A dynamic model for vibration studies of dry and lubricated deep groove ball bearings considering local defects on races

Measurement 137 (2019) 535–555

Contents lists available at ScienceDirect

Measurement journal homepage: www.elsevier.com/locate/measurement

A dynamic model for vibration studies of dry and lubricated deep groove ball bearings considering local defects on races Dipen S. Shah a,b,⇑, V. N. Patel c a

S. V. I. T, Vasad 388306, Gujarat, India C. U. Shah University, Wadhwan City 363030, Gujarat, India c G. H. Patel College of Engineering and Technology, V. V. Nagar 388120, India b

a r t i c l e

i n f o

Article history: Received 1 July 2017 Received in revised form 28 November 2018 Accepted 31 January 2019 Available online 2 February 2019 Keywords: Deep groove ball bearing Elastohydrodynamic lubrication Localized defect Vibration

a b s t r a c t A dynamic model has been developed to predict vibration generated by healthy and defective deep groove ball bearing. The masses of shaft, raceways, ball and housing have been considered in the present model. Moreover, stiffness and damping due to non-linear Hertzian contact and lubricant film have been included in the dynamic model. The additional deflection of rolling elements in presence of local defect has been considered during simulation of defect in dynamic model. The derived equations of motions have been solved analytically using fourth order Runge-Kutta method in MATLAB software. The vibrations generated by dry and lubricated contact bearings having local defects on their races have been studied theoretically and experimentally. It has been observed that the vibration amplitude of characteristic defect frequency is affected by presence of lubricant, shaft rotational speed, radial load, defect location and its size. Good relation between theoretical and experimental results proves the effectiveness of the present model. The authors believe that the present dynamic model can be used with confidence to predict the amplitude and frequency generated by healthy and defective bearings. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Rolling element bearing is an important part of any rotating machinery. These bearings are manufactured by sophisticated machinery with great quality standards. Even though these rolling element bearings may possess local defects like cracks, pits, spalls, etc. on their components (races and rolling elements) either in the beginning itself as a new piece or during its use. The existence of even tiny defects on the mating surfaces of the bearing components can lead to failure through passage of time. The failure of bearing causes economical loss and some time it may harm machine operator. Therefore, detection of defects particularly local defects in their early stages through vibrations analysis of shaft bearing system is an imperative task. Even healthy bearings generate vibrations during motion due to time varying contact stiffness at rolling elements and races contact. The dynamic behaviour of shaft bearing system changes in presence of local defects on bearing races. Therefore, dynamic model of shaft bearing system has a great significance for condition monitoring and predictive maintenance of machinery in the industry. ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (D.S. Shah), [email protected] (V.N. Patel). https://doi.org/10.1016/j.measurement.2019.01.097 0263-2241/Ó 2019 Elsevier Ltd. All rights reserved.

Several researchers have proposed dynamic models to study vibrations generated by bearings in presence and absence of local defect. The authors of review papers [1–3] have discussed various modelling techniques for shaft bearing system in presence of local defect on bearing elements. McFadden and Smith [4,5] have presented primary dynamic model to describe the vibration characteristics of rolling element bearings in presence of single and multiple point defects on the inner races. Inner race defects have been modelled as a product of impulse series. The series of impulse is generated due to the interaction of local defect with rolling element. [4,6]. Tandon and Choudhary [6] extended the concept of McFadden and Smith [4,5] to predict the vibration response of localized defective ball bearing. The authors have simulated interaction of balls and local defect as a series of impulses having regular shape like rectangular, triangle and half sine pulse. The frequency of generated impulses depends on location of defect i.e. inner race, outer race or rolling element. Choudhury and Tandon [7] have also proposed simple lumped-mass model to predict vibration response of defective bearing. In their model masses of inner race and outer race have been added into masses of shaft and housing, respectively. Moreover, linear stiffness at rolling elements and races contact have been considered. Arslan and Aktürk [8] have developed a shaft-bearing model to investigate the vibrations of an angular contact ball bearing considering the masses of the rolling elements

536

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

Nomenclatures C Cr D d din dout ds E EHL ek G K k Kd Mbi Mh Ms Nb n Pd Q Rx t U W

damping coefficient, N-sec/mm Internal radial clearance pitch diameter of the bearing, mm ball diameter, mm diameter of inner race, mm diameter of outer race, mm diameter of shaft, mm Young’s modulus, N/mm2 elastohydrodynamic lubrication ellipticity parameter non-dimensional material parameter stiffness, N/mm Contact stiffness for ball-raceway deformation constant mass of ith ball, gm total Mass of housing with mass of outer race, Kg total mass of shaft with mass of inner race, Kg number of balls in bearing load deflection exponent Diametral clearance, mm Radial load applied on the test bearing, N radius of curvature in the principal plane X time coordinate, sec non-dimensional speed parameter non-dimensional load parameter

and shaft. Patil et al. [9] have studied the effect of defect size on vibration response of bearing. Authors have noticed that the vibration amplitude increases with defect size. Patel et al. [10,11] have developed 6 DOF dynamic model for deep groove ball bearing in presence of local defect on either race of bearing. The additional deflection of balls in presence of local defects have been considered during defect simulation. While Khanam et al. [12,13] have derived the impact force based dynamic model to predict the vibration characteristics of shaft bearing system in presence of local defects. Researchers [14–19] have used various tools like Simulink, Multibody dynamics, Finite Element Analysis to analyse vibrations generated by shaft bearing system in presence of local defect on bearing races. Moazen-ahmadi and Carl Howard [14] have proposed the defect size estimation method and algorithm for outer race defect considering the effect of inertia and centrifugal force which act on a rolling element in a defective bearing. Authors have also studied the effect of speed on the position of the highfrequency events on the vibration response. While Ahmadi et al. [15] have developed multi body dynamic model with considering finite size of rolling element which can predict the contact forces and vibration generated by defective bearing. Moreover, Kogan et al. [16] have developed a multi body non-linear dynamic model to study interactions between a rolling-element and outer race spall. Moreover, Patel and Upadhyay [17] have derived 9 DOF dynamic model of cylindrical roller bearing-rotor system to study the vibration in presence of local defect on bearing races. The simulated and experimental dynamic behaviour shows instability and chaos in the dynamic system due to presence of combined local defect. A more sophisticated 6-DOF model of a rotor-bearinghousing system have been developed by Liu and Shao [18]. Authors have included rotor compliance, housing compliance, elastic interface between the housing and outer race, and time-dependent excitations generated by a local defect in presented dynamic model. Authors found that the rotor compliance and timedependent contact stiffness coefficient due to local defect have

X, Y

Rq

deflection along the axes, mm defect angle with reference to X axis surface pattern parameter hydrodynamic roughness parameter additional deflection of ball centre in defected zone, mm contact deflection in radial direction, mm dimensionless contact deflection loss factor (depends on the material) Poisson’s ratio angular velocity, rad/s angular position of the ball, rad load distribution factor curvature sum

Subscript b c eq h i l in out s

ball cage equivalent housing ball number lubricant inner race outer race shaft

a c k

w d d*

g m x h

e

major influence on the dynamic behaviour of rotor-bearinghousing system. Mishra et al. [19] have developed mathematical model using tools like MATLAB Simulink Model, Bond Graph Model and Multi body dynamics Model for rotor- bearing system in presence of local defects on bearing races. The authors have validated theoretical results with experimental work for development and testing of bearing fault diagnostic tools. Many researchers [20–26] have used Finite Elements Analysis (FEA) as a tool for simulation of bearing vibrations in presence of defect. The relationship between pulse waveform and localized defect size and shape for ball bearings have been established using the explicit dynamic FEA method by Liu et al. [20]. The effects of size and shape of localized defects on the pulse waveform characteristics at different load and speed have been studied. A nonlinear finite element model of a rolling element bearing having spall on its outer race has also been presented by Singh et al. [21]. Authors have discussed generation of multiple impulses due to de-stressing and re-stressing of rolling elements when it passes through outer race defect. While, Kiral and Karagulle [22,23] have presented a model to study loading mechanism of rolling bearing in presence of defect on bearing elements and unbalanced force using Finite Element (FE) approach. The effects of number of the outer ring defects and defect location on bearing vibrations have been studied in time domain and frequency domain. In another study, Kiral and Karagulle [23] have presented an impulse force based dynamic loading models for defect-free and defective rolling element bearing structures and obtained the vibration of bearing structure using the FE method. Liu et al. [24] have also developed a six degree of freedom dynamic model of shaft bearing system considering effect of local defect on races and contact stiffness between outer race and housing on ball bearing vibration response. The authors have concluded that contact stiffness between outer race and housing, vibration transmission characteristics of impulse due to local defect and vibration amplitude of bearing system is affected by material properties of housing structure. Moreover, Utpat [25] has proposed numerical model of shaft bearing system

537

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

including local defect on either of bearing races which has been solved using FE software. While, Chouksey et al. [26] have updated the Finite Element (FE) model of an actual rotor system supported on ball bearings and to explore its effectiveness in the prediction of rotor response and balancing. According to theory of elastohydrodynamic lubrication (EHL) lubricant film is assumed equally stiff as the contacting solids [27]. The equations to find minimum or central lubricant film thickness of smooth surface have been derived by Hamrock and Dowson [27]. Dietl et al. [28] have measured and calculated the damping capability of rolling element bearings. They have concluded that the rolling bearing damping is strongly influenced by the lubricated contacts between rolling elements and raceways. Kraus et al. [29] investigated the effect of speed and preload on radial and axial bearing damping in deep-groove ball bearings and found that damping decreases with increase of speed and preload. A dynamic model for a deep groove ball bearing in presence of local and distributed defect considering non-linear Hertzian contact deformation and effect of lubrication has been presented by Sopanen and Mikkola [30]. The thickness of lubricant film depends on surface velocities, geometry and material property of solids and viscosity of lubricant [30]. The empirical relations for equivalent stiffness and damping due to presence of lubricant at contact region is formulated by Sarangi et al [31]. Babu et al. [32] have proposed 6 DOF dynamic model of rigid rotor supported on the lubricated angular contact ball bearing considering combined influences of frictional moments and waviness on ball and races. The friction due to presence of lubricant in a bearing depends on rotational speed, grease consistency and oil viscosity. Vibration amplitude at waviness frequency has been reduced in presence of lubricant friction due to viscous damping of lubricant. While, a 12 DOF dynamic model for rigid rolling bearing system in presence of localized defect on races has been developed by Liu and Shao [33]. This model analyse the effect of housing support stiffness, sharp edges of local defect and its sizes and lubricating oil film. However, Shah and Patel [34] have studied the effect of governing parameters like radial load, shaft rotational speed, waviness order, waviness location and waviness amplitude on vibration amplitude of waviness defect frequency in presence of lubricant through theoretical and experimental vibration study. The literature review reveals that the most of the researchers [4–13,19,26,35] have developed the dynamic models considering nonlinear Hertzian contact stiffness in absence of lubricant to study the vibrations generated by defective bearings. However, elastohydrodynamic lubrication is used during normal bearing operations. Even though, very few researchers [30,32–34] have considered lubricant film stiffness and damping in bearing modeling. The authors of the present paper could not notice any dynamic model for the vibration study of a defective deep groove ball bearing which include masses of the shaft, balls, and housing, along with stiffness and damping due to non-linear Hertzian contact and lubricant film. Therefore, it is necessary to develop more realistic dynamic model which includes the effect of non-linear contact stiffness, lubricant film stiffness and damping to study the bearing vibrations precisely. The objective of the present paper is to study the influence of radial load, presence of lubricant, size and location of defect on vibration amplitude at bearing defect frequency. The additional deflection of rolling elements in presence of local defect has been considered during simulation of defect in dynamic model. The derived equations of motions have been solved analytically using fourth order Runge-Kutta method in MATLAB software. The frequencies predicted by dynamic model have been validated by experimentations. All the expected defect frequencies, side bands and their harmonics have been noticed in simulated and experimental vibration spectra of test bearings. A similarity between

simulated results and experimental results has proved the effectiveness of presented model. 2. Dynamic model of Shaft-Bearing system The photographic view of experimental setup for shaft-bearing system under investigation is shown in Fig. 1(a). A polymeric cage deep groove ball bearing (SKF BB1B420206) was mounted on left end of the shaft. A split type test bearing housing has been designed and developed for protection and rigid support to this test bearing. The Fig. 1(b) shows free body diagram of shaftbearing system to study vibration generated by defective ball bearing. To study the vibration characteristics of rolling element bearing the ball-raceway contact has been considered as a spring-mass system. In the proposed model it has been assumed that outer race of bearing is stationary, and it is fixed rigidly in bearing housing, while, inner race is rigidly mounted on shaft. The following realistic assumptions and considerations have been incorporated in proposed dynamic model:
Friction between ball and races is neglected.
Mass of shaft, mass of housing and mass of rolling element have been considered during modeling.
The mass of the inner race is included in the mass of the cantilever portion of shaft, and the mass of the housing includes the mass of outer race.
Effect of thermal aspect is neglected.
Forces act in radial direction along X axis only.
Consider pure rolling motion of ball along the raceway surface.
Effects of centrifugal forces acting on balls are neglected.
Effect of damping and stiffness due to lubrication film at ballraceway contact is accounted.
Non-linear Hertzian contact theory is considered for finding deformation at ball-raceway contacts.
Balls are located at equal distance around the inner raceway and there is no interaction between them.
Inner and outer races are rigidly fitted with shaft and housing respectively. 2.1. Radial deflection of rolling element The elastic deformation due to interaction of rolling element and raceway generates non-linear contact force and deformation which has been obtained using Hertzian theory of elasticity. A steady radial load was applied on the test bearing housing. A non-linear relationship for load-deformation proposed by Harris [36] has been used to compute deformation.

W ¼ K:dn

ð1Þ

The value of load–deflection exponent (n) = 3/2 has been used for ball bearing. The load–deflection factor (K) depends on the ball-raceway contact geometry. The equivalent load–deflection factor between two raceways is the sum of the contact stiffness between rolling element and each raceway.

3n

2

6 1 1 7 7 K¼ 6 4
n1 þ
n1 5 1 Kin

ð2Þ

1 Kout

The ball-inner race contact stiffness (Kin) and ball-outer raceway contact stiffness (Kout) have been obtained using following relation [36].

Kin=out ¼ 2:15  105 

X




 q1=2 in=out  din=out

3=2

ð3Þ

538

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

Fig. 1. (a) Photographic view of experimental setup (b) Free body diagram of shaft-bearing system under study [34].

Refer the reference [36] for more details about the computations for curvature sum and non dimensional contact deflections for inner and outer raceway of rolling bearing. The radial deflection of ith ball at any angle hi has been computed by following relation (Refer Fig. 2):

dr ¼ ðxcoshi  Cr Þ

ð4Þ

where, internal radial clearance (Cr) has been obtained using Eq. (5), provided by Patel et al. [10].

Cr ¼

Pd 2  ð1  coshÞ

ð5Þ

The position of any rolling element ‘hi’ at any time, ‘t’ with respect to its initial position ‘h0’ depends on angular velocity of cage (xc) and number of rolling elements (Nb) in bearings:

hi ¼

2pi þ xc t þ h 0 Nb

ð6Þ

where, the angular velocity of cage, xc is represented in term of angular velocity of shaft (xs) and computed by Eq. (7).



xc ¼ 1 

  d xs  D 2

ð7Þ

Deformation of ith ball between inner race and ball in radial direction depends on its angular position and radial difference of inner race position and ball position which is provided by Eq. (8).

 din i ¼ Xs  Xbi  coshi  Cr

ð8Þ

Deformation of ith ball between outer race and ball in radial direction depends on its angular position and radial difference of outer race position and ball position which is provided by Eq. (9).

 dout ¼ Xbi  Xh  coshi i

ð9Þ

The total deflection of ith ball in radial direction has been computed by Eq. (10).

  out dtotal ¼ din ¼ Xs  Xbi  coshi þ Xbi  Xh  coshi  Cr i i þ di ð10Þ The load distribution and angular position of rolling elements for a test bearing is shown in Fig. 3. The load zones (2600, 1200 and 1400) and maximum rolling element loads (60.81 kg, 155.8 kg and 216.4 kg) have been computed using Eq. (11). Based on the geometry of test bearing a radial load of 12 kg, 22 kg and 32 kg were applied. The test bearing load distribution for 22 kg radial load can be observed in Fig. 3. The bearing load distribution factor (e) is obtained using relation e = (1/2) (1  Pd / 2dr). As per the load distribution curve shown in Fig. 3(b) the maximum load acts at 0° ball position, results in maximum deformation of ball no. 8, while ball no. 4 has minimum deformation.

Fig. 2. Schematic representation for radial deflection of ball bearing [34].

   1:5 1 ð1  coshÞ Q h ¼ Q max 1  2e

for  h1 < h < h1 ¼ 0

ð11Þ

539

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

Fig. 3. Load distribution in test bearing (for radial load of 22 kg) (a) Representation of load distribution (b) Angular position of ball.

2.2. Contact stiffness and damping coefficients in Shaft-Bearing system The stiffness and damping properties of each element of any system plays vital role in its dynamic characteristics. Thus, it is prime requirement to estimate the correct value of stiffness and damping coefficients of each component of shaft-bearing system under study. 2.2.1. Stiffness coefficients The stiffness of cantilever portion of the shaft (on which the test bearing is mounted) and stiffness of bearing housing have been computed by means of finite element analysis. The computed value of shaft stiffness is 2.638  104 N/mm under actual loading condition. The static structural analysis of test bearing housing has been carried out using ANSYS Workbench R15 commercial software. During analysis of test bearing housing 3284 nodes and 1785 elements were considered. Structural steel material having density of 7850 kg/m3 and yield strength of 250 MPa was considered for test bearing housing. The coarse meshing quality have been considered in present simulation. Radial load at the center of outer bottom surface of housing was applied. During application of radial load inner bore surface was kept fixed to determine deformation of housing. The mesh model of test bearing housing is shown in Fig. 4. The calculated value for stiffness of test bearing housing is 2.42  107 N/mm. It is essential to determine the minimum film thickness to confirm the amount of lubricant present at contact surfaces. The minimum lubricant film thickness is found at outlet zone of rolling

contact. The minimum lubricant film thickness for elliptical contact has been calculated through empirical relationship (Eq. (12)) provided by Hamrock and Dowson [27]. 0:49 hin=out ¼ 3:63  Rxin=out  U0:68  W0:073 in=out  G in=out

 ð1  eð0:68  ekin=out Þ Þ

ð12Þ

where, Rx, U, G, W, and ek are the effective radius of curvature in the principal plane, dimensionless speed parameter, dimensionless material parameter, dimensionless load parameter, and ellipticity parameter respectively (refer Appendix). To determine lubricant film stiffness, ‘kEHLin=out ’ for mixed elastohydrodynamic lubrication, a non-dimensional relationship derived by Sarangi et al. [31], has been adopted here.


 kEHLin=out ¼ klin=out þ kcin=out  Ed  Rxin=out

ð13Þ

It is worth to mention here that the stiffness due to lubricant film thickness (kEHLin=out ) and asperity contact stiffness (kc in/out) are added due to consideration of parallel connections. The stiffness of lubricated film in contact (kl in/out) is determined by Eq. (14) provided by Sarangi et al. [31]. The surface pattern parameter (c), hydrodynamic roughness parameter (k), geometrical parameter (rin/out) and deformation constant (Kd) have been computed using equations mentioned in Appendix provided by Sarangi et al. [31]. 1:1678 klin=out ¼ 0:4053  G0:2521  U0:6995 in=out  Win=out  ð0:5891 0:3102

 ðekin=out  eekin=out Þ  eðk

0:928

Þ

Þ  K0:0358  k0:3368 in=out d

 c0:0399

ð14Þ

If elliptical parameter ‘ekin/out’, for inner and outer race is less than or equal to 4, the asperity contact stiffness (kc in/out) is determined by Eq. (15) provided by Sarangi et al. [31]: 0:7239 kCin=out ¼ 0:1828  G0:1065  U0:4931 in=out  Win=out  ð0:1263 0:2377

k  k0:5396  ðekin=out  eðekin=out Þ0:2462 ÞÞ  K1:0418 in=out  e d

1:6221

 c0:0861

ð15Þ

Otherwise, 0:7239 kCin=out ¼ 0:1828  G0:1065  U0:4931 in=out  Win=out  ð0:0588 0:4420

 ðekin=out  eðekin=out Þ0:1081 ÞÞ  Kd1:0418  k0:5396 in=out  ek

Fig. 4. Mesh Model of Test bearing Housing [34].

1:6221

 c0:0861

ð16Þ

The non-linear stiffness for Hertzian contacts (Kin/out) due to elastic deformation has been computed through Eq. (3). The total equivalent contact stiffness at contact (between rolling element

540

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

and either race) has been computed with consideration of parallel connection between Hertzian contacts due to elastic deformation and elastohydrodynamic lubrication film. The equivalent stiffness for lubricated contact has been determined by Eq. (17),

Kin=out eq

¼ KEHLin=out þ Kin=out

gs;h  Ks;h Cs;h ¼ xext

ð18Þ

where, xext = Shaft rotational frequency Damping due to elastohydrodynamic lubrication between inner and outer raceway-ball contact has been computed by Eq. (19) provided by Sarangi et al. [31].

ðEd  Rxin=out Þ  0:5  d  ð0:7300  G0:7511 ðUin=out  rin=out Þ 0:1275

 ðe

ÞÞÞ 

    width of defect width of defect  h i  xs t þ di di

ð20Þ

K0:0013 d



k0:3521 in=out

 c0:0366 Þ

e

2.3.2. Defect on outer raceway surface Normally local defect on stationary outer race is observed in load zone. The relation to determine angular position of ith ball when it passes through outer race defect is determined by following Eq. (21).

aouter 

    width of defect width of defect  hi  aouter þ do do ð21Þ

where, aout = Angle of outer raceway defect at bearing centre The additional deflection of ith ball, when it passes through defect is computed by Eq. (22) provided by Patel et al.[10].

w¼

1:1963  U0:5871 in=out  Win=out  ð0:5188  ðekin=out ðek0:4548 in=out Þ

xs t 

ð17Þ

2.2.2. Damping Coefficients The hysteretic and equivalent viscous damping coefficient for shaft and housing has been computed using following Eq. (18) provided by Genta [37].

Cin=out ¼

matically the angular position ‘hi’ of ith ball when it passes through inner race defect can be determined by following relation [10]:

  d  0:5dcosð0:5uball Þ 2

ð22Þ

where, uball = Width of defect/radius of ball Hence, when ith ball passes through defect zone of outer race its

ðk0:8472 Þ in=out

ð19Þ

2.3. Simulation of local raceway defect A fatigue crack on either race of test bearing has been simulated as a rectangular defect. The severity of the defect/crack can simulated by varying the width of the defect as discussed by Liu et al. [38]. Fig. 5(a) and (b) represent the schematic diagram for inner race and outer race local defect respectively. The additional deflection has been computed, when the moving rolling elements enter in defect zone. This additional deflection has been considered in equation of motions (Refer Eqs. (23)–(25)). The additional deflection depends on width of the defect.

total deflection = dtotal + w, while, it passes through defect zone of i  w. inner race its total deflection = dtotal i 2.4. Equations of motion Equations of motion derived (based on the assumptions made in Section 2) for shaft, balls and housing have been presented in subsection (Refer Fig. 1(b)). 2.4.1. For shaft Equation of motion for shaft in radial (X) direction is as follow:

€ S þ Cs X_ S þ KS XS þ Ms X

Nb X


3=2 in Kin coshi eq di

i¼1

2.3.1. Defect on inner raceway surface The inner race of a test bearing is rigidly mounted on a shaft. The inner race defect rotates at shaft angular velocity (xs). Mathe-

þ

Nb X

Cin




 X_ s  X_ bi coshi ¼ 0

i¼1

Fig. 5. Representations of ball-raceway defect interaction (a) Ball-inner raceway defect contact (b) Ball-outer raceway defect contact.

ð23Þ

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

2.4.2. For rolling element (Ball) Equation of motion for ith ball in radial (X) direction is as follow:

€ bi  Mbi X

Nb X

Nb
 X in 3=2 Cin X_ s  X_ bi coshi  Kin coshi eq ðdi Þ

i¼1

þ

Nb X

Kout eq

i¼1 Nb
 X  out 3=2 di coshi þ Cout X_ bi  X_ h coshi ¼ 0

i¼1

ð24Þ

i¼1

2.4.3. For bearing housing Equation of motion for bearing housing in radial (X) direction is as follow:

€ h þ Kh Xh þ Ch X_ h  Mh X

Nb X


 Cout X_ bi  X_ h coshi

i¼1



Nb X

 out 3=2 Kout coshi ¼ Q eq di

ð25Þ

i¼1 out The deflections of rolling elements din at inner race and i and di outer raceway defect contacts are computed through Eqs. (26) and (27) respectively.

din i ¼ ðXs  Xbi Þcoshi  Cr  bi w

ð26Þ

¼ ðXbi  Xh Þcoshi þ bi w dout i

ð27Þ

where, bi = 1 When ith ball passes through defect, otherwise bi = 0. Where, Cr is radial clearance and w is additional deflection of ball.

541

vibrations signals generated by test bearing during their experimentations. The captured vibrations signal were sampled with sampling frequency of 1000 Hz with sample size of 4096. These vibrations data were acquired using vibration analyser, model IMVA 440. The stored time domain data have been analysed using MATLAB software. The artificial rectangular defects having defect width of 0.3 mm, 0.5 mm, 1 mm and 1.5 mm and depth of 0.1 mm on either inner race or outer race were generated using electric discharge machining (EDM). The images of inner and outer raceway defect generated on test bearing are shown in Fig. 7(a) and (b) respectively. Vibration responses of fresh bearings were captured to monitor their health before defect generation on them. Two separate sets of experiments on same test bearing have been conducted to measure vibrations response of dry (metal to metal contact) and lubricated contact. The equal amount of lithium-based grease (NLGI-3) was applied manually at the ball-raceway groove contact on all tests bearing to create lubricated contact. Before application of grease, the bearing raceway groove was cleaned properly to avoid the contaminations of foreign particles in lubricant. 5. Results and discussions The vibration velocity spectra of defect free and defective test bearings in presence and absence of lubricant have been presented here. The bearing specifications and other necessary data used during the present study have been mentioned in Table 1. 5.1. Vibration response of defect-free (healthy) bearing

3. Computational procedure To determine the vibration response of shaft-bearing system, the governing equations of motion (Eqs. (23)–(25)) for angular position hi of ith ball have been solved iteratively for each step of time. The displacements in X directions and velocities X_ at time (t + dt) have been computed using equations (23–25) by 4th order Runge-Kutta numerical method. The time step ‘dt’ of 10-6 s have been considered during simulation. The governing equations of motion have been iteratively solved for specified time with help of MATLAB software. A flowchart for numerical computation is shown in Fig. 6. 4. Experimentations As shown in Fig. 1(a) the shaft was supported on rollers while, power was supplied by AC motor through belt and pulley mechanism. The shaft speed variations were controlled by means of variable frequency drive (VFD) of electric motor. The test bearing was mounted on the right side of shaft at cantilever end and constrained by means of lock nut. The test bearing was fixed in split type housing. The radial load was applied at the bottom of test bearing housing by means of attached hook and hanger arrangement. A permanent magnet type uniaxial piezoelectric accelerometer CTC make (Model Number AC-102-1A) having magnetic sensitivity of 100 mv/g and frequency range of 0–30 kHz was mounted on top of the test bearing housing to capture vibration signals generated by test bearing. Significant vibration in direction of applied radial load have been noticed as compared to other two directions during actual experimentations in case of deep groove ball bearings. Variation in contact forces in presence of dents occurs in direction of applied radial load (along X axis) while, due to transmission error amplitude along Y axis reduces [39]. Hence authors of references [9–13] also mounted accelerometer on top of test bearing housing (radial direction only) to capture

The velocity spectra of healthy test bearing at shaft rotational speed of 1500 rpm (fs = 25 Hz) and 12 kg radial load has been obtained through numerical solution of Eqs. (23)–(25). Vibration spectra of (simulated data) healthy bearings for dry and lubricated contact have been plotted in Fig. 8(a) and (b) respectively. Frequency peaks at shaft rotational frequency (fs = 25 Hz) along with their harmonics (2  fs = 50 Hz), (3  fs = 75 Hz) can be noticed in Fig. 8(a) and (b). These theoretical results have been validated through experimental vibration spectra shown in Fig. 8(c) and (d). The same shaft rotational frequency (fs = 25 Hz) along with multiple harmonics (2  fs = 50 Hz), (3  fs = 75 Hz) are also visible in Fig. 8(c) and (d). Dominant vibration peaks at shaft rotations frequency (fs) have been observed in both simulated and experimental spectra of Fig. 8. As compared to dry contact the vibration amplitude at shaft rotational frequency has reduced in presence of lubricated contact. This reduction in vibration amplitude occurs due to damping effect provided by lubricant film. This results are in line with the result presented by Liu et al. [18]. However, in experimental results (Fig. 8(c and d)) little variation in frequency amplitude have been observed at first harmonic of shaft rotational frequency in presence and absence of lubricant. While, the velocity amplitude diminished faster in subsequent harmonics of fs in presence of lubricant. It is necessary to mention here that the percentage difference computed using equation, |Simulated shaft rotational frequency (Hz) – Experimental shaft rotational frequency (Hz)|  100/Experimental shaft rotational frequency (Hz) for dry contact and lubricated contact are 0.08% and 3.76% respectively. 5.2. Vibration response of defective bearing To study the effect of radial load on amplitude of vibration velocity, the radial load of 0 kg, 12 kg, 22 kg and 32 kg was applied. The shaft speed was kept 1500 rpm (fs = 25 Hz) during all simula-

542

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

Fig. 6. Flowchart for numerical simulation.

tions and experimentation. Theoretical and experimental results for dry contact and lubricated contact have been presented here to study the effect of lubricant on vibration amplitude of defective bearing. This section discusses the theoretical and experimental vibration results of bearing having varying defect sizes on inner race and outer raceway of bearing. The results comparison for

simulated and experimental work for inner and outer raceway defect has been reported in the Sections 5.2.1 and 5.2.2 respectively. 5.2.1. Defect on inner raceway The vibrations generated by bearing having local defects on its inner race are complicated in nature due to the rotation of the

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

543

Fig. 7. Image of localized Defects (a) Inner raceway defect (b) Outer raceway defect.

Table 1 Input Data. Single raw deep groove ball bearing Bore of bearing, mm Inner race diameter (di), mm Outer race diameter (do), mm Pitch diameter (D), mm Ball diameter (d), mm Diametral clearance (Pd), lm Inner raceway grooves radius, mm Outer raceway grooves radius, mm Number of balls (Nb) Speed of shaft, rpm Radial load applied on test Bearing, Kg Contact angle (a), degree Mass of each ball (Mb), gm Mass of inner race, gm Mass of outer race, gm Mass of shaft (Ms), kg Mass of test bearing housing (Mh), kg Mass of hanger, kg Overhung length of shaft, mm

(SKF BB1B 420206) 30 39.38 51.82 45.6 10.36 10 5.38 5.48 8 1500 0, 12, 22 and 32 0 4.49 40.05 85.28 1.79 1.820 0.5 250

defect at the shaft speed. Due to the change of the position of defect the load acting at defect varies and it results in variation of vibration amplitudes. Fig. 9(a–d) shows simulated and experimental vibration response of test bearing for 0.3 mm inner race defect at shaft rotation speed of 1500 RPM and 12 kg radial load for dry and lubricated contact. It is necessary to mention here, that computed value of

Ball Pass Inner Race Defect Frequency obtained by (BPFI = (Nb/2)  (Ns/60)  (1 + (d/D))) is 122.72 Hz. The peaks at shaft rotational frequency (fs = 25 Hz), BPFI = 122.72 Hz and its harmonics are clearly observed in Fig. 9(a–d). The side band frequencies at BPFI ± fs (99 Hz and 147 Hz) are also visible due to modulation. The good matching of vibration amplitude at defect frequency for experimental and theoretical results proves the effectiveness of present model. However, under lower load and small defect size, the shaft frequency fs is merged with other low frequencies. A large peak observed between fc and fs in Fig. 9(c), is a frequency peak of fs– fc (15.37 Hz). The same kinds of simulation and experimentations have been carried out for 0.3 mm defect width with varying radial load. Fig. 10(a) and (b) represent the comparisons of vibration amplitudes at defect frequency under varying radial load for dry and lubricated bearing respectively. It can be observed that the amplitude of vibration continuously varies with increase of radial load on test bearing for both theoretical and experimental study. The similar methodology has been adopted during simulation and experimentation, for inner race defect size of 0.5 mm, 1 mm and 1.5 mm under 12 kg radial load for dry and lubricated contact. The results have been presented in Figs. 11–13 to authenticate the present dynamic model. Babu et al. [32] have observed that the vibration amplitude in rolling elements bearings are influenced by load dependent friction moment and load independent friction moment. Vibration

Fig. 8. Vibration velocity spectra for healthy bearing at shaft rotational speed 1500 rpm, 12 kg radial load (a) Theoretical result, dry contact (b) Theoretical result, lubricated contact (c) Experimental result, dry contact (d) Experimental result, lubricated contact.

544

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

Fig. 9. Vibration velocity spectra for shaft rotational speed 1500 rpm, Inner race defect size 0.3 mm, 12 kg radial load (a) Theoretical result, dry contact (b) Experimental result, dry contact (c) Theoretical result, lubricated contact (d) Experimental result, lubricated contact.

Fig. 10. Vibration amplitude comparison for shaft rotational speed 1500 rpm, Inner race defect size 0.3 mm with varying radial load (a) Dry contact (b) Lubricated contact.

amplitude reduces significantly in presence of load independent frictional moment compared to load dependent moment. Load independent frictional moment occurs due to lubricant friction. Due to the effect of load independent frictional moment in presence of lubricant at lower speed and small load, the BPFI amplitude has reduced in Figs. 9(c), 11(c) and 12(c). Exhaustive theoretical and experimental study has been carried out to understand the effect of defect size and radial load on the vibration amplitude at defect frequency. Miscellaneous frequency peaks can be observed in experimental vibration spectra of Figs. 9–13. These peaks are identified as 2  fs = 49.80 Hz, 3  fs = 74.70 Hz, 2  BPFI  fs = 220.44 Hz and 2  BPFI  2  fs = 195.44 Hz. These additional sideband peaks are representation of defect position variation due to its rotation at shaft speed [1]. Apart from these frequencies, other frequency peaks due to unwanted sources of vibration (noise) are also visible in experimental vibration spectra. These noise sources have not been considered in present dynamic model. 5.2.1.1. Effect of radial load on vibration amplitude. The vibration amplitudes at defect frequency by varying radial load and defect size have been presented in Figs. 14–16. The variation in vibration

amplitude between experimental and theoretical results may occur due to the dynamic unbalance and misalignment [10,11]. It is worth to mention here, that the effect of unbalance, misalignment, centrifugal force and gyroscopic couple have not been considered in present model, while during experimentation these parameters cannot be ignored. Moreover, from Figs. 14–16 authors of this paper could not identify direct relationship between radial load and vibration amplitudes for inner race defect. It may take place because of variation of contact forces at moving defect. While, in case of lubricated bearing lubricant film thickness decreases with increase of radial load to support more amount of load which results in enhancement of film stiffness and damping capability. 5.2.1.2. Effects of inner race defect on vibration amplitude. The amplitude of impact force generated due to interaction of defect and rolling element increases with defect size. Due to this increased disturbing force, the vibration amplitudes are expected to increase with defect size. Figs. 11–13 shows the change in vibration amplitudes at defect frequency due to change in the inner race defect size under same radial load. From Figs. 17–20, it has been noticed

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

545

Fig. 11. Vibration velocity spectra for shaft rotational speed 1500 rpm, Inner race defect size 0.5 mm, 12 kg radial load (a) Theoretical result, dry contact (b) Experimental result, dry contact (c) Theoretical result, lubricated contact (d) Experimental result, lubricated contact.

Fig. 12. Vibration velocity spectra for shaft rotational speed 1500 rpm, Inner race defect size 1 mm, 12 kg radial load (a) Theoretical result, dry contact (b) Experimental result, dry contact (c) Theoretical result, lubricated contact (d) Experimental result, lubricated contact.

that the vibration amplitude increases with increase of defect size for both dry and lubricated contact. The radial load has not affected the trend. Moreover, the vibration amplitudes for lubricated contact are found less as compared to dry contact. This decrease of amplitude is due to increase of stiffness and damping at contact. 5.2.2. Defect on outer raceway Static or dynamic external load acted on bearing is normally taken care by rolling elements present in the load zone of bearing. Hence, the defect on non-moving outer race is normally expected

in load zone of bearing. Even the variation in angular defect position in load zone also affects the vibration amplitude. The significant vibration amplitude produced when defect is exactly below the top centre of test bearing housing if radial load applied at the bottom of housing. An impulse train of constant amplitude is expected due to interaction of rolling elements and defect. As the defect move away from load zone the amplitude decreases. Fig. 21(a–d) shows simulated (theoretical) and experimental vibration response of dry and lubricated contact deep groove ball bearing having 0.3 mm defect width on its outer race at shaft

546

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

Fig. 13. Vibration velocity spectra for shaft rotational speed 1500 rpm, Inner race defect size 1.5 mm, 12 kg radial load (a) Theoretical result, dry contact (b) Experimental result, dry contact (c) Theoretical result, lubricated contact (d) Experimental result, lubricated contact.

Fig. 14. Vibration amplitude comparison for shaft rotational speed 1500 rpm, Inner race defect size 0.5 mm with varying radial load (a) Dry contact (b) Lubricated contact.

Fig. 15. Vibration amplitude comparison for shaft rotational speed 1500 rpm, Inner race defect size 1 mm with varying radial load (a) Dry contact (b) Lubricated contact.

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

547

Fig. 16. Vibration amplitude comparison for shaft rotational speed 1500 rpm, Inner race defect size 1.5 mm with varying radial load (a) Dry contact (b) Lubricated contact.

Fig. 17. Vibration amplitude comparison for shaft rotational speed 1500 rpm, radial load 0 kg with varying inner race defect size (a) Dry contact (b) Lubricated contact.

Fig. 18. Vibration amplitude comparison for shaft rotational speed 1500 rpm, radial load 12 kg with varying inner race defect size (a) Dry contact (b) Lubricated contact.

rotation speed of 1500 RPM and 22 kg radial load. It is necessary to mention here, that computed value of Ball Pass Outer Race Defect Frequency obtained by (BPFO = (Nb/2)  (Ns/60)  (1  (d/D))) is 77.28 Hz for shaft rotational speed of 25 Hz. In experimental results vibration peak at defect frequency BPFO is observed at 78.89 Hz which is close to theoretical calculated value (77.28 Hz). Moreover, harmonics of characteristic defect frequency at 2  BPFO, 3  BPFO are also visible clearly in Fig. 21(a–d). The similar methodology has been adopted during simulation and experimentation, for outer race defect size of 0.5 mm, 1 mm and 1.5 mm under 22 kg radial load for dry and lubricated contact.

The results have been presented in Figs. 22–24 to authenticate the present dynamic model. Moreover, same kinds of simulation and experimentations have been also carried out for 0.3 mm, 0.5 mm, 1 mm and 1.5 mm outer race defect width under 0 kg and 32 kg radial load. 5.2.2.1. Effects of radial load on vibration amplitude. The vibration amplitudes at defect frequency under different radial loads and defect size have been presented in Figs. 25–28. The vibration amplitude comparison for both theoretical and experimental results (Refer Figs. 25–28) indicates that vibration amplitude at

548

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

Fig. 19. Vibration amplitude comparison for shaft rotational speed 1500 rpm, radial load 22 kg with varying inner race defect size (a) Dry contact (b) Lubricated contact.

Fig. 20. Vibration amplitude comparison for shaft rotational speed 1500 rpm, radial load 32 kg with varying inner race defect size (a) Dry contact (b) Lubricated contact.

Fig. 21. Vibration velocity spectra for shaft rotational speed 1500 rpm, Outer race defect size 0.3 mm, 22 kg radial load (a) Theoretical result, dry contact (b) Experimental result, dry contact (c) Theoretical result, lubricated contact (d) Experimental result, lubricated contact.

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

549

Fig. 22. Vibration velocity spectra for shaft rotational speed 1500 rpm, Outer race defect size 0.5 mm, 22 kg radial load (a) Theoretical result, dry contact (b) Experimental result, dry contact (c) Theoretical result, lubricated contact (d) Experimental result, lubricated contact.

Fig. 23. Vibration velocity spectra for shaft rotational speed 1500 rpm, Outer race defect size 1 mm, 22 kg radial load (a) Theoretical result, dry contact (b) Experimental result, dry contact (c) Theoretical result, lubricated contact (d) Experimental result, lubricated contact.

defect frequency has decreased as radial load on test bearing increased. But at the same time, it has been observed that vibration amplitude at shaft frequency increases in both dry and lubricated bearing contact. Similar trends in both theoretical and experimental results have proved the reliability of the present model. Reduction in vibration velocity amplitude at defect frequency (BPFO) have been noticed with increase of applied radial load. This might

have occurred due to reduction of lubricant film thickness to support more load which enhanced lubricant film stiffness and damping along with non-linear contact stiffness. The authors of present paper believe that internal radial clearance between ball-raceway contacts decreases with increase of applied radial load. At the same moment the fluctuation of ball speed decreased and reduced the vibration at outer race defect frequency.

550

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

Fig. 24. Vibration velocity spectra for shaft rotational speed 1500 rpm, Outer race defect size 1.5 mm, 22 kg radial load (a) Theoretical result, dry contact (b) Experimental result, dry contact (c) Theoretical result, lubricated contact (d) Experimental result, lubricated contact.

Fig. 25. Vibration amplitude comparison for shaft rotational speed 1500 rpm, Outer race defect size 0.3 mm with varying radial load (a) Dry contact (b) Lubricated contact.

5.2.2.2. Effects of outer race defect sizes on vibration amplitude. The outer race defect is kept in bearing load zone during experimentation for varying radial load and local defect size. The vibration spectra have been plotted for different defect size in Figs. 20–23. The defect frequency, BPFO (77.28 Hz) and their harmonics are clearly visible in both simulated (theoretical) and experimental results. The experimental and simulated vibration amplitude has increased as defect size on outer race increased gradually at a varying radial load for both dry and lubricated bearing. This increase of vibration amplitude occurred because of additional deflection produced due to interaction between increased size of defect and balls. From Figs. 20–23, it is clearly visible that the vibration amplitude for lubricated bearing has decreased as compared to dry contact bearings. The Figs. 29–31 provides comparison of theoretical and experimental velocity amplitude for different outer race defect size under different radial loads. These results show that theoretical and experimental vibration amplitude at defect frequencies increased with increase of defect size. Also, the velocity amplitudes at defect frequencies in dry contacts are more as compared to

lubricated contact in both simulated and experimental vibration response. The higher vibration amplitudes have been noticed for simulated result as compared to experimental result for defect size 1 mm and 1.5 mm [Refer Figs. 29–31]. This dissimilarity of amplitude has also been noticed by Babu et al. [32] for simulated and experimental results. The probable reasons are as below.
Vibration transmission ratio for outer race defect is adversely affected by defect size in case of rotor bearing system [18].
The angular difference of defect position from load zone during experimentations also affect the captured vibration amplitude.
Vibration signals picking location, which is at the bearing housing far away from the bearing centre [32]. However, the overall trend of increase in vibration amplitude with increase of defect size has been observed for both simulated and experimental vibration study. Authors of the present paper have also noticed that the overall trend of vibration amplitude under different radial load and defect

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

551

Fig. 26. Vibration amplitude comparison for shaft rotational speed 1500 rpm, Outer race defect size 0.5 mm with varying radial load (a) Dry contact (b) Lubricated contact.

Fig. 27. Vibration amplitude comparison for shaft rotational speed 1500 rpm, Outer race defect size 1 mm with varying radial load (a) Dry contact (b) Lubricated contact.

Fig. 28. Vibration amplitude comparison for shaft rotational speed 1500 rpm, Outer race defect size 1.5 mm with varying radial load (a) Dry contact (b) Lubricated contact.

size is in line with the theoretical and experimental results available in literatures [13,40,41]. The percentage difference of defect frequencies for different defect size, radial loads, defect location (inner race or outer race) in presence and absence of lubricant has been presented in Table A.1. It is necessary to mention here that the percentage

difference has been computed with equation, |Simulated defect frequency (Hz)  Experimental defect frequency (Hz)|  100/ Experimental defect frequency (Hz). From Table A.1 it can be observed that the maximum frequency difference between experimental and simulated results in case of dry contact is 5.24%, while it is 3.51% in case of lubricated contact.

552

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555

Fig. 29. Vibration amplitude comparison for shaft rotational speed 1500 rpm, radial load 0 kg with varying outer race defect size (a) Dry contact (b) Lubricated contact.

Fig. 30. Vibration amplitude comparison for shaft rotational speed 1500 rpm, radial load 22 kg with varying outer race defect size (a) Dry contact (b) Lubricated contact.

Fig. 31. Vibration amplitude comparison for shaft rotational speed 1500 rpm, radial load 32 kg with varying outer race defect size (a) Dry contact (b) Lubricated contact.

Which can give validation to present model. Thus, authors of present papers claim that this dynamic model can be used to predict defect frequency with minimum error. However, remarkable difference in experimental and theoretical vibration velocity amplitude (mm/s) can be noticed from vibration amplitude comparison presented for all discussed cases. Possible reasons for variation in experimental

and theoretical vibration velocity amplitude (mm/s) are as follows:
The simulated results obtained with considering the mass of test bearing housing is concentrated in bottom of housing end but actually bearing vibration measured during experiment at the top of housing [10].

553

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555


Reduction in vibration energy during transfer of vibration signal from bearing ball-defect contact to housing and finally to accelerometer.
The effect of damping due to frictional forces at ball-race contact in presented simulated results has been ignored.


In case of lubricated defective test bearing the simulation has been performed with certain assumed surface roughness value whereas in experiment the bearing was lubricated once during each operation in experiment work [31].

Table A.1 Percentage difference between experimental and simulated defect frequency (%). Figure No.

Fig. 9 Fig. 10

Fig. Fig. Fig. Fig.

11 12 13 14

Fig. 15

Fig. 16

Fig. 17

Fig. 18

Fig. 19

Fig. 20

Fig. Fig. Fig. Fig. Fig.

21 22 23 24 25

Fig. 26

Fig. 27

Fig. 28

Fig. 29

Fig. 30

Fig. 31

Description

Inner race defect size 0.3 mm, 12 kg radial load Inner race defect size 0.3 mm for 0 kg radial load Inner race defect size 0.3 mm for 12 kg radial load Inner race defect size 0.3 mm for 22 kg radial load Inner race defect size 0.3 mm for 32 kg radial load Inner race defect size 0.5 mm, 12 kg radial load Inner race defect size 1 mm, 12 kg radial load. Inner race defect size 1.5 mm, 12 kg radial load Inner race defect size 0.5 mm for 0 kg radial load Inner race defect size 0.5 mm for 12 kg radial load Inner race defect size 0.5 mm for 22 kg radial load. Inner race defect size 0.5 mm for 32 kg radial load Inner race defect size 1 mm for 0 kg radial load Inner race defect size 1 mm for 12 kg radial load Inner race defect size 1 mm for 22 kg radial load Inner race defect size 1 mm for 32 kg radial load Inner race defect size 1.5 mm for 0 kg radial load Inner race defect size 1.5 mm for 12 kg radial load Inner race defect size 1.5 mm for 22 kg radial load Inner race defect size 1.5 mm for 32 kg radial load Radial load 0 kg for 0.3 mm inner race defect size Radial load 0 kg for 0.5 mm inner race defect size Radial load 0 kg for 1 mm inner race defect size Radial load 0 kg for 1.5 mm inner race defect size Radial load 12 kg for 0.3 mm inner race defect size Radial load 12 kg for 0.5 mm inner race defect size Radial load 12 kg for 1 mm inner race defect size Radial load 12 kg for 1.5 mm inner race defect size Radial load 22 kg for 0.3 mm inner race defect size Radial load 22 kg for 0.5 mm inner race defect size Radial load 22 kg for 1 mm inner race defect size Radial load 22 kg for 1.5 mm inner race defect size Radial load 32 kg for 0.3 mm inner race defect size Radial load 32 kg for 0.5 mm inner race defect size Radial load 32 kg for 1 mm inner race defect size Radial load 32 kg for 1.5 mm inner race defect size Outer race defect size 0.3 mm, 22 kg radial load Outer race defect size 0.5 mm, 22 kg radial load Outer race defect size 1 mm, 22 kg radial load Outer race defect size 1.5 mm, 22 kg radial load Outer race defect size 0.3 mm for 0 kg radial load Outer race defect size 0.3 mm for 22 kg radial load Outer race defect size 0.3 mm for 32 kg radial load Outer race defect size 0.5 mm for 0 kg radial load Outer race defect size 0.5 mm for 22 kg radial load Outer race defect size 0.5 mm for 32 kg radial load Outer race defect size 1 mm for 0 kg radial load Outer race defect size 1 mm for 22 kg radial load Outer race defect size 1 mm for 32 kg radial load Outer race defect size 1.5 mm for 0 kg radial load Outer race defect size 1.5 mm for 22 kg radial load Outer race defect size 1.5 mm for 32 kg radial load Radial load 0 kg for 0.3 mm outer race defect size Radial load 0 kg for 0.5 mm outer race defect size Radial load 0 kg for 1 mm outer race defect size Radial load 0 kg for 1.5 mm outer race defect size Radial load 22 kg for 0.3 mm outer race defect size Radial load 22 kg for 0.5 mm outer race defect size Radial load 22 kg for 1 mm outer race defect size Radial load 22 kg for 1.5 mm outer race defect size Radial load 32 kg for 0.3 mm outer race defect size Radial load 32 kg for 0.5 mm outer race defect size Radial load 32 kg for 1 mm outer race defect size Radial load 32 kg for 1.5 mm outer race defect size

Percentage difference between experimental and simulated defect frequency (%) Dry contact

Lubricated contact

2.16 2.16 2.16 0.64 3.38 2.16 1.36 0.16 4.08 2.16 1.36 1.44 0.16 4.01 1.44 0.72 0.16 2.16 0.08 0.16 1.99 4.08 0.16 0.08 3.32 2.16 3.22 0.16 0.64 1.36 1.46 0.08 1.44 1.44 0.72 0.16 2.42 2.43 0.01 1.88 2.53 2.43 4.24 0.01 2.79 2.42 2.49 5.24 2.42 2.49 1.88 0.73 0.23 1.44 0.04 0.56 0.37 2.42 0.69 0.85 2.25 1.88 2.43 0.98

2.24 2.88 2.24 1.52 2.08 1.99 0 0 0.72 1.99 1.52 3.45 0.00 2.96 0.00 1.99 0.72 2.08 1.52 0.16 1.43 0.72 1.52 1.76 1.76 1.99 0.72 0.56 1.20 0.72 1.52 0.00 2.55 3.45 1.99 0.16 2.25 3.51 0.23 2.25 2.48 2.25 2.48 1.47 3.51 2.25 2.25 0.23 0.23 2.45 2.25 0.98 0.23 1.47 0.56 1.47 0.99 3.51 0.23 2.25 2.25 2.25 0.23 0.98

554

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555


It may be due to difference in predicted value and actual value of lubricant film thickness, film stiffness and damping coefficient.
Slip of balls during rotation has been neglected during simulation.
Variation in value of bearing radial clearances not considered during simulation which occurs during actual bearings operations.
Rotor compliances play significant role in rotor bearing housing system [18], but rotor compliances have been included in present study. 6. Conclusions A dynamic model to study vibration generated by healthy and defective deep groove ball bearings in presence and absence of EHD lubrication has been presented here. Vibration amplitude of healthy and defective bearings have been analysed theoretically and experimentally. The vibration peak only at shaft rotational frequency (fs) along with its harmonics were visible for defect free bearing. The vibration peaks at characteristics defect frequencies were recorded for both dry and lubricated defective races. However, side bands at shaft rotational frequency were also visible in case of inner race defect. No specific relationship between vibration amplitude and radial load could be established for inner race defect. However, it has been noticed that BPFO amplitude decreases with increase of radial load for both dry contact and lubricated contact. Moreover, the amplitude at shaft rotation frequency increased with increase of radial load. Also, the vibration amplitude enhances in presence of outer race local defect as compared to inner race defect for same defect size, shaft speed and radial load. The vibration amplitude at bearing defect frequency for both dry and lubricated contact increases with increase of defect size on either of races for same radial load or rotational speed. The presence of lubricant film stiffness and damping has reduced the vibration amplitudes (velocities) at defect frequency as compared to dry contact. The nature of vibrations generated through present dynamic model is in line with the theoretical and experimental results available in literature. The maximum frequency difference between experimental results and theoretical results, obtained through derived dynamic model is 5.24% and 3.51% for dry contact and lubricated contact, respectively. Which can give validation to present model. Therefore, authors believe with confidence that the present dynamic model can be used to predict defect frequency and to analyse vibration amplitude of shaft-bearing system in presence and absence of lubricant. Acknowledgements The corresponding author acknowledges the sincere thanks to my research supervisor Dr. V. N. Patel and Co-author of this manuscript for their valuable support and guidance. Moreover, D. S. Shah acknowledges and thanks the management of his parental institute Sardar Vallabhbhai Patel Institute of Technology, Vasad, Gujarat, India for granting the needful permission to pursue his Ph.D. program. Appendix Equivalent

radius

of

inner

race

in X plane

Rxin ¼ dðDdÞ ð2DÞ

T. A. Harris [36] Equivalent radius of outer race in X plane Rxout ¼ dðDþdÞ ð2DÞ T. A. Harris [36]

Equivalent radius of inner race/outer race in y plane d

f

RY in=out ¼ 2fin=out T. A. Harris [36] in=out 1 Ellipticity ekin=out

parameter for inner race/outer  0:636 Rxin=out ¼ 1:0339  RY Hamrock and Dowson [27]

race

in=out

Equivalent elasticity modulus Ed ¼
ð1t2 Þ 2 ð1t2 Þ T. A. Harris [36] E1

1

þ

E2

2

Dimension less material parameter G ¼ ap  Ed Hamrock and Dowson [27]
2 2 d  xs Average surface velocity of inner race and cage u ¼ D4D T. A. Harris [36] Dimensionless speed parameter for
 g0 u U in=out ¼ E Rx Hamrock and Dowson [27] in=out

inner/outer

race

inner/outer

race

d

Dimension less load parameter for W in=out ¼ Ed  QRx Hamrock and Dowson [27] in=out

Hydrodynamic roughness parameter kin=out ¼ et al. [31] Geometrical Parameter r

in=out

hin

= out 0:5

ðf 2r þf 2b Þ

Sarangi

¼ Rx0:5d Sarangi et al. [31] in=out

References [1] N. Tandon, A. Choudhury, A review of vibration and acoustic measurement methods for the detection of defects in rolling element bearings, Tribol. Int. 32 (8) (1999) 469–480. [2] M.S. Patil, J. Mathew, P.K. RajendraKumar, Bearing signature analysis as a medium for fault detection: a review, J. Tribol. 130 (1) (2007), 014001– 014001-7. [3] D.S. Shah, V.N. Patel, A review of dynamic modeling and fault identifications methods for rolling element bearing, Procedia Technol. 14 (2014) 447–456. [4] P.D. McFadden, J.D. Smith, Model for the vibration produced by a single point defect in a rolling element bearing, J. Sound Vib. 96 (1) (1984) 69–82. [5] P.D. McFadden, J.D. Smith, The vibration produced by multiple point defects in a rolling element bearing, J. Sound Vib. 98 (2) (1985) 263–273. [6] N. Tandon, A. Choudhury, An analytical model for the prediction of the vibration response of rolling element bearings due to a localized defect, J. Sound Vib. 205 (3) (1997) 275–292. [7] A. Choudhury, N. Tandon, Vibration response of rolling element bearings in a rotor bearing system to a local defect under radial load, J. Tribol. 128 (2) (2005) 252–261. [8] H. Arslan, N. Aktürk, An investigation of rolling element vibrations caused by local defects, J. Tribol. 130 (4) (2008), 041101–041101-12. [9] M.S. Patil, J. Mathew, P.K. Rajendrakumar, S. Desai, A theoretical model to predict the effect of localized defect on vibrations associated with ball bearing, Int. J. Mech. Sci. 52 (9) (2010) 1193–1201. [10] 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, J. Tribol. 132 (4) (2010) 041101. [11] V.N. Patel, N. Tandon, R.K. Pandey, Vibration studies of dynamically loaded deep groove ball bearings in presence of local defects on races, Procedia Eng. 64 (2013) 1582–1591. [12] S. Khanam, N. Tandon, J.K. Dutt, Multi-event excitation force model for inner race defect in a rolling element bearing, J. Tribol. 138 (1) (2015), 011106– 011106-15. [13] S. Khanam, J. Dutt, N. Tandon, Impact force based model for bearing local fault identification, J. Vib. Acoust. 137 (5) (2015) 051002. [14] A. Moazen-ahmadi, C.Q. Howard, A defect size estimation method based on operational speed and path of rolling elements in defective bearings, J. Sound Vib. 385 (2016) 138–148. [15] A. Moazen-ahmadi, D. Petersen, C. Howard, A non-linear dynamic vibration model of defective bearings – The importance of modelling the finite size of rolling elements, 2014. [16] G. Kogan, J. Bortman, R. Klein, A new model for spall-rolling-element interaction, Nonlinear Dyn. 87 (1) (2017) 219–236. [17] U.A. Patel, S.H. Upadhyay, Non-linear dynamic response of cylindrical roller bearing–rotor system with 9 degree of freedom model having a combined localized defect at inner–outer races of bearing, Tribol. Trans. 60 (2) (2017) 284–299. [18] J. Liu, Y. Xu, Y. Shao, Dynamic modelling of a rotor-bearing-housing system including a localized fault, Proc. Inst. Mech. Eng., Part K: J. f Multi-body Dyn. 232 (3) (2018) 385–397. [19] C. Mishra, A.K. Samantaray, G. Chakraborty, Ball bearing defect models: a study of simulated and experimental fault signatures, J. Sound Vib. 400 (2017) 86– 112.

D.S. Shah, V.N. Patel / Measurement 137 (2019) 535–555 [20] J. Liu, Y. Shao, M.J. Zuo, The effects of the shape of localized defect in ball bearings on the vibration waveform, Proc. Inst. Mech. Eng., Part K: J. f Multibody Dyn. 227 (3) (2013) 261–274. [21] S. Singh, Uwe G. Kopke, Carl Q. Howard, Dick Petersen, Analyses of contact forces and vibration response for a defective rolling element bearing using an explicit dynamics finite element model, J. Sound Vib. 333 (21) (2014) 5356– 5377. [22] Z. Kıral, H. Karagülle, Vibration analysis of rolling element bearings with various defects under the action of an unbalanced force, Mech. Syst. Sig. Process. 20 (8) (2006) 1967–1991. [23] Z. Kiral, H. Karagülle, Simulation and analysis of vibration signals generated by rolling element bearing with defects, Tribol. Int. 36 (9) (2003) 667–678. [24] J. Liu, Y. Shao, T.C. Lim, Impulse vibration transmissibility characteristics in the presence of localized surface defects in deep groove ball bearing systems, Proc. Inst. Mech. Eng., Part K: J. f Multi-body Dyn. 228 (1) (2014) 62–81. [25] A. Utpat, Vibration Signature analysis of defective deep groove ball bearings by Numerical and Experimental approach, Int. J. Sci. Eng. Res. 4 (6) (2013) 592– 598. [26] M. Chouksey, J.K. Dutt, S.V. Modak, Model updating of rotors supported on ball bearings and its application in response prediction and balancing, Measurement 46 (10) (2013) 4261–4273. [27] B.J. Hamrock, D. Dowson, Ball bearing Lubrication, John Wiley and Sons, New York, 1981. [28] P. Dietl, J. Wensing, G.C.V. Nijen, Rolling bearing damping for dynamic analysis of multi-body systems -experimental and theoretical results, Proc. Inst. Mech. Eng., Part K: J. f Multi-body Dyn. 214 (1) (2000) 33–43. [29] J. Kraus, J.J. Blech, S.G. Braun, In situ determination of rolling bearing stiffness and damping by modal analysis, J. Vibr., Acoust., Stress, Reliab. Des. 109 (3) (1987) 235–240. [30] J. Sopanen, A. Mikkola, Dynamic model of a deep-groove ball bearing including localized and distributed defects. Part 1: theory, Proc. Inst. Mech. Eng., Part K: J. Multi-body Dyn. 217 (3) (2003) 201–211.

555

[31] M. Sarangi, B.C. Majumdar, A.S. Sekhar, On the dynamics of elastohydrodynamic mixed lubricated ball bearings. Part I: formulation of stiffness and damping coefficients, Proc. IMechE Part J: J. Eng. Tribol. 219 (2005) 411–421. [32] C.K. Babu, N. Tandon, R.K. Pandey, Vibration modeling of a rigid rotor supported on the lubricated angular contact ball bearings considering six degrees of freedom and waviness on balls and races, J. Vib. Acoust. 134 (1) (2011), 011006–011006-12. [33] J. Liu, Y. Shao, Dynamic modeling for rigid rotor bearing systems with a localized defect considering additional deformations at the sharp edges, J. Sound Vib. 398 (2017) 84–102. [34] D.S. Shah, V.N. Patel, Theoretical and experimental vibration studies of lubricated deep groove ball bearings having surface waviness on its races, Measurement 129 (2018) 405–423. [35] D.S. Shah, V.N. Patel, Study on excitation forces generated by defective races of rolling bearing, Procedia Technol. 23 (2016) 209–216. [36] T.A. Harris, Rolling Bearing Analysis, John Wiley, New York, 2001. [37] G. Genta, On a persistent misunderstanding of the role of hysteretic damping in rotor dynamics, J. Vib. Acoust. 126 (3) (2004) 459–461. [38] J. Liu, Y. Shao, T.C. Lim, Vibration analysis of ball bearings with a localized defect applying piecewise response function, Mech. Mach. Theory 56 (2012) 156–169. [39] A. Ashtekar, F. Sadeghi, L.E. Stacke, A new approach to modeling surface defects in bearing dynamics simulations, J. Tribol. 130 (4) (2008), 041103– 041103-8. [40] H. Hong, M. Liang, Fault severity assessment for rolling element bearings using the Lempel-Ziv complexity and continuous wavelet transform, J. Sound Vib. 320 (1) (2009) 452–468. [41] V.N. Patel, D. Prajapati, Experimental study of vibrations generated by defective deep groove ball bearing, Int. J. Des. Eng. 6 (4) (2016) 309–325.