advantageous. However, there are disadvantages
with inductive sensors : (a) There are very few variety of
inductive sensors available in the market as it is very costly, (b)
The shaft target ring has to be made from laminated silicon
steel.
In eddy current sensors, further improvements of the output
voltage are possible: (a) Two eddy current sensors can be placed on
one axis and the output voltages are subtracted to give
differential operation. The second harmonic and even harmonics are
reduced and temperature drift is decreased.
-
917
(b) The target material can be replaced with non-magnetic
stainless steel (or copper) to avoid magnetic imperfections.
However, sensor amplifier linearity should be taken care of. (c)
The sensor head diameter should be small so that the target ring
does not interfere with the two-axis movement. For example, the
sensor head diameter should be 5 mm or less for a target ring
diameter of 50 mm. On the other hand, the sensor head diameter
should be large compared to the air gap length for better
sensitivity and linearity. For example, a head diameter of 5 mm
should be used for 1 mm air gap length or less, and (d) the
excitation frequency of the x-, y- and z-axis sensors should be set
far apart to avoid interference. The difference in excitation
frequency should be greater that the frequency range of the sensor.
For example, for a sensor response range of 20 kHz, when the x-axis
sensor is
excited at 2 MHz the frequencies of the other sensors should be
less than 1.96 MHz or higher than 2.04 MHz to provide enough
frequency separation.
(ii) Capacitive displacement transducers: In capacitive
proximity sensors, the sensed object changes the dielectric
constant between two plates. The capacity of a plate capacitor
varies with its clearance.
Therefore, a good isolation between the sensor and the shaft is
necessary. In addition, the air must be clean, and the oil and
other particles should not be present because this will affect the
dielectric. Using the capacitive measuring method, the sensor and
the opposing object to be measured form one electrode of a plate
capacitor each (Figure 15.13). Within the measuring system, an
alternating current with a constant frequency runs through the
sensor. The voltage amplitude at the sensor is proportional to the
clearance between the sensor electrode and the object to be
measured, and is demodulated and amplified by a special
circuit.
A proximity sensor has a range, which is usually quoted relative
to water. Because changes in
capacitance take a relatively long time to detect, the upper
switching range of a proximity sensor is about 50 Hz. The proximity
sensor is often found in bulk-handling machines, level detectors,
and package detection. One advantage of capacitive proximity
sensors is that they are unaffected by dust or opaque containers,
allowing them to replace optical devices. In addition, the air must
be clean, and the oil and other particles should not be present
because this will affect the dielectric. A typical capacitive
proximity sensor has a 10-mm sensing range and is 30 mm in
diameter. The proximity sensor incorporates a potentiometer to
allow fine tuning of the sensing range and can repetitively detect
objects within 0.01 mm of the set point. Switching frequency is 10
Hz, and operating temperature range is 30 to 70C. A proximity
sensor that measures current flow between the sensing
electrode and the target provides readouts in appropriate
engineering units. Usually, one side of the voltage source or
oscillator connects to the sensing electrode, and the other side
connects through a current-measuring circuit to the target, which
generally is a metal part at earth or ground potential.
-
918
Figure 15.13 Capacitance displacement sensor
Probes used with a capacitive proximity sensor have either a
flat disc or rectangular sensing element surrounded by a guard
electrode that provides electrical isolation between the proximity
sensor and its housing. The guard also ensures that the lines of
electrostatic field emanating from the probe are parallel and
perpendicular to the surface of the proximity sensor. Capacitance
proximity sensor
systems can make measurements in 100 sec with resolutions to
0.001 micron, however, the
ccommercially available capacitive displacement measuring
systems are expensive. The bandwidth of the output signal ranges
between approximately 5 kHz and 100 kHz. The electrostatic charging
of the contactless rotor may cause interferences too. The sensors
are sensitive to dirt which modifies the dielectric constant in the
air gap.
(iii) Optical displacement transducers: The simplest principle
of an optical displacement sensor consists of covering a light
source opposite to a light-sensitive sensor by the object to be
measured (Figure 15.14). The resulting difference in the light
intensity is converted into an electric signal and serves as a
measurement for the position of the object. By selecting
appropriate light sources, light sensors and suitable apertures, we
obtain a nearly linear displacement signal. A similar approach
consists of reflecting light by the object to be measured. The
fraction of light received by the sensor changes according to the
motion of the object (Figure 15.15). For this kind of system photo
diodes, photo transistors, photo resistors, and photo-electric
cells can be used as sensors. The wavelength of the light source
should be adjusted to the sensor.
-
919
Figure 15.14 Light barrier principle Figure 15.15 Light
reflecting principle
Another possibility is the application of an image sensor.
Charge-coupled device (CCD) is an electronic memory that records
the intensity of light as a variable charge. Widely used in still
cameras, camcorders and scanners to capture images, CCDs are analog
devices. Their charges equate to shades of light for monochrome
images or shades of red, green and blue when used with colour
filters. Take for example a line array camera (CCD sensor) in a
rotor system (Figures 15.16). The rotor image is reflected both for
the x- and the y- direction over a mirror on a CCD sensor. The
picture of the rotor, tinted black in front of a lit-up background,
is converted into a video signal. By counting the pixels
(light-sensitive dots) until the light-dark boundary is reached one
obtains a digital displacement signal. However, optical
displacement measuring systems are not appropriate for many
application fields, since they are very sensitive to dirt, and the
resolution is limited due to defraction effects.
Figure 15.16 An optical displacement sensor
Since the advent of the laser in the early 1960s the field of
optical metrology has provided accurate experimental data in
situations in which, previously, it would have been considered
unattainable. The technique of laser Doppler velocimetry (LDV) is
now well established and was initially applied to obtain
non-intrusive measurements in fluid flows by laser Doppler
anemometry (LDA; Durst et al.,
-
920
1981 and 1988). Although the use of LDV for solid surface
velocity measurement was recognized at an early stage, its
development in this area received little attention compared with
the effort in fluid mechanics. Accordingly, the measurements of
vibration was still extensively achieved with accelerometers or
other forms of transducer which rely on contact with the
measurement surface for successful operation. There are, however,
many cases of engineering interest where this approach is
either impractical or impossible. Typical examples are the
measurement of very hot or light surfaces, such as exhaust pipes or
loudspeakers, and measurement on rotating surfaces which prevent
their use.
In the area rotating surfaces the measurement of torsional
vibration of rotating components presented a particularly difficult
measurement problem. When designing rotating machinery components,
an engineer must be careful to suppress torsional oscillations,
since incorrect or insufficient control may lead to fatigue
failure, rapid bearing wear, gear hammer, fan belt slippage and can
produce associated excessive noise problems. Torsional oscillations
are a particular problem in engine crankshaft design where
torsional dampers are commonly used to maintain oscillations at an
acceptable level over the
working speed range of the engine. Torsional transducers have
formerly included optical, seismic and mechanical torsiographs,
strain gauges and slotted discs. The latter system has found common
use in the automotive industry and consists of a slotted disc fixed
to the end of the crankshaft. A proximity transducer monitors the
slot passing frequency, which is then demodulated to provide a
voltage analogue of the crankshaft speed and hence torsional
oscillations, but within a limited frequency range. Strain gauges
and associated telemetry or slip ring systems are disreputably
difficult to fix, calibrate and use successfully. In summary, the
measurement of torsional oscillations presented difficult problems
for contacting transducer technology and, of course, necessitated
machinery downtime and special arrangements being made for fitting,
calibration, etc. Very often, the cost of this
machinery stoppage would prevent a measurement being attempted,
even though the vibration engineer had concluded that it was vital
if a design improvement is to be made. There was therefore a real
need for a torsional vibration transducer which was user friendly
and could provide data immediately in on-site situations. It was
not until the advent of laser technology that a solution was found.
It allows the engineer to point low powered laser beams at a
rotating target component and obtain torsional vibration
information (Halliwell, 1996).
Laser optical range transducers: It operates on the principle
triangulation (Figure 15.17). A laser light beam reflected from the
surface of a structure is focused onto an internal photo-sensitive
device.
As the structure moves, the position of the focused spot on the
photo-sensitive device moves. The photo-sensitive device generates
a signal according to the position of the focused spot. This output
is then conditioned and linearised to give an analogue signal
proportional to the range of the surfaces motion. The displacement
is detected by reflecting laser light so that a uniform target
surface is required to prevent the noise.
-
921
Figure 15.17 The basic principle of the laser optical
sensors
A laser vibrometer is an optical system that can be used to
measure the instantaneous velocity of a
point (or points) on a structure. The instrument is a
non-contact device in which the velocity measured is the velocity
components in the direction of incident laser beam. The velocity is
measured by the
detection of the Doppler frequency shift (is the change in
frequency and wavelength of a wave for an observer (e..g, the
surface of shaft) moving relative to the source of the waves.) of
light scattered from the moving surface. Sophisticated optics and
signal processing mean that these devices are expensive. Scanning
systems are now available in which the laser beam can be moved
rapidly over a grid of measurement points on a structure. It is
possible to make finely detailed measurements on complex structures
that are not amenable to, or accessible for, conventional
transducers. Further development in laser measurement techniques
now enable the measurement of rotational responses (Tiwari et al.,
2005).
15.3.2 Accelerometers: The most widely used types of seismic
transducers give an output signal proportional to the acceleration.
Accelerometers contain usually piezo-electric crystals, which
are
loaded with a small inertia weight and rigidly mounted in a
casing. They produce a voltage output, which is proportional to the
acceleration over a wide frequency range, up to the point where the
output/(unit acceleration) starts to rise due to natural frequency
of the inertia weight supported on the crystal.
The primitive type of accelerometer had very high source
impedance (1010 Ohms) with all the cabling problems associated with
this. In recent years, accelerometers have become available with
the matching charge amplifier inbuilt within the accelerometer
casing. Often the size of these low-
impedance accelerometers is not very much larger then the
original version. They are no longer self-generating and need a dc
power supply to drive them (typically 18 V DC).
-
922
The primitive type of velocity transducer in the form of a
spring-mounted coil (resonant frequency in the region of 10 Hz),
producing a signal proportional to velocity, has become virtually
absolute. This is because of their limited frequency range, the
relatively large size and weight, and mounting problems, together
with problems of maintaining optimum damping necessary to obtain a
flat frequency response. Instead of this inductive type of velocity
transducer are available, some
manufacturers supply a piezoelectric velocity transducer with
the internal integration and the low impedance output (the
impedance is defined as the ratio of applied SHM force to resulting
velocity).
An accelerometer can have several parameters, which can be used
for the selection of the transducer. For example, sensitivity,
frequency range, residual noise level in the measuring range,
temperature range, maximum operational and shock levels, weight,
connectors, mountings, type of out put (charge /voltages), etc.
Accelerometers are available based on applications, e.g., the
general purpose, high sensitivity, high temperature, high frequency
(very small size), shock, human vibration, under water, modal
analysis, industrial, aerospace and flight test, special purpose
like the tri-axial and rotational
measurements, etc.
Figure 15.18 Simple spring-mass Figure 15.19 Schematic of
typical seismic instrument -damper system
The seismic instrument is a device that has the functional form
of the system shown in Figure 15.18, which is a single-DOF
spring-mass-damper system with the support motion. A schematic of a
typical instrument is shown in Figure 15.19. The mass is connected
through the parallel spring and damper arrangement to the housing
frame. This frame is than connected to the vibration source (e.g.,
bearing housing) whose characteristics are to be measured. From
Figure 15.18 using Newtons law of motion, we have
2 2 2 1 1my cy ky cy ky+ + = + (15.12)
-
923
where 1y and 2y are the absolute displacements of the housing
and the suspended mass, respectively.
It is assumed that the damping force is proportional to the
velocity. We assume that a harmonic motion is applied on the
instrument such that
1 1 cosy Y t= (15.13)
where Y is the displacement amplitude. The aim is to obtain an
expression for the relative
displacement ( )2 1y y in terms of this base motion. The
relative displacement is that which is detected by the transducer
shown in Figure 15.19. On substituting equation (15.13) into
equation (15.12) and rearranging it gives
2 2 2 cos cosc k k cy y y Y t tm m m m
+ + =
(15.14)
The solution to equation (15.14) is
( ) ( ) ( ) ( )( ){ }2
122 1 1/ 222 2 2
coscos sin
d d
cm
nf nfmY t
y y e A t B tk m c
= + +
+
(15.15)
where the damped natural frequency is given by
1/ 22
2dnfk cm m
=
for 1.0c
c
c . For low values of damping ratio the displacement
amplitude may become quite large. The output becomes essentially
a linear function of input at high-
frequency ratios (curve becomes relatively straight). Thus a
seismic-vibration pickup for measurement of displacement amplitude
should be utilized for measurement of frequencies substantially
higher than
its natural frequency. The instrument constants ccc and nf
should be known or obtain from the
calibration. The anticipated accuracy of measurement may than be
calculated for various frequencies.
The acceleration amplitude of the input vibration is
21 1 1a y Y= = (15.22)
We may thus use the measured output of the instrument as a
measure of acceleration. However, there
are restrictions associated with this application. In equation
(15.18) the bracketed term is the one that governs the linearity of
the acceleration response, since nf will be fixed for a given
instruments. In
Figure 15.21 we have a plot ( ) 22 1 1/nfY Y a versus / nf ,
which indicates the non-dimensionalised acceleration response.
Figure 15.21 The acceleration response of a seismic instrument
as given equation (15.22)
Thus by measurement ( )2 1Y Y , we can calculate the input
acceleration 1a . Generally inadequate performance is observed at
frequency ratio above 0.4. Thus for acceleration measurements we
need to
operate at frequencies much lower than the natural frequency, in
contrast to the desirable region of
operation for displacement measurements. In view of instrument
construction we need to have a low
0/ =ccc
Frequency ratio
Acc
eler
atio
n pa
ram
eter
-
926
natural frequency (soft spring, large mass) for displacement
measurements and a high natural frequency (stiff spring, small
mass) for acceleration measurements in order to be able to operate
over a wide range of frequencies and still linear response. The
seismic instrument may also be used for
velocity measurements by employing a variable-reluctance
magnetic pickup as the seismic transducer.
The output of such a pickup will be proportional to the relative
velocity amplitude, i.e., the quantity
( )2 1V V . From the above discussion it may be seen the seismic
instrument is a very versatile device that may be used for
measurement of a variety of vibration parameters. This encourages
to operate
many commercial vibration and acceleration pickups on the
seismic instrument. The seismic
instrument may be used for either the displacement or
acceleration measurements by proper selection
of the mass, spring and damper combinations. In general, as
large mass and soft spring are desirable
for vibrational displacement measurements, while a relative
small mass and stiff spring are used for
acceleration indicator.
The transient response of the seismic instrument is governed
partially by the exponential decay term
in equation (15.15). The time constant for this term could be
taken as
2mTc
= (15.23)
or, in terms of the natural frequency and critical damping
ratio
1n
T = (15.24)
The specific transient response of the seismic-instrument system
is also a function of the type of input
signal, i.e., whether it is a step function, harmonic function,
ramp function, etc. The linearity of a
vibration transducer is thus influenced by the frequency-ratio
requirements that are necessary to give
linear response as indicated by equations (15.15) and (15.18).
The design of a transducer for particular response characteristics
must involve a compromise between these two effects, combined with
a
consideration of the sensitivity of the displacement sensing
transducer and its transient response
characteristics.
-
927
Figure 15.22 A stud mounting on an accelerometer on the
vibrating surface
Example 15.4 For measurement of displacement using the amplitude
ratio equation (15.18) and 0.72 = , calculate the value of / nf
such that ( )2 1 1 0.98Y Y Y = ; that is, the measurement error
is
2 percent.
Solution : We have
( ) ( ){ }2
1/ 22 220.98
1 2 0.72
=
+ (a)
Rearranging this equation gives the quadratic relation
4 21.78 24.25 0 = (b)
which yields / 2.427nf = . It is evident from this example that
the natural frequency of the
instrument should be low.
Example 15.5 For measurement of acceleration the amplitude ratio
equation (15.18) and 0.72 = , calculate the value of / nf such that
( ) ( )2 22 1 1 0.98nfY Y Y = ; that is, the error is 2
percent.
( )( ) ( ){ }
2 11/ 22 2 221
10.981 2 0.72
Y YY
= =
+ (a)
Rearranging this equation gives the quadratic relation
-
928
4 20.0736 0.0412 0 + = (b)
which yields / 0.412nf = . It is evident from this example that
the natural frequency of the
instrument should be high.
15.4 Signal Conditioning & Analysis Equipments: The raw
signal from the vibration transducer may need to be transformed
into the right form, e.g., signals from accelerometers may need to
be
integrated to provide a velocity or displacement signal.
Furthermore, signals may need to be amplified
before being fed to the metering and alarm circuits, or in some
cases passed through a filter system to
eliminate unwanted portions of the frequency spectrum, and
finally the system impedance may to be
reduced. All of these processes are known as signal conditioning
and this can be defined as the
transformation of the transducer signal into a form, which is
suitable for the analysis, metering, or
feeding into an alarm or advance signal processing system.
15.4.1 Filters: Filters are probably the most widely used of all
vibration analysis equipment once the signal is
available from transducers. It can be there in-built in the
conditioning amplifier (which amplify the weak signal usually
available from transducers) or as a separate device in the form of
hardware or software. A filter limits a vibration signal in some
predictable fashion such that a single frequency or
group of frequencies may be isolated for the measurement or
study. Filters can be classified mainly
two different ways:
(i) Frequencies passed or passband: Under this category filters
are further classified based on the frequencies that to be allowed
or rejected.
(a) High pass: It passes all frequencies above some specified
frequency; generally it is required whenever the signals from
accelerometers are double integrated to displacement.
(b) Low pass: It passes all frequencies below some specified
frequency; it is often used with shaft displacement signals to
eliminate high frequencies generated by shaft scratches.
(c) Bandpass: It passes a band of frequencies while eliminating
all frequencies both above and below the desired passband.
(d) Band (notch) reject: The reverse of a bandpass filter,
eliminating all frequencies within a specified band while allowing
all others both above and below to pass; it permits a rapid
assessment of the total vibration energy present, exclusive of a
specific frequency.
-
929
(ii) Method of tuning: Under this categories filters are further
classified based on the method of tuning.
(a) Manual tracking: In manual tuning filters are of two types
namely, the constant bandwidth (pass a constant frequency band
regardless of where the filter center frequency is positioned hence
it provides uniform resolution) and the constant percentage
bandwidth (the frequencies passed are some fixed percentage of the
filter central frequency such that as the filter is tuned
to higher frequencies, the bandwidth becomes larger with a
corresponding reduction in
resolution) filters are in use. (b) Automatic tracking: In the
automatic or tracking filter the tuning signal is generated by
and
synchronized with the shaft under study (i.e., at rotating
frequency or multiple of running frequency). It is widely used in
balancing applications, and for tracking phase and amplitude
response during a startup or coast-down of heavy rotating
machineries (e.g., turbines and generators).
15.4.2 Measurement amplifier Generally, an amplifier is any
device that changes, usually increases, the amplitude of a signal.
The
signal is usually voltage or current. The relationship of the
input to the output of an amplifier
usually expressed as a function of the input frequency is called
the transfer function of the
amplifier, and the magnitude of the transfer function is termed
the gain. A The measurement amplifier
is used to convert the charge signal output from the transducers
(accelerometer or force transducers) to voltage signal. The
amplifier can be used for amplification of the one signal and the
sensitivity of
the transducer has to be matched (or fed) with the amplifier.
Different level of amplification could be achieved depending upon
the requirement and quite often the amplifier also have provisions
for
filtering.
15.4.3 Oscilloscope, Spectrum analyzer and Data Acquisition
System An oscilloscope (commonly abbreviated to scope or O-scope)
is a type of electronic test equipment that allows signal voltages
to be viewed, usually as a two-dimensional graph of one or more
electrical
potential differences (vertical axis) plotted as a function of
time or of some other voltage (horizontal axis). Oscilloscope can
have several functions that helps in capturing and analysis the
vibration signal. Depending upon the level of the signal can be
amplified or reduced. The time base can also be varied
to have better visulaising of the signal on the screen before
capturing. One important feature of the
trigger level setting provides capturing of the singal when it
exceeds certain level. This feature helps
in capturing relevant signal expecially during modal testing
using the impact hammer to synchronise
the time of hitting and the caturing of the signal.
-
930
A spectrum analyzer is an instrument that displays signal
amplitude (strength) as it varies by signal frequency. The
frequency appears on the horizontal axis, and the amplitude is
displayed on the
vertical axis. A spectrum analyzer looks like an oscilloscope
and, in fact, some instruments can
function either as oscilloscopes or/and as spectrum analyzers.
In spectrum analyzer various in-built
functions for statistical processing of periodic or random
signals are available. It includes FFT, power
spectrum, autocorrelation, cross-correlations, spectral density,
probability density function, ensemble
or temporal averages, etc. Multi-channel spectrum analyzers are
very expensive and that has led to
the development of various software for the analysis of the
vibration signal. Such multi channel
analyzer system consists of a PC with LAN interface and data
acquisition hardware. The system can
possess time capture and FFT analyzers. It also has provision
for setting of a project is to ensure that a measurement is set up
exactly according to individual specifications.
A data acquisition system is a device designed to measure and
log some parameters. The purpose of
the data acquisition system is generally the analysis of the
logged data and the improvement of the
object of measurements. The data acquisition system is normally
electronics based, and it is made of hardware and software. The
hardware part is made of sensors, cables and electronics
components
(among which memory is where information are stored). The
software part is made of the data acquisition logic and the
analysis software (and some other utilities that can be used to
configure the logic or to move data from data acquisition memory to
a laptop or to a mainframe computer).
15.5 Vibration Exciter Systems: In order to apply a test item
(e.g., a rotor system) to a specific vibration, a source of motion
is required. Devices used for supplying vibrational excitation are
usually referred to simply as shakers or exciters. In most cases,
simple harmonic motion is provided, but systems supplying
complex
waveforms (two or multi-frequency, random, impulse, etc.) are
also available. There are various forms of shakers and the
variation is depending on the source of driving force. In general,
the primary source
of motion may be electromagnetic, mechanical, or
hydraulic-pneumatic or in certain cases, acoustical,
aerodynamic. Each is subjected to inherent limitations, which
usually dictate the choice.
15.5.1 Electromagnetic Systems A sectional view of an
electromagnetic exciter is shown in Figure 15.23. This consists of
a field coil, which supplies a fixed magnetic flux across the air
gap and a driver coil supplied from a variable-
frequency source. Permanent magnets are also sometimes used for
the fixed field (or the biased field), which reduces the power
consumption. The support of the driving coil is by means of
springs, which
permit the coil to reciprocate when driven by the force
interaction between the two magnetic fields. It
can be seen that the electromagnetic driving head is very
similar to the field and voice (moving) coil arrangement in the
ordinary radio loudspeaker.
-
931
Figure 15.23 A sectional view of the electromagnetic exciter
An electromagnetic shaker is rated according to its force
capacity, which in turn is limited by the
current-carrying ability of the moving coil. Temperature
limitations of the insulation basically
determine the shaker force capacity. The driving force is
commonly simple harmonic (complex waveforms are also used) and may
be thought as a rotating vector. The force used for the rating is
the vector force exerted between the moving and field coils. The
rated force is never completely available
for driving the test item. It is the force developed within the
system, from which must be subtracted
the force required by the moving portion of the shaker system
proper. It may be expressed as
rt t aF F F= (15.25)
in which Frt is the net force available to shake the test item,
Ft is the manufacturers rated capacity, or
the total force produced by the magnetic interaction of the
moving and field coils; and Fa is the force
required to accelerate the moving parts of the shaker system,
including the moving coil, table and
appropriate portions of the moving coil flexure beam.
Specification for a typical electromagnetic
exciter systems contain (i) maximum rated force, (200 2105) N
(ii) frequency range, (0 10000) Hz (iii) Peak-to-peak amplitude (up
to 25 mm), (iv) cooling requirement and (v) weight of the moving
armature (0.35 110 kg), (vi) type of excitation (sinusoidal,
multi-frequency, random, impulse, sine-sweep, etc.). While using
sine-sweep excitation, it is often required to cross the resonance
condition. In advanced electromagnetic exciters a feedback control
based on the vibration level provide variable
force so that at the resonance the force applied is very small
to avoid catastrophic failure of the test
item.
-
932
15.5.2 Mechanical-Type Exciters There are two types of
mechanical shakers: the directly driven and the inertia. The
directly driven
shaker consists of a test table that is forced to reciprocate by
some form of mechanical linkage. Crank
and connecting rod mechanisms, Scotch yokes, or cams may be used
for this purpose. Another
mechanical type uses counter-rotating masses to apply the
driving force. The force adjustment is provided by relative offset
of the weights and the counter-rotation cancels shaking forces in
one
direction, say the x-direction, while supplementing the y-force.
The frequency is controlled by a
variable speed motor. There are two primary advantages in such
inertia systems. In the first place,
high force capacities are not difficult to obtain. Secondly, the
shaking amplitude of the system
remains unchanged by frequency cycling. Therefore, if a system
is set to provide a 1 mm amplitude at
30 Hz, changing the frequency to 40 Hz will not change the
amplitude (since the stroke of the linkage remains the same for a
particular setting, however, it could be changed by changing the
linkage
dimensions). It should be noted that both the available
excitation force and the required accelerating force are harmonic
functions of the square of the exciting frequency; hence as the
requirement
changes with frequency, it also changes the available force.
15.5.3 Hydraulic and Pneumatic Systems Important disadvantages
of the electromagnetic and mechanical shaker systems are limited
load
capacity and limited frequency, respectively. As result, the
search for other sources of controllable
excitation has led to investigation in the areas of hydraulics
and pneumatics. In this arrangement an
electrically actuated servo valve operates a main control valve,
in turn regulating flow to each end of a
main driving cylinder. Large capacities (up to 2 MN) and
relatively high frequencies (to 400 Hz), with amplitudes as great
as 46 mm, have been attained. Of course, the maximum values cannot
be attained simultaneously. As would be expected, a primary problem
in designing a satisfactory system of this
sort has been in developing valving with sufficient capacity and
response to operate at the required
speeds.
Relative Merits and Limitations of Each System Excitation
Frequency: The upper frequency ranges are available only through
use of the
electromagnetic shaker. In general, the larger the force
capacity of the electromagnetic exciter, the
lower its upper frequency will be. However, even the 2105 N
shaker boasts an upper frequency of
2000 Hz. To attain this value with a mechanical exciter would
require spin speeds of 1 20 000 rpm.
The maximum frequency available from the smaller mechanical
units is limited to approximately 120
Hz (7200 rpm) and for the larger machines to 60 Hz (3600 rpm).
Hydraulic units are presently limited to about 2000 Hz with very
high peak load.
-
933
Force Limitations: Electromagnetic shakers have been built with
the peak force ratings of 2 MN.
Variable-frequency power sources for shakers of this type and
size are very expensive. Within the
frequency limitations of mechanical and hydraulic systems,
corresponding or higher force capacities
may be obtained at lower costs by hydraulic shakers.
Maximum Excursion: The upper limit of peak-to-peak displacement
for the electromagnetic exciter
may be considered as 25 mm or slight more. Mechanical type may
provide displacement of the order of 150 mm, whereas the hydraulic
exciter can provide displacement of the order of 450 mm.
Magnetic Fields: Because the electromagnetic shaker requires
relatively intense fixed magnetic field, special precautions are
sometimes required in testing certain items such as solenoids or
relays, or any
device in which, induced voltages may be a problem. Although the
flux is rather completely restricted
to the magnetic field structure, relatively high stray flux is
nevertheless present in the immediate
vicinity of the shaker. Operation of items sensitive to magnetic
fields may therefore be affected,
degaussing coils are sometimes used around the table to reduce
flux level.
Non-sinusoidal Excitation: The shaker head motion may be
sinusoidal or complex, periodic or completely random. Although
sinusoidal motion is by far the most common, other waveforms
and
random motions are sometimes specified. The electromagnetic
shaker offers most of waveforms.
Although, the hydraulic type may produce non-harmonic motion,
precise control of a complex
waveform is not easy. Here again, future development of valving
may alter the situation. The voice
(moving) coil of the ordinary loudspeaker normally produces
complex random motion, depending on the sound to be reproduced.
Complex random shaker head motions are obtained in essentially
the
same manner. Instead of using a fixed-frequency harmonic
oscillator as the signal source, either a
strictly random or a predetermined random signal source is used.
Electronic noise sources are
available, or a record of the motion of the actual end use of
the device may be recorded on magnetic
tape and used as the signal source for driving the shaker. As an
example, the electronic gear may be
subjected to combat-vehicle motions by first tape-recording the
output of motion transducers, then using the record to device a
shaker.
15.5.4 Impact hammer One of the popular methods of excitation is
through use of an impact or hammer. It is a relatively
simple means of exciting the structure into vibration. The
equipment consists of an hammer, usually
with a set of different heads and tips, which serve to extend
the frequency and force level ranges for
testing a variety of different structures. The hammer tips can
be of rubber, aluminum, steel, etc. Using
different sizes of hammer may also extend the useful range.
Integral with the impact is a force
transducer, which detects the magnitude of the force, felt by
the impact, and which is assumed to be
-
934
equal and opposite to that experienced by the structure. The
impact incorporates a handle to form a
hammer as shown in Figure 15.24, so that impact can be applied
manually. Basically, the hammerhead and the acceleration with which
it is moving when it hits the structure determine the
magnitude of the impact. The frequency range, which is
effectively excited by this type of device, is
controlled by the stiffness of the contacting surface and the
mass of the impact head. The stiffer the tip
materials, the shorter will be the duration of the pulse and the
higher will be the frequency range
covered by the impact. It is for this purpose that a set of
different hammer tips and heads are used to
permit the regulation of the frequency range to be encompassed.
Care should be taken while
impacting so that multiple impacts or hammer bounce does not
occur, otherwise these would create
difficulties in the signal processing stage.
Figure 15.24 Impact hammer
A typical specifications of the impact hammer are as follows:
Sensitivity at output of hammer: 0.95 pC/N, Rubber tip
specifications: Frequency range 0-500 Hz, Duration range 5-1.5 ms,
Force range 0-700 N, Physical: Weight of the hammer 280 g,
Materials of Tips: Anodized aluminum, stainless
steel, titanium, neoprene rubber. When using the above type of
the hammer, the actual impact force
applied to the test structure will always be greater than the
force measured across the transducer
because of the inertia of the tip. These forces are related as
follows
( )a m tF F m m m= (15.26)
where Fa is actual force input to structure, Fm is measured
force, m weight of the hammer plus tip and
mt weight of the tip.
Determining Natural Frequencies of the Rotor Bearing System
Using Impact Hammer Test Natural frequencies of the rotor bearing
system are important parameters to be determined prior to any
investigation. For a two rotor system two natural frequencies
are obtained by using the impact test.
Impact is applied at one of the rigid disks while the rotor is
stationary (non-rotating). Displacement to impulse force is
measured at the bearing end both in the horizontal and vertical
directions using
-
935
proximity probe transducer. The FFT of the measured impulse
response then gives frequency domain
impulse response. In the frequency domain response natural
frequencies appear as higher amplitude
peaks. Figures 15.25 and 15.26 show the absolute value of the
FFT of the measured impulse response in the horizontal and vertical
directions, respectively. These plots indicate the first and second
natural
frequencies, and these are equal to 38 Hz and 125 Hz, for the
present configuration of the rotor bearing system.
Figure 15.25 Natural frequencies of the rotor bearing system in
the horizontal direction
Figure 15.26 Natural frequencies of the rotor bearing system in
the vertical direction
Example 15.6 An electromagnetic-type sinusoidal vibration
exciter has a rated force capacity of 25 N is to be used to excite
a test item weighing 3 kg. If the moving parts of the shaker have a
mass of 0.75 kg and the amplitude of the vibration is 0.15 mm
(peak-to-peak amplitude = 0.30 mm), determine the maximum
excitation frequency that can be applied.
Solution: We have, the amplitude of vibration a = 0.15 mm, the
rated force capacity Ft = 25 N, mass of the test item m = 3 kg,
mass of the moving parts of shaker mc = 0.75 kg, and the
maximum
frequency of excitation is to be determined. The dynamic force
can be expressed as
-
936
( ) 2t cF m m a= + (a)
On substituting into equation (a), we have
25 = (3 + 0.75) 2 0.00015 (b)
which gives the maximum excitation frequency = 210.82 rad/s =
33.6 Hz. It could be seen that
because of relatively heavy mass as compared to the capacity of
exciter, the maximum frequency of
excitation is relatively low.
Example 15.7 Suppose a vibration test requires a sinusoidal
excitation force for a 10-kg test item at 100 Hz with a
displacement amplitude of 2 mm (peak-to-peak amplitude = 4 mm) What
will be the capacity of the exciter required? If support fixtures
are required, they too must be shaken along with
the moving coil of the shaker itself. Suppose these items (the
fixture and moving-coil assembly) have a mass of 5 kg.
Solution: The maximum force will correspond to the maximum
acceleration, and the maximum acceleration can be calculated as
follows:
The circular frequency = = 2 pi 100 = 628 rad/s
The maximum acceleration = the displacement amplitude 2 = (2
10-3) (628)2 = 789 m/s2
The maximum force = m a = 10 789 = 7890 N
This, of course, is the force amplitude required to shake the
test item only. An additional force of 5
789 = 3945 N is required towards the fixture and moving-coil
assembly. The rated capacity of the
shaker must therefore be a maximum of about 7890 + 3945 = 11835
N 12 kN.
15.6 Sound Measurements Sound waves are a vibratory phenomenon.
Acoustic effects also give rise to harmonic pressure
fluctuations that they produce in a liquid or gaseous medium.
They also characterized by an energy
flux per unit area and per unit time as the acoustic waves moves
through the medium. A mathematical
description of different acoustic will be given in this section.
It is standard practice in acoustic
measurements to relate the sound intensity and the sound
pressure to certain reference values 0I and
0p , which correspond to the intensity and mean pressure
fluctuations of the faintest audible sound at
a frequency of 1000 Hz. These reference levels are
-
937
120 10I
= W/m2 (15.27) and
50 2 10p
= N/m2 (15.28)
The intensity and pressure levels are measured in decibels.
Thus
Intensity level (dB) = ( )010log /I I (15.29) and
Pressure level (dB) = ( )020log /p p (15.30)
When pressure fluctuations and particle displacements are is
phase, such as in plane acoustic wave,
these levels are equal,
0 010log 20log pI I p=
The magnitudes of the particle velocity and pressure
fluctuations created by a sound wave are small.
For example, a plane sound wave having an intensity of 90 dB is
considered the maximum
permissible level for extended human exposure. In many
circumstances we shall be interested in the
sound intensity that results from several sound sources. This
calculation could of course, be
performed with equation (15.29) and (15.30). Sound pressure is
the local pressure deviation from the ambient (average, or
equilibrium) pressure caused by a sound wave. Sound pressure can be
measured using a microphone in air and a hydrophone in water. The
SI unit for sound pressure is the Pascal
(symbol: Pa or N/m2).
In machinery analysis the dB scale is used to express the ratio
between two voltages - an output to an
input, for example (mathematically a dB is a 20-log voltage
ratio). When the output equals the input, 20 log 1 = 0 or 0 dB.
Thus, the zero or reference on the dB voltage scale occurs when
input and
output are equal. It follows that perfect reproduction of an
input signal could be stated by specifying a
0 dB permissible tolerance or deviation, which is an ideal case.
In practice, it is customary to specify
the output of an instrument as a tolerance, 20 dB for example
(ten-to-one attenuation across the instrument: 20log10 = 20;
similarly a 100-to-1 attenuation corresponds to 40 dB and 1000-to-1
= 60 dB), over a given frequency range as a measure of the
instruments deviation in voltage output compared to the voltage
input.
-
938
The dB scale can be used equally well for the gain or an
increase in the voltage. Using previous
example a gain of 20 dB equals an increase in voltage by a
factor of 10, 40 dB equals an increase by a
factor of 100 etc. The dB scale is nothing more than a method to
express the ratio between two
quantities. As pointed out earlier, 20 dB represents a voltage
ratio of 10 with multiple of 20 dB (e.g., 20, 40, 60, 80, 100)
equal to powers of 10 (e.g., 10, 102, 103, 104, 105). Similarly
voltage ratio, 2 dB = 1.26, 3 dB = 1.41, 5dB =1.78. Next, 6 dB
represents a voltage ratio of approximately 2:1 while 10 dB
approximates 3:1 and 14 dB, 5:1. As a voltage ratio 50 dB = 40 dB
(nearest multiple of 20) + 10 dB. Since adding algorithms is
equivalent to multiplying numbers, 50 dB equals a voltage ratio of
100
(40dB) 3 (10 dB) 300 (actual 316).
Example 15.8 Calculate the total sound intensity from two sound
sources at 40 and 50 dB.
Solution: T