Abstract
Fatigure in rolling element bearings results in spalling of the raceways or rolling elements which is the most common bearing fault. The size of the spalling area, however, has close relationship with dualimpulse behavior produced by a localized surface spall. Based on the Hertzian contact theory and investigation of the excitation mechanism of dualimpulse behavior excited by a localized surface spall, a new coupled nonlinear dynamic model of the ball bearing with a localized surface spall on outer race is developed, which considers the timevarying displacement and contact force excitation. The solution to the differential equations is obtained applying the RungeKutta numerical integral method. The time interval of dualimpulse behavior is investigated in this paper. Comparisons of the stimulated and theoretical results and the experimental and theoretical results show the effectiveness of the proposed model.
1. Introduction
Rolling element bearings are widely applied in the most majority of rotary machinery due to their low friction coefficient, efficient load transmission capacity, and reliability. And the failure of bearings are the most common reason for machine breakdowns. A dualimpulse response will happen when a rolling element moves through the spalled area under cyclic loads [1]. Importantly, this dualimpulse behavior provides a completely reliable and novel method for spalling fault detection, fault diagnosis and life prediction. And therefore, it is crucial to investigate the dualimpulse behavior produced by a spall on inner or outer races of a ball bearing.
Over the past decades, some studies have been carried out for the dualimpulse behavior produced by a spall. The earliest research of bearing fault characterization and quantification using vibration analysis was the Ph.D. study by Epps [2] in 1991. He reported that the vibration characteristics associated with rolling elements passage over a raceway spall could consist of a step response for the balls entry into the spall and an impulse response for the balls impact on the trailing edge of the spall. The step response is associated with lower frequency content than the impulse response in which broad frequency content including highfrequency resonance oscillations can be evident. Whilst he pointed out that the size of the flaking area could be estimated if each response event (i.e., step and impulse response) was visible in the timedomain series collected from a rolling element bearing in presence of a localized surface fault. He only did analyze, however, the two responses in vibration signal, and didn’t carry out profound investigations for the generation mechanism of the two responses and develop a dynamic model which valued the two impulse responses. On this basis, Sawalhi and Randall [1] indicated that the acceleration signals resulting from the entry of the rolling element into the spall zone and exit from it were of different characteristics and the amplitudes of the two events were always greatly different. Furthermore, they developed an analytical model involving the two events to describe the observed phenomena. Unfortunately, the spectral characteristics of the vibration signal involving the two events were not further investigated and they did not, simultaneously, proposed a dynamic model for describing dualimpulse behavior. Ming et al. [3] did a further analysis for dualimpulse behavior. In their paper, the acoustic emission (AE) signal was applied to evaluate the size of the single spall. Considering the sensitivity of the AE signal to the incipient fault detection and related theories. Based on the hypothesis that the AE signal is composed of two events for each passage of the rolling element across the spall area, they proposed an analytical model, i.e., dualimpulse response model to describe the AE signal. Then, an averaged dualimpulse interval determining method is developed to estimate the size of the spall. However, a dynamic model based on the generation mechanism of dualimpulse behavior was not developed. These abovementioned studies involve only mechanism analyses of dualimpulse behavior given rise to by a spall on raceways of a bearing, however, it is worth mentioning here that a dynamic model for dualimpulse behavior has not been proposed and established.
Considering the abovementioned current problems. As a result, it is great important and practical to conduct a research on dynamic modeling for describing dualimpulse behavior, which provides a theoretical basis for the study of dualimpulse behavior exited by a localized defect in the ball bearing and spall size assessment.
2. Timevary deflection excitation
When a spall occurs on the outer race of the rolling element, the total deflection within the loaded zone of the rolling element bearing at the contact of the rolling element and local surface spall will increase, i.e., the additional displacement ${\delta}_{d}$ will be considered into the total deflection. When the center of ball approaches to the spall area, the additional displacement of the ball remains zero. However, when it passes over the spall, the additional displacement increase. It reaches its maximum value when the ball reaches at the center of the spall zone. And then, the additional displacement decreases from maximum to zero as the ball rolls from the spall center to other edge of the spall. And thereafter the additional displacement finally becomes zero as the ball exits the spall area.
Fig. 1a) Sketch of the additional deflection when the rolling element passes across the spall area, b) top view of the spall area
a)
b)
Based on the abovementioned analysis and the geometry of the rolling element bearing, considering the mathematical model of Liu et al. [6]. As a result, a timevary deflection model which the spall is modeled as a circumferential half sinusoidal wave of the defective ball bearing presented in this paper:
where ${D}_{b}$, is the diameter of the ball, ${L}_{s}$ is the spall length, $\vartheta $ is the ballspall contact angle as shown in Fig. 1(a); ${\varphi}_{s}$ denotes the angular position of the spall center with respect to the $x$ axis, it can be considered 1.5$\pi $ in this paper; ${\theta}_{l}$ is the angular position of the rolling element with respect to the $x$ axis, can be defined as follows:
where ${\theta}_{0}$ is the initial angular offset of the first rolling element to the $x$ axis, $l$ denotes the $l$th rolling element of the bearing, ${N}_{b}$ is the total number of the rolling elements; ${\omega}_{c}$ is the cage speed, which can be written by the following relation [5]:
where ${\omega}_{s}$ is the rotor speed, $\gamma ={D}_{b}\mathrm{c}\mathrm{o}\mathrm{s}\alpha /{D}_{m}$ denotes the intermediate variable.
Considering the abovementioned mathematical formulation of additional deflection, the timevarying deflection excitation model can be given as:
where ${C}_{d}$ is the internal diametral clearance of the rolling element bearing, ${\lambda}_{l}$ denotes the ballspall contact parameter of the $l$th rolling element, can be written as follows:
where ${\phi}_{d}$ is the size of the spall with respect to the outer race center, as illustrated in Fig. 1(a).
Fig. 2 illustrates the simulated result of timevarying displacement excitation model as mentioned above within 1 second, it shows the motion locuses of a rolling elements when it rolls over the spall area. It is worth noting that the simulated result only presents the variation of the additional deflection. For the purpose of observing clearly the change tendency of the motion locus of the ball, as a result, Fig. 2(b) presents the enlarged view of the encircled zone of Fig. 2(a).
Fig. 2a) Timevarying displacement excitation of the first ball, b) the zoomedin view of the encircled zone of a)
a)
b)
3. Dynamic modeling
Based on the Hertzian elastic contact theory, the nonlinear relation of loaddeformation is written as follows [5]:
where $n$ is the loaddeflection exponent, $n=$ 1.5 for a ball bearing and 10/9 for a roller bearing. $\delta $ is the radial deflection or contact deformation, $K$ denotes the total contact stiffness coefficient between the rolling element and the bearing races or constant for Hertzian contact elastic deformation, it can be given by [5]:
where ${K}_{in}$ and ${K}_{ou}$ denote the contact stiffness’s between the ball and the inner race, and that between the ball and the outer race, respectively. For ball bearing, the contact stiffness can be calculated by the following relation [5]:
in which $\sum \rho $ denotes the curvature sum of the contact of a ball and the inner race or a ball and the outer race, ${E}_{e}$ is the equivalent elastic modulus, ${\zeta}_{1}$ and ${\zeta}_{2}$ denote the first and second kind of elliptical integral, respectively; and $e$ is the elliptical eccentricity parameter. $\sum \rho $, ${E}_{e}$, ${\zeta}_{1}$, ${\zeta}_{2}$ and $e$ can be calculated by the methods presented in Ref. [5].
Since the Hertzian forces occur only when there is a contact deformation between the rolling element and the bearing races within loaded zone of the bearing. Consequently, the contact force between a ball and races can be defined as follows:
As a result of the total contact force which between a rolling element and the races is the sum of the contact force from each of the rolling elements, so, resolving the total contact force along the $x$ and $y$ axis, the components of contact forces can be determined by the following relation:
To investigate the dualimpulse behavior of the rolling element bearing with a localized surface spall on the outer race, the contact between the ball and the bearing races is considered as a springmass system. Based on the hypotheses made in Ref. [7, 8] and the model which was first developed by Sunnersjӧ [9], a novel dynamic model which considers the timevarying deflection excitation is proposed in this paper to study the dualimpulse behavior produced by the spall located on the outer race of the bearing. The nonlinear differential equations of describing dualimpulse behavior are obtained as follows:
where parameter ${M}_{eq}$ denotes the equivalent mass of the shaft and the inner race of the rolling element bearing, $C$ is the damping coefficient; ${W}_{x}$ and ${W}_{y}$ are the components of external force acting on the shaft, respectively, $x$ and $y$ are the displacement response in the $x$ and $y$ directions; ${\chi}_{l}$ is the loading zone parameter of the lth rolling element, ${\chi}_{l}=$ 1 for ${\delta}_{l}\ge $ 0 and 0 for ${\delta}_{l}<$ 0.
4. Results and discussion
4.1. Simulated result
The nonlinear dynamic differential equations are solved numerically applying the RungeKutta fourthorder numerical integration scheme. The time step $\mathrm{\Delta}t=$ 5×10^{6} s, the specified time period for solution of Eqs. (12) and (13) is 1 s, the equivalent mass ${M}_{eq}$ is 0.56 kg, the damping coefficient $C$ of the massspring system was chosen to be 2200 N∙s/m, the speed ${\omega}_{s}$ of the rotor is 439.01 RPM, the components of radial force acting on the rotor ${W}_{x}$ and ${W}_{y}$ are 45 N and 0 N, respectively. Assuming the circumferential length ${L}_{s}$ of the spall is 1 mm. The initial velocities are assumed to be zero, i.e., $\dot{x}=$ 0 and $\dot{y}=$ 0, initial displacements are set to the following values: ${x}_{0}=$ 1 µm and ${y}_{0}=$ 1 µm. And the input parameters of the dynamic model proposed in this paper are listed in Table 1.
Table 1Dimensions of the hybrid ceramic ball bearing
Parameters  Value 
Pitch diameter (${D}_{m}$), mm  38.500 
Ball diameter (${D}_{b}$), mm  7.938 
Number of ball (${N}_{b}$)  9.000 
Inner race diameter (${D}_{in}$), mm  30.562 
Outer race diameter (${D}_{ou}$), mm  46.438 
Internal diametrical clearance (${C}_{d}$), µm  1.000 
Contact angle ($\alpha $), degree  0.000 
4.2. Experimental result
To invalidate the effectiveness of the proposed dynamic model for describing the duanimpulse behavior, experiment is carried out on the fault analog experimental test rig QPZZII, as shown in Fig. 3.
Fig. 3a) Test rig of QPZZII, b) faulty rolling element bearing with bearing housing
a)
b)
Fig. 4 illustrates the experimental result, from which a typical dualimpulse phenomenon can be observed. For the purpose of observing clear the dualimpulse phenomenon and calculating the time interval of dualimpulse behavior, a enlarged view of dualimpulse behavior of the encircled zone of Fig. 4(a) is shown in Fig. 4(b).
Fig. 5 shows the simulated result, from which we can observe the typical dualimpulse phenomenon in Fig. 5(b).
Table 2Comparison of dualimpulse time interval (DTI)
Comparison of simulated and theoretical DTI  Comparison of experimental and theoretical DTI  
Simulated DTI  Theoretical DTI  Relative error (%)  Experimental DTI  Theoretical DTI  Relative error (%) 
0.00117  0.00120  2.5 %  0.001464  0.00120  22 % 
Fig. 4a) Measured vibration signal, b) zoomedin view of the encircled zone of a)
a)
b)
Fig. 5a) Simulated vibration signal, b) enlarged view of the encircled zone of a)
a)
b)
5. Conclusions
In this paper, a novel dynamic model for describing dualimpulse behavior is developed which considers the timevarying deflection excitation excited by the spall on outer race of bearing. Comparisons of simulated and theoretical DTI and experimental and theoretical DTI show the effectiveness of the proposed dynamic model. As a result, the developed dynamic model provides a theoretical basis for monitoring and fault diagnosis of rolling element bearing.
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About this article
Project supported by the National Science Foundation of China (Grant No. 51365023).