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
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.
Test rig
and processing
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
with a 5 cm (or 60o) scale for the same range of a measuring parameter (e.g., 100 µm 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
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 µm displacement dial gauge with an accuracy of 1 percent would be accurate
within ±1 µm 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 instrument’s 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 (50×100/1000), while a precision of ±1 percent
(10×100/1040 = 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.
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 instrument’s 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 µm 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 transducer’s output per unit change in the measured
quantity. A more sensitive instrument’s 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/µm. 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 µC/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
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
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
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
E AB = E
R + Ri
1 + ( Ri / R)
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
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
R R + Ri
( R + Ri )
and the maximizing condition
dP d
dR dR
( R + Ri )
= 0 is applied. It results in
( R + Ri )
−2 E 2 R
( R + Ri )
= 0;
( R + Ri ) − 2 R = 0;
R = Ri
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
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
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
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 x1 , x2 ,
, xn (e.g., the rotor speed, inlet pressure,
pressure drop, diameter, length, clearance, temperature, force, excitation frequency, displacement,
acceleration, etc.). Thus,
R = R ( x1 , x2 ,
, xn )
Let wR be the uncertainty in the result and w1 , w2 ,
, wn be the uncertainties in the independent
variables. Then the uncertainty in the result is given as
wR =
1/ 2
∂R R ( x1 + ∆x1 ) − R ( x1 )
where ∆x1 , ∆x2 ,
∂R R ( x1 + ∆x1 ) − R ( x1 )
, ∆xn are the small perturbations of the independent variables. It should be noted
that the uncertainty propagation in the results wR predicted by equation (15.7) depends on the squares
of the uncertainty in the independent variables wk ( k = 1, 2,
, 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
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
made, while both current and voltage will be measured in (2). The register has a nominal stated value
of 5 Ω ± 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 wE = 50 × 0.01 = 0.5 V and wI = 5 × 0.01 = 0.05 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
wp =
∂P ( E , R )
wE2 +
∂P ( E , R )
1/ 2
∂P ( E , R )
∂P ( E , R )
=− 2
On substituting equation (b) into equation (a), the uncertainty in the power could be written as
wp =
wE2 + − 2
1/ 2
= 4 E
+ R
1/ 2
Inserting numerical values for the uncertainty, we get
100 = 100 4 ( 5 / 50 ) + ( 0.5 / 5 )
1/ 2
= 100 4 ( 0.01) + ( 0.01)
1/ 2
= 2.24%
(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 , I )
∂P( E , I )
On using equation (e), the uncertainty in the power could be written as
wp =
∂P ( E , I )
wE2 +
∂P ( E , I )
1/ 2
wP =
(I )
wE2 + ( E ) wI2
1/ 2
+ I
1/ 2
On substituting numerical values for the uncertainty, we get
100 = 100 ( 0.01) + ( 0.01)
1/ 2
= 1.414%
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 Ω, Rm = 1200 Ω ± 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: wE = 500 × 0.01 = 5 V, wI = 5 × 0.01 = 0.05 A, and
wRm = 1200 × 0.05 = 60 Ω. Let I1 and I2 are currents flowing through registers R and Rm, respectively. A
current balance on the circuit gives
R Rm
I1 + I 2 = I
I1 = I −
The power dissipated in the resistor R is give as
P = EI1 = EI −
so that
=− 2
The nominal value of the power is thus calculated as
P = 500 × 5 −
= 2292 W
In terms of known quantities the power has the functional from P = f ( E , I , Rm ) and so the
uncertainty for the power is now written as
wp =
w +
1/ 2
w +
w + (E) w + 2
1/ 2
On substituting the given numerical values in equation (d), we get
wp =
2 × 500
52 + ( 500 ) ( 0.05 ) +
= [ 434 + 625 + 108.5]
1/ 2
1/ 2
( 60 )
= 34.2 W
100 =
100 = 1.49%
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
equation (15.1) that this results from the fact that Rm
R ( Rm = 10 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.
(b) Excitation other than ω n
(a) No excitation
(c) Excitation at ω 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
ωnf = 3.52
where ω n is the natural frequency in rad/s, E is the Young’s 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
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
−7.04 EI
The uncertainty in the natural frequency is given by
wωnf =
1/ 2
where wL is the uncertainty in length measurement in m. On substituting equation (15.9) into
equation (15.10), and after simplification we obtain
wL =
wωnf L3
7.04 EI / m
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 2.1 × 1011 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
E = 2.1 × 1011 N/m2;
d4 =
(0.6) 4 = 6.362 × 10−3 mm4 = 6.362 × 10−15 m4
d 2 7800 × π × ( 0.6 ) × 10
m = ρπ
= 2.205 × 10−3 kg/m
For L = 60 mm, from (15.8), we have
ωnf = 3.52
= 3.52
( 2.1×10 ) × ( 6.362 ×10 )
1/ 2
2.205 × 10−3 × 0.064
= 761.1 rad/s
Similarly, for L = 200 mm, we will have ωn = 68.50 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 wωnf = 0.02 × 68.50 = 1.37 . From equation (15.11), we have the allowable
uncertainty in the measurement of length
wL =
wωnf L3
7.04 EI / m
1.37 × 0.23
7.04 2.1 × 10 × 6.362 × 10
/ 2.205 × 10
= 1.999 × 10−3 m = 2.0 mm
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
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
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.
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 airgap.
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.
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
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 µm gap, current standards
require either a 4 mV/µm or 8 mV/µm proportionality between gap and voltage. Thus, a 250 µm
change in gap should produce a voltage change of 1 volt at 4 mV/µm or 2 volts at 8 mV/µm (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/µm (or 8.0×103 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 target’s 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/µm , linear range of typical non-contact measuring systems observing 4140 steel will vary
from approximately 1525 µm with 5 mm tip diameter and -18 VDC supply to 2160 µm 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.
(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 70°C. 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.
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 µsec 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.
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
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.,
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 surface’s
motion. The displacement is detected by reflecting laser light so that a uniform target surface is
required to prevent the noise.
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
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 lowimpedance accelerometers is not very much larger then the original version. They are no longer selfgenerating and need a dc power supply to drive them (typically 18 V DC).
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
-damper system
Figure 15.19 Schematic of typical seismic instrument
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 Newton’s law of motion,
we have
my2 + cy2 + ky2 = cy1 + ky1
where y1 and y2 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
y1 = Y1 cos ω t
where Y is the displacement amplitude. The aim is to obtain an expression for the relative
( y2 − y1 )
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
y2 +
y2 + y2 = Y
cos ωt − ω cos ωt
The solution to equation (15.14) is
− c
( y2 − y1 ) = e ( 2m ) ( A cos ωnf t + B sin ωnf t ) +
mY1ω 2 cos (ωt − ϕ )
{( k − mω ) + c ω }
1/ 2
where the damped natural frequency is given by
1/ 2
< 1.0
and the phase angle by
ϕ = tan −1
k − mω 2
where A and B are constants of integration determined from the initial conditions (transient vibration
part). Note that equation (15.15) is composed of two terms (i) the transient term involving the
exponential function and (ii) the steady-state term. This means that after the initial transient has died
out a steady-state harmonic motion is established in accordance with the second term. The frequency
of this steady-state motion is the same as that of the base motion, and its amplitude ratio is (from
equation (15.15)).
(Y2 − Y1 )
{(1 − ω ) + ( 2ζω ) }
1/ 2
ω = ω / ωnf ,
ζ = c / cc
where ωnf d is the damped natural frequency, ωnf is the natural frequency (undamped) and ω is the
base motion frequency. It should be noted that the denominator of equation (15.18) for ω ≈ 0
becomes 1 and measured displacement amplitude becomes proportional the acceleration of the
vibrating object. Whereas, for ω
1 the amplitude ratio becomes 1, that is measure displacement
amplitude becomes equal to the displacement of the vibrating object. The natural frequency ωnf and
critical damping coefficient cc are given by
ωnf = k / m
cc = 2 mk
The phase angle may also be written as (equation (15.17))
ϕ = tan −1
1− ω2
A plot of equation (15.18) is given in Figure 15.20
Amplitude ratio
c cc = 0
Frequency ratio
Figure 15.20 Displacement response of a seismic instrument as given by equation (15.18)
It may be seen that the output displacement amplitude is very nearly equal to the input displacement
amplitude when c cc = 0.7 and ω ω n > 2 . For low values of damping ratio the displacement
amplitude may become quite large. The output becomes essentially a linear function of input at highfrequency 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 c c c and ωnf 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
a1 = y1 = ω 2Y1
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 ωnf will be fixed for a given instruments. In
Figure 15.21 we have a plot (Y2 − Y1 ) ωnf2 / a1 versus ω / ωnf , which indicates the non-dimensionalised
acceleration response.
Acceleration parameter
c / cc = 0
Frequency ratio
Figure 15.21 The acceleration response of a seismic instrument as given equation (15.22)
Thus by measurement ( Y2 − Y1 ) , we can calculate the input acceleration a1 . 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
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
(V2 − V1 ) . 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
or, in terms of the natural frequency and critical damping ratio
ω nζ
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
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
ζ = 0.72 , calculate the value of ω / ωnf such that (Y2 − Y1 ) Y1 = 0.98 ; that is, the measurement error is
2 percent.
Solution : We have
0.98 =
{(1 − ω ) + ( 2 × 0.72ω ) }
2 2
1/ 2
Rearranging this equation gives the quadratic relation
ω 4 − 1.78ω 2 − 24.25 = 0
which yields ω / ωnf = 2.427 . 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 ζ = 0.72 ,
calculate the value of ω / ωnf such that (Y2 − Y1 ) ωnf2
(Y2 − Y1 )
= 0.98 =
(Y ω ) = 0.98 ; that is, the error is 2 percent.
{(1 − ω ) + ( 2 × 0.72ω ) }
Rearranging this equation gives the quadratic relation
1/ 2
ω 4 + 0.0736ω 2 − 0.0412 = 0
which yields ω / ωnf = 0.412 . 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.
(ii) Method of tuning: Under this categories filters are further classified based on the method of
(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
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
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.
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 variablefrequency 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.
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
Frt = Ft − Fa
in which Frt is the net force available to shake the test item, Ft is the manufacturer’s 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 – 2×105) 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, sinesweep, 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
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
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 2×105 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.
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
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 0700 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
Fa = Fm m (m − mt )
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
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 ω is to be determined. The dynamic force can be expressed as
Ft = ( m + mc ) ω 2 a
On substituting into equation (a), we have
25 = (3 + 0.75) ×ω2 × 0.00015
which gives the maximum excitation frequency ω = 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 = ω = 2 π × 100 = 628 rad/s
The maximum acceleration = the displacement amplitude × ω2 = (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 I 0 and
p 0 , which correspond to the intensity and mean pressure fluctuations of the faintest audible sound at
a frequency of 1000 Hz. These reference levels are
I 0 = 10−12 W/m2
p0 = 2 × 10−5 N/m2
The intensity and pressure levels are measured in decibels. Thus
Intensity level (dB) = 10 log ( I / I 0 )
Pressure level (dB) = 20log ( p / p0 )
When pressure fluctuations and particle displacements are is phase, such as in “plane acoustic wave”,
these levels are equal,
10log I
= 20log 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 instrument’s deviation in voltage output
compared to the voltage input.
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: The sound intensity for the two sources can be calculated as
40 = 10log
I1 = 104 I 0 W/m2
50 = 10log
I 2 = 105 I 0 W/m2
Hence, the total sound intensity would be IT = I1 + I 2 = (104 + 105 ) I 0 = 1.1 × 105 I 0 W/m2. Now, we can
obtain the sound intensity in dB would be
Intensity level (dB) = 10log
1.1× 105 I 0
= 50.41 dB
It should be noted that, the sound intensity difference from the maximum intensity would be 50.41 –
50 = 0.41 dB.
Example 15.9 Calculate the total sound intensity resulting from three sound sources at 70, 75 and 80
Solution: Let us first calculate the sound intensity of the individual sound sources as follows
70 = 10log
I1 = 107 I 0 W/m2,
75 = 10log
I 2 = 107.5 I 0 W/m2
80 = 10log
I 3 = 108 I 0 W/m2
On combing all the intensity, we get
IT = I1 + I 2 + I 3 = (107 + 107.5 + 108 ) I 0 = (0.100 + 0.316 + 1.000) × 108 I 0 = 1.416 × 108 I 0 W/m2
Now, we can obtain the sound intensity in dB would be
Intensity level (dB) = 10log
1.416 × 108 I 0
= 81.511 dB
It should be noted that, the sound intensity difference from the maximum intensity would be 81.511 –
80 = 0.511 dB.
Table 15.1 Typical sound pressure level
Source of sound
Sound pressure
Theoretical limit for undistorted sound at
Sound pressure level
(dB ref. 20 Pa)
101,325 Pa
191 dB
Jet engine at 30 m
630 Pa
150 dB
Rifle being fired at 1 m
200 Pa
140 dB
Threshold of pain
100 Pa
130 dB
20 Pa
120 dB
6 – 200 Pa
110 – 140 dB
2 Pa
100 dB
6×10−1 Pa
85 dB
Major road at 10 m
2×10−1 – 6×10−1 Pa
80 – 90 dB
Passenger car at 10 m
2×10−2 – 2×10−1 Pa
60 – 80 dB
2×10−2 Pa
60 dB
Normal talking at 1 m
2×10−3 – 2×10−2 Pa
40 – 60 dB
Very calm room
2×10−4 – 6×10−4 Pa
20 – 30 dB
Leaves rustling, calm breathing
6×10−5 Pa
10 dB
Auditory threshold at 2 kHz
2×10−5 Pa
0 dB
1 atmosphere environmental pressure
Hearing damage (due to short-term exposure)
Jet at 100 m
Jack hammer at 1 m
Hearing damage (due to long-term exposure)
TV (set at home level) at 1 m
Sound level measurements are performed with some type of microphone, which may be considered a
type of seismic vibration instrument. The electric output of the microphone is proportional to soundpressure level, which may be used to calculate the sound intensity according to equation (15.29).
Appropriate power amplifiers and readout meters or recorders indicate the sound level. In general, the
microphone must be calibrated in a test facility with a source of known frequency and intensity. Even
with careful calibration, accuracies of better than ±1 dB may not be expected in sound-pressure-level
measurements. A typical practical application of sound-level measurements may call for an analysis
of the noise spectrum in a certain sound source. For many commercial noise and vibration meters
calibration may be achieved with a simple whistle or tuning fork in quiet room. Sound-level readings
will generally follow the inverse square law with respect to distance. Table 15.1 gives typical
examples of sound pressure level in practice.
Example 15.10 A certain microphone has an open-circuit sensitivity of –80dB referenced to 1 V for a
sound-pressure excitation of 10 N/m2. Calculate the voltage output when exposed to a sound field of
(a) 90 dB and (b) 45 dB.
Solution: We first calculate the sound pressure level from equation (15.30). We have, for the 90-dB
90 = 20log
2 × 10−5
p = 0.632 N/m2
The reference voltage for an excitation of 10 N/m2 is calculated as
−80dB = 20log
E = 10−4 mV
Because we assume the output voltage varies in a linear manner with the impressed sound field, the
output voltage at 90-dB sound level is
E = (0.632 /10) × 10−4 = 6.32 × 10−6 V
For a sound level of 45 dB the sound pressure is obtained from
45dB = 20log
2 × 10−5
p = 3.557 × 10−3 N/m2
The output voltage is therefore
E = {(3.557 /10) × 10−3 }(10−4 ) = 3.557 × 10−11 V
Final Remarks
In the present chapter, we studied basic principles based on which various kinds of transducers are
designed for measurement of vibrations and acoustics. These transducers are based on eddy current,
capacitance, inductance, magnetic, piezoelectric, optical, laser, etc. Various performance parameters
used in transducers are described, e.g., readability, least account, calibration, working range,
sensitivity, accuracy, precision, resolution, frequency range, impedance, etc. Various signal
conditioning, amplifying and analyses instruments are described like filters, amplifiers, oscilloscopes,
spectrum analyzers, etc. The electromagnetic, mechanical and pneumatic exciters are described and
compared in respect to their merits and demerits. In acoustics or sound measurement basic principles
of the measurements are described. Various parameters to be measured are described, e.g., sound
pressure, sound intensity, etc.
Exercise Problems
Exercise 11.1: A sinusoidal forcing function is impressed on the spring-mass damper system
subjected to a force input. The natural frequency is 100 Hz, and the damping ratio ζ is 0.7. Calculate
the amplitude ratio and time lag of the system for an input frequency of 40 Hz (the time lag is the time
interval between the maximum force input and maximum displacement output).
Exercise 11.2: For a natural frequency of 100 Hz and a damping ratio of 0.7, compute the frequency
range for which the system of example 11.1 will have an amplitude ratio of 1.00±0.01.
Exercise 11.3: Two-terminal device shown in Figure 11.2 has an internal resistance of 5000 Ω. A
meter with an impedance of 20,000 Ω is connected to the output to perform a voltage measurement.
What is the percent error in determination of the internal voltage?
Exercise 11.4: The device in problem 11.3 has an internal voltage of 100 V. Calculate the power
output for the loading conditions indicated. What would be the maximum power output? What power
output would result for a load resistance of 1000 Ω?
Exercise 11.5: The vibrating wedge shown in Figure E11.5 is used for an amplitude measurement.
The length of the wedge is 15 cm ± 0.5 mm, and the thickness is 2.5 cm ± 0.2 mm. The x distance is
measured as x = 5.6 cm ± 1.3 mm. Calculate the vibration amplitude and its uncertainty in percent.
x = a cos ωt
Figure E11.5 Simple wedge as a device for amplitude-displacement measurements (a) at rest (b) in
Exercise 11.6: A small cantilever vibrometer is available for measurement of vibration frequency, but
the specification sheet is lost, so that the properties of the device are not known. The instrument is
calibrated by placing it on a large compressor in the laboratory, which is rotating at 300 ± 2.0 rpm.
The measured length for resonance conditions is 5.6 cm ± 0.2 mm. Calculate the frequency that the
measurement will indicate when L = 10 cm ± 0.5 mm. Also calculate the uncertainty in the
measurement at this length. Calculate the phase angle for the conditions of the above case.
Exercise 11.7: A seismic accelerometer is to be used to measure the linear acceleration over a range
from 30 and 300 m/s2. The natural frequency of the instrument is 200 Hz, and this value may vary by
±2 Hz owing to temperature fluctuations. Calculate the allowable uncertainty in the relative
displacement measurement in order to ensure an uncertainty of no more than 5 percent in the
acceleration measurement. Calculate the value of the time constant for the instrument if ζ = 0.65.
Exercise 11.8: Plot the error in acceleration measurement of a seismic instrument for ζ = 0.7 versus
the frequency ratio i.e.
( x2 − x1 )0 ωn2
Exercise 11.9: A large seismic instrument is constructed so that m = 45 kg and ζ = 0.707. A spring
with k = 2.9 kN/m is used so that the instrument will be relatively insensitive to low-frequency signals
for displacement measurements and relatively insensitive to high-frequency signals for acceleration
measurements. Calculate the value of linear acceleration that will produce a relative displacement of
2.5 mm on the instrument. Calculate the value of ω /ωn such that
( x2 − x1 )0
= 0.99
Exercise 11.10: A seismic accelerometer is to be designed so that the time constant T = 1/ ζωn is equal
to the period of the maximum acceptable frequency for a 1 percent error in measurement; that is,
( x2 − x1 )0 / a0 = 0.99 . Plot ωn versus ζ in accordance with this condition.
Exercise 11.1: An acceleration is used for measuring the amplitude of a mechanical vibration. The
following data are obtained (i) waveform: simple sinusoidal (ii) period of vibration = 0.0023 s (iii)
output voltage from accelerometer = 0.214 V rms (iv) acceleration calibration = 0.187 V/standard g.
What vibrational displacement (amplitude in mm) is sensed by the accelerometer?
Exercise 11.2: An accelerometer is designed to have a maximum practical error of 4% for
measurements having frequencies in the range of 0 to 10 kHz. If the damping constant is 50 Ns/m,
determine the spring constant and suspended mass.
Exercise 11.2: For a simple frequency meter consist of a uniformly sectioned cantilevered beam the
undamped first-mode natural frequency may be calculated from the expression
f = 3.52
where E is the Young’s modulus for the material of the beam, m is the mass per unit length, I is the
area moment of inertia of the beam section and L is the length of the beam. For a steel wire of 1.5 mm
diameter, plot the resonance frequencies over a range of L = 10 to 25 cm. Use the density of steel =
7900 kg/m3.
Example 11.2: The waveform from a mechanical vibration is sensed by a velocity-sensitive
vibrometer. The CRO (Cathode-ray oscilloscope) trace indicates that the motion is essentially simple
harmonic. A 1-kHz oscillator is used for time calibration and 4 cycles of the vibration are found to
correspond to 24 cycles from the oscillator. Calibrated vibrometer output indicates a velocity
amplitude (half of the peak-to-peak) of 3.8 mm/s. Determine (a) the displacement amplitude in mm
and (b) the acceleration amplitude in standard g’s. (Answer: 0.41 g).
Example 11.2: A vibrometer is used to measure the time-dependent displacement of a machine
vibrating with the motion
y = 0.5sin(3π t ) + 0.8sin(10π t ),
where y is in cm and t is in s. If the vibrometer has an undampled natural frequency of 1 Hz and a
critical damping ratio of 0.65, determine the vibrometer time-dependent output and explain any
discrepancies between the machine vibration and the vibrometer readings.
Example 11.2: At full rated load of 5000 N a load cell deflects 0.1 mm. If it is used to measure the
thrust of a small (200 kg) jet engine, what maximum frequency component of the thrust may be
accurately measured if the inherent error is limited to 2%? Damping is negligible. What is the basis
that you use for determining the limiting frequency?
Durst F, Melling A and Whitelaw J H 1981 Principles and Practice of Laser-Doppler Anemometry,
London: Academic.
Durst F; Muller R; Jovanovic J (1988) Determination of the measuring position in laser-Doppler
anemometry. Exp Fluids 6: 105–110.
Halliwell N. A., 1996, Journal of Sound and Vibration, 190(3), 399-418. The laser torsional
vibrometer: a step forward in rotating machinery diagnostics.
On order tracking software for rotors. The old way of doing it was to
synchronise the sampling to the rotational speed which required lots
of hardware. The standard way these days is to sample at a relatively
high rate (including the key phasor or other angle measure) and then
post-process the data the obtain the different orders (not only 1X but
all the harmonics too). We have the Vold software, but LMS, B&K and
other data acquisition systems have their own tracking filters. If you
use a stand alone ADC system then you would need to sort this out
yourself. I remember Mark Smart wrote his own order tracking software
in MATLAB (I don't think it was too much trouble). I've just taken a
look on MATLAB Central and there are some scripts available - try
There are some order tracking algorithms that are supposed to cope
with fast run-downs and run-ups (I think there are many papers on
this, although I haven't looked at them), although the transients will
always be a problem. Can you run-up / down the machine more slowly? We
tend to make sure the machine is quasi-steady and so we do not excite
the transients.
Phase measurement between two components which are supposed to rigidly connected to each other
could give indication of looseness of the fasteners.
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