Damage Detection of Roller Bearing System Using Experimental Data

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Procedia Engineering 144 ( 2016 ) 202 – 207

Peer-review under responsibility of the organizing committee of ICOVP 2015 doi: 10.1016/j.proeng.2016.05.025

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* Corresponding author. Tel.: +91 8793538605; E-mail address: [email protected]

12th International Conference on Vibration Problems, ICOVP 2015

Damage Detection of Roller Bearing System Using Experimental

Data

R.G. Desavale

a

, V.G. Salunkhe

b*

a_{Department of Mechanical and Automotive Engineering, ADCET, Ashta, Maharashtra, India, India-416 301} b _{PG Student, Department of Mechanical Engineering, ADCET, Ashta, Maharashtra, India, India-416 301}

Abstract

Roller bearings are widely used in many rotating systems of automobiles, process industries namely cement, sugar, textile, petrochemical etc., where supported loads coming on the bearings are relatively high as well as rotating speeds are generally high. Being critical member of the entire system, these bearings play an important role in the overall performance of the system. Failure of such bearings in any one of the critical machine of process industry disrupts entire production process and loads to heavy production loss. Early detection of faults in the bearing is, therefore, highly essential to avoid catastrophic. Monitoring and analysis of vibration signal generated by faults in the bearings helps to diagnose exact fault in the bearings. This paper deals with the study of vibration response of roller bearings by using mathematical model. Mathematical model considers the influence of the bearing structure parameters on the vibration of the system. The experimental data technique is used to find out bearing vibration amplitude and defect frequencies. The method proposed in this paper for vibration characteristics calculation of a roller bearing is credible and will save time and costs by timely detection of imminent bearing failure.

Peer-review under responsibility of the organizing committee of ICOVP 2015.

Keywords: experimental data based analysis, dimensional analysis, bearing vibrations

1.Introduction

It is necessary to analyze the effect of all these parameters on the vibration of bearing. Experimental data (ED) can be used to investigate the nature of the solution of such vibration problem [1,2]. Various models have been developed in the past few decades to study the vibration response of the rotor-bearing system with defects in the bearing elements [3-8]. Researchers [9-10] modeled the rolling element bearing system response in the presence of defects. In comparison with the literature cited above, so far, the concept of vibration detection in terms of velocity

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which was not measured in [1]. With its aid, the maximum amount of useful information is obtained from a given number of experiments. ED establishes the conditions for the validity of experiments on models and the law of comparison of models with their prototypes. The results of investigations can always be presented in non-dimensional form and are then immediately applicable, irrespective of the sizes of the fundamental units. By combining variables into smaller number of dimensionless parameters, the work of experimental data analysis is considerably reduced.

Nomenclature

ED experimental data EDM electric discharge machine

2.Mathematical Formulation

The commonly used method of experimental data analysis is Buckingham’s

ૈ

-

theorem. It states that for each dimensional homogeneous and complete relationship ܎of

ܖ

physical variablesܠܑ, ܎ሺܠ૚ǡ ܠ૛ǡ ǥ Ǥ ǡ ܠܖሻ ൌ ૙ there exists

a corresponding relationship ܎ሺܠ૚ǡ ܠ૛ǡ ǥ Ǥ ǡ ܠܖሻ ൌ ૙ of only ܕ ൏ ܖ dimensionless parameters

X

D

ji

C

jk

R

m

n

r

D k n r i C k r k j jk jk

3

;

,

;

1 1

S

(1)

where r denotes the rank of the dimensional matrix, i.e. the number of independent dimensions. The dimensional matrix can be established from the knowledge of the problems parameters and their respective dimensions. The columns of the dimensional matrix correspond to the parameters while the rows correspond to the value of the dimension exponent of the variable. The elements of the dimensional matrix are therefore the exponents of the dimensions of all the parameters.

> @

»

¼

º

«

¬

ª

»

¼

º

«

¬

ª

»

¼

º

«

¬

ª

rk k r nk k r n r k

m

A

m

B

m

Dim

BA

M

1 1 11 1 1 1 1 1 (2)

The dimensional matrix M, shown in Eq. (2) on the left, is formed by the relevant parameters _୧ as the columns and the corresponding dimensional exponents _୧୨ as the rows of the matrix. The dimensional matrix M can be partitioned into the sub matrix ሺ ൈ ሻ which contains the ൌ ሺሻ base variables and the sub matrix ൫ ൈ

ሺ െ ሻ൯ which contains the െ dependent variables. The dimensional matrix can be extended to a dimensional set which contains the additional sub matrices C and D.

ܦ݅݉ ሾܤሿ ሾܣሿ (3) ߨ௜ ሾܦሿ ሾܥሿ

In this dimensional set the sub matrix ൫ሺ െ ሻ ൈ ሺ െ ሻ൯ can be nearly freely chosen with the only limitation of D being regular. The sub matrix ൫ሺ െ ሻ ൈ ൯ is the determined by.

T

B

A

D

(3)

The dimensionless groups are then determined as jl ji D l n r l C i r i j

3

x

3

x

1 1

S

(5)

These dimensionless groups form a minimal set of parameters for the given problem. Due to this property, data plots in dimensionless groups are more compact then plots in dimensional variables and often allow better visualization of physical effects.

3.Models by experimental data

The function dependence of bearing vibrations on the parameters can be estimated by performing an experimental data on the Buckingham’s Ɏ-theorem described in Table 1.

Table 1. Parameters and their corresponding dimensions.

Parameter Symbol Unit Dimensions

Vibration amplitude V mm/sec ିଵ

Bore diameter D m L

Roller diameter ୠ m L

Inner race diameter ୧ m L

Outer race diameter ୭ m L

Pitch circle diameter ୮ m L

Mass of the rotor ୰୭୲୭୰ kg ିଵଶ

Mass of inner race ୧ kg ିଵଶ

Mass of outer race ୭ kg ିଵଶ

Mass of the roller ୰୭୪୪ୣ୰ kg ିଵଶ

Young’s modulus E N/ଶ ିଶ

Density of bearing material ɏ kg/ଷ ିସଶ

Roller defect frequency ୰ୢ Hz ିଵ

Outer race defect frequency ୭ Hz ିଵ

Surface defect ୢ

Speed of the rotor N rpm ିଵ

Radial load P N F

The vibration amplitude in RMS (mm/s) of the bearing can be given by the equation,

D

d

m

E

f

S

N

P

f

V

,

b

,

i

,

o

,

p

,

rotor

,

i

,

o

,

roller

,

U

,

rd

,

o

,

d

,

(6)

It may be assumed that the vibration amplitude of bearing depends on the speed, load, defect size, defect frequencies and density of material. In rotor bearing system, speed, load and volume of defect size has significant role to play in changing the vibration amplitude and defect frequencies. The dimensions of all these quantities are reported in

ǡ ǡ ǡ Ʌ system are shown in Table 1

All the above variables considered for the problem are assembled using Buckingham’s Ɏ-theorem in a number of dimensionless products ൫Ɏ୨൯ as

1

,

2

,...

...,

m

0

f

S

(7)

Dimension formula for relation (6) is

(8)

where ǡ ǡ ǡ ǡ ǡ ǡ ǡ ǡ ǡ ǡ ǡ ǡ ǡ ǡ ǡ are indices of the variables in Eq. (8).

By equating the powers of the fundamental units on both sides of the Eq. (8), set of simultaneous linear equations are obtained which can later be solved to obtain the magnitudes of these constants. To utilize the algebraic approach to experimental data analysis, it is convenient to write the dimensions of the variables in the matrix form.

> @

> @ > @ > @ > @ > @

>

@ >

@>

@ > @ >

@

> @ > @

1 1

> @

1

> @

>

0 0 0 0

@

2 4 2 2 1 2 1 2 1 2 1 1

T

L

F

T

L

T

FL

T

FL

T

FL

T

FL

T

FL

L

LT

q p o n m l k j i h g f e d c b a

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Table 2. Matrix of reference units. V D ܌܊ ܌ܑ ܌ܗ ܌ܘ ܕܚܗܜܗܚ ܕܑ ܕܗ ܕܚܗܔܔ܍ܚ E ૉ ܎ܚ܌ ܎ܗ ܁܌ N P F 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 L 1 1 1 1 1 1 -1 -1 -1 -1 -2 -4 0 0 1 0 0 T -1 0 0 0 0 0 -2 -2 -2 -2 0 2 -1 -1 0 -1 0

For finding out number of dimensionless parameters to be formed the rank of matrix r is three. Therefore, number of dimensionless products m is

݉ ൌ ݊ െ ݎ ൌ ͳ͹ െ ͵ ൌ ͳͶ(9)

Thus the numbers of dimensionless products needed to characterize the complete set of variables are fourteen. Therefore the complete set of dimensionless products resulting from the matrix is obtained as listed in Table 3.

Table 3.Dimensional Ɏ-products. πଵ ୢ π଼ ୧ୢଶ πଶ ୢ πଽ ୭ୢଶ πଷ ୠ ୢ πଵ଴ ୰୭୪୪ୣ୰ୢଶ πସ ଴ ୢ πଵଵ ୰ୢ πହ ୧ ୢ πଵଶ ୭ π଺ ୢ ୮ πଵଷ ୢଶ π଻ ୰୭୲୭୰ୢ ଶ πଵସ ρଶ ୢସ

Out of fourteen higher level groups some of them

¸¸

¹

·

¨¨

©

§

11 12 10 9 8 7 5 4 3 2

,

S

_a _b _c _d _e may

be considered to form a constant Ⱦ as

S

a

S

b

S

c

S

d

S

e

E

u

(10)

It is to be noticed that Ⱦ contains several constant parameters of bearing. A change in the value of Ⱦ can be due to appropriate change in any of the constant parameters involved inȾ. According to Buckingham’s Ɏ-theorem, the relation between the dimensionless products can be given as

S

1

,

S

2

,

S

3

0

or

S

1

f

S

2

,

S

3

f

(11)

Combining ɎଵǡȾǡ Ɏ଺ǡ Ɏଵଷǡ Ɏଵସ into the above equation gives the following

¸

¹

·

¨

©

§

¸¸

¹

·

¨¨

©

§

u

¸¸

¹

·

¨¨

©

§

u

¸

¹

·

¨

©

§

u

c d b d a p d d

P

S

N

P

ES

d

S

f

NS

V

E

2

U

2 4 (12) 4.Experiments

A schematic diagram of the rotor-bearing for experimental validation is shown in Fig. 1. In order to evaluate the proposed model, experiments are carried out for six different speeds. The experiments are performed at shaft

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rotational speed of 500 rpm to 3000 rpm with 200 N radial loading on the test bearing having surface defects (single) in outer races is shown in Fig. 2. The defect size at outer race for single defect are kept as 1.0 mm. Defect are created on races of the test bearings by electric discharge machining (EDM). A Bruel & Kjaer type 4368 accelerometer has been used for measuring the vibration signals.

Fig.1

.

Experimental setup under study: schematic view.

Fig.2

.

Defect on outer race.

5.Results and discussion

In order to obtain the values of constant, experiments are conducted with all the variables and the vibration amplitude is measured. The standard error of estimation and adj. R2_{value are found to be 0.120 and 0.92}

respectively. The high coefficient of correlation indicates a good fit. The values of constants ઺ǡ ܉ǡ ܊܉ܖ܌܋ are to found to be ૙Ǥ ૙ૢ૜ǡ െ૙Ǥ ૢ૙૚૞ǡ ૙Ǥ ૠ૞૙૟܉ܖ܌૙Ǥ ૚ૡ૚ૡ respectively. Figure 3 present the variations of vibration amplitude with speed of the proposed model. It is observed that amplitude of vibration is increasing with speed of rotor. Therefore, the term ૈ_૚૝ which involving N2_{in the numerator is justified in Eq. (12). However, reasonably fair}

correlations are also visible for vibrational amplitude. Overall good matching of model and experimental results develops good confidence in the proposed experimental data based model. Hence the experimental and theoretical results shows good agreement may be seen in Fig 3.Therefore it is justified to have ૈ૚૝ with N2 in its numerator, in

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Fig.3

.

Vibration spectra

The relation shows there exit a non linear relation of all parameters on vibration amplitude. The Eq. (12) shows that apart from material properties, surface defect, load and speed have a reasonable effect on the vibration amplitude. It is obvious that as surface defect increases the vibration amplitude produced will be bigger. The increase in surface defect, speed and load causes the vibration amplitude to increases and decreases respectively. Thus it can be noticed that the expression obtained reasonably agree with the experimental results.

6.Results and discussion

An experimental data relation was obtained for vibration amplitude as a function of surface defect, load and speed, material properties. The dimensionless constant has been determined by the regression analysis data from the experiment. Experimental and theoretical modal analysis showed a good fit which indicates the versatility of the mathematical equation obtained using dimensional analysis.

References

[1] R.G. Desavale, R. Venkatachalam, S.P. Chavan, Detection of rotor-bearing damage by a new experimental data based models and multivariable regression analyses (MVRA) approach, ASME Journal of Vibration and Acoustics., 136 , 021022-1-10 (2014).

[2] R.G. Desavale, R. Venkatachalam, S.P. Chavan, Antifriction bearings damage analysis using experimental data based models, ASME Journal of Tribology., 135 (4) , 041105-1-12 (2013).

[3] A. Ashtekar, F. Sadeghi, L.E. Stacke, A new approach to modeling surface defects in bearing dynamics simulations, Journal of Tribology., 130 (4), 041103 (2008).

[4] M.Cao, A.Xiao, A comprehensive dynamic model of double-row spherical roller bearing—model development and case studies on surface defects, preloads, and radial clearance, Mechanical Systems and Signal Processing., 22, 467–489 (2007).

[5] A. Choudhury, N. Tandon, A theoretical model has been developed to obtain the vibration response due to a localized defect in various bearing elements in a rotor-bearing system under radial load conditions, Journal of Tribology., 128, 252-261 (2006).

[6] Z. Kıral, H. Karagulle, Vibration analysis of rolling element bearings with various defects under the action of an unbalanced force, Mechanical Systems and Signal Processing., 20, 1967–1991 (2006).

[7] J. Sopanen, A. Mikkola, Dynamic Model of a Deep-Groove Ball Bearing Including Localized and Distributed Defects. Part 1: Theory, Proceedings of Institute of Mechanical Engineering, Part K: J. Multi-body Dynamics., 217, 201–211 (2003).

[8] N. Tandon, A. Choudhury, A theoretical model to predict the vibration response of rolling element bearing in a rotor bearing system to distributed defects under radial load, Journal of Tribology., 122, 609-615 (2000).

[9] D. Brie, Modeling of the Spalled Rolling Element Bearing Vibration Signal: An Overview and Some New Results, Mechanical System and Signal Processing., 14, 353–369 (2000).

[10] N. Tandon, A. Choudhury, A review of vibration and acoustic measurement methods for the detection of defects in rolling element bearings, Tribology International., 32, 469–480 (1999).