A THEORETICAL MODEL OF DEEP GROOVE BALL … · Various sources of vibration in any machine are...

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International Journal of Mechanical Engineering and Technology (IJMET)Volume 8, Issue 6, JuneAvailable online at ISSN Print: 0976 © IAEME

A THEORETICAL MODEL

ABSTRACTThis paper discusses the modelling of ball bearing for predicting the presence of

defects on outer and inner races. A computer program in Matlab is developed and DOF equations are solved. The output of the program is time domain signal of the bearing. The program takes into consideration the size of the defects, location of the defect and also the effect of changing the load on the bearing. The vibration signal obtained through modelling is analysed for frequencies. The peaks at the characteristic defect frequencies in the spectrum indicate the location of the fault on races. Key words:frequencies, spectrum indicator, fault on racesCite this ArticleDeep Groove Ball Bearing for Predicting the Effect of Localized Defects on Vibrations2017, pp. 760http://www.i

1. INTRODUCTIONVarious sources of vibration in any machine are unbalance, misalignment, faulty gear and faulty bearing etc. Among these the If a bearing fails, the entire production line is affected and ultimately, it will affect the productivity. Hence it is important that the rolling element bearings are monitored regularly for checkand theoretical. In theoretical methods, a model is developed and equations of motion are solved. Through this the vibration behaviour of bearings can be predicted.

As beardeveloped a model for studying vibrations produced by bearings carrying spalling etc.


International Journal of Mechanical Engineering and Technology (IJMET)Volume 8, Issue 6, JuneAvailable online at http://www.iaeme.com/IJMEISSN Print: 0976-6340 and IS

© IAEME Publication

A THEORETICAL MODEL BALL BEARING FOR PREEFFECT OF LOCALIZED

M. Tech., BVDU

Professor, BVDU

ABSTRACT This paper discusses the modelling of ball bearing for predicting the presence of

defects on outer and inner races. A computer program in Matlab is developed and DOF equations are solved. The output of the program is time domain signal of the bearing. The program takes into consideration the size of the defects, location of the defect and also the effect of changing the load on the bearing. The vibration signal btained through modelling is analysed for frequencies. The peaks at the

characteristic defect frequencies in the spectrum indicate the location of the fault on races. Key words: ball bearing, size of defects, location of defects, vibration signals, defectfrequencies, spectrum indicator, fault on racesCite this ArticleDeep Groove Ball Bearing for Predicting the Effect of Localized Defects on Vibrations. International Journal of Mechanical E2017, pp. 760–76http://www.iaeme.com/IJME

INTRODUCTIONVarious sources of vibration in any machine are unbalance, misalignment, faulty gear and faulty bearing etc. Among these the If a bearing fails, the entire production line is affected and ultimately, it will affect the productivity. Hence it is important that the rolling element bearings are monitored regularly for checking their health. Mainly the methods of monitoring are categorised as experimental and theoretical. In theoretical methods, a model is developed and equations of motion are solved. Through this the vibration behaviour of bearings can be predicted.

As bearing rotates, the rolling elements change their positions. developed a model for studying vibrations produced by bearings carrying spalling etc.

http://www.iaeme.com/IJMET/index.

International Journal of Mechanical Engineering and Technology (IJMET)Volume 8, Issue 6, June 2017, pp.

http://www.iaeme.com/IJME6340 and ISSN Online: 0976

Publication


M. Tech., BVDU -

Professor, BVDU -

This paper discusses the modelling of ball bearing for predicting the presence of defects on outer and inner races. A computer program in Matlab is developed and DOF equations are solved. The output of the program is time domain signal of the bearing. The program takes into consideration the size of the defects, location of the defect and also the effect of changing the load on the bearing. The vibration signal btained through modelling is analysed for frequencies. The peaks at the

characteristic defect frequencies in the spectrum indicate the location of the fault on

ball bearing, size of defects, location of defects, vibration signals, defectfrequencies, spectrum indicator, fault on racesCite this Article: Arundhati Garad and Prof. V. J. ShindeDeep Groove Ball Bearing for Predicting the Effect of Localized Defects on

International Journal of Mechanical E769.

aeme.com/IJME

INTRODUCTION Various sources of vibration in any machine are unbalance, misalignment, faulty gear and faulty bearing etc. Among these the If a bearing fails, the entire production line is affected and ultimately, it will affect the productivity. Hence it is important that the rolling element bearings are monitored regularly

ing their health. Mainly the methods of monitoring are categorised as experimental and theoretical. In theoretical methods, a model is developed and equations of motion are solved. Through this the vibration behaviour of bearings can be predicted.

ing rotates, the rolling elements change their positions. developed a model for studying vibrations produced by bearings carrying spalling etc.

IJMET/index.asp

International Journal of Mechanical Engineering and Technology (IJMET)2017, pp. 760–769, Article ID: IJM

http://www.iaeme.com/IJMESN Online: 0976

Scopus Indexed


VIBRATIONSArundhati Garad

- Bharati Vidyapeeth Deemed University

Pro- Bharati Vidyapeeth Deemed University, Pune, India



ball bearing, size of defects, location of defects, vibration signals, defectfrequencies, spectrum indicator, fault on races

Arundhati Garad and Prof. V. J. ShindeDeep Groove Ball Bearing for Predicting the Effect of Localized Defects on

International Journal of Mechanical E

aeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=6

Various sources of vibration in any machine are unbalance, misalignment, faulty gear and faulty bearing etc. Among these the rolling element bearing is If a bearing fails, the entire production line is affected and ultimately, it will affect the productivity. Hence it is important that the rolling element bearings are monitored regularly



asp 760

International Journal of Mechanical Engineering and Technology (IJMET)Article ID: IJM

http://www.iaeme.com/IJMET/issues.asp?JType=IJMESN Online: 0976-6359

Indexed


VIBRATIONSArundhati Garad

Bharati Vidyapeeth Deemed University

Prof. V. J. Shinde Bharati Vidyapeeth Deemed University, Pune, India



ball bearing, size of defects, location of defects, vibration signals, defectfrequencies, spectrum indicator, fault on races.


International Journal of Mechanical E

asp?JType=IJMET&VType=8&IType=6

Various sources of vibration in any machine are unbalance, misalignment, faulty gear and rolling element bearing is

If a bearing fails, the entire production line is affected and ultimately, it will affect the productivity. Hence it is important that the rolling element bearings are monitored regularly



International Journal of Mechanical Engineering and Technology (IJMET)Article ID: IJMET_08_06

asp?JType=IJME

A THEORETICAL MODEL OF DEEP GROOVE BALL BEARING FOR PREDICTING THE EFFECT OF LOCALIZED DEFECTS ON

VIBRATIONS Arundhati Garad

Bharati Vidyapeeth Deemed University

f. V. J. Shinde Bharati Vidyapeeth Deemed University, Pune, India



ball bearing, size of defects, location of defects, vibration signals, defect


International Journal of Mechanical Engineering and Technology


Various sources of vibration in any machine are unbalance, misalignment, faulty gear and rolling element bearing is the important machine element.




[email protected]

International Journal of Mechanical Engineering and Technology (IJMET) 06_080


OF DEEP GROOVE DICTING THE DEFECTS ON

Bharati Vidyapeeth Deemed University, Pune

Bharati Vidyapeeth Deemed University, Pune, India




Arundhati Garad and Prof. V. J. Shinde. A Theoretical Model of Deep Groove Ball Bearing for Predicting the Effect of Localized Defects on

ngineering and Technology


Various sources of vibration in any machine are unbalance, misalignment, faulty gear and the important machine element.



ing rotates, the rolling elements change their positions. In 1978 developed a model for studying vibrations produced by bearings carrying spalling etc.

[email protected]

T&VType=8&IType=6


Pune, India

Bharati Vidyapeeth Deemed University, Pune, India




A Theoretical Model of Deep Groove Ball Bearing for Predicting the Effect of Localized Defects on

ngineering and Technology, 8(6),




ing their health. Mainly the methods of monitoring are categorised as experimental and theoretical. In theoretical methods, a model is developed and equations of motion are

In 1978 Sunnersjo [1]

developed a model for studying vibrations produced by bearings carrying spalling etc.

[email protected]

T&VType=8&IType=6


This paper discusses the modelling of ball bearing for predicting the presence of defects on outer and inner races. A computer program in Matlab is developed and 2 DOF equations are solved. The output of the program is time domain signal of the bearing. The program takes into consideration the size of the defects, location of the defect and also the effect of changing the load on the bearing. The vibration signal btained through modelling is analysed for frequencies. The peaks at the



A Theoretical Model of Deep Groove Ball Bearing for Predicting the Effect of Localized Defects on

, 8(6),



ing their health. Mainly the methods of monitoring are categorised as experimental and theoretical. In theoretical methods, a model is developed and equations of motion are

Sunnersjo [1] developed a model for studying vibrations produced by bearings carrying spalling etc. Tandon


and Choudhury [2]bearings with defects those may occur on the surface of bearing. McFadden and Smith [4], [5] in their models have discussed the effect of single as well as multi point defects obearing. Tandon and Choudhury [6] have developed an analytical model considering the defects on races and ball. The load was simulated on bearing in axial and radial direction. The different shapes of pulses are considered for modelling the considered mass of the shaft along with mass of bearing to develop the vibration model. Patil et al. [8] have considered the bearing vibrations in the form of sine wave in the dynamic model. The theoretical models developed for monitoring health of bearings. Kulkarni and Sahasrabudhe [9] have simulated the defect pulse produced by the striking of defect by balls as cubic hermite spline.

2. MATHEMATICAL MODEL OWhile devmounted at the end of the shaft. The bearing is supported in bearing casing on which radial load is applied. The magnitude of the load is changed in the program to understand the of load on the vibrations of the bearing.

The bearing is subjected to a rotating force which causes the system to undergo vibrations in the dynamic state. In the mathematical model, the ball bearings are considered as a nonlinear spring

The model presented here considers the fixed outer race, loaded radially with a rotating force and the imbalanced force acting as a combination. The imbalanced force considered is the result of residual imba

As per ISO 1940Permissible unbalance in gm

quality grade in mm/sec=0.4 for spindles and drives of highin kg = 4 kg

While developing the model, the assumptions considered are mentioned bel The ball bearing model has equi

There is no Slipping of the balls


and Choudhury [2]bearings with defects. A book by Harris [3] discusses a detailed study on different types of defects those may occur on the surface of bearing. McFadden and Smith [4], [5] in their models have discussed the effect of single as well as multi point defects obearing. Tandon and Choudhury [6] have developed an analytical model considering the defects on races and ball. The load was simulated on bearing in axial and radial direction. The different shapes of pulses are considered for modelling the considered mass of the shaft along with mass of bearing to develop the vibration model. Patil et al. [8] have considered the bearing vibrations in the form of sine wave in the dynamic model. The theoretical models developed for monitoring health of bearings. Kulkarni and Sahasrabudhe [9] have simulated the defect pulse produced by the striking of defect by balls as cubic hermite spline.

MATHEMATICAL MODEL OWhile developing the mathematical model of bearing, it is assumed that the bearing is mounted at the end of the shaft. The bearing is supported in bearing casing on which radial load is applied. The magnitude of the load is changed in the program to understand the of load on the vibrations of the bearing.

The bearing is subjected to a rotating force which causes the system to undergo vibrations in the dynamic state. In the mathematical model, the ball bearings are considered as a nonlinear spring-mass syste

The model presented here considers the fixed outer race, loaded radially with a rotating force and the imbalanced force acting as a combination. The imbalanced force considered is the result of residual imba

As per ISO 1940-1:2003(E)Permissible unbalance in gm

quality grade in mm/sec=0.4 for spindles and drives of highin kg = 4 kg

While developing the model, the assumptions considered are mentioned belThe ball bearing model has equi



and Choudhury [2] discussed various measurement methods for diagnosis of health of defects. A book by Harris [3] discusses a detailed study on different types of

defects those may occur on the surface of bearing. McFadden and Smith [4], [5] in their models have discussed the effect of single as well as multi point defects obearing. Tandon and Choudhury [6] have developed an analytical model considering the defects on races and ball. The load was simulated on bearing in axial and radial direction. The different shapes of pulses are considered for modelling the considered mass of the shaft along with mass of bearing to develop the vibration model. Patil et al. [8] have considered the bearing vibrations in the form of sine wave in the dynamic model. The theoretical models developed for monitoring health of bearings. Kulkarni and Sahasrabudhe [9] have simulated the defect pulse produced by the striking of defect by balls as cubic hermite spline.

MATHEMATICAL MODEL Oeloping the mathematical model of bearing, it is assumed that the bearing is

mounted at the end of the shaft. The bearing is supported in bearing casing on which radial load is applied. The magnitude of the load is changed in the program to understand the of load on the vibrations of the bearing.

The bearing is subjected to a rotating force which causes the system to undergo vibrations in the dynamic state. In the mathematical model, the ball bearings are considered as a non

mass system. Fig. 1 shows a bearing subjected to loading. The model presented here considers the fixed outer race, loaded radially with a rotating

force and the imbalanced force acting as a combination. The imbalanced force considered is the result of residual imbalance left in the system.

Figure

1:2003(E) Permissible unbalance in gm

quality grade in mm/sec=0.4 for spindles and drives of high

While developing the model, the assumptions considered are mentioned belThe ball bearing model has equi


Arundhati Garad and Prof. V. J. Shinde

IJMET/index.asp

discussed various measurement methods for diagnosis of health of defects. A book by Harris [3] discusses a detailed study on different types of

defects those may occur on the surface of bearing. McFadden and Smith [4], [5] in their models have discussed the effect of single as well as multi point defects obearing. Tandon and Choudhury [6] have developed an analytical model considering the defects on races and ball. The load was simulated on bearing in axial and radial direction. The different shapes of pulses are considered for modelling the considered mass of the shaft along with mass of bearing to develop the vibration model. Patil et al. [8] have considered the bearing vibrations in the form of sine wave in the dynamic model. The theoretical models developed for monitoring health of bearings. Kulkarni and Sahasrabudhe [9] have simulated the defect pulse produced by the striking of defect by balls as cubic hermite spline.

MATHEMATICAL MODEL Oeloping the mathematical model of bearing, it is assumed that the bearing is

mounted at the end of the shaft. The bearing is supported in bearing casing on which radial load is applied. The magnitude of the load is changed in the program to understand the of load on the vibrations of the bearing.


m. Fig. 1 shows a bearing subjected to loading. The model presented here considers the fixed outer race, loaded radially with a rotating

force and the imbalanced force acting as a combination. The imbalanced force considered is lance left in the system.

Figure 1 Bearing subjected to radial load

Permissible unbalance in gm-mm Uper= 1000 (eper.Ω) M/Ωquality grade in mm/sec=0.4 for spindles and drives of high

While developing the model, the assumptions considered are mentioned belThe ball bearing model has equi-spaced



asp 761


defects those may occur on the surface of bearing. McFadden and Smith [4], [5] in their models have discussed the effect of single as well as multi point defects obearing. Tandon and Choudhury [6] have developed an analytical model considering the defects on races and ball. The load was simulated on bearing in axial and radial direction. The different shapes of pulses are considered for modelling the considered mass of the shaft along with mass of bearing to develop the vibration model. Patil et al. [8] have considered the bearing vibrations in the form of sine wave in the dynamic model. The theoretical models developed for bearing defect modelling gives accurate results for monitoring health of bearings. Kulkarni and Sahasrabudhe [9] have simulated the defect pulse produced by the striking of defect by balls as cubic hermite spline.

MATHEMATICAL MODEL OF SYSTEMeloping the mathematical model of bearing, it is assumed that the bearing is

mounted at the end of the shaft. The bearing is supported in bearing casing on which radial load is applied. The magnitude of the load is changed in the program to understand the

The bearing is subjected to a rotating force which causes the system to undergo vibrations

in the dynamic state. In the mathematical model, the ball bearings are considered as a nonm. Fig. 1 shows a bearing subjected to loading.

The model presented here considers the fixed outer race, loaded radially with a rotating force and the imbalanced force acting as a combination. The imbalanced force considered is

lance left in the system.

Bearing subjected to radial load

mm Uper= 1000 (eper.Ω) M/Ωquality grade in mm/sec=0.4 for spindles and drives of high

While developing the model, the assumptions considered are mentioned belspaced



defects those may occur on the surface of bearing. McFadden and Smith [4], [5] in their models have discussed the effect of single as well as multi point defects obearing. Tandon and Choudhury [6] have developed an analytical model considering the defects on races and ball. The load was simulated on bearing in axial and radial direction. The different shapes of pulses are considered for modelling the defects. Patel et al. [7] have considered mass of the shaft along with mass of bearing to develop the vibration model. Patil et al. [8] have considered the bearing vibrations in the form of sine wave in the dynamic

for bearing defect modelling gives accurate results for monitoring health of bearings. Kulkarni and Sahasrabudhe [9] have simulated the defect pulse produced by the striking of defect by balls as cubic hermite spline.

F SYSTEM eloping the mathematical model of bearing, it is assumed that the bearing is

mounted at the end of the shaft. The bearing is supported in bearing casing on which radial load is applied. The magnitude of the load is changed in the program to understand the



force and the imbalanced force acting as a combination. The imbalanced force considered is


mm Uper= 1000 (eper.Ω) M/Ωquality grade in mm/sec=0.4 for spindles and drives of high-precision systems

While developing the model, the assumptions considered are mentioned bel


[email protected]


defects those may occur on the surface of bearing. McFadden and Smith [4], [5] in their models have discussed the effect of single as well as multi point defects obearing. Tandon and Choudhury [6] have developed an analytical model considering the defects on races and ball. The load was simulated on bearing in axial and radial direction. The

defects. Patel et al. [7] have considered mass of the shaft along with mass of bearing to develop the vibration model. Patil et al. [8] have considered the bearing vibrations in the form of sine wave in the dynamic

for bearing defect modelling gives accurate results for monitoring health of bearings. Kulkarni and Sahasrabudhe [9] have simulated the defect pulse produced by the striking of defect by balls as cubic hermite spline.

eloping the mathematical model of bearing, it is assumed that the bearing is mounted at the end of the shaft. The bearing is supported in bearing casing on which radial load is applied. The magnitude of the load is changed in the program to understand the



force and the imbalanced force acting as a combination. The imbalanced force considered is


mm Uper= 1000 (eper.Ω) M/Ω eper.Ω= selected balance precision systems

While developing the model, the assumptions considered are mentioned bel

[email protected]


defects those may occur on the surface of bearing. McFadden and Smith [4], [5] in their models have discussed the effect of single as well as multi point defects on the races of bearing. Tandon and Choudhury [6] have developed an analytical model considering the defects on races and ball. The load was simulated on bearing in axial and radial direction. The


for bearing defect modelling gives accurate results for monitoring health of bearings. Kulkarni and Sahasrabudhe [9] have simulated the defect

eloping the mathematical model of bearing, it is assumed that the bearing is mounted at the end of the shaft. The bearing is supported in bearing casing on which radial load is applied. The magnitude of the load is changed in the program to understand the



eper.Ω= selected balance precision systems M=Rotor mass

While developing the model, the assumptions considered are mentioned below:

[email protected]


defects those may occur on the surface of bearing. McFadden and Smith [4], [5] in their n the races of

bearing. Tandon and Choudhury [6] have developed an analytical model considering the defects on races and ball. The load was simulated on bearing in axial and radial direction. The


for bearing defect modelling gives accurate results for monitoring health of bearings. Kulkarni and Sahasrabudhe [9] have simulated the defect

eloping the mathematical model of bearing, it is assumed that the bearing is mounted at the end of the shaft. The bearing is supported in bearing casing on which radial load is applied. The magnitude of the load is changed in the program to understand the effect

The bearing is subjected to a rotating force which causes the system to undergo vibrations in the dynamic state. In the mathematical model, the ball bearings are considered as a non-


eper.Ω= selected balance M=Rotor mass

A Theoretical Model of Deep Groove Ball Bearing for Predicting the Effect of Localized Defects


The motions of race and balls occur in the plane of the bearing only.

The outer race is fixed in a rigid support.

There is no change in temperature of bearing

2.1. Motions and Load Different expressions for developing the bearing model are derived in book by Harris. A sketch of bearing with rotating inner race is shown in Fig. 2

Bearing load distribution with equation [3]:

=

where

=

In general the deflection ball located at any angular position given by:δ =where

x is deflection along X axis y is deflection along Y axi C is clearance

The elastic modulus for the contact of a ball with the inner race is

=The elastic modulus for the contact of a ball with the inner race is

=The effective elastic modulus K for the bearing system is written as






otions and Load Different expressions for developing the bearing model are derived in book by Harris. A sketch of bearing with rotating inner race is shown in Fig. 2

Bearing load distribution with equation [3]:

= 1 −

where

=

1 −

In general the deflection ball located at any angular position given by:= xcosθ + ysin

where θ is location of the ball.x is deflection along X axisy is deflection along Y axiC is clearance


= 3.12 × 10The elastic modulus for the contact of a ball with the inner race is

= 3.12 × 10The effective elastic modulus K for the bearing system is written as






otions and Load Distribution in a Ball BDifferent expressions for developing the bearing model are derived in book by Harris. A sketch of bearing with rotating inner race is shown in Fig. 2

Figure

Bearing load distribution with

− (1 −

In general the deflection ball located at any angular position given by:ysinθ − C

is location of the ball.x is deflection along X axisy is deflection along Y axiC is clearance


10 ∑ (The elastic modulus for the contact of a ball with the inner race is

10 ∑The effective elastic modulus K for the bearing system is written as


IJMET/index.asp




Distribution in a Ball BDifferent expressions for developing the bearing model are derived in book by Harris. A sketch of bearing with rotating inner race is shown in Fig. 2

Figure 2 Bearing with rotating inner race

Bearing load distribution with respect to angular position of ball is calculated by the

Ψ)

In general the deflection ball located at any angular position given by:

is location of the ball. x is deflection along X axis y is deflection along Y axis

The elastic modulus for the contact of a ball with the inner race is ( ∗)

The elastic modulus for the contact of a ball with the inner race is ( ∗)

The effective elastic modulus K for the bearing system is written as

A Theoretical Model of Deep Groove Ball Bearing for Predicting the Effect of Localized Defects on Vibrations

asp 762




Distribution in a Ball BDifferent expressions for developing the bearing model are derived in book by Harris. A sketch of bearing with rotating inner race is shown in Fig. 2

Bearing with rotating inner race

respect to angular position of ball is calculated by the



N/mm The elastic modulus for the contact of a ball with the inner race is

N/mm The effective elastic modulus K for the bearing system is written as



Distribution in a Ball Bearing Different expressions for developing the bearing model are derived in book by Harris. A sketch of bearing with rotating inner race is shown in Fig. 2








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Different expressions for developing the bearing model are derived in book by Harris. A








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(1)

(2)

(3)

(4)

(5)


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= (6)

To compute effective elastic modulus (K) for bearing system, ( ) and ( ) are referred from Table 2 and ∗ and ∗ are based on Table 1.

Table 1 Dimensionless Contact Parameters

( ) ∗ 0 1

0.1075 0.9974 0.3204 0.9761 0.4795 0.9429 0.5916 0.9077 0.6716 0.8733 0.7332 0.8394 0.7948 0.7961 0.83495 0.7602 0.87366 0.7169 0.90999 0.6636 0.93657 0.6112 0.95738 0.5551 0.97290 0.4960 0.983797 0.4352 0.990902 0.3745 0.995112 0.3176 0.997300 0.2705 0.9981847 0.2427 0.9989156 0.2106 0.9994785 0.17167 0.9998527 0.11995

1 0

Table 2 Input Data for the model

Inner race diameter 52 mm Outer race diameter 25 mm Pitch diameter 39 mm Ball diameter 7.94 mm Normal Clearance 20 µm Radial load 212 N Mass of rotor 4 kg No of balls 9 Speed of rotor 2400 rpm Damping factor 200 Ns/m Contact angle 00

Knowing above parameters, the contact force (F) is

= {3.12 × 10 (∑ ) ( ∗) } (N) (7)



2.2. Simulation of Defective BMathematical model is developed or defective bearing defect on inner and outer race. Fig. 3 shows a ball striking the defect on the race

While deriving the mathematical model, the presence of defect on the races is accounted for by modelling the defect as a sinusoidal wave.

respectively and describes the angular extent of the fault measured about the outer race centre. The cage angle at which the ball enters the fault

Similarly the cage angle at the exit edge is expressed as:

Δ =

Thus the additional deflection outer race can be determined as:

where h is depth of the defect. The defect on the outer race doesand is defined as:

=

The defect on inner race rotates with the speed of the shaft

2.3. Equation of MTaking x and y as the displacements along X and Y directions, the governing equations for a two degree of freedom system are formed.



. Simulation of Defective BMathematical model is developed or defective bearing defect on inner and outer race. Fig. 3 shows a ball striking the defect on the race


and are the location of entry edge and exirespectively and describes the angular extent of the fault measured about the outer race centre. The cage angle at which the ball enters the fault

=


= ℎ


Δ = ℎ

where h is depth of the defect. The defect on the outer race doesis defined as:= +

The defect on inner race rotates with the speed of the shaft= ( −

. Equation of MTaking x and y as the displacements along X and Y directions, the governing equations for a two degree of freedom system are formed.

+ + ∑

+ +



. Simulation of Defective BMathematical model is developed or defective bearing defect on inner and outer race. Fig. 3 shows a ball striking the defect on the race

Figure


are the location of entry edge and exirespectively and describes the angular extent of the fault measured about the outer race centre. The cage angle at which the ball enters the fault


(


−where h is depth of the defect. The defect on the outer race does

is defined as: ( − )

The defect on inner race rotates with the speed of the shaft) + (

. Equation of Motion Taking x and y as the displacements along X and Y directions, the governing equations for a two degree of freedom system are formed.

[(

[(


IJMET/index.asp

. Simulation of Defective BearingMathematical model is developed or defective bearing defect on inner and outer race. Fig. 3 shows a ball striking the defect on the race

Figure 3 Ball in the defective region of race [9]




( − )

Thus the additional deflection Δ for the travel of the ball in the defectiveouter race can be determined as:

( −

where h is depth of the defect. The defect on the outer race does

The defect on inner race rotates with the speed of the shaft( − )

otion Taking x and y as the displacements along X and Y directions, the governing equations for a two degree of freedom system are formed.

+

+


asp 764

earing Mathematical model is developed or defective bearing defect on inner and outer race. Fig. 3 shows a ball striking the defect on the race.

Ball in the defective region of race [9]




for the travel of the ball in the defective

)


The defect on inner race rotates with the speed of the shaft

Taking x and y as the displacements along X and Y directions, the governing equations for a two degree of freedom system are formed.

) − ( + ∆

) − ( + ∆


Mathematical model is developed or defective bearing defect on inner and outer race. Fig. 3


While deriving the mathematical model, the presence of defect on the races is accounted

are the location of entry edge and exit edge of the fault on the outer race respectively and describes the angular extent of the fault measured about the outer race centre.

is expressed as:




The defect on inner race rotates with the speed of the shaft(

Taking x and y as the displacements along X and Y directions, the governing equations for a

∆)]

∆)]


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t edge of the fault on the outer race respectively and describes the angular extent of the fault measured about the outer race centre.

is expressed as:


where h is depth of the defect. The defect on the outer race does not change its position

( ). Thus ϕ


= +

=


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(8)

(9)

for the travel of the ball in the defective region of the

(10)

not change its position

(11)

ϕt is defined as: (12)


(1

(1


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region of the

not change its position

)

is defined as: )


(13)

14)


In above equations, over the defect. The unbalance force The damping in this system is represented by an equivalent viscous damping

3. COMPUTATIONAL PROCEDUsing Euler’s method Eqs. (13) andand their time derivatives are obtained. Initial conditions of x and y are 10later modified according to acceleration values to account for steady state condition. The time step of MATLAB is developed for solving the equations.

4. RESULTS AND DISCUSSI

4.1. Variation of Defect Size on Outer Race Defect with Constant L


In above equations, over the defect. The unbalance force The damping in this system is represented by an equivalent viscous damping

COMPUTATIONAL PROCEDUsing Euler’s method Eqs. (13) andand their time derivatives are obtained. Initial conditions of x and y are 10later modified according to acceleration values to account for steady state condition. The time step of 6.95x10-6 MATLAB is developed for solving the equations.

RESULTS AND DISCUSSI

. Variation of Defect Size on Outer Race Defect with Constant L


In above equations, ∆ term corresponds to additional deflection for the coover the defect. The unbalance force The damping in this system is represented by an equivalent viscous damping

COMPUTATIONAL PROCEDUsing Euler’s method Eqs. (13) andand their time derivatives are obtained. Initial conditions of x and y are 10later modified according to acceleration values to account for steady state condition. The time

sec has been considered in the computation. A computer program in MATLAB is developed for solving the equations.

RESULTS AND DISCUSSI



IJMET/index.asp

term corresponds to additional deflection for the coover the defect. The unbalance force accounts for the residual imbalance left in the system. The damping in this system is represented by an equivalent viscous damping

COMPUTATIONAL PROCEDUsing Euler’s method Eqs. (13) and (14) are solved and displacements in X and Y directions and their time derivatives are obtained. Initial conditions of x and y are 10later modified according to acceleration values to account for steady state condition. The time


RESULTS AND DISCUSSION


0.1

0.4


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term corresponds to additional deflection for the coaccounts for the residual imbalance left in the system.

The damping in this system is represented by an equivalent viscous damping

COMPUTATIONAL PROCEDURE (14) are solved and displacements in X and Y directions

and their time derivatives are obtained. Initial conditions of x and y are 10later modified according to acceleration values to account for steady state condition. The time


ON


0.1 mm defect

0.4 mm defect

1 mm defect




(14) are solved and displacements in X and Y directions and their time derivatives are obtained. Initial conditions of x and y are 10later modified according to acceleration values to account for steady state condition. The time

sec has been considered in the computation. A computer program in


mm defect

mm defect

1 mm defect


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(14) are solved and displacements in X and Y directions and their time derivatives are obtained. Initial conditions of x and y are 10later modified according to acceleration values to account for steady state condition. The time



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term corresponds to additional deflection for the contact of each ball accounts for the residual imbalance left in the system.

The damping in this system is represented by an equivalent viscous damping c.

(14) are solved and displacements in X and Y directions and their time derivatives are obtained. Initial conditions of x and y are 10-6 are chosen and later modified according to acceleration values to account for steady state condition. The time


. Variation of Defect Size on Outer Race Defect with Constant Load

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ntact of each ball accounts for the residual imbalance left in the system.

(14) are solved and displacements in X and Y directions are chosen and

later modified according to acceleration values to account for steady state condition. The time sec has been considered in the computation. A computer program in



It is clear based on different cases shown in Fig. 4 that as defect size increases the peak at the outer race defect frequency increases. This correspobearings.

4.2. Variation of

Fig. 5 shows the time domain signal obtained through matlab program.



It is clear based on different cases shown in Fig. 4 that as defect size increases the peak at the outer race defect frequency increases. This correspobearings.

Variation of

Fig




Figure 4

It is clear based on different cases shown in Fig. 4 that as defect size increases the peak at the outer race defect frequency increases. This correspo

Variation of Load for

Figure 5 Simulated vibration signal with defect on inner race.


Figure


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1.5

4 Effect of defect size on vibrations of bearing.


oad for Inner Race

Simulated vibration signal with defect on inner race.


6 Spectrum of


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1.5 mm defect

Effect of defect size on vibrations of bearing.


ace Defect



Spectrum of bearing with inner race defect.


mm defect


It is clear based on different cases shown in Fig. 4 that as defect size increases the peak at the outer race defect frequency increases. This corresponds to the progressive degradation in



bearing with inner race defect.


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It is clear based on different cases shown in Fig. 4 that as defect size increases the peak at nds to the progressive degradation in



bearing with inner race defect.


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The spectrum of the bearing with faulty inner race is shown in Fig. 6. The peak at 220 Hz indicate that the defect is present on the inner race.

Various cases discussed in Fig. 7 are with respect to what happens when the load on bearing is changed.

The simulated study changes the intensity of load on the bearing. It is observed that as load is decreased, the peaks at inner race dehigh peak and at lower loads the peaks are low at the characteristic defect frequency.







Figure




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ure 7 Effect of load on vibratio

Various cases discussed in Fig. 7 are with respect to what happens when the load on



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600 N load

400 N load

200 N load

Effect of load on vibratio


The simulated study changes the intensity of load on the bearing. It is observed that as load is decreased, the peaks at inner race defect frequency reduces. . At higher loads, there is high peak and at lower loads the peaks are low at the characteristic defect frequency.


The spectrum of the bearing with faulty inner race is shown in Fig. 6. The peak at 220 Hz

600 N load

Effect of load on vibrations of bearing


The simulated study changes the intensity of load on the bearing. It is observed that as fect frequency reduces. . At higher loads, there is

high peak and at lower loads the peaks are low at the characteristic defect frequency.


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ns of bearing




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4.3. Effect of

The effect of unbalance present in the rotor is studied by unbalance of 100 gm at 2400 rpm. Fig. 9 shows significant peak at the rotational frequency which 40 Hz. This high peak at 1X corresponds to the unbalance

5. CONCLUSIONBased on the results presented in above section, it is concluded that the model based studies for assessing health of bearing provides reasonably good results. The model developed in the paper focuses on the effect of chanvibration behaviour o bearing. Additionally results are presented for effect of unbalanced rotor on vibrations of bearing.

REFERENCES[1]

[2]

[3]

[4]

[5]



Effect of Unbalanced

Fig


CONCLUSIONBased on the results presented in above section, it is concluded that the model based studies for assessing health of bearing provides reasonably good results. The model developed in the paper focuses on the effect of chanvibration behaviour o bearing. Additionally results are presented for effect of unbalanced rotor on vibrations of bearing.

REFERENCES C. S. Sunnersjo, “Varying compliance vibrations of rolling bear

and Vibration”, 58(3) (1978) 363

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

T. A. Harris, “Rolling Bearing Analysis”, Third Ed. John Wiley and Son’s, New York, USA (2001).

P. D. McFadden and J. D. Smith, defect in a rolling element bearing”, Journal of Sound and Vibration,

P. D. McFadden and J. D. Smith, defects in a rolling element bearing”, Journal of Sound and Vibration,73.



nbalanced F

Figure 8 Time domain signal for unbalanced mass in the rotor

Figure


CONCLUSIONS Based on the results presented in above section, it is concluded that the model based studies for assessing health of bearing provides reasonably good results. The model developed in the paper focuses on the effect of chanvibration behaviour o bearing. Additionally results are presented for effect of unbalanced rotor on vibrations of bearing.

REFERENCES C. S. Sunnersjo, “Varying compliance vibrations of rolling bearand Vibration”, 58(3) (1978) 363

N. Tandon, A. Choudhury, “A review of vibration and acoustic measurement methods for the detection of defects in rolling element bearings”, Tribology International, 32 (1999)

480.

A. Harris, “Rolling Bearing Analysis”, Third Ed. John Wiley and Son’s, New York, USA (2001).


P. D. McFadden and J. D. Smith, defects in a rolling element bearing”, Journal of Sound and Vibration,


IJMET/index.asp

Force in the

Time domain signal for unbalanced mass in the rotor

ure 9 Spectrum for unbalance in the rotor

The effect of unbalance present in the rotor is studied by unbalance of 100 gm at 2400 rpm. Fig. 9 shows significant peak at the rotational frequency which 40 Hz. This high peak at 1X corresponds to the unbalance present in the rotor.

Based on the results presented in above section, it is concluded that the model based studies for assessing health of bearing provides reasonably good results. The model developed in the paper focuses on the effect of change in defect size and effect of change if the load on vibration behaviour o bearing. Additionally results are presented for effect of unbalanced rotor on vibrations of bearing.

C. S. Sunnersjo, “Varying compliance vibrations of rolling bearand Vibration”, 58(3) (1978) 363


A. Harris, “Rolling Bearing Analysis”, Third Ed. John Wiley and Son’s, New York,


P. D. McFadden and J. D. Smith, defects in a rolling element bearing”, Journal of Sound and Vibration,


asp 768

orce in the Rotor


Spectrum for unbalance in the rotor

The effect of unbalance present in the rotor is studied by unbalance of 100 gm at 2400 rpm. Fig. 9 shows significant peak at the rotational frequency which 40 Hz. This high peak at

present in the rotor.

Based on the results presented in above section, it is concluded that the model based studies for assessing health of bearing provides reasonably good results. The model developed in the

ge in defect size and effect of change if the load on vibration behaviour o bearing. Additionally results are presented for effect of unbalanced

C. S. Sunnersjo, “Varying compliance vibrations of rolling bearand Vibration”, 58(3) (1978) 363-373.



P. D. McFadden and J. D. Smith, “Model for the vibration produced by a single point defect in a rolling element bearing”, Journal of Sound and Vibration,

P. D. McFadden and J. D. Smith, “Model for the vibration produced by multiple point defects in a rolling element bearing”, Journal of Sound and Vibration,





present in the rotor.



C. S. Sunnersjo, “Varying compliance vibrations of rolling bear



Model for the vibration produced by a single point defect in a rolling element bearing”, Journal of Sound and Vibration,

Model for the vibration produced by multiple point defects in a rolling element bearing”, Journal of Sound and Vibration,


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C. S. Sunnersjo, “Varying compliance vibrations of rolling bearings”, Journal of Sound



Model for the vibration produced by a single point defect in a rolling element bearing”, Journal of Sound and Vibration,

Model for the vibration produced by multiple point defects in a rolling element bearing”, Journal of Sound and Vibration,


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ings”, Journal of Sound



Model for the vibration produced by a single point defect in a rolling element bearing”, Journal of Sound and Vibration, 96(1)(1984) 69

Model for the vibration produced by multiple point defects in a rolling element bearing”, Journal of Sound and Vibration, 98(2) (1985) 263


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ings”, Journal of Sound



Model for the vibration produced by a single point 96(1)(1984) 69-82.

Model for the vibration produced by multiple point 98(2) (1985) 263–


http://www.iaeme.com/IJMET/index.asp 769 [email protected]

[6] N. Tandon, A. Choudhury, “An analytical model for the prediction of the vibration response of rolling element bearings due to a localized defect”, Journal of Sound and Vibration, 205(3) (1997)275–92.

[7] V. N. Patel, N. Tandon, R. K. Pandey, “A dynamic model for vibration studies of deep groove ball bearings considering single and multiple defects in races”, Journal of Tribology, 132 (2010) 041101-1-10.

[8] M.S.Patil, Jose Mathew, P.K.Rajendrakumar, Sandeep Desai, “A theoretical model to predict the effect of localized defect on vibrations associated with ball bearing,” International Journal of Mechanical Sciences, 52 (2010) 1193–1201.

[9] P. G. Kulkarni, A. D. Sahasrabudhe, “A dynamic model of ball bearing for simulating localized defects on outer race using cubic hermite spline”, Journal of Mechanical Science and Technology, 28 (9) (2014) 3433-3442.

[10] Prof. Dr. Zena K. Kadhim and Hadi O. Mery. Free Convection From Optimum Sinusoidal Surface Exposed to Vertical Vibrations, International Journal of Mechanical Engineering and Technology, 7 (1), 2016, pp. 214-224.

[11] Prince Kumar and Sandeep Nasier, An Analytic and Constructive Approach to Control Seismic Vibrations in Buildings. International Journal of Civil Engineering and Technology, 7(5), 2016, pp.103–110.

A THEORETICAL MODEL OF DEEP GROOVE BALL … · Various sources of vibration in any machine are...

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