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1. INTRODUCTION

Rolling bearings are rated to prevent the initiation of rolling contact fatigue, (RCF).

However, nowadays, due to material and technology improvements, the RCF

comprises only a small fraction of common failure types. Unfortunately, most of

failures are caused by bearing operations outside of recommended practice:

bad mounting procedures, misalignment, poor lubrication, contamination, rolling

bearings can develop prematurely failures. These ahead of time failures are

usually accompanied by an increase in bearing vibration and therefore the

condition monitoring was used for many years do detect degrading bearings

before they catastrophically break down.

The sources of bearing vibration are discussed along with the characteristic

vibration frequencies that are.

Based on the characteristic vibration signatures which rolling bearings exhibit as

its rolling surfaces deteriorate, nowadays vibration monitoring has become a part

of many planned maintenance regimes. However, in most of practical situations

the bearing vibration cannot be measured directly. The signal provided by the

bearing travels through a mechanical structure with structural resonances which

may significantly alters it before being captured by the measuring transducer.

Even worse, the acquisitioned signal incorporates vibration data from other

transmission parts (gears, chains, belts, etc.) and mechanical devices (electric

motors, hydraulics). All these make the interpretation data difficult other than by

a trained specialist and in some situations lead to wrong diagnosis

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2. VIBRATION BASICS

2.1 DEFINITIONS. CLASSIFICATIONS

In the absence of any external action, the elements of a mechanical system are

positioned in the reference states. Mechanical vibrations are alternating

movements of the component masses of mechanical systems with respect to their

reference states.

Vibration data are acquired by appropriate transducers that generate analog

electrical signals representing instantaneous values of the parameters of motions

(accelerations, velocities and displacements), forces and specific strains, as

functions of time. A sample record, representing a single vibration measurement

x(t) over a duration T is called time-history.

A stationary vibration is one whose basic proprieties do not vary with time.

Mechanical machines running in their normal regimes, with constant speed and

loading are accompanied by stationary vibrations. Stationary vibrations may

have a deterministic or a random evolution in time.

A deterministic vibration follows an established pattern so that the value of the

vibration at any future time is completely predictable from the past history.

A random vibrations is one whose future basic proprieties are unpredictable

except on the basis of probability.

A non-stationary vibration is one whose basic proprieties vary with time, but slowly

relative to the lowest component frequency of the vibration. Mechanical

equipments running in transient regimes, as speed up or speed down are

accompanied by non-stationary vibrations. Non-stationary vibrations may have

a continuous time evolution or a transient one.

From an energetic point of view vibrations are classified as free vibrations and

forced vibrations.

In free vibrations, the vibration movement is the result of an initial disturbance

and there is no energy supplied to system to maintain the vibratory movement.

The damping exists in any real system and causes a fast diminishing to zero of the

free vibrations amplitudes.

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In forced vibration there is a continuously energy supply toward the vibratory

system that compensate the damping losses and maintain the amplitudes at a

energetically balanced value. The values of the basic proprieties of the forced

vibrations depends on the excitation that introduces the energy in mechanical

structure as well as on elastic proprieties of the system, the later being expressed

mathematically by frequency response function.

Shock is a transient vibratory motions induced by an excitation having the form

of a pulse or step that acts theoretically over an infinite short time. Practically are

considered shocks any transient excitation acting over a time shorter than the

fundamental period of natural free vibration of the system. The vibration induced

by a shock excitation includes both the frequencies of the excitation and the

natural frequencies of the system. The shorter the shock is the more pregnant are

the natural frequencies of the system in the resultant vibratory movement. The

popular SPM testing method is based on this particularity.

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2.2 HARMONIC VIBRATION.

2.2.1 Sinusoidal representation

The pure sinusoidal movement, (Eq. (2.1) and Figure 2.1), is used to define the

basic descriptors of a vibratory movement in both time and frequency domains.

() = sin( + ) (2.1)

where A is called the amplitude of displacement, is the angular frequency and

represents the initial phase. The sinus movement is a continuous periodic motion

having the period T, Figure 2.1. The circular frequency and angular frequency are

defined as:

=1

, [] , and = 2, [rad/s] (2.2)

In the time domain the sinus vibratory movement has continuous harmonic

variations with the same frequency for the displacement, velocity and

accelerations. In the frequency domain the sine movement has a discrete

representation with only one frequency component (Figure 2.1).

Fig. 2.1- Basic descriptors of a sinus vibratory movement

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2.2 2 Complex representation of harmonic vibration

The graphical representation of a vibration as a linear time dependent sinusoidal

motion has the first disadvantage that both time and phase are represented

along x-axis and the second one that mathematical concept of negative

frequency appears meaningless.

Another way of representing harmonic oscillations is using the complex numbers.

Eulers formulae allow to write any complex number can as:

= [cos + sin], = 2 (2.3)

and,

cos =

2( + ), A sin =

2 ( ) , = 2 (2.4, 2.5)

The complex number carries amplitude and angle information and is called

phasor of the harmonic motion. Figure 2.2(b) illustrates the sinusoidal motion

Fig. 2.2.- Basic descriptors of a sinus vibratory movement by contra-rotating

vectors (phasors).

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as a vector sum of two contra-rotating vectors, each with amplitude A/2, and the

same absolute values for the angular frequencies and initial phases but different

signs. From Figure 2.2b it can be seen that when the contra-rotating vectors

rotate with time, the imaginary parts cancel out so the vector sum will always be

real and will trace out the harmonic curve illustrated in Figure 2.2(a).

2.2.3 Beating phenomenon.

If the vectors represented asynchronous vibration, vector addition applies, but at

any instance of time the resulting vector has different magnitude because the

two vectors rotate with different angular frequency. If the two harmonic motions,

hence phasors, have the same amplitude but slightly different frequencies there

are time instances when the two phasors will have a phase difference of

= (2 + 1), , that gives a zero value for the resultant amplitude. Using

the trigonometric representation we have:

1() = cos(), and 2 = cos[( + )] (2.5)

() = 1() + 2() = 2cos (

2) cos [( +

2) t] = () cos [( +

2) t]

(2.6)

The resultant motion x(t) is a cosine vibration of angular frequency equal to

( + /2), but having a time dependent amplitude, Figure 2. 3.

Fig. 2.3 Beating phenomenon.

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Whenever the amplitude reaches a maximum, there is said to be a beat, and the

time evolution is called beating phenomenon. The beat frequency and period

are:

=/2

2=

4, (Hertz); =

4

, (seconds) (2.7)

2.3 PERIODIC VIBRATORY MOVEMENT

A vibratory movement is periodic if there is a time period T that fulfills the equation:

() = ( + ), 0 (2.8)

Most of real vibrations have periodic evolutions, but very few of them are pure

harmonics.

2.3.1 Time domain description

Because the amplitude A and period T are not sufficient to characterize the time

evolution of a non-harmonic motion along one period, new parameters

represented by:

the arithmetic average RA-av,

the root mean square average RRMS ,and

the Crest Factor

were needed to be considered, Figure 2.1.

For the case of pure harmonic motion the Crest Factor = 2.

The elastic energy accumulated by the linear spring along one period is:

=

2 2()d

0 (2.9)

Divided this energy by one period the average power along one period is found

as being proportional with the square of XRMS revealing an important physical

significance which explains the large utilization of XRMS versus XA-av.

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2.3.2 Frequency domain description. Harmonic analysis and frequency spectrum

To evaluate both

the effect of vibration on the mechanical structures, and

the necessary measures to limit the possible damages

it is very useful to have a clear description of the frequencies content of the

vibratory movement. That is achieved by frequency analysis method.

In conditions of Dirichlets restrictions the theorem of Fourier establishes that a

periodic vibration can be represented with arbitrary accuracy as a finite or infinite

sum of harmonic movements which have their angular frequencies as multiples

of the fundamental one:

=2

, = (2.10)

() = 0

2+ [cos() + sin()]

=1 (2.11)

The procedure is called the harmonic analysis of the periodic function x(t).

The angular frequency Is called fundamental and the movement x(t) is

considered as sum of harmonic movements that have frequencies equal to the

fundamental and its integer multiply.

The sum from equations (2.11) is called the Fourier series, where the constants Ar ,

Br and are called Fourier coefficients and are mathematically formulated as:

0 =2

()d

0 (2.12)

() =2

()cos()

0d, () =

2

()sin()d

0 (2.13)

The motion expressed by the equation (2.11) can be easily written as a sum of

sinusoidal motions having angular frequencies multiplies of the fundamental one,

(Eq. (2.10)):

() = 0

2+ [sin ( + )]

1 (2.14)

In Eq. (2.14) the amplitudes Xr(r) and initial phases () are:

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() = 2 + 2 , () = tan1 (

) (2.15)

This procedure is called the harmonic analysis of a periodic motion (or, more

generally, periodic functions).

The harmonics can be plotted as vertical lines on the amplitude versus frequency

diagram called a frequency spectrum or a spectral diagram.

The spectrum of squared amplitudes is known as the power spectrum, and offers

information on how the vibration power is divided on different harmonics.

However, the power spectrum does not contain information regarding the initial

phases.

A reasonable accuracy is obtained even in the sum from Eq. (2.10) the first terms

are considered only. This statement will be sustained by two examples.

Example 2.1 Fourier series analysis of a rectangular wave.

The function x(t) of the hypothetic square motion is expressed as:

() = {1 when < < ( + 0.5)

1 when ( + 0.5) < < ( + 1)} , = 0,1,2,3,

The Fourier series coefficients are obtained from Eqs. (2.12) and (2.13):

() =2

() cos() d =

2

() cos() d +

2

0

0

2

() cos() d =

2

=2

1

sin()

/2

0

2

1

sin()

/2= 0

() =2

() sin() d =

2

() sin() d +

2

0

0

2

() s in() d =

2

= 1

cos()

/2

0 +

1

cos()

0

/2

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= 0 , for = (2 + 1), , ( is odd)

=4

, for = 2, , ( is even)

Fig. 2.4 Frequency spectrum for a rectangular wave

Exemple 2.2 The motion of the piston exemplified in Figure 2.5 is described

analytically by the equation:

() = [1 cos() +

2sin2(t) +

3

8sin4() +

5

32sin6() + ]

Only the first two terms attain significant values, and consequently the piston

acceleration becomes:

() = 2[cos () + cos (2)]

The two components of the sum from Eq.(2.15) represent the frequency spectrum

of the piston motion; a suggestive description is got if amplitudes of acceleration

are presented as function of frequency( Figure 2.5).

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Fig. 2.5 Periodic non-harmonic motion of a piston and its harmonic components.

2.3.3 Complex form of Fourier series

The complex representation of Fourier series presents any periodic motion as a

sum of contra-rotating vectors at equally spaced frequencies:

= 1, ( 1 =1

, ).

() = [()2]== (2.16)

The amplitude of the r component is obtained from the integral:

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() =1

()2

/2

/2d (2.17)

In Eq. (2.17) the quantity to be integrated is a product between the periodic

movement x(t) and the unit phasor 2 which rotates at the circular frequency

. If the periodic movement x(t) contains component rotating at a circular

frequency of + then its product with the mentioned phasor annulus the rotation

of this movement component such that it integrates with time at a finite value,

Figure 2.6(a). All components at other frequencies will still rotate even after

multiplication by the mentioned unit phasor and thus integrate to zero over the

periodic time, Figure 2.6(b).

Fig. 2.6 (a) Integration of a non-rotating vector to a finite value;

- (b) Integration of a phasor to zero

The Eq. (2.10) has the effect of extracting from the movement x(t) the

components it contains which rotate at each frequency , and also freezes the

phase angle of each as that existing at time zero (when 2 = 1). The actual

position of each vector at any other time t can thus be obtained by multiplying

its initial value () by the oppositely rotating unit vector 2. Consequently,

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the movement x(t) will be the sum vector of all these vectors in their instantaneous

positions. That is the physical meaning of Eq. (2.16).

The series of complex values of () represents the complex spectrum

components of the vibratory movement x(t). Because each frequency

component () contains information relative to amplitude and phase

(equivalent real and imaginary part) the complex spectrum needs a 3D

representation, Figure 2.7.

Fig. 2.7 3D representation of the complex spectrum of a periodic movement.

As in the trigonometric Fourier series analysis, the complex Fourier series analysis

points out that a motion that is periodic in the time domain has a discrete

frequency spectrum that has all its spectrum components fall at frequencies

which are integral multiples of the fundamental frequency. However, the phasor

representation offers an intuitive explanation: the time for one rotation of the

phasor at the fundamental frequency 1 is the one time period. All the other

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phasors rotate at speeds which are integer multiples of 1 so each of them rotate

its own integer number of turns during the movement period and all have

returned to their starting positions, and the whole process will begin to repeat

exactly.

Because the time movement x(t) is a real-valued function, each component at

frequency must be matched by a component at which has equal

amplitude but opposite phase. In the complex plane that means equal real part

and opposite imaginary part that represent two complex conjugate complex

numbers:

() = ()

In this way the imaginary parts of all frequencies will always cancel and the

resultant will be always real.

() = 0 + 2Re[ (2)=1 ] (2.18)

Because the series of imaginary parts (or equivalently phase angles) is anti-

symmetric around zero frequency, the zero frequency (or DC) component has

zero (or ) phase angle and is always real.

2.3.4 Power of a time periodic motion. Power spectrum and Parcevals theorem.

Time domain analysis. The instantaneous power of the motion () is equal to

[()]2. The mean power over one period is given by integrating the instantaneous

value over one period (that represents the energy along one period) and dividing

it by the periodic time:

=1

[()]2d

0 (2.19)

For a typical harmonic component () = sin(21 + ) this results in:

_ = 1

2

sin2(21 + )d =

2

[

1

2

1

2cos(21 + )]

0=

2

2

(2.20)

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_ = [

2]

2= [_]

2 (2.21)

The power content at each frequency is obtained directly by the square of the

amplitude of the Fourier series component. The large of usage of the root mean

square value which, directly connected with mean power, becomes clear. The

distribution with frequency of the power content in the vibratory movement

represents its power spectrum.

Frequency domain analysis. In the frequency domain, except for the DC

component the amplitude of any () is 2 , and thus the square of this is 2/4.

The amplitude spectrum is even and the negative frequency component (from

() so the square of its amplitude is also

2/4. The total mean power

associated with the frequency will be 2 2 , the same as obtained in the time

domain.

Parcevals theorem. The total power obtained by integrating the squared

instantaneous motion amplitude with time and dividing by this time are equal with

the total power obtained by summing the squared amplitudes of all frequencies

of the frequency component. This is called Parcevals theorem.

2.3.5 Fourier transform

Letting the period the Fourier series can be extended to non periodic

motions. In the case of , the spacing 1/T between the harmonics tends to

zero and the amplitudes Cr(f) become a continuous function of linear frequency

= 1/ = /(2).

Also, in the assumption of infinite period, the equations (2.17) and (2.16) tend to:

() = ()2+

d = (()) ) (2.22)

() = ()+12+

d = (()) (2.23)

The equations (2.22) and (2.23) represent the Fourier Transform Pair:

- the Eq. (2.22), called the forward Fourier transform, converts the motion

x(t) from time domain into the frequency domain, whereas

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- the Eq.(2.23), called the inverse Fourier transform, converts the frequency

spectrum X(f) from frequency domain into the time domain.

The Fourier transform decomposes a wave form into harmonics.