Vibration Measurement of Rotating Machinery

Dr. R. Tiwari ([email protected]) 1

ByDr. Rajiv Tiwari

Department of Mechanical EngineeringIndian Institute of Technology Guwahati 781039

Under AICTE Sponsored QIP Short Term Course on

Theory & Practice of Rotor Dynamics(15-19 Dec 2008)

IIT Guwahati

VIBRATION MEASUREMENTING EQUIPMENTS AND SIGNAL PROCESSING


IntroductionAim of the presentation is to describe some of the means of measuring and recording data and to discuss the diagnosis of fault conditions.

Rotating machinery vibration is measured to monitor the condition of the machine and enable to detect machine fault and correct as soon as possible.

High levels of vibration are indicative of high levels of component stress, high noise levels and reduced machine fatigue life.

Measurements are usually taken of the system vibration amplitude, phase and frequency.

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.


Signal measurement and display

The transducers from where the signal originates will usually be either (i) a non-contacting proximity transducer,(ii) a velocity transducer mounted on the machine or(iii) an accelerometer mounted on the vibrating machine.


Proximity displacement measuring transducer

Accelerometer

Microphones Sound meter

Vibration meter

Vibrations and acoustics measurement sensors, meters and display units


Oscilloscope Spectrum analyzer


In applications where phase is to be measured, a reference signal indicating a particular location on the shaft will be required.

This ‘key phasor’ signal might be obtained by using(i) a proximity transducer or (ii) an optical transducer

The key phasor signal is used in conjunction with the vibration signal, the phase angle recorded being that between the peaks of the two signals.(see Figure 1).

The instruments most frequently used for signal measuring of rotating machine vibration are oscilloscopes, tracking filters, and spectrum analysers.

Some oscilloscopes enable a phase-indicating pulse of extra-bright beamto be displaced on the orbit once per revolution of the shaft. The shape of the orbit, or Lissajous figure, can itself be a useful tool in monitoring machine health.


A typical whirl orbit for a machine subject to a small amount of imbalance is shown in Figure 2. (as displayed on an oscilloscope).

The tracking filter is a device which accepts two input signals, one being the vibration signal under consideration and the other being a phase reference signal and it removes from the vibration signal any components which are not of the same frequency as the reference signal.

The output from a tracking filter can be used to construct Bode plots(Figure 3) and Nyquist diagrams (Figure 4) used, for example, during balancing operations.

The spectrum analyser is used to separate out the incoming vibration signal into all of frequencies from which the total signal is composed.


The data is thus said to be displayed in the frequency domain. Many spectrum analyzer have the facility to plot several such graphs ‘in cascade’ or ‘waterfall’ diagrams.

Once the vibration signals have been collected and measured, they are used to judge whether or not the machine in question is operating properly.

Spectrum analysers have various convenient functions, such as

o tracking analysis,

o Campbell diagram

o Waterfall diagram.

In tracking analysis, dynamic characteristics of a rotating machine are investigated by changing the rotational speed.

A waterfall diagram is a 3-dimensional plot of spectra at various speeds. Figure 11.19 shows these diagrams for illustration.

Dr. R. Tiwari ([email protected]) 10Fig 19 Functions of spectrum analyzer


Description of the rotor test rig at IIT GuwahatiThe rotor kit is a precision high-speed rotating machine that may be assembled and operated in various configurations. The rotor kit is designed and manufactured by Bently Nevada, USA Company. The rotor test rig is shown in Fig. 32.

Fig. 32 Rotor bearing test rig


The rotor is supported by a servo fluid control bearing at one end and bush bearing at other end. The rotor is driven through flexible coupling by an electric motor, whose speed is adjusted by a frequency controller. Bearing load is adjusted by shafting the masses mounted on the rotor.

The rotor test rig consists of the following parts and part numbers are in consistence with the number mentioned Figure 32: (1) Motor (2) Rotor base (3) Probe mount (4) Servo fluid control bearing (5) Preload frame (6) Rotor (7) Rotor masses (rigid disks) (8) Journal (9) Oil fill port (10) Oil plug (11) Oil seal.

A brief description of the various parts of the test rig is given below.

� Motor:It closely holds the desired speed with changes in loading conditions. This has been accomplished by incorporating a direct current motor and high performance control circuitry.

The motor can run in either a clockwise or counter-clockwise direction and has adjustable slow roll speed capability.


It can be controlled remotely by using a ± 5-Volts control input, such as a signal generator or Dc power supply, to drive the motor speed control device. Rotor speed is displayed on a digital tachometer with a large LCD readout.

� Rotor base:The Rotor base has a V-frame design that has been developed by the manufacturers to provide better control of the housing dynamic stiffness properties. The mechanical tolerances have also been tightened, resulting in more accurate machine behavior modeling. It houses motor, bearing and rotor.

� Probe mount: It is used in holding the proximity sensor probes, when response is taken off the bearing center.

� Servo fluid control bearing:The servo fluid control exhibits certain characteristics of both bearing types, hydrostatic and hydrodynamic, and combines them in an innovative way that gives new and significant improvement over these other designs.


Figure 33 Servo fluid control bearing

Figure 33 shows the servo fluid control bearing. It contains the following sub parts (the digit after the decimal point represent the part number given in Figure 33) (4.1) Probe mounting holes (4.2) Bearing (4.3) Retainer (4.4) Oil reservoir (4.5) Bearing support (4.6) Main pressure valve (4.7) Pocket pressure gauge (4.8) Oil seal. A brief description of the servo fluid control bearing is given below.


(4.1) Probe mounting holes:Probes are mounted in these holes to measure the response in terms of voltage.

(4.2) Bearing:It houses the journal of the rotor and proximity probe transducers using which response is taken in two transverse directions of the rotor axis.

(4.3) Retainer:It holds the bearing to the bearing support, which is further supported on rotor base

(4.4) Oil reservoir:It stores the oil, which would be circulated during rotation of the journal. In the present work oil Chevron GST 32 oil is used.

(4.5) Bearing support:It holds the bearing and encloses the oil reservoir.

(4.6) Main pressure valve: It is used in controlling amount of oil circulatingto the bearing and also in controlling the pressure of oil flow at bearing.


(4.7) Pocket pressure gauge:It quantifies the pressure of the oil flow at the bearing.

� Preload frame:Preload frame is used in applying pretension to the rotor. In the present work preload frame is used to position the journal at the center of the bearing clearance while determining the natural frequencies of the rotor bearing system by impact test.

� Rotor: It is the rotating part of the rotor-bearing system. It consists of a circular shaft and rigid masses. Solid steel shafts of diameter 10 mm and length 780 mm were used, along which balance disks of inside diameter 10 mm, outside diameter 78mm and thickness 25 mm could be fixed at any desired axial position on to the shaft. There were 16 equally placed M4-threaded holes in each disks at radii of 30 mm, to allow for addition of balance weights, however in the present work additional weights were not used. A flexible coupling connects the rotor to the motor driver shaft.

� Oil seal:It prevents the leakage of oil from the servo fluid control bearing housing, when the rotor is in motion.


� Machinery Fault Simulator (MFS)The MFS is the most comprehensive Laboratory Scale Machine on the for performing rotor dynamics experiments and learning vibration signatures of the most common machinery faults in a controlled manner without compromising your quality production/profit.

The bench-top system has spacious modular design featuring versatility, operational simplicity, and robustness.

In depth studies of a variety of faults can be conducted using over 28 application specific option kits supplied by Spectra Quest.

� Benefitso Improve machinery performance and minimize unscheduled downtime by

sharpening the skills of the analyst.

o Optimize/establish the training program; reduce training cost and time.

o Expedite the learning of machinery condition monitoring and fault diagnosis techniques.

o Learn to diagnose common machinery malfunctions: alignment, balancing, bearing, gearbox, belt drive, variable speed machines, resonance, pump, motors and many more.


� Featureso Simple methods for introducing controlled, calibrated

faults. o Study the vibration spectra of common faults and

learn fault signatures. o Bench top machine for hands-on training and

sharpening skills. o Manual with exercises for individually paced study. o Learn resonance, variable speed and foundation

design issues. o Study correlation between vibration and motor current

spectra. o Model rotor dynamics and develop advanced

diagnostic techniques.

APPLICATIONS


� Description of the instruments� Impact hammer

One of the popular methods of excitation is through use of an impact or hammer (Ewins, 1984). It is a relatively simple means of exciting the structure into vibration.

The equipment consists of an impact, 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. Using different sizes of impact may also extend the useful range.

In the present work rubber tip is used in the hammer. 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 21, so that impact can be applied manually.

Dr. R. Tiwari ([email protected]) 20Modal hammers, force transducers, and impedance heads for impact testing


Fig. 34 Impact hammer

• 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.

Specifications of the impact hammer are as follows:� Attenuation factor: 4.06, � Sensitivity at output of hammer: 0.95 pC/N, � Rubber tip specifications:

o Frequency range 0-500 Hz, o Duration range 5-1.5 ms, o Force range 0-700 N,

� Physical:o Weight of the hammer 280 g, o Weight of the rubber tip 4.1g, o Materials: Anodized aluminium, 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


(39)

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.

� Measurement amplifierA Bruel and Kjaer make measurement amplifier is used in the present work to convert the charge signal out put from the impact hammer to voltage signal.

This amplifier can be used for amplification of the one signal. In the present application, sensitivity at output of the amplifier is 10 mv/N.

� Proximity probe transducerBently Nevada make proximity probe transducers are used to capture the displacement signals.

The proximity probe transducers are eddy current type and the sensitivity of the proximity probe is 200 mv/mil (7.87 × 103 mv/mm).

( )a m tF F M M M= −


� Pulse analyzer (Data acquisition system)Pulse analyzer 3560 C (make Bruel and Kajer) is a multi channel analyzer system, and it consists of a PC with LAN interface, Pulse software, Windows NT/2000, Microsoft office and data acquisition hardware.

The system possesses time capture and FFT analyzers, it can be used for sampling and data acquisition of displacement signals.

Setting of a project is to ensure that a measurement is set up exactly the following organisers are included:

o Configuration organizer:o Measurement organizer: o Function organizer: o Display organizer:


Measurements and analysis� Determining the bearing center

Bearing center is required to be known, to calculate the eccentricity under running conditions.

In the present case it is found as described in the subsequent lines. Clearance circle is drawn initially by manually rotating the shaft against the bearing’s internal wall.

The average of two extreme outputs of each probe is considered as the bearing center and is used to calculate the eccentricity under running conditions.

The bearing center and clearance circle are shown in the Figure 35. Diametral clearance is determined, from the clearance circle and is found to be 0.16 mm.

Figure 35 Experimental bearing clearance circle


� Determining natural frequencies of the rotor bearing systemNatural frequencies of the rotor bearing system are important parameters to be determined prior to any investigation.

In the present work two natural frequencies are obtained by using the impact test.

Prior to impact test journal position is shifted to the center of the bearing clearance circle by applying pretension to the rotor, as journal would be at the bottom of the clearance circle initially.

Impact is then applied at either of the rigid disks while the rotor is stationary. Displacement to impulse force is measured at the bearing end both in horizontal and vertical direction using proximity probe transducer.

FFT of the measured impulse response then gives frequency domainimpulse response. In the frequency domain response natural frequencies appear as higher amplitude peaks.

In the present work two natural frequencies were observed.

Figure 36 and Figure 37 show the absolute value of the FFT of the measured impulse response in both horizontal and vertical directions respectively.


These figures also 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 23 Natural frequencies of the rotor bearing system in horizontal direction

Figure 24 Natural frequencies of the rotor bearing system in vertical direction

s


A variety of vibration exciters (shakers) and associated amplifiers:


Vibration measuring InstrumentDepending upon frequency range utilized, displacement, velocity or acceleration is indicated by the relative motion of the suspended mass with respect to the case.

The equation of motion of a mass suspended in a seismic unit as shown in Figure 15.

(5)

where x is the displacement of seismic mass and y is the displacement of vibrating body. (i.e. machine surface).

)()( yxkyxcxm −−−−= ��


Both measured with reference to inertial reference. Let z = ( x – y ) and assuming sinusoidal motion of the vibration body, the EOM becomes

; (6)

EOM is same as support motion. The steady state solution , then

with(7)

and

(8)

tYmkzzczm ωω sin2=++ ��

[ ] [ ]222

2

222

2

)/(2)/(1

)/(

)()(nn

nY

cmk

YmZ

ωωζωω

ωωωω

ω

+−=

+−=

)sin( φω −= tZz

[ ] ])/(1/[)/(2)/(tan 221nnmkc ωωωωζωωφ −=−= −


Figure 16 shows characteristics of equation (7) and (8).

Seismometer-Instrument with low natural frequency (velometer)When the natural frequency �n of the instrument (seismometer) is low in comparison to the vibration frequency � to be measured, �/�n approaches large value and relative displacement ratio |Z/Y| approaches one or displacement Z approaches Y regardless of the value of damping � (as shown in Figure 16 for Z/Y vs �/�n).


Since Z�Y , the mass m then remain stationary while the supporting case moves with the vibrating body, such instruments are called seismometers. One of the disadvantages of the seismometer is its large size.

Since Z = Y, relative motion of the seismic mass must be of the same order of magnitude as that of the vibration to be measured.

The relative motion z is usually converted to an electric voltage by making the seismic mass (a magnet) moving relative to coils fixed in the case as shown in Figure 17.


Since the voltage generated is proportional to the rate of cutting of the magnet field, the output of the instrument will be proportional to the velocity of the vibrating body. Such instruments are called velometers. For such instruments the value of �n is 1Hz to 5 Hz. Useful frequency range of � is 10Hz to 2000 Hz. Sensitivity of such instruments (output (mV)/input (cm/s)) are 20 mV/ cm/s to 350mV/ cm/s with maximum displacement limited to 0.5 cm (peak to peak).

Both displacement or acceleration are available from the velocity type transducer by means of the integrator or differentiator provided in most signal conditioning units.

Accelerometer-Instrument with high natural frequencyWhen the natural frequency of the instrument is high compared to that of the vibration to be measured, the instrument indicates acceleration. From equation (7) the factor

approaches unity for �/�n � 0, so that[ ] [ ]222 )/(2)/(1 nn ωωζωω +−


or Z = acceleration/ �n2 (9)

Thus Z becomes proportional to the acceleration of the motion to be measured with a factor of 1/ �n

2. The useful range of the accelerometer can be seen form Figure 18. From Figure 18 it can be seen that the useful frequency range of the undamped accelerometer is somewhat limited.

c

2222 /// nn YZYZ ωωωω =→=

[ ] [ ]222 2)(1

1

RR ζ+−

ζ=0

ζ=1ζ=0.7


However with � = 0.7 the useful frequency range is 0 � �/�n �0.2 with maximum error less than 0.01 percent. Electromagnetic type accelerometers generally utilize damping around � = 0.7, which not only extends the useful frequency range but also prevent phase distorsion for complex waves.

Piezoelectric crystal accelerometers (they have very high natural frequency) have almost zero damping and operate without distortion up to frequencies of 0.06 fn.


However, to use these equipments correctly, we must have some background knowledge of signal processing.

Further, if we have to construct a specific data analysis system that fits our research, we must have sufficient understanding of the fundamental of signal processing.

The vibration of the rotor is a whirling motion and therefore not only the frequencies but also the directions of the whirling motions are important enough to pursue their causes.

However, since the usual fast Fourier transform (FFT) theory gives information about magnitudes of frequencies and phases only, we cannot know the whirl direction using the conventional FFT-analyser.

For this purpose, Ishida (1997) and Lee (2000) proposed a signal processing method where the whirling plane of a rotor is overlapped to the complex plane. This method is called the complex-FFT (or directional-FFT) method, enables us to know the directions of whirling motion besides the magnitudes of the frequencies.

We will discuss fundamental ideas necessary to understand signalprocessing by computer.

In addition, applications of the complex-FFT method of studying stationary and non-stationary vibrations are explained


�Measurement and Sampling Problem

Measurement system and Digital Signal: A measurement system is shown in Fig 20a.

Fig. 20(a) A measurement system

Fig. 20(b) Analog and digital signals


Necessary data are detected from the vibrating structure by sensors.

In a rotating machine, rotor displacements in two directions forming a right angle and a rotating speed are detected as voltage variations.

The output signal x(t) from the sensor is an analog signal that is continuous with time. But the signal is discretised when it is acquired by computer through an interface. This digital signal is a series of discrete data {xn} obtained by measuring (called sampling) an analog signal instantly at every time interval ∆∆∆∆t and is given as xn=x(n∆∆∆∆t) where n is an integer. This interval ∆∆∆∆t is called a sampling interval.

A digital signal is descretised in both time and magnitude. Discretization in magnitude is called quantization, and the magnitude is represented by binary numbers (unit: bits).

Digital data in a personal computer are processed into various forms using software programs.

In this operation, two representative processing are performed.

o One is signal extraction, where unnecessary signal components are abandoned in the acquired data, and

o the other is data transformation, where the data are converted to a convenient form (i.e. from binary to actual numbers).


Problems in Signal Processing:

When an analog signal x(t) is changed into a sequence of digital data {xn} (n = 0, 1, 2, …, N) a virtual (or imaginary) wave is obtained if a fast signal is sampled slowly.

For example, when a signal illustrated by the full line is sampled as shown in Fig. 21, although it is not contained in the original signal. This phenomenon is called aliasing.

Fig. 21 Aliasing


It says: when a signal is composed of the components whose frequencies are all smaller than fc (i.e. fc is the maximum frequency component contained in the signal) , we must sample it with a frequencies higher than 2 fc for the sake of not losing the original signal’s information. The frequency is called Nyquist frequency.

However, this theorem teaches us that digital data with more than two points during one period can express the original signal correctly. For example, if we sample the signal having components of 1, 2 and 6 kHz (i.e. fc=6 kHz and fnq=12 kHz) with a sampling frequency of 10 kHz, we have an imaginary spectrum, which does not exist practically. But, if we sample it with a frequency of more than 12 kHz ( kHz), such an alising problem does not occur.

In practical measurements, we do not commonly determine the sampling frequency by trial measurement. Instead, we use a low-pass filter to eliminate the unnecessary high-frequency components in the signal and sample with the frequency higher than twice the cutoff frequency. By such a procedure, we can prevent aliasing.


� Fourier Series In data processing, we must first know the frequency components contained in a signal. The fundamental knowledge necessary for it is the Fourier series. We will briefly summaries it from the point of view of signal processing. One type of Fourier series is expressed by real numbers, while the other is by complex number.

(i) Real Fourier Series: A periodic function with period T can be expanded by trigonometric functions which belong to the orthogonal function systems as follows

where . This series is called the Fourier series or real Fourier series. Its coefficients are given by

(ii) Complex Fourier Series: Fourier series can be expressed by complex numbers using Euler’s formula. Complex numbers make it easier to treat the expressions. As will be mentioned later, complex representation makes it possible to represent a whirling motion on the complex plane. Substituting Euler’s formula into equation (10), we have

( ) ( )�∞

=

++=1

0 sincos2

n

nn tnbtnaa

tx ωω

�−

=2/

2/

cos)(2

T

T

n tdtntxT

a ω �−

=2/

2/

sin)(2

T

T

n tdtntxT

b ω

T/2πω =

�∞

−∞=

=n

tjnnectx ω)(

(10)

(11)

(12)


where the complex coefficients are given by

Equation (12) is called the complex Fourier series. Between the real and complex Fourier coefficients, the relationships

hold, where n > 0. From this we know the following relationship

Therefore, if these quantities are illustrated in the figure taking the orderas the abscissa, the real part is symmetric about the ordinate axis, and the imaginary part is skew-symmetric about the origin. These complex Fourier coefficients can also be represented by

where the absolute value is called an amplitude spectrum, the angle a phase spectrum and a power spectrum. As an example, the

complex coefficients of the square wave with period T = 8 are shown in Fig. 22This wave is defined as

�−

−=2/

2/

)(1

T

T

ntjn dtetx

Tc ω ( )�,2,1,0 ±±=n

2nn

n

jbac

−=

20

0

ac =

2nn

n

jbac

+=−

nn cc −=( )�,2,1,0 ±±=nn

njnn ecc θ−=

222nnn bac +=

( )nnnn abc 1tan −=∠=θ2

nc

(13)

(14)

(15)

(16)


(a) Time History (b) SpectrumFig.22 Spectrum of a square wave (complex form)

{ }1 1 40 0 00 4 1 1

0 1 0 / 2 /c e dt e dt e dt T T−

− −= × + × + × =� � �

2sinj 0n

nc

Tnω

ω= + × 0≠n

)(tx )()( xfxf −=

nc

For this square wave, we have the following from equation (13)

,

for

Since is an ever function i.e. , the imaginary part of

is zero

Fig23 (a) An odd function (b) An even function

(17)

cn

n

and


� Fourier Transform

When x(t) is an isolated pulse, it cannot be converted to a discrete spectrum since it is not periodic. However, let us consider that this interval is extended to infinity. Then the spectra obtained will represent the spectra of the isolated pulse. Substituting equation (13) into equation (12), we get

If we make , we have

This can be expressed in separate forms as follows

Equation (11.20) is called the Fourier transform of x(t) and equation (11.21) is called the inverse Fourier transform of X(ωωωω).

0 0

/ 2j j

/ 2

1 2( ) ( )

2

Tn t n t

n T

x t x t e dt eT

ω ω ππ

∞−

=−∞ −

� �= � �

� ��

∞→T

j j1( ) ( )

2t tx t x t e dt e dtω ω

π

∞ ∞− −

−∞ −∞

� �= � �

� ��

j( ) ( ) tx t X e dtωω∞

−∞

= �( ) j1

( )2

tX x t e dtωωπ

∞−

−∞

= �

(18)

(19)

(20)

(21)


Fig. 24 shows a square pulse defined as x(t) = 1 for

= 0 for all other tby the Fourier transformation

Now, let us compare the spectrum of a square wave of period T as shown in Fig.22 and that of a square pulse shown in Fig.24.

From equation (17) and (23) that the Fourier coefficients in Figure 24 have the following relationship to .

where is the fundamental frequency. Therefore, the envelope of the quantities obtained by multiplying to the line spectra of the Fourier coefficients of the square wave gives the continuous spectra of the Fourier transform of the square pulse. As shown in Figure 11.25, multiplying the spectrum for the square wave with period T = 8 in Fig. 24 gives the spectrum for the square pulse in Fig. 25.

� Example:

11 ≤≤− t

πωωω sin

)( =X

)(ωX

0ωπ2/T

)(ωX

(22)

(23)

)(2 0ωπ

nXTcn =


Fig. 24. Spectrum of a square pulse

Fig.25. Comparison between spectra for a square wave and square pulse

(Wave)


This DFT is defined as follows:Given N data sampled with the interval , the DFT is defined as a series expansion on the assumption that the original signal is periodic function with the period (although the original signal is not necessary periodic).

However,various problems occur in the course of this processing.The first is the aliasing problem. When the signal is sampled with interval , information about the components with frequencies higher than is lost. Therefore, we must pay attention to valid range of the spectra obtained.

The second is the problem of the coincidence of periods. It is impossible to know the correct period of the original signal before the measurement. Therefore, the period of the original signal and the period of DFT do not coincide, and this difference produces the leakage error.

The third is the problem about the length of measurement.

� Discrete Fourier Transform

t∆tN∆

t∆1

2 t∆


How to compute DFT.Let us assume that we obtained data sequence by sampling. These data are extended periodically, as shown by the

dashed curve in Fig. 27.

The fundamental period is and the fundamental frequency is . (24)In the case of a discrete signal, the integral of equation (13) must be calculated by replacing t, T, x(t) and � with and �respectively. By such replacements, we have

(25)We represent the right-hand side of this expression by , that is

(26)

and call this descrete Fourier transform of the discrete signal .

Inverse discrete Fourier transform (IDFT)

1210 ,,,, −Nxxxx �

tNT ∆=Tπωω 20 =∆=

1 1j j (2 / ) ( / )

0 0

1 1N Nn k t n T k T N

n k kk k

c x e t x eT N

ω π− −

− ∆ ∆ −

= =

= ∆ =� �

1j2 /

0

1 Nnk N

n kk

X x eN

π−

−

=

= � (n = 0, 1, 2, …, N-1)

nx

12 /

0

Nj nk N

k nn

x X e π−

−

=

=� (k = 0, 1, 2, …, (N-1)) (27)

, , kk t N t x∆ ∆


These transformations map the discrete signal of a finite number on the time axis to the discrete spectra of a finite number on the frequency axis, or vice versa. These expressions using complex numbers are called complex discrete Fourier transform and the complex inverse discrete Fourier transform.We also have transformation using only real numbers. One is the real discrete Fourier transform, given by

Further, the inverse real discrete Fourier transform is given by

We explain the characteristics of the spectra obtain by DFT using an

example in the following slide.

1

0

1

0

1 2cos

1 2sin

N

n kk

N

n kk

nkA x

N N

nkB x

N N

π

π

−

=

−

=

=

=

�

�

(n = 0, 1, 2, …, N-1)

�−

=

��

� −=1

0

2sin

2cos

N

n

nnk Nnk

BNnk

Axππ (n = 0, 1, 2, …, N-1) (30)

(28)

(29)


Fig. 27(a) shows a square wave with period T = 8 and sixteen sampled data: and obtained by sampling with interval

In this example, the signal is sampled intentionally in the range that coincides with the period of the original square wave to avoid the leakage error.

Fig. 27(b)-(e) shows spectra representing the real part of ,the imaginary part of ,the amplitude ,and the phase .These spectra have the following characteristics:� The spectra is periodic with period N.

� The same spectra as those of the negative order n = -N/2, …, -1 also appear in the range n = N/2, …, (N-1).

� The spectra of the real part and those of the amplitude are both symmetric about n = N/2.

� The spectra of the imaginary part and those of the phase are skew symmetric about n = N/2.

� The spectra in the left half of the zone n = 0, …, (N-1) are valid. The spectra in the right half are virtual and are too high compared to the sampling frequency. If the sampling interval is narrowed the number of spectra increases, and therefore such a spectra diagram written in the interval extends to the right.

If the sampling frequency is shortened, the sampled data become substantially equal to the continuous wave, and therefore its spectra will approach those of the Fourier series shown in Figure 24.

140 === xx � 0155 === xx � 5.0=∆t

nXnX nX nX∠

T/2πω =∆

Dr. R. Tiwari ([email protected]) 53Fig. 27 Digital signal in the time and frequency domains


Fig. 24. Spectrum of a square pulse


• Different types of definition of DFT and IDFT• Definition I:

• Definition II:(MATLAB uses this)

� Discrete Fourier Transform

1j2 /

0

Nnk N

n kk

X t x e π−

−

=

= ∆ �1

j2 /

0

1 Nnk N

k nn

x X eT

π−

=

= �

(n = 0, 1, 2, …N-1) (31)

(k = 0, 1, 2, …N-1) (32)

1j2 /

0

Nnk N

n kk

X x e π−

−

=

=�1

j2 /

0

1 Nnk N

k nn

x X eN

π−

=

= �

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(k = 0, 1, 2, …N-1) (34)


� Fast Fourier Transform In 1965, Cooley and Tukey proposed an algorithm that enabled the fast computation of DFT. The algorithm is called Fast Fourier Transform (FFT), has made real-time spectrum analysis a practical tool.

In the calculation of DFT given by equation (16), we must perform many multiplications and additions. However, the same calculation appears repeatedly since the function glgkgdlgkdgdgldfkgldfgkdgldkgdlhas a periodic characteristic.

The FFT algorithm eliminated such repetition and allowed the DFT to be computed with significantly fewer multiplications than directevaluation of DFT.

The FFT algorithm has the restriction that the number of data must be dsajkldjasdjasldjasd.kWhen the number of data N is ,DFT needs multiplications and FFT needs multiplications.

For example, when ,about 1,050,000 multiplications are necessary in DFT and about 20,480 in FFT. If N increases this difference increases extremely large. MATLAB has FFT function name sfjslkfsfjwhere .

( ){ } ( ){ }NnkjNnke Nnkj πππ 2sin2cos/2 −=−

( )2 1, 2, , n n = ∞�nN 2=

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1024210 ==

)( nxfft 1}{ ×= nn xx


� Leakage Error and Countermeasures� Leakage Error:

In FFT or DFT, computations are based on the assumption that the data sampled over a time period are repeated before and after data measurement.

Fig. 28 shows the assumed signals and their spectra for two types of measurement of a sinusoidal signal . Both cases have 32 sampled data, but their sampling intervals are different.

In case A, the sampling interval is and the range measured is exactly twice the fundamental period. The computation of FFT or DFT is performed for the wave as shown by the dotted line. In this case the assumed wave is same as the original signal and therefore we get a correct signal spectrum.

In case B, the sampling interval is , and the range measured is about 2.5 times the period of the original signal. In this case, the assumed wave shown in Fig. 28(c) is not smooth at the junction and differs fro the original signal in time domain.

ttx sin)( =

3926.016/2 ==∆ πt

236.0=∆t

Dr. R. Tiwari ([email protected]) 58Fig. 28. Leakage error

Case A


As a result, the magnitude of the correct spectrum decreases andspectra that do not exist in the original signal appear.

As seen in this example, if the time duration measured and the period of the original signal do not coincide, the magnitude of the correct spectrum decreases and spectra that do not exist in the original signal appear on both sides of the correct spectrum. This phenomenon is called leakage error.


� Countermeasures for leakage error (Window Function):To decrease the leakage error due to discrepancy between the time duration measured and the period of the original signal, we must connect the repeated wave smoothly. For this purpose we multiply a weighting function that decrease gradually at both sides. This weighting function is called time window. Representative time windows: the Hanning window, Hamming window and Blackman-Harris window are shown in Fig. 29. These windows are defined in the range: as o Hanning windowo Hamming windowo Blackman-Harris window

and outside for all three cases.

10 −≤≤ Nn( )( ) 0.5 0.5cos 2 /w n n Nπ= −

( )( ) 0.54 0.46cos 2 /w n n Nπ= −

( ) ( )( ) 0.423 0.498cos 2 / 0.0792cos 4 /w n n N n Nπ π= − +

10 −<≤ Nn ( ) 0w n =

(35)(36)

(37)

Fig. 29 Window function


� Preventation of leakage by coinciding periods:

We can obtain the correct result if the time duration measured coincides with the integer multiple of the period of the original signal. If we can attain this by some means, it is better than the use of window functions, which distorts the original signal.

For example, for numerical calculations that can be repeated in exactly the same way and whose sampling interval can be adjusted freely, we can determine the measurement duration after we know the period of the original signal by trial simulation, and then execute the actual numerical simulation.

On the contrary, for experiments, fine adjustment of sampling intervals is generally impossible using practical measuring instruments.

However, if the phenomenon appears within a speed range, we can change the rotational speed little by little and adopt the best result where the period, often determined by the rotational speed, and the sampling interval fit.


� Applications of FFT to Rotor Vibrations

• In FFT (or DFT), elements of data sequence obtained by sampling are considered as real numbers and those of data sequence obtained by discrete Fourier transform are considered as complex numbers.

• In the following, we introduce a method that can distinguish between whirling directions utilizing the revised FFT. In this FFT, rotor whirling motion is represented by a complex number by overlapping the whirling plane on the complex plane and applying FFT to these complex sampled data.

• Let us assume that a disc mounted on a elastic shaft is whirling in the y-z-plane. We get sampled data and by measuring the deflections dddfandin the y and z directions respectively. Taking the y-axis as real axis and the z-axis as imaginary axis, we overlap the whirling plane on the complex plane.

• Using sampled data and , we define the complex numbers as follows:

{ }kx

{ }ky { }kz )(ty )(tz

ky kz

kkk jzyS += (k = 0, 1, 2,…., N-1)

and apply FFT (DFT) to them. We call such a method the complex-FFT method.


Fig. 30. Spectra of the sub-harmonic resonance of ½ order of a forward whirling mode

Fig. 31. Spectrum of the combination resonance (complex FFT method


Shaft ImbalanceThe vibration occurs at 1× machine rotational frequency in general, but sometimes higher harmonics of rotational speed are excited.


The amplitude peaks at the critical speed, and phase changes by 1800

passing through about 900 at the critical and it can be seen in the Bode plot (see Figure 6).

The shaft whirl orbit takes (generally) on an ellipse shape (unless the shaft support impedance is isotropic, in which case it is circular), the orientation of the ellipse changs as the critical speed is passed through (see Fig. 7).

Example of fault in same categories are(i) hot spot on the shaft causing a thermal bow,(ii) components mounted loosely on the shaft and (iii) moist entering a hollow shaft.

Misalignment, pre-loaded shaftShafts with a heavy pre-load carried by the bearings, as distinct from the out-of-balance load, can show variation characteristics similar to those caused by bearing misalignment.


Misalignment may be present because of improper machine assembly or as a consequence of thermal distortion, and itself results in additional loads being applied to the bearing.

The vibration associated with misalignment occurs at 1× machine running speed but, unlike imbalance case, there is usually a substantial component in the axial direction, which may be greater than radial direction.

Substantial amounts of misalignment (or pre-load) can also cause vibrations at frequencies of 2 × machine running speed, and sometimes higher multiplies.


For moderate misalignment the shaft amplitude is reduced in the sense of the applied pre-load, so that if it were originally circular it might become elliptical; that is because there is more resistance to motion in the direction opposing the preload so the rotor ‘feels’ a higher support stiffness in this direction.

For more severe cases the orbit may become banana-shaped or Figure-8shaped.

Coupling alignment may be checked by taking dial gauge readings and gap measurements. The variations in these measurements should berecorded as both shafts rotated simultaneously, and averaged values used to calculate the misalignment present.

Alignment of bearings may be checked using optical methods (one bearing with another) or using feeler gauges in the clearance between journal and bush.


Other indicators of alignment in fluid bearings are measurements of the bearing temperature and lubricant pressure distribution.


Diagnostics of Misalignments


Rubs : Rubs are produced when the rotating shaft comes into contact with the stationary components of the machine and these may be classified as

(i) partial rubs,

(ii) full rubs or

(iii) may take the form of a stick-slip action.

A partial rub is that where contact between the shaft and stationary component exists only for part of the cycle time, for example high spot on a shaft rubs against part of a seal and its frequency spectra always show some synchronous vibration (1× shaft rotational shaft) together with some sub-harmonics which are related to free lateral vibrations of the rotor.

In case of a full rub, the shaft may no longer make contact with the stationary member only at local high spots, but instead bounce its way all around its orbit.


A rub may also give rise to stick-slip action of the shaft against the stationary components. It is a vibration set-up as a consequence of the friction coefficient between shaft and stationary component changing with relative velocity.

In the Figure 11 the rotor rotation initially causes it to roll over the stationary component so that its center moves from left to right.

The vibrational frequency is usually much higher than the shaft rotationalfrequency, and the fault frequently shows up as a torsional vibration as well as a lateral one.


In case where rubs cannot be avoided by increasing clearance then either the rotor surface or that of stationary component should be manufactured from a soft material which will easily wear away, this will help to avoid backward precession.

Loose Components In some machines vibration levels may be excessive as a consequence of components being assembled too loosely, for example in the case of a bearing, which is not properly secured.


The vibration signal detected from a transducer is normally a sine wave, but is truncated at one extremity as shown in Figure 12(a).

The truncation of the sine wave occurs because of the non-linearity of, for example, the bearing mount which suddenly becomes very stiff when the loose component reaches the limit of its allowable levels. In frequency domain these truncation (impulses) show up as a series of harmonics of rotational frequency (Figure 12(b)).

The 2× machine running speed component is usually the harmonic most easily detected. In cases where it is the bearing, which is not properly secured, the shaft average support stiffness over one cycle is effectively reduced.

When the shaft speed is less than 2× normal resonant frequency, the reduction in shaft support stiffness acts to lower the resonant frequency to ½ × shaft rotational speed so that vibration at this frequency becomes significant.


A similar effect may occur with other sub-harmonics of rotational speed. (i.e. � , ¼ etc.).

Shaft cracksShaft cracks are potentially the cause of catastrophic machine failures. They are particularly likely to occur in instances where shaft stresses are high and where machine has endured many operation cycles throughout its life, so that material failure has occurred as a consequence of fatigue.

If the shaft cracks can be detected before catastrophic failure occurs then the machine can be temporally taken out of service and repaired before situation gets out of hand.

The presence of a transverse shaft crack sometimes is detected by monitoring changes in vibration characteristics of the machine

The shaft stiffness at the location of the crack is reduced, by an amount depending on the crack size.


This in turn affects the machine natural frequencies, so that changes in natural frequencies may be symptomatic of a shaft crack.

A transverse crack results in significant changes in both1× and 2×rotational speed vibration frequencies. The 1 × rotational speed component may change in both amplitude (which may either grow ordecay) and phase, as a consequence of the change in rotor bending stiffness.

A transverse crack also results in a bigger rotor bow due to a steady load (for example gravity) which may be detected under ‘slow roll’ conditions.

The classical symptom of a cracked shaft is the occurrence of a vibration component at 2 × shaft rotational frequency as a consequence of the shaft asymmetry at the crack location in the presence of a steady load. The 2 × rotational frequency component is usually prevalent when the machine is running at a critical speed or ½ × any critical speed (similar to sub-critical due to gravity).


Other types of crack, in particular those which originate from the center of the shaft, may not cause a change in shaft bending stiffness which is significant enough to enable the crack to be detected by 2 × rotational speed components in the vibration spectrum.


Crack Diagnostics


Signal Generator Module

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Rolling element bearing faultsFaults in rolling bearings may occur prematurely as a consequence of operating the bearing under inappropriate loading conditions (including misalignment) and at excessive speeds.

Alternatively they may be produced simply as part of the normal wear process during the life of the bearing.

The growing trend, however, is to monitor the condition of rolling element bearings continually so that bearing wear may be detected at an early stage, and enable the engineer to ensure that the bearing is replaced at a convenient time before the bearing fails completely


Condition monitoring of rolling element bearings has enabled cost saving of over 50%; as compared with traditional maintenance method.

The most common method of monitoring the condition of rolling element bearings is to measure the vibration of the machine at the bearing at regular intervals using a velocity transducer or an accelerometer mounted on the machine casing.

More recently, observation of the bearing outer-race deformations using fiber optics and high sensitivity proximity transducers has also been used to monitor bearing conditions.

Defects in the bearing may develop on either raceway, on the rolling elements themselves, or on the cage; subsequent vibrations are forced as a consequence of impact between the fault and other bearing components, so that the frequency of the resulting vibrations is largely dependent on the frequency of impacting.


For example, a defect in a bearing outer raceway would set up vibrations corresponding to the frequency with which rolling elements passed over the defects.

Consideration of the bearing kinematics (geometry) enables calculation of the frequencies associated with defects in different bearing components.

Outer race frequency =

Inner race frequency =

Rolling element frequency =

Case frequency =

where N is the shaft frequency in rev/min and n is the number of rolling elements.

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For example, a defect in a bearing outer raceway would set up vibrations corresponding to the frequency with which rolling elements passed over the defects.

Consideration of the bearing kinematics (geometry) enables calculation of the frequencies associated with defects in different bearing components.

Outer race frequency =

Inner race frequency =

Rolling element frequency =

Case frequency =

where N is the shaft frequency in rev/min and n is the number of rolling elements.

(1)

(3)

(4)


Whilst the characteristic frequencies can be easily calculated, the process of diagnosing a fault can be complicated by a number of factors.

Some of the characteristics frequencies may be very close to harmonics of rotational speed; this means that a narrow-band spectrum analyzer is required in order to distinguish vibration components caused by a bearing failure those caused by for example, imbalance.

As wear of the bearing progresses the frequency spectrum changesfurther. Sometimes higher-order harmonics of the defect frequency become present, sometimes with their own sidebands, and can dominate the spectrum.


In addition, wear particles are transported around the bearing and accelerate the development of further defects at other locations, leading to high level of vibration at many frequencies so that peaks which are characteristic of particular defect difficult to distinguish.

A most important feature of condition monitoring of bearings (and rotating machinery in general), is the collection of ‘baseline’ reference measurements of vibration taken when the machine is first commissioned (or re-commissioned after overhaul).

An alternative method of monitoring rolling element bearing condition is the ‘shock pulse method’ (SPM). This methods depends upon the signal input to the measuring instrument being passed through a narrow-band filter whose center frequency corresponds to the accelerometer natural frequency.


As part of the bearing collides with a defect, a shock wave is transmitted through the bearing and machine casting thus exciting the accelerometer by means of a pulse input.

The accelerometer output is a damped transient waveform whose frequency is much higher than the frequencies characteristic of specific bearing faults, and whose magnitude is dependent upon the magnitude of the defect in the bearing.

Again this method relies heavily upon establishing reliable baseline data and on monitoring changes in output level rather than on absolute values. SPM can not indicate the cause of the vibration levels, but certainly it is a less expensive technique.


Experimental Set-up at IIT Kanpur during PhD


The rotor-bearing set-up

Dr. R. Tiwari ([email protected]) 109General view of experimental rotor rig (Edwards, PhD Thesis , 1999, UWS)

Experimental Set-up at University of Wales Swansea, UKduring post doctoral work

Dr. R. Tiwari ([email protected]) 110General view of experimental rotor rig (Edwards, PhD Thesis , 1999, UWS)




(Edwards, PhD Thesis , 1999, UWS)



(Edwards, PhD Thesis, 1999, UWS)


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The procedure for carrying out the experiment is as outlined below.

Start the servo fluid control bearing, and set the pressure in the pocket

pressure gauge.

Run the rotor at a particular speed, sample and save the initial

displacements at the bearings in both the vertical and horizontal directions

Keep the rotor running at the same speed; hit the rotor horizontally with the

impact hammer (or alternately unbalance can be put at discs) andrecord

the response at bearings in both horizontal and vertical directions and

impact force.

Keep the rotor running at the same speed; hit the rotor vertically with the

impact hammer (or alternately unbalance can be put at discs) and record

responses at bearings in both the horizontal and vertical directions and the


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Faults in gearsVibration measurements taken at the bearings can also indicate the

condition of the gearbox.

A narrow band spectrum analyzer is very useful for this purpose,because the monitoring process involves the detection of discrete frequency components which must be distinguished from frequencies generated through other mechanisms (i.e. rolling element bearings).


• The characteristic frequency of a particular gear set is the gear mesh frequency (the number of teeth on gear × the shaft rotational frequency) which will be evident in the spectrum relating to any gearbox, in good condition or otherwise.

• When one gear becomes damaged the gear mesh frequency component of vibration may increase substantially as compared to the base line vibration measurements, but this is not always the case.

• Harmonics of gear mesh frequency may also become more apparent. Another frequency, which is often excited by gear defects, is the resonant frequency of geared shaft itself. This frequency can usually be measured by impulse testing.


Both the natural frequency of the geared shaft and the gear meshfrequency may have accompanying side bands; sometimes side bandsthemselves may be the main indicator of a defective gearbox.


Conveyor Belt Gallery Fault


Protection against spurious signalsWhen studying vibration signals it is important to be sure that the signal under consideration represents vibration, which is actually present.

In some situations it is possible for the electrical signal received by the measuring system to indicate the presence of vibration which does not in fact exist.

Alternatively, a spurious signal may be superimposed on a genuine vibration signal and thereby upset its measurement. The two most common causes of the spurious signals are ‘noise’ and ‘run out’.


� 1. Electrical noise :Electrical noise may arise from a number of different sources :(i) random electron motion,(ii) local magnetic fields arcing and (iii) earth loop faults.

Noise created by the random motion of electrons is known as ‘thermal’noise. It is generally only of the order of �V however, and is not normally significant in measurements of rotating machinery vibrations.

The noise set up by local magnetic fields is usually of the order of mV and is more significant. It is set up as a consequence of magnetic fields in nearby electrical apparatus introducing noise signals in electrical leads conveying the signal which is to be measured.

Some protection against this type of noise-generating mechanism can be obtained by screening the leads with a high-conductivity material which is earthed.


A similar effect can also arise when one piece of instrumentation is sited close to another, because the magnetic fields inside one can induce signals in the circuit of the other.

In this instant instruments should be screened by placing it inside a metal case.

Another source of electromagnetic radiation which can induce noise signals is that emitted as a consequence of electrical arcing in switches and commutators.

This type of noise-generating mechanism can also be protected against by screening as described above. Earth loop noise can occur when there are too many earth connections in the instrumentation.

� 2. Runout: Proximity transducers used to monitor rotating machinery vibrations depend for their operation on a change in transducer reactance. Changes in transducer reactance may be present as a consequence of either mechanical or electrical runout.


Mechanical runout is present when the shaft section being monitored by the transducer is eccentric to the axis of rotation, or has significant surface undulations, alternatively, the shaft may be bent.

Each of these conditions results in motion of the shaft surface towards or away from the transducer tip when the shaft is rotated, motion which is not caused by shaft vibration.

To remove mechanical runout the shaft must be re-machined; if the shaft is bent it must be straightness prior to re-machining.

Electrical runout is present when, around the shaft circumference, there are variations in the permeability of the shaft material to the electrical field set up by the transducer.

This might be caused by residual magnetic fields in the shaft surface (due for example to particular methods of manufacture or non-destructive testing), by material in-homogeneity or by local residual stresses.


Residual magnetic fields can be removed by degaussing, if residual magnetism is not the cause of the electrical runout then specialtreatments such as burnishing, micropeening, or electroplating may be necessary.

The amount of mechanical runout present can be determined by mounting a dial test indicator next to the transducer, and noting the variation in reading when the shaft is rotated slowly.

The electrical runout present may be determined by noting the transducer reading when the shaft is rotated at low speed (sufficient to obtain a reading on the measuring instrument but not so high as to cause the shaft to vibrate).

The subtraction of the mechanical runout vector from the measured vector then gives the electrical runout vector.

All runout vectors should be removed from signal measurements to enable true vibration vectors to be recorded.


Such ‘nulling’ is carried out by subtracting the total residual runout vector from the measured vector, as indicated below.

� Removing runout from a vibration signalSome types of tracking filter enable runout vectors to be automatically removed from the incoming signal, so that the true vibration vector is displaced immediately.

This facility can also be programmed into data logging equipment.


The nulling is also carried out between critical speeds when producing a Bode plot, as discussed earlier, to obtain information about phase to be used in the balancing.

In this case it is not only the runout that is removed from the incoming signal, but also components of vibration associated mainly with modes whose characteristics frequencies are lower than the frequency at which the nulling is being carried out.

� Electronic differentiation and integrationIn many instances the vibration signal is to be differentiated or integrated prior to being measured.

For example, many signal conditioning devices enable accelerometer output signals to be integrated twice to provide a signal which is proportional to displacement, or alternatively they might differentiate a velocity transducer output signal to provide a measurement of acceleration.


Generally, passive electronic differentiators and integrators results in a loss of signal accuracy of the order of several per cent because their mathematical foundation involves an approximation.

Active devices, requiring a power source for internal amplifiers, are more accurate but are also more expensive.

It is particularly important to guard against spurious signals when differentiation or integration is necessary.

Since the differentiation requires that the signal is multiplied by its frequency, so that if the spurious signal is of a higher frequency than the vibration signal then after differentiation the spurious signal may appear to be far more significant.

A similar effect occurs with integration when the spurious signal is of a lower frequency than the vibration signal.


Thank you

Vibration Measurement of Rotating Machinery

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