Dynamic response and contact characteristics of shaft-bearing-pedestal system with localized defect using 2-D explicit dynamics finite element model

2.1. 2-D FEM model of shaft-bearing-pedestal system

In order to build FEM model of the shaft-bearing-pedestal system conveniently, these simplifications and assumptions are given as follows:

(1) The chamfers on the inner and outer rings, shaft and pedestal have slight effect on the dynamic response of the bearing, only are important for assembly process, therefore these chamfers are ignored.

(2) The threaded holes on the pedestal are ignored.

(3) Since the temperature of bearing is almost constant in a short simulation time, thermal effect of the model can be ignored.

(4) Because deep groove ball bearing ER-16k is usually applied radial load, the influences of axial load would be neglected in this model.

(5) It is assumed that all components are isotropic materials.

Based on the simplifications mentioned above, a 2-D FEM model of shaft-bearing-pedestal system is built as shown in Fig. 3. The morphology characteristics of the defect is shown in Fig. 4, which is located at the 6 o'clock position on the outer raceway. The defect angle is 45° and the defect size (width × depth) is 3 mm×0.3 mm.

Fig. 32-D FEM model

Fig. 4Morphology characteristics of defect

2.2. Material properties

In this FEM model, all types of components are regarded as deformable. The rings and rolling elements are made of GCr15. The materials of cage, bearing pedestal and the shaft are nylon, HT150 and 40Cr, respectively. The properties of the materials are shown in Table 2.

2.3. Loads and boundary conditions

There are three sequential explicit dynamic steps. The first one is named loading step (0.001 s), which is used to exerted the force. The second one is named speed step (0.001 s), which is used to apply the speed of inner ring. The last one is named propagated step (0.030 s), which would maintain the previous force and speed applied on the model in the subsequent simulation time. The sampling frequency is 100 kHz.

In general, bearing dynamics are very complicated with interactions among the rolling elements, cage, and raceways. Therefore, the loads and boundary conditions below are given to simulate the running of bearing under typical operating conditions in the FEM model:

(1) The gravitational acceleration of 9.8 m/s² is applied to the system.

(2) A radial load of 25 N on the center of the shaft in the negative global $y$-direction is applied.

(3) The cage is created as nine discrete parts, which are constrained as a whole by coupling constraints. The rotational speed is not applied to the outer ring and inner ring directly, the inner ring will rotate with the rotating shaft. The shaft rotates at 1500 r/min in a clockwise direction.

2.4. Discretization of FEM model

Mesh is an important step for discretization of FEM model. In order to obtain high quality elements, the rolling elements are meshed with mixture of quadrilateral and triangular elements (qua-dominated). The appropriate element size is critical for analysis accuracy and time in FEM model simulation. Because the vibration of bearing is mainly caused by interactions between rolling elements and raceways, the surfaces of the inner and outer raceway are meshed by the smaller element size 0.1 mm. The element size 0.5 mm is chosen to mesh the rolling elements, cage and other zones of inner and outer rings. The element sizes of pedestal and shaft are chosen as 1 mm. The mesh of the FEM model is shown in Fig. 5.

Fig. 5Images of meshed model: a) whole meshed model, b) enlarged meshed part in red zone

a)

b)

3.1. Results and analysis of simulated signals in time domain

The FEM model of bearing with localized defect is established and simulated, the stress contour of the FEM model is shown in Fig. 6. There are three rolling elements in the loading zone of the bearing. It is worthy to note that the stress field near the contact surface between outer ring and pedestal is non-zero when the rolling element passes over the loading zone. Because a rigid pedestal would enhance the stiffness of FEM model, and result in an unrealistic increase in the amplitude of vibration signals, it is more suitable to define pedestal as a deformable rather than a rigid body in FEM modeling. The vibration acceleration signals of the bearing are sampled from point P1 shown as Fig. 6 for analysis.

Fig. 6Stress contour

It is shown in Fig. 7 that the acceleration signals of vibration in time domain. When the loading and speed of the bearing are applied during 0-0.002 s, which would excite the first and second pulses, marked as L and S in the red dotted boxes, respectively. While the rolling elements 1, 2 and 3 pass through the defect, the three dual-impulse signals would be excited sequentially. Therefore, the FEM model is reasonable by analysis of the acceleration signals of vibration in the time domain.

Fig. 7Simulated acceleration signals of vibration in time domain

3.2. Analysis of simulated signals in frequency domain

The frequency spectrum of acceleration signals obtained by FFT is shown in Fig. 8, and the characteristic frequency of the outer raceway ($f_{bpo}$) and its harmonic frequencies $n\times f_{bpo}$ are obvious. The $f_{bpo}$ can be calculated by Eq. (1):

where, $f_s$ is the rotation frequency of the shaft, $z$ describes the number of balls, $D_r$ denotes the ball diameter, $D_p$ refers to the bearing pitch diameter, and $\alpha$ represents the contact angle.

Fig. 8Frequency spectrum of FEM model

The parameters of bearing ER-16k related to the Eq. (1) are given in Table 1 and contact angle $\alpha =0$. According to Eq. (1), the theoretical $f_{bpo}$ is 89.24 Hz. The $f_{bpo}$ of simulated signals is 90.03 Hz. The FEM simulated $f_{bpo}$ fits the theoretical $f_{bpo}$ very well. The difference between them is 0.88 %, Therefore, the FEM model is reliable and can be used to analyze the response of vibration and contact characteristics in further.

4.1. Experimental test rig

The MFS test rig produced by Spectra Quest Incorporation (SQI) in USA is used to test the localized defect on outer raceway surface of bearing, which is shown in Fig. 9.

Fig. 9MFS test rig

In this experiment, the shaft was driven by a 1 HP 3-phase AC motor at 1500 r/min. The motor was connected with shaft by a flexible coupling. The shaft was supported by two deep groove ball bearings ER-16k, one is localized defect bearing and the other is healthy bearing. An ICP tri-axial accelerometer is mounted on top of the pedestal of the defective bearing to sample the acceleration signals of vibration. A 50 N radial load was put on the shaft by a 5.1 kg rotor disk installed in the middle of the two bearings, therefore each bearing supports a 25 N radial load. The AC controller was used to control the start and stop of the motor, set the speed of the motor and the safety protection of the system, the digital tachometer display was used to show the actual speed of the motor. A DAQ acquisition card of portable spectra PAD connected between the accelerometer by a 3-channel data cable and a computer by USB port, was used to collect acceleration signals of vibration, which limit sampling frequency is 102.4 kHz. After AC motor was running stable, three groups of the acceleration signals of vibration were collected with a sampling frequency of 66 kHz, and the duration was 4 s.

It is shown as Fig. 10, the bearing with localized defect was installed near the motor, which parameters are width 3 mm, depth 0.3 mm and defect angle 90°.

Fig. 10Experimental bearing, a) defective bearing ER-16k, b) fabricated defect on outer raceway

a)

b)

4.2. Model verification by experimental results

One group of signals is randomly selected from the three groups of acceleration signals of vibration collected from the test rig to verify FEM model, which is shown in Fig. 11(a). The acceleration signals of vibration during 1.3 to 1.6 s are taken to analyze the faulty information, as shown in Fig. 11(b). Furthermore, it is shown in Fig. 11(c) that the acceleration signals of vibration in 1.48 to 1.51 s are enlarged. It is obvious that the dual-impulse are excited while the rolling element passing across the defect, and the time interval between two dual-impulse approximates 0.011 s, which fits with the result of FEM model as shown in Fig. 7.

Fig. 11Experimental acceleration signals of vibration, a) 0-4 s, b) 1.3-1.6 s, c) 1.48-1.51 s

a)

b)

c)

Fig. 12Frequency spectrum of experimental acceleration signals

The frequency spectrum of experimental acceleration signals is obtained by FFT, as shown in Fig. 12. The characteristic frequency $f_{bpo}$ and its harmonic frequency can be clearly found. It is obvious that the $f_{bpo}$ of experimental results is 89.88 Hz, while the $f_{bpo}$ of the simulated results is 90.03 Hz. The difference between the results of the experiment and simulation is only 0.2 %, which is almost consistent. The small difference of $f_{bpo}$ between simulation and experiment comes from the slight difference between the actual speed and the set speed of the motor in the experiment. Therefore, the FEM model is almost accurate and the data from the FEM model could be analyzed through other methods to get more information.

5.1. Contact force and multi-event process

As shown in Fig. 4, the three rolling elements on the right side of the 6 o'clock of bearing are marked as ball 1, ball 2 and ball 3 counterclockwise. The contact forces between these rolling elements and the inner and outer raceway are extracted from the FEM models of a healthy bearing and a defective bearing, respectively. The defect angle $\beta =$45° and size is 3 mm×0.3 mm. The contact forces $CFO$ between the rolling elements and the outer raceway in the $x$-direction and $y$-direction are represented as $CF{O}_x$ and $CF{O}_y$, respectively. Similarly, the contact forces $CFI$ between the rolling elements and the inner raceway are expressed as $CF{I}_x$ and $CF{I}_y$, respectively.

The contact force $CFO$ and $CFI$ of healthy bearing in the $x$-direction and $y$-direction are shown as Fig. 13 and Fig. 14, respectively. The curve in the black box 'A' is the loading process of the simulation. The green box 'B' is the position where the rolling element runs to the 6 o'clock of the bearing, which is the center of the load zone. The boxes 'E' and 'C' are the start and end points of load zone, respectively.

Fig. 13Contact force of healthy bearing in x-direction, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

Fig. 14Contact force of healthy bearing in y-direction, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

The contact characteristics of the bearing could be clearly reflected by contact force between the rolling elements and raceway in the running process. The loading zone of bearing is shown in Fig. 15, it is visually shown that rolling elements are entering or exiting the load zone, the range of the load zone, and the time from the rolling element running to the bottom of the load zone. When the bearing starts to rotate, the $CF{O}_x$ of ball 1 appears a rapid increase, which is the loading process (box 'A' in Fig. 13).

Ball 1 is at the lower left side of the outer raceway, which is located at the load zone. The $CF{O}_x$ will increase sharply, while the bearing starts running and loading. Then, the $CF{O}_x$ and $CF{I}_x$ of ball 1 decrease at the same time and finally intersect on the zero lines, because ball 1 has reached the 6 o'clock of the outer ring (box 'B' in Fig. 13), the component force of the contact force in the $x$-direction is zero, all concentrated in the y-direction ($CF{O}_y$ and $CF{I}_y$ have been kept above 1000 N in this period), as shown in Fig. 14. With the rotation of the inner ring, ball 1 gradually exits from the load zone, and the contact force between ball 1 and raceways gradually reduces to zero. Ball 1 exits the load zone when $CF{O}_x$, $CF{I}_x$, $CF{O}_y$ and $CF{I}_y$ decrease to zero at the same time. The time interval from ball arriving at the bottom of raceway to exiting the load zone is half of the time when ball passes the load zone. Since the load zone can be calculated by Eqs. (2) and (3):

where $L_{bearing}$ is the arc length of load zone, $\delta$ is the center angle of load zone, $\mathrm{\Delta }t$ is the time from roller running to the bottom of the raceway to exiting point of the load zone, $\omega$ is the angular speed of inner raceway, $D$ is the diameter of outer raceway.

Fig. 15Load zone of bearing

Noteworthy, it is shown in Fig. 13(c) and Fig. 14(c), that the $CF{O}_x$, $CF{I}_x$, $CF{O}_y$ and $CF{I}_y$ are zero in theory, because the ball 3 is not located at the load zone before the box 'E'. However, the $CF{O}_x$, $CF{I}_x$ in zone 'D' shown as Fig. 13(c) present a series of small pulses during some times. Before ball 3 enters into the load zone, it is located near the $x$-direction. Since the centrifugal force of ball 3 would cause slight impacts, the contact force caused by these impact acts on the horizontal direction ($x$-direction), the component of the contact force in the vertical direction ($y$-direction) is 0.

The contact force $CFO$ and $CFI$ of defective bearing in the $x$-direction and $y$-direction are shown in Fig. 16 and Fig. 17, respectively. When rolling element enters or exits the defect, it is obvious that the contact force between the rolling element and the inner, outer raceway presents a multi-event process [23]. Taking ball 1 passing through the defect as an example, the multi-event process is described as following.

At the beginning of applying load, the absolute value of the contact force in $x$ and $y$ directions between ball 1 and raceway would increase, which are shown in Fig. 16(a) and Fig. 17(a), respectively. During the stage of applying speed, because ball 1 moves near the 6 o'clock of the outer ring, the contact force is mainly in the vertical component at this time, which leads to the $CF{O}_x$ being a downward trend and the $CF{O}_y$ being a slow-increasing trend. At 0.004 s, ball 1 runs to the edge of the defect, the $CF{O}_y$ reaches the maximum. As ball 1 enters the defect with the 45° slope, the $CF{O}_x$, $CF{I}_x$, $CF{O}_y$ and $CF{I}_y$ of ball 1 will decrease gradually. When ball 1 enters the defect completely, contact forces $CF{O}_x$, $CF{I}_x$, $CF{O}_y$ and $CF{I}_y$ would decrease to 0 because of the unloading of ball 1. The process of the rolling element entering the defect is regarded as “event-1”, which is shown in Fig. 16 and Fig. 17.

Fig. 16Contact force of defective bearing in x-direction, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

Fig. 17Contact force of defective bearing in y-direction, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

When ball 1 has entered the defect completely (0.005-0.006 s), this process is considered as event-2, as shown in “event-2” in Fig. 16 and Fig. 17. When the depth of the defect is small, ball 1 will impact on the bottom of the defect due to inertia force, which causes slight pulses on its contact force. But while the depth of the defect is large, the cage carries ball 1 in its pocket hole so that it does not approximately impact the bottom of the defect.

As the bearing rotates continuously, ball 1 begins to contact the defect edge and gradually exits the defect after 0.006 s. The contact forces in horizontal direction $CF{I}_x$ and $CF{O}_x$ increase, contrarily the contact force in vertical direction $CF{I}_y$ and $CF{O}_y$ increase rapidly to the maximum value in a short time. Since the defect is located on the outer raceway, ball 1 would impact the edge of the defect, then the bearing will reload, the $CF{O}_x$ increases faster than $CF{I}_x$. The process of the rolling element exiting the defect and impacting with the edge of the defect is defined as “event-4”, which is shown in Fig. 16 and Fig. 17.

When ball 2 crosses over the defect, it will endure a process of load redistribution. As ball 2 is at the unloading stage, the original load supported by ball 2 will be shared by the other rolling elements within the load area, which causes the contact force of ball 1 and ball 3 to increase rapidly. As ball 2 is at the reloading stage, the contact force of ball 1 and ball 3 will decrease to the normal level. The process is called load compensation and is defined as “event-3”, which is shown in Fig. 16 and Fig. 17.

5.2. Effect of defect sizes on contact force

It is shown from the above studies that the amplitude trend of the contact force in the defective bearing is similar to that in the healthy bearing, except when the rolling elements pass over the defect. In order to investigate the influence of defect sizes on contact force, the indicator $sc{f}_{defect-health}$ is constructed, which is as Eq. (4):

where, $c{f}_{defect}$ and $c{f}_{health}$ are the contact force in the defective and healthy bearing, respectively.

Since the defect is located in the 6 o'clock direction on outer raceway, the contact force between rolling elements and raceway is obviously shown in the $y$-direction near the defect region. A series of cases with different defect width and depth listed in Table 3 are simulated to calculate $sc{f}_{defect-health}$ by the contact force between rolling elements and outer raceway. The indicator in $y$-direction $sc{f}_{defect-health-y}$ between ball 1, 2, 3 and raceway corresponding to different defect widths 1.0, 1.4, 1.8, 2.2, 2.6 and 3.0 mm are shown in Figs. 18-23(a), (b) and (c), respectively.

Fig. 18scfdefect-health-y of different defect depths when defect width is 1.0 mm, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

Fig. 19scfdefect-health-y of different defect depths when defect width is 1.4 mm, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

Fig. 20scfdefect-health-y of different defect depths when defect width is 1.8mm, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

Fig. 21scfdefect-health-y of different defect depths when defect width is 2.2 mm, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

Fig. 22scfdefect-health-y of different defect depths when defect width is 2.6 mm, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

Fig. 23scfdefect-health-y of different defect depths when defect width is 3.0 mm, a) ball 1, b) ball 2, c) ball 3

a)

b)

c)

Fig. 24scfdefect-health-y between ball 2 and outer raceway during 0.0155-0.0175 s when defect width is 1.0 mm

It is shown from Figs. 18-23 that the indicator $sc{f}_{defect-health-y}$ would increase during event-2 and event-3, and is close to zero in the other periods. When $CF{O}_y$ and $CF{I}_y$ would be close to the maximum during event-2, the peak of indicator $sc{f}_{defect-health-y}$ during event-2 is higher than that during event-3. Therefore, the indicator $sc{f}_{defect-health-y}$ can reflect the difference of contact force between healthy bearing and defective bearing when ball enters defect. In order to study the peak of indicator $sc{f}_{defect-health-y}$ further, Figs. 18-23(b) from 0.0155-0.0175 s are enlarged as Figs. 24-29, respectively.

Fig. 25scfdefect-health-y between ball 2 and outer raceway during 0.0155-0.0175 s when defect width is 1.4 mm

Fig. 26scfdefect-health-y between ball 2 and outer raceway during 0.0155-0.0175 s when defect width is 1.8 mm

Fig. 27scfdefect-health-y between ball 2 and outer raceway during 0.0155-0.0175 s when defect width is 2.2 mm

In order to investigate the effect of defect sizes on the maximum of indicator $sc{f}_{defect-health-y}$, the maximum of indicator $sc{f}_{defect-health-y}$ corresponding to different defect widths and depths are performed via cubic spline interpolations, which is shown in Fig. 30.

Fig. 28scfdefect-health-y between ball 2 and outer raceway during 0.0155-0.0175 s when defect width is 2.6 mm

Fig. 29scfdefect-health-y between ball 2 and outer raceway during 0.0155-0.0175 s when defect width is 3.0 mm

Fig. 30Cubic spline interpolation to maximum of indicator scfdefect-health-y when defect width is from 1.0 to 3.0 mm

As shown in Fig. 30, when the defect depth is the same, maximum of indicator $sc{f}_{defect-health-y}$ increases with the defect width increases. However, when the width of defect reaches a certain value, the indicator $sc{f}_{defect-health-y}$ is close to the maximum value. When the defect width is small (1.0-2.2 mm), rolling elements would not impact the bottom of the defect as passing over the defect, therefore the contact force of rolling element bearings with different defect widths is quite different. With increasing of defect width (2.2-3.0 mm), rolling elements could completely collide with the bottom of defect when they pass through the defect, therefore the maximum of indicator $sc{f}_{defect-health-y}$ would not change significantly.

When the defect width is the same, the maximum values of indicator $sc{f}_{defect-health-y}$ would be higher with deeper defect during ball 2 passing through the defect. When the defect depth is small, the rolling element would not unload completely as entering the defect, therefore contact force of defective bearing grows gradually. Distinctively, the maximum value of indicator $sc{f}_{defect-health-y}$ in red is defect depth being 0.1 mm with different defect widths. It would reach the maximum value very quickly, because when the defect depth is small, a slight increase with the defect width would lead to the rolling element impacting the bottom of defect, at which stage the contact force does not change significantly even if the defect widens further.

Fig. 31Effects of defect sizes on maximum of indicator scfdefect-health-y

Fig. 31 shows the effects of the defect width and depth on the maximum of indicator $sc{f}_{defect-health-y}$. Here, the defect width are 1.0, 1.4, 1.8, 2.2, 2.6 and 3.0 mm; the defect depth are 0.1, 0.14, 0.18, 0.22, 0.26 and 0.3 mm, respectively. A polynomial fitting approach is applied to obtain the mathematic relationship between the maximum of indicator $sc{f}_{defect-health-y}$ and defect sizes. According to the simulation results in Fig. 31, the relationship between the maximum of indicator $sc{f}_{defect-health-y}$ and defect sizes is given as:

where, $w$ is the defect width, $d$ is the defect depth.

According to the above analysis, the contact force between rolling elements and raceways could be greatly affected by the defect sizes. The established approach can give a useful numerical method for analyzing the contact characteristics of a rolling element bearing with localized defect, and provide innovative theoretical guidance for bearing fault diagnosis and defect prediction.