The theoretical modeling and experimental validation for distributed defect on inner race of ball bearing under radial load

Abstract

This article presents the theoretical model to study the vibration characteristics of ball bearing. The vibration analysis is accounted for peripheral motions of rolling elements as well as inner and outer races using the Lagrangian approach. In this mathematical model, the contact between the balls and the bearing races is considered as nonlinear springs, whose stiffness is obtained using the Hertzian elastic contact deformation theory. The nonlinear equations of motions are solved by the Runge–Kutta method iteratively. The effects of extended defect on the inner race of the bearing at different speeds and defect sizes have been studied for predicting the vibration response of the bearing. The fast Fourier transformation shows that the vibration characteristic of the ball bearing changes when the ball interacts with the defect as a result of nonlinear load–deflection relation. The analysis implied that defect size and speed of ball bearing are the influencing parameters affecting the dynamic behavior of the ball bearing.

Keywords

Hertzian contact ball bearing theoretical model distributed defect fast Fourier transform vibration

Introduction

Rolling element bearings are one of the most extensively used components in the mechanical rotating machinery and a variety of industries such as sugar industry, construction, mining, paper mills, railway, and renewable energy. These are specifically used to allow rotary motion and support a significant load.¹ The low-speed, high-load rotor bearing system often shows the unpredictable dynamic response during running condition. In sugar industries, the deep-groove ball bearings used in crane applications for carrying sugar bags and canes are subjected to fatigue spalling. In such type of bearing, the inception of micro-scale subsurface fatigue cracks commences below highly stressed rolling region. These micro-structural discontinuities generate micro-plastic deformation in the region of maximum stresses.² Due to the continuous and cyclic loading during bearing operation, such type of fatigue crack continues to spread along the periphery of the rolling surface. The damage and failure of such type of bearings lead to increase in the machinery breakdown, consequently costing significant economical losses.

Bearing defects may be classified as local and distributed faults. The local faults include pits, spalls, and cracks on rolling elements caused by fatigue on the rolling surface. The distributed faults encompass surface roughness, waviness, and misaligned races and off-size balls. These defects may be due to manufacturing errors and operating conditions.³ Various techniques have been adopted,⁴ such as high-frequency resonance technique, localized current method, sound measurement technique, and acoustic emission technique for bearing fault detection. Vibration-based detection techniques for monitoring the health of bearings have been widely used in both time and frequency domains.^5–7

Early studies on the development of mathematical models of bearing with the local defect have been proposed by McFadden and Smith⁸ and Choudhury and Tondon.⁹ They have predicted the vibration frequencies and the amplitude of significant frequencies of rolling element bearing due to local defects under radial and axial loads. It has been concluded that the position-dependent characteristic defect frequencies are useful to detect the existence of a defect and also diagnose its location. Many theoretical formulations have also been developed based on Newton’s equations.^10,11 In addition to this, Sopanan and Mikola¹² have presented a dynamic model for deep-groove ball bearing with local and distributed defects to analyze the effect of internal radial clearance and unbalance excitation of the system. Some of the researchers^8–13 have shown that the detection of outer race defect is much easier than inner race defect, whereas vibration response shows the confusing spectrum with many additional spectral components for the bearings with inner race defect. Hence, in line with conclusions of early studies, this study focuses on theoretical and experimental investigation of defect on inner race of the ball bearing.

In case of distributed defect analysis, Meyer et al.¹⁴ have proposed an analytical model for ball bearing vibrations. The distributed faults are simulated, and the spectral components resulting from these defects have been predicted. Tondon and Choudhury¹⁵ have also prepared the theoretical model to predict the vibration response of bearing due to distributed defects and showed that discrete spectrum was observed with a specific frequency component for each order of waviness. Recently, Tiwari et al.¹⁶ and Harsha and Kankar¹⁷ investigated the stability of a rigid rotor supported by deep-groove ball bearings, which has been noted that the increase in the number of balls will reduce the effect of the ball pass frequency and the amplitude of the vibration increased due to waviness on the inner race of the bearing. But in contrast to the model development of local and distributed faults, extended distributed defects have received very less attention.¹⁸ Only Sawalhi and Randall¹⁹ and Petersen et al.²⁰ have contributed their works for the development of multi-body dynamic model to predict the vibration response of bearing with extended raceway defects. A new stochastic model was also introduced by Behzad et al.²¹ for predicting the vibration response generated due to the defective ball bearing and concluded that when a defect grows in the bearing, the roughness of the contacting surface grows locally and stochastic excitation becomes stronger in defective areas.

This work focuses on the study of the effect of extended defects on the inner race of the bearing at different speeds and defect sizes. The efforts have been made to develop a theoretical model based on the Lagrangian approach for analyzing the vibration response. Furthermore, the results are compared. In this article, next section discusses the characterization of fault. Section “Model analysis” explains the theoretical model with 2 degree-of-freedom system, whose nonlinear equations are solved by the Runge–Kutta method iteratively. The results of theoretical analysis are presented in the form of time and frequency spectra in section “Numerical results”. Section “Experimental analysis” describes the experimental setup, instrumentation, and results. The article ends with discussion and conclusion.

Characterization of fault

Earlier studies have been carried out in the development of model of ball bearing vibration generated due to local defect and waviness. Fatigue is the major mode of failure in rolling element bearing.¹ Singh et al.¹⁸ have also noticed that bearing frequently develops localized defects in the raceway and reported that during cyclic operation of rolling element, the leading and trailing edge of localized fault wears and causing it to spread the defect size. In addition to this, because of variable loading, such local defects grow along the periphery of raceway generally considered as an extended raceway defect, which is termed as a distributed type of defect.

An extended distributed defect on the raceway has been characterized as a defect size that is larger than a localized defect, whose length is less than spacing between two balls. It can be also categorized as defect size smaller than the waviness. Surface defect has been described by a sinusoidal function as shown in Figure 1, signified by amplitude R_a and wavelength “λ”.

Figure 1.

Sinusoidal wave of inner race defect.

Referring to Figure 1, the amplitude of surface defect at contact angle is given by

w_{a m p i} = R_{a} \times \sin (N_{w} {\times θ}_{i n})

(1)

where N_w is the number of waves on the surface defect of inner race and R_a is the amplitude of surface defect on inner race. Hence, the contact angle is

θ_{i n} = w_{i n} \times t + θ_{d} \times \frac{(i - 1)}{N_{b}}

(2)

where w_in is the angular velocity of inner race, $θ_{d} = 15 \times π / 180$ is the defect angle on the surface of inner race, and $i = 1, 2, \dots, N_{b}$ .

Model analysis

Earlier, researchers have developed mathematical models to identify the bearing vibration signature frequency. In line with the objective of the research work reported in this article, the model has been developed to analyze the structure vibration response of the ball bearing in which the outer race of the bearing is fixed and inner race is to be rigidly fixed to the shaft. The geometrical and physical properties used for model analysis of bearing as indicated in Table 1. The governing differential equations are obtained by taking stiffness variation into account. During theoretical analysis, balls were considered as a mass spring system whose contact act as nonlinear contact spring. The Hertzian forces arise only during contact deformation when the springs were required to act under only in compression. A schematic diagram of ball bearing is shown in Figure 2.

Table 1.

Ball bearing properties for model.

Property	Value
Mass of rolling elements (m_b)	0.0293 kg
Mass of inner race (m_in)	0.18 kg
Mass of outer race (m_out)	0.215 kg
Mass of rotor (M)	3.0 kg
Bearing outside diameter (D_o)	85 mm
Bearing bore diameter (d_i)	30 mm
Ball diameter (d)	17.463 mm
Number of balls (N_b)	7
Pitch diameter (D)	57.5 mm
Radial load (W)	50 N

Figure 2.

A schematic diagram of ball bearing.

The assumptions made in theoretical analysis are as follows:

Deformation occurs according to the Hertzian contact theory, whereas all the bearing elements and rotor are considered as rigid.

The rolling elements, inner race, outer race, and the rotor have motions in the plane of bearing only.

There is no slipping of the balls as they roll on the surface of races.

It is assumed that the balls have constant angular separation (β), hence there is no interaction between rolling elements.

Equivalent contact stiffness

According to Hertzian contact theory, the stress and deformation occur in the perfectly smooth, ellipsoidal, contacting elastic solid. The classical theory of elasticity is applicable for ball and roller element bearing. Therefore, point of contact between the raceway and ball develops into a contact area which has the shape of an ellipse. The curvature sum and curvature differences are needed in order to obtain the contact force of the ball. In Figure 3, the upper body is denoted by I and lower body is denoted by II. The principal planes are denoted by 1 and 2.

Figure 3.

Geometry of contact bodies.²²

The radius of curvature of body I in plane 2 is denoted by r_I2, where “r” denotes the radius of curvature. To describe the contact between mating surfaces of revolution, the following parameters are obtained from Harris.²²

If inner race is considered

Curvature sum

\in ρi = \frac{1}{db} (4 - 2 \times β + (\frac{2 \times γ}{1 - γ}))

(3)

Curvature difference

F (ρi) = \frac{2 \times β + (\frac{2 \times γ}{1 - γ})}{4 - 2 \times β + (\frac{2 \times γ}{1 - γ})}

(4)

If outer race is considered

Curvature sum

\in ρo = \frac{1}{db} (4 - 2 \times β - (\frac{2 \times γ}{1 + γ}))

(5)

Curvature difference

F (ρo) = \frac{2 \times β - (\frac{2 \times γ}{1 + γ})}{4 - 2 \times β - (\frac{2 \times γ}{1 + γ})}

(6)

The parameters are calculated from the table in Harris.²²

Load deformation constant of bearing

The load deformation constant “K_p” between ball and raceway depends on contact geometry.¹¹ The total deflection between two raceways is the sum of the approaches between rolling elements and each raceway. Using this we get

K_{p} = {[\frac{1}{{(\frac{1}{k_{inr}})}^{\frac{1}{1.5}} + {(\frac{1}{k_{outr}})}^{\frac{1}{1.5}}}]}^{1.5}

(7)

where k_inr and k_outr are the inner and outer raceways to ball contact stiffness, respectively, which can be obtained using

\begin{array}{l} k_{inr} = 3.12 \times 10^{7} \times {(\in ρi)}^{- 0.5} \times {({δi}^{*})}^{- 1.5} \\ k_{outr} = 3.12 \times 10^{7} \times {(\in ρo)}^{- 0.5} \times {({δo}^{*})}^{- 1.5} \end{array}

(8)

where ${δi}^{*} and {δ o}^{*}$ are the dimensionless contact deformations based on the curvature differences obtained from the table in Harris.²²

Equation of motion

The equation of motion describes that the dynamic behavior of the ball bearing as complete model can be derived using Lagrange’s equation for a set of independent generalized coordinates, as given by

\frac{d}{dt} (\frac{\partial T}{\partial \dot{x}}) - (\frac{\partial T}{\partial x}) + (\frac{\partial V}{\partial x}) = (f)

(9)

where “T” is the kinetic energy, “V” is the potential energy, “x” is a vector with generalized degree-of-freedom coordinate, and “f” is a vector of generalized force. The kinetic and potential energies can be derived separately for rolling elements, inner race, and outer race. Figure 4 shows the schematic view of a spring–mass system used to describe the kinetic energy of the bearing. The total kinetic energy (T) of the rotor bearing system is the sum of the kinetic energies of the rolling elements, inner race, and outer race. The subscripts i-race and o-race refer to “the inner race” and “the outer race”, respectively

T = (T_{b} + T_{ir} + T_{or})

(10)

Figure 4.

Spring–mass system of bearing.

The potential energy is provided by deformation of the balls with raceways, and deformation occurs according to Hertzian contact theory of elasticity. Potential energy formulation is performed taking datum as horizontal plane through global origin “O” as shown in Figure 4. Total potential energy (V) of the rotor bearing system is the sum of the balls, inner race, and outer race as

V = (V_{b} + V_{or} + V_{ir})

(11)

where V_b, V_or, and V_ir are potential energies due to elevation of balls, inner, and outer races, respectively.

Contribution of inner race

The inner race is considered as a rigid body except local deformation of elements in contact with inner race. The kinetic energy of the inner race about its center of mass has been evaluated in x and y outlines. The kinetic energy expression of the inner race is

T_{ir} = \frac{1}{2} m_{in} z_{in} dot \times z_{in} dot + \frac{1}{2} I_{in} \times ω_{in}^{2}

(12)

where I_in and ω_in are the moment of inertia and angular velocity of inner race, respectively. The displacement vector z_in showing the location of the inner race center with respect to the outer race center is given by

z_{in} = (x_{in} + x_{out}) i + (y_{in} + y_{out}) j

(13)

Since the position of the inner race is defined from outer race center, the potential energy for the inner race is

V_{ir} = \frac{1}{2} K_{inx} ({\overset{\cdot}{x}}_{in}^{2}) + \frac{1}{2} K_{iny} ({\dot{y}}_{in}^{2})

(14)

Contribution of outer race

The outer is considered as a rigid body and is assumed to be stationary. The outer race of the bearing has assumed to be fixed, whose angular and radial velocities are zero. Therefore, the kinetic energy of the bearing contributed by the outer race will be zero.

The kinetic energy of the outer race is

T = T_{or} = 0

(15)

The potential energy of the outer race is

V_{or} = m_{out} {\times g \times y}_{out}

(16)

Contribution of rolling element (ball)

The rolling elements are also being considered as rigid bodies. For describing their kinetic energy, the position of the each rolling element is described by three transitional degrees of freedom, ${\dot{z}}_{j}, v_{j}, and ω_{j}$ , which are the velocity of the ball along the contact, the velocity of the ball along the center of gravity, and the angular velocity of the ball along its axis, respectively. The total kinetic energy of the balls is obtained by the summation of each ball, which is given as

T_{b} = \sum_{j = 1}^{N_{b}} \frac{1}{2} m_{b} \times {\dot{z}}_{j}^{2} + \frac{1}{2} m_{b} \times v_{b}^{2} + \frac{1}{2} I_{b} \times ω_{j}^{2} + \frac{1}{2} m_{b} ({\dot{x}}_{in}^{2} + {\dot{y}}_{in}^{2})

(17)

The contacts between the balls and the race are treated as nonlinear springs, whose stiffnesses are derived by the Hertzian theory of elasticity. Potential energy stored in the balls is given as

V_{b} = \int_{0}^{δ} {k \times δ}^{3} dδ = \frac{{k \times δ}^{4}}{4}

(18)

where k and δ are the stiffnesses due to the Hertzian contact effects of the ball and spring deflection, respectively. For contacting bodies made of steel, the stiffness of the bearing between two contacting and deforming surfaces is given as

\begin{array}{l} k_{inr} = k_{inx} \times \sqrt{δ_{inr}} N / mm \\ k_{outr} = k_{outx} \times \sqrt{δ_{outr}} N / mm \end{array}

(19)

where δ_inr is the deformation at contact points between the rolling element and the inner race, and δ_outr is the deformation at contact points between the rolling element and the outer race.

The total potential energy of the balls considering defect contributed to the bearing system, which is given as

V_{b} = \frac{1}{4} k_{inr} {(z_{j} - w_{{amp}_{i}})}^{4} + \frac{1}{4} K_{outr} {(z_{j})}^{4}

(20)

where z_j is the deflection of the ball and w_ampi is the amplitude of defect.

The potential energy and kinetic energy contributed by the inner race, outer race, and balls from equations (13) and (14) can be differentiated with respect to generalized coordinates z_j, x_in, and y_in to obtain the equation of motion. For the generalized coordinates z_j, where j = 1, 2,…, N_β, the equations are

m_{b} \times {\dot{z}}_{j} + k_{inr} \times {(z_{j} - w_{ampi})}_{+}^{3} + k_{outr} {\times z}_{j}^{3} = 0

(21)

where j = 1, 2,…, N_b. Here, k_inr, k_outr, and w_ampi have to be taken to the right-hand side of the equation, which represent the forcing function of this equation.

Equation of motion for the generalized coordinate x_in

(m_{b} + m_{ir}) \times {\overset{\cdot\cdot}{x}}_{in} + k_{inx} \times x_{in} = {f}

(22)

Equation of motion for generalized coordinate y_in

(m_{b} + m_{ir}) \times {\overset{\cdot\cdot}{y}}_{in} + k_{iny} \times y_{in} = {f}

(23)

Here, the excitation force of the inner race equation along the y-direction is assumed to be the result of an imbalance force of the rotor, which is given by the expression

{f} = f_{u} \times \sin (ω_{in} \times t)

(24)

where f_u is the rotor imbalance force

f_{u} = W + m_{rotor} \times e \times ω_{ir}^{2}

(25)

where ω_in is the angular velocity of the inner race and W is the applied radial load.

This system is of second-order nonlinear differential equations. The constant radial load is applied in y-direction on the ball bearing. The “+” sign as subscripts in equation (21) signifies that if expression inside the bracket is greater than zero, then the balls are in the load zone giving rise to restoring force, and if inside the bracket is negative or zero, force is set to zero, then balls are not in the load zone.

Numerical results

Equations (21)–(23) were used to predict the motion of each ball and response of the inner ring in x- and y-direction. These equations are solved by the fourth-order Runge–Kutta method to obtain the vibration acceleration amplitude of balls and inner race. The theoretical and experimental results at different speeds and defect sizes have been plotted and comparison has been made. It has been observed that theoretical results obtained by the Lagrangian approach have shown close agreement with experimental results.

The time step for investigation is ΔT = 10⁻³ s. At t = 0, the shaft’s initial conditions and initial displacements are Z₁, Z₂, Z₃, Z₄, Z₅, Z₆, and Z₇ = 10⁻³ mm. The unbalance time history has been examined for periodic behavior. The shaft is held at the center of the bearing and all balls are assumed to have equal axial preload. The single-row, deep-groove ball bearing of DFM-85 with a single extended distributed defect on its outer race is considered in the analysis. At time t = 0, the defect is in contact with one of the rolling elements and lies at the center of the loaded region in the direction of the applied radial load. In order to obtain numerical results for acceleration amplitude, ball bearing has been considered with normal internal radial clearance. The program then computes the amplitudes of acceleration of the vibrations of the housing for significant frequency components with the help of the Lagrangian method. The results revealed variation in the amplitude of vibration with the variation in speed and the defect size.

Effect of speed

Figures 5 –8 show the theoretical frequency-domain plots of vibration due to inner race defect with Ra₂ amplitude at 300, 600, 900, and 1200 r/min speed, respectively. The peaks at ball pass frequency inner race (BPFI) and its harmonics are seen in the frequency spectrum. The impulses at different speeds are obtained at frequencies 23.62, 49.61, 61.42, and 92.13 Hz, respectively.

Figure 5.

Theoretical Frequency spectrum of inner race defective bearing at 300 r/min.

Figure 6.

Theoretical Frequency spectrum of inner race defective bearing at 600 r/min.

Figure 7.

Theoretical Frequency spectrum of inner race defective bearing at 900 r/min.

Figure 8.

Theoretical Frequency spectrum of inner race defective bearing at 1200 r/min.

It is observed that there is a significant increase in the amplitude of vibration response with an increase in the speed. The amplitude values of acceleration observed in the spectrum are indicated in Table 2. The inertia effects of the ball may explain the reduction in amplitude for speed of 1200 r/min.

Table 2.

Comparison of model and experimental acceleration response.

Speed	Calculated frequency (Hz)	Model response			Experimental response
		Frequency (Hz)	Amplitude (mm/s²)	Amplitude (m/s²)	Frequency (Hz)	Test 1	Test 2	Test 3
		Frequency (Hz)	Amplitude (mm/s²)	Amplitude (m/s²)	Frequency (Hz)	Amplitude (m/s²)	Amplitude (m/s²)	Amplitude (m/s²)
300	22.81	23.62	76.59	0.07659	21.97	0.24	0.347	0.415
600	45.63	49.61	250.5	0.2505	49.13	0.345	0.541	0.373
900	68.44	63.78	603.9	0.6039	66.83	0.578	0.639	0.402
1200	91.25	92.13	160.3	0.1603	89.11	0.151	0.282	0.289
1500	114.07	108.7	155	0.155	111.7	1.27	1.56	1.4

Effect of inner race defect size

Figures 9 –11 show the acceleration response of the first ball when it strikes with defect located on the inner race of bearing. All the observations are plotted at 900 r/min under 50 N radial load when the amplitude of the defect was increased from Ra₁ to Ra₃. Peaks at BPFI and its harmonics are seen in fast Fourier transform (FFT) plot. The dominant frequency is 70.87 Hz (BPFI). It is observed that the frequency obtained from the model is close to experimentally obtained BPFI (66.83 Hz) and theoretical BPFI (68.44 Hz). There is a significant increase in the amplitude of acceleration at this frequency with increase in defect size. These figures show that the radial vibration of the balls moving over the surface defect tends to increase as the defect amplitude of the bearing is increased from Ra₁ to Ra₃.

Figure 9.

Theoretical acceleration response of first ball at Ra₁ amplitude.

Figure 10.

Theoretical acceleration response of first ball at Ra₂ amplitude.

Figure 11.

Theoretical acceleration response of first ball at Ra₃ amplitude.

Experimental analysis

For radially loaded ball bearings, the frequency spectra for an inner race defect show many additional components other than spectral components at defect frequencies. Therefore, from the detection point of view, the defects on the inner race are more serious concern. Several investigations ^10
–13,23 have shown that the detection of an outer race defect is much easier in comparison with an inner race defect. Hence, in this study, experimental investigation is restricted to bearing with inner race defect.

Experimental setup and instrumentation

To validate the results from a mathematical model, experimental results were obtained from designed test rig to study bearing failure and vibration monitoring. To fully utilize the production capacity and to reduce the machinery downtime in the industry, it is necessary to predict the degradation of bearing working condition before it reaches to hazardous failure. In the present experimental investigation, the faults are introduced in the inner race of the ball bearing by artificially creating extended distributed fault. A sketch of the test rig on which the experiments have been conducted is shown in Figure 12.

Figure 12.

Experimental test rig.

The special-purpose bearings DFM-85²⁴ are used as test bearings, normally used in four-wheeler engines. It was mounted in bearing housing on the shaft and loaded by turn buckle in radial direction. The vibrations of bearing were recorded using piezoelectric accelerometer with the DEWOSOFT FFT analyzer. Vibration signals were acquired at different speeds up to 1200 r/min of the system for both healthy and defective bearings. The defect was created on the outer race of bearing by electric discharge machining.

The vibration signal was acquired from the analysis of four test bearings at different speeds. The signals are sampled at 5000 Hz with 2048 samples. The experiments have been carried out on separate test bearings having defect sizes Ra₁ = 4.1135 µm, Ra₂ = 4.6764 µm, and Ra₃ = 5.2853 µm artificially induced on the inner race of bearing separately. Electric discharge machining was used to create the defect on bearing race (Figure 13). The defect sizes are measured from DELUX bearings Pvt Ltd, Pune.

Figure 13.

Defect on bearing race.

Experimental results

Figures 14 –17 depict the results for the bearing with inner race defect at 300, 600, 900, and 1200 r/min speed, respectively, under constant radial load of 50 N. The peak amplitudes are observed at frequencies 21.97, 49.61, 61.42, and 89.11 Hz (BPFI) and its harmonics are seen in the frequency spectrum. The discussion in line with these results is made in the next section.

Figure 14.

Experimental Frequency spectrum of inner race defective bearing at 300 r/min.

Figure 15.

Experimental Frequency spectrum of inner race defective bearing at 600 r/min.

Figure 16.

Experimental Frequency spectrum of inner race defective bearing at 900 r/min.

Figure 17.

Experimental Frequency spectrum of inner race defective bearing at 1200 r/min.

Discussion

The analytical values of acceleration amplitudes along with their rotational frequencies have been normalized with respect to the experimental values of amplitude at different inner race defect frequencies as indicated in Table 2. Figure 18 shows that the amplitude variation of acceleration between the model responses has a fair agreement with experimental response. The amplitude of acceleration increases with increase in the speed of shaft. However, this comparison is restricted to the components at inner race defect frequencies and sidebands about components at multiples of shaft frequency. Other significant components of experimental analysis, especially the peaks at multiples of shaft frequency, cannot be predicted by this model.

Figure 18.

Comparison of analytical values of acceleration amplitude of inner race defect with experimental values.

However, the response of vibration amplitude obtained at 300 and 1500 r/min may be due to the resonance-generated instability in the ball bearing in the presence of defects. Since the defect present on inner race is movable, it produces strong vibrations when it is in the load zone. Moreover, the percentage variation between the theoretical and experimental defect frequencies is found to be 2%–7%.

Conclusion

The defect in the ball bearing is modeled as sinusoidal wave, which helps to simulate the effect of defect size and predicts the spectral components of vibration. The effect of speed and defect size variation in amplitude of vibration indicates better agreement between theoretical and experimental results. The accuracy of predictions using mathematical model in terms of the amplitude of vibration confirms with experimental results to higher end.

The model also predicts the peaks of frequency spectrum observed at characteristic defect frequencies, and the amplitude at these frequencies originated from the bearings. The frequency spectrum of bearing vibration due to defect comprises mainly at BPFI and its harmonics and combination of BPFI and shaft frequency. The frequency of components due to defects and amplitude of vibration obtained from theoretical analysis is in close agreement with results obtained from experimental results.

The satisfactory performance of model has been confirmed by experiments on bearing with inner race defect. The prediction of actual (system) amplitude of vibration is not possible with this model because it is difficult to incorporate the effect of the entire rotor bearing system. Hence, speed and defect size are the influencing parameters of the vibratory behavior of the bearing.

Footnotes

Appendix 1 Acknowledgements

The authors are grateful to the R&D department of DELUX bearings Pvt Ltd, Sanaswadi, Pune, for providing the test facility and bearings for the present investigation.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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