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Vibration Characteristics of Deep Groove Ball Bearing Based on 4-DOF Mathematical Model
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 174 (2017) 808 – 814
13th Global Congress on Manufacturing and Management, GCMM 2016
Vibration characteristics of deep groove ball bearing based on 4-DOF mathematical model Guangwei Yu*, Meng Su, Wei Xia, Rui Wu, Qing Wang School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China
Abstract A 4-DOF bearing dynamic mathematical model is established for the deep groove ball bearing 6201. The vibrating characteristics of deep groove ball bearing such as displacement, velocity and acceleration are solved by four order Runge-Kutta Method and Matlab. The frequency of rolling element passing outer-circle and the simulation value based on the model are calculated, and they have a good coherence. The vibrating acceleration of deep groove ball bearing is measured by a vibrating measurement instrument, and the measured value is consist with the simulation value based on the mathematical model. It shows that the analyzing method is feasible and can be used for the design, manufacturing and measurement of deep groove ball bearing. 2016The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license © 2017 © Published by Elsevier Ltd. This Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management Keywords: Deep groove ball bearing; Matlab; Vibrating characteristics
1. Introduction Rolling bearings are essential parts of most rotating machineries, widely used in the industry. They are important supporting components for transmitting motion and bearing loads. When a rolling bearing rotates, its movement and mechanical relationship between the parts are complex, and the dynamic analysis and numerical simulation of the rolling bearing are becoming a trend. The result of dynamic analysis and numerical simulation can be used for the design, manufacturing of rolling bearing [1]. A 4-DOF bearing dynamic mathematical model is established for the deep groove ball bearing 6201. The vibrating characteristics of deep groove ball bearing such as displacement, velocity and acceleration are solved by four order Runge-Kutta Method and Matlab. The frequency of rolling element passing outer-circle and the
* Corresponding author. Tel.: +86-1362-1725-946. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management
doi:10.1016/j.proeng.2017.01.226
809
Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814
simulation value based on the model are calculated, and they have a good coherence. The effects of rotating speed and radial load on the vibration acceleration of the bearing is analyzed, and the vibrating acceleration of the bearing is also measured by a vibrating measurement instrument. 2. The 4-DOF bearing dynamic mathematical model The 4-DOF rolling bearing dynamic model [2], is shown as Fig. 1. In the model, it is assumed that the axial clearance and lubrication at various interfaces are ignored [3], and the bearing inner ring is fixed on the rigid shaft. The chamfer has little effect on the bearing internal stress, so it can be ignored.
Fig. 1. Four-DOF rolling bearing dynamic mode.l
The general equations of motion for the 4-DOF bearing mathematical dynamic model can be expressed as formula (1). m x c x ° 1 1 1 1 Fx wx ° m y c y ° 1 1 1 1 Fy wy ® °m2 x2 c2 x2 Fx 0 ° °m y c y F 0 ¯ 2 2 2 2 y
(1)
In which, m1, m2, c1 and c2 respectively represent the mass of shaft and inner race, the mass of outer race, damping factor of inner race and outer race; x1 and y1 respectively represent the displacements of the mass center of inner race; x2 and y2 respectively represent displacements of the mass center of outer race; wx and wy respectively represent the horizontal force and vertical force applied to the inner race of the bearing; F x and Fy respectively represent the contact forces of the bearing in the x-direction and y-direction. The function for total deformation of the jth ball depends on the angular position of the jth ball θj, displacements of the mass center of inner and outer races and bearing clearance z, as expressed by the following equation: G j ( x1 x2 ) cos T j ( y1 y2 )sin T j z (2) In which, j=1,2,...N, N is the number of balls. Only in the bearing zone, the ball will be deformed. The switch function λj is introduced as:
Oj
1 ° ® ° ¯0
Gj ! 0
Gj d 0
The angular positions of the jth ball θj is a function of time increment t, and the equation is given as:
(3)
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Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814
Tj
2S ( j 1) Zct To N
(4)
In which, θo is the previous cage position, and ωc is the cage speed. The total nonlinear contact forces in the x- and y-directions for a rolling bearing with N balls are, respectively,
° Fx ° ® °F °¯ y
N
K ¦ O jG 1.5 j cos T j j 1
(5)
N
K ¦ O jG
1.5 j
sin T j
j 1
In which, K is the contact stiffness amid inner-race and outer-race, which depends on the contact geometry and the properties of material. 3. Analysis on simulated results MATLAB is an efficient computing solution that integrates computing, visualization, and programming into an easy-to-use environment. The vibrating characteristics of rolling bearing are solved by four order Runge-Kutta method and Matlab [4]. The damping factor of inner race and outer race are chosen to be 200Ns/m. The 6201 deep grove ball bearing parameters are shown in Table 1. Table 1. Geometric properties of the rolling bearing 6201. Diameter of inner race (Di) Diameter of outer race(Do) Number of balls(N) Diameter of rolling element (Dw) Contact angle(¢) Radius of curvature of inner ring raceway(ri) Radius of curvature of outer ring raceway(ro) Pitch diameter(dm)
12mm 32mm 7 5.556mm 0o 2.834mm 2.945mm 22mm
3.1. Vibration characteristics of the mass centre of outer race When the rotating speed is 1500r/min, the wx is 0N and wy is 10N, which is similar to the test condition; the vibration characteristics of mass center of outer race can be obtained and shown as Fig. 2. a
b
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Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814
c
Fig. 2. (a) vibration displacement response; (b) vibration velocity response; (c) vibration acceleration response.
According to Fig.2, it is shown that the acceleration signal varies periodically and the period is 0.0158s, which is equal to the period of displacement and velocity. From the calculation of formula (6), the ball passing frequency at the outer race is 65.4 Hz.
fbo
Nfi D (1 w ) dm 2
(6)
In which, fbo is the frequency of ball passing outer race, fi is the rotational frequency of the bearing inner race. Table 2. Simulation value and theoretical value. Simulation value of period
Theoretical value of period
Error
0.0158
0.0153
3.27%
The displacement frequency spectrum can be obtained by Matlab and shown as Fig. 3. The frequency spectrum of displacement response shows that the frequency of ball passing outer race is 66Hz. The frequency of rolling element passing outer race and the simulation value based on the model are calculated, and they have a good coherence. It shows that the analyzing method is feasible.
Fig. 3. FFT of the Y direction displacement.
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Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814
3.2. The influence of rotating speed on vibration acceleration of bearing When the wx is 0N and wy is 100N, the rotating speed parameters changes from 500r/min to 2000r/min, the acceleration responses can be obtained and shown as Fig. 4. a
b
c
d
Fig. 4. Vibration acceleration response: (a) rotating speed is 500r/min; (b) rotating speed is 1000r/min; (c) rotating speed is 1500r/min; (d) rotating speed is 2000r/min.
The vibration acceleration response changes under the same load and different rotating speeds, which is shown as Fig. 5. It can be concluded that the vibration acceleration of bearing constantly changes and increase with the increase of rotating speed.
Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814 Fig. 5. Variation of vibration acceleration with rotating speed.
3.3. The influence of load on vibration acceleration of bearing In order to study the relationship between the load and vibration acceleration, the rotating speed is kept at 1500r/min. The change of radial load parameter is divided into two groups. The first group of load parameters change from 30N to 100N, and the step is 10N; and the other group of load parameters changes from 200N to 2000N, and the step is 200N. The variation of vibration acceleration with radial load can be obtained and shown as Fig. 6. It can be seen from Fig. 6 (a) that when the load is less than 60N, the vibration acceleration of bearing decreases sharply with the increase of the load, and when the load is greater than 60N, there is no significant change for the vibration acceleration of bearing with the increase of load. Fig. 6 (b) shows that when the load is more than 200N and less than 1000N, the vibration acceleration of bearing increases sharply with the increase of load, and when the load is greater than 1000N, the vibration acceleration of bearing decreases with the increase of the load. a
b
Fig. 6. Variation of vibration acceleration with radial load: (a) the load is set between 30N and 100N; (b) the load is set between 200N and 2000N.
4. Test verification The rolling bearing vibration testing system is shown as Fig. 7 (a), including the hardware and the software, which is developed according to the Chinese standard GB/T 24610-2009 ‘rolling bearings-measuring methods for vibration(acceleration)’. Root mean square, peak, crest factor, time domain and frequency domain waveform are obtained by the developed rolling bearing vibration testing system, which is used for analyzing the vibration characteristics of rolling bearing. The software system is developed based on LabVIEW8.5 [5], and the main interface is shown as Fig. 7 (b). In the study, ten sets of 6201 deep groove ball bearings are used for test. When the rotating speed is 1500r/min and the load is 10N, each is measured twice for front and back side, so the 20 sets of vibration acceleration data are obtained, and the average value is 21.99 dB. a
b
813
814
Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814 Fig. 7. (a) the rolling bearing vibration testing system; (b) the main interface.
The measured value L of vibration acceleration of bearing is obtained and calculated according to the equation(7).
L
20 lg
a g
(7)
In which, a is the root mean square value of vibration acceleration, and g is referenced acceleration, which is equal to
9.81u103 m / s 2 . Table 3. Comparison between simulated and measured values. Simulation value of vibration acceleration
Measured value of vibration acceleration
Error
20.79 dB
21.99 dB
5.46%
The calculated value is in consistent with the measured value. However, the presumed condition of dynamic mathematical model is not the same as the actual measurement condition completely, so the simulation value is slightly different from the measured value. 5. Conclusion The 4-DOF bearing dynamic mathematical model and the vibrating characteristics of deep groove ball bearing such as displacement, velocity and acceleration are solved by four order Runge-Kutta Method and Matlab. And the period of the model is consist with the theoretical value. By using the dynamic mathematical model, the influence of load on vibration acceleration of bearing is analyzed. It is concluded that the vibration response of the bearing is different under the same rotating speed and different load, and the effect of load on vibration acceleration is non-monotonic. Based on the mentioned model, the influence of rotating speed on vibration acceleration of bearing is analyzed. The results show that the vibration response of the bearing is different under the same load and different rotating speed. And the vibration acceleration of bearing increases with the increase of rotating speed and it is consistent with people's intuitive understanding. Acknowledgement Thanks for all authors of the references. Special thanks for the members of the research team and senior engineer Mr. Wang Jialiang who helps to develop the vibration measurement system of rolling bearings. References [1] C. Tao, P. Zhong, R. Wang, Failure Analysis and Prevention for Rotor in Aero-engine, National Defend Industry Press, Beijing, 2000. [2] P. Wang, Vibro-acoustic Coupling Algorithm of Bearing and Simulation Research on Vibration and Acoustic Features with Bearing Waviness, Master Dissertation, Chongqing University, Chongqing, 2012. [3] J. Liu, Y. Shao, Dynamic Modeling on Localized Defect of Cylindrical Roller Bearing Based on Non-Hertz Line Contact Characteristics, Journal of Mechanical Engineering, 50(2014) 91-97. [4] P. Zhou, MATLAB: Application of numerical analysis, Publishing House of Electronics Industry, Beijing, 2014. [5] F. Chen, Dynamic Characteristics of Ball Bearing Based on Explicit Dynamics, Master Dissertation, Shanghai University, Shanghai, 2015. [6] GB/T 24610-2009/ ISO 15242-2004 Rolling bearings-Measuring methods for vibration (acceleration).
ScienceDirect Procedia Engineering 174 (2017) 808 – 814
13th Global Congress on Manufacturing and Management, GCMM 2016
Vibration characteristics of deep groove ball bearing based on 4-DOF mathematical model Guangwei Yu*, Meng Su, Wei Xia, Rui Wu, Qing Wang School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China
Abstract A 4-DOF bearing dynamic mathematical model is established for the deep groove ball bearing 6201. The vibrating characteristics of deep groove ball bearing such as displacement, velocity and acceleration are solved by four order Runge-Kutta Method and Matlab. The frequency of rolling element passing outer-circle and the simulation value based on the model are calculated, and they have a good coherence. The vibrating acceleration of deep groove ball bearing is measured by a vibrating measurement instrument, and the measured value is consist with the simulation value based on the mathematical model. It shows that the analyzing method is feasible and can be used for the design, manufacturing and measurement of deep groove ball bearing. 2016The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license © 2017 © Published by Elsevier Ltd. This Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management Keywords: Deep groove ball bearing; Matlab; Vibrating characteristics
1. Introduction Rolling bearings are essential parts of most rotating machineries, widely used in the industry. They are important supporting components for transmitting motion and bearing loads. When a rolling bearing rotates, its movement and mechanical relationship between the parts are complex, and the dynamic analysis and numerical simulation of the rolling bearing are becoming a trend. The result of dynamic analysis and numerical simulation can be used for the design, manufacturing of rolling bearing [1]. A 4-DOF bearing dynamic mathematical model is established for the deep groove ball bearing 6201. The vibrating characteristics of deep groove ball bearing such as displacement, velocity and acceleration are solved by four order Runge-Kutta Method and Matlab. The frequency of rolling element passing outer-circle and the
* Corresponding author. Tel.: +86-1362-1725-946. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management
doi:10.1016/j.proeng.2017.01.226
809
Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814
simulation value based on the model are calculated, and they have a good coherence. The effects of rotating speed and radial load on the vibration acceleration of the bearing is analyzed, and the vibrating acceleration of the bearing is also measured by a vibrating measurement instrument. 2. The 4-DOF bearing dynamic mathematical model The 4-DOF rolling bearing dynamic model [2], is shown as Fig. 1. In the model, it is assumed that the axial clearance and lubrication at various interfaces are ignored [3], and the bearing inner ring is fixed on the rigid shaft. The chamfer has little effect on the bearing internal stress, so it can be ignored.
Fig. 1. Four-DOF rolling bearing dynamic mode.l
The general equations of motion for the 4-DOF bearing mathematical dynamic model can be expressed as formula (1). m x c x ° 1 1 1 1 Fx wx ° m y c y ° 1 1 1 1 Fy wy ® °m2 x2 c2 x2 Fx 0 ° °m y c y F 0 ¯ 2 2 2 2 y
(1)
In which, m1, m2, c1 and c2 respectively represent the mass of shaft and inner race, the mass of outer race, damping factor of inner race and outer race; x1 and y1 respectively represent the displacements of the mass center of inner race; x2 and y2 respectively represent displacements of the mass center of outer race; wx and wy respectively represent the horizontal force and vertical force applied to the inner race of the bearing; F x and Fy respectively represent the contact forces of the bearing in the x-direction and y-direction. The function for total deformation of the jth ball depends on the angular position of the jth ball θj, displacements of the mass center of inner and outer races and bearing clearance z, as expressed by the following equation: G j ( x1 x2 ) cos T j ( y1 y2 )sin T j z (2) In which, j=1,2,...N, N is the number of balls. Only in the bearing zone, the ball will be deformed. The switch function λj is introduced as:
Oj
1 ° ® ° ¯0
Gj ! 0
Gj d 0
The angular positions of the jth ball θj is a function of time increment t, and the equation is given as:
(3)
810
Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814
Tj
2S ( j 1) Zct To N
(4)
In which, θo is the previous cage position, and ωc is the cage speed. The total nonlinear contact forces in the x- and y-directions for a rolling bearing with N balls are, respectively,
° Fx ° ® °F °¯ y
N
K ¦ O jG 1.5 j cos T j j 1
(5)
N
K ¦ O jG
1.5 j
sin T j
j 1
In which, K is the contact stiffness amid inner-race and outer-race, which depends on the contact geometry and the properties of material. 3. Analysis on simulated results MATLAB is an efficient computing solution that integrates computing, visualization, and programming into an easy-to-use environment. The vibrating characteristics of rolling bearing are solved by four order Runge-Kutta method and Matlab [4]. The damping factor of inner race and outer race are chosen to be 200Ns/m. The 6201 deep grove ball bearing parameters are shown in Table 1. Table 1. Geometric properties of the rolling bearing 6201. Diameter of inner race (Di) Diameter of outer race(Do) Number of balls(N) Diameter of rolling element (Dw) Contact angle(¢) Radius of curvature of inner ring raceway(ri) Radius of curvature of outer ring raceway(ro) Pitch diameter(dm)
12mm 32mm 7 5.556mm 0o 2.834mm 2.945mm 22mm
3.1. Vibration characteristics of the mass centre of outer race When the rotating speed is 1500r/min, the wx is 0N and wy is 10N, which is similar to the test condition; the vibration characteristics of mass center of outer race can be obtained and shown as Fig. 2. a
b
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Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814
c
Fig. 2. (a) vibration displacement response; (b) vibration velocity response; (c) vibration acceleration response.
According to Fig.2, it is shown that the acceleration signal varies periodically and the period is 0.0158s, which is equal to the period of displacement and velocity. From the calculation of formula (6), the ball passing frequency at the outer race is 65.4 Hz.
fbo
Nfi D (1 w ) dm 2
(6)
In which, fbo is the frequency of ball passing outer race, fi is the rotational frequency of the bearing inner race. Table 2. Simulation value and theoretical value. Simulation value of period
Theoretical value of period
Error
0.0158
0.0153
3.27%
The displacement frequency spectrum can be obtained by Matlab and shown as Fig. 3. The frequency spectrum of displacement response shows that the frequency of ball passing outer race is 66Hz. The frequency of rolling element passing outer race and the simulation value based on the model are calculated, and they have a good coherence. It shows that the analyzing method is feasible.
Fig. 3. FFT of the Y direction displacement.
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Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814
3.2. The influence of rotating speed on vibration acceleration of bearing When the wx is 0N and wy is 100N, the rotating speed parameters changes from 500r/min to 2000r/min, the acceleration responses can be obtained and shown as Fig. 4. a
b
c
d
Fig. 4. Vibration acceleration response: (a) rotating speed is 500r/min; (b) rotating speed is 1000r/min; (c) rotating speed is 1500r/min; (d) rotating speed is 2000r/min.
The vibration acceleration response changes under the same load and different rotating speeds, which is shown as Fig. 5. It can be concluded that the vibration acceleration of bearing constantly changes and increase with the increase of rotating speed.
Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814 Fig. 5. Variation of vibration acceleration with rotating speed.
3.3. The influence of load on vibration acceleration of bearing In order to study the relationship between the load and vibration acceleration, the rotating speed is kept at 1500r/min. The change of radial load parameter is divided into two groups. The first group of load parameters change from 30N to 100N, and the step is 10N; and the other group of load parameters changes from 200N to 2000N, and the step is 200N. The variation of vibration acceleration with radial load can be obtained and shown as Fig. 6. It can be seen from Fig. 6 (a) that when the load is less than 60N, the vibration acceleration of bearing decreases sharply with the increase of the load, and when the load is greater than 60N, there is no significant change for the vibration acceleration of bearing with the increase of load. Fig. 6 (b) shows that when the load is more than 200N and less than 1000N, the vibration acceleration of bearing increases sharply with the increase of load, and when the load is greater than 1000N, the vibration acceleration of bearing decreases with the increase of the load. a
b
Fig. 6. Variation of vibration acceleration with radial load: (a) the load is set between 30N and 100N; (b) the load is set between 200N and 2000N.
4. Test verification The rolling bearing vibration testing system is shown as Fig. 7 (a), including the hardware and the software, which is developed according to the Chinese standard GB/T 24610-2009 ‘rolling bearings-measuring methods for vibration(acceleration)’. Root mean square, peak, crest factor, time domain and frequency domain waveform are obtained by the developed rolling bearing vibration testing system, which is used for analyzing the vibration characteristics of rolling bearing. The software system is developed based on LabVIEW8.5 [5], and the main interface is shown as Fig. 7 (b). In the study, ten sets of 6201 deep groove ball bearings are used for test. When the rotating speed is 1500r/min and the load is 10N, each is measured twice for front and back side, so the 20 sets of vibration acceleration data are obtained, and the average value is 21.99 dB. a
b
813
814
Guangwei Yu et al. / Procedia Engineering 174 (2017) 808 – 814 Fig. 7. (a) the rolling bearing vibration testing system; (b) the main interface.
The measured value L of vibration acceleration of bearing is obtained and calculated according to the equation(7).
L
20 lg
a g
(7)
In which, a is the root mean square value of vibration acceleration, and g is referenced acceleration, which is equal to
9.81u103 m / s 2 . Table 3. Comparison between simulated and measured values. Simulation value of vibration acceleration
Measured value of vibration acceleration
Error
20.79 dB
21.99 dB
5.46%
The calculated value is in consistent with the measured value. However, the presumed condition of dynamic mathematical model is not the same as the actual measurement condition completely, so the simulation value is slightly different from the measured value. 5. Conclusion The 4-DOF bearing dynamic mathematical model and the vibrating characteristics of deep groove ball bearing such as displacement, velocity and acceleration are solved by four order Runge-Kutta Method and Matlab. And the period of the model is consist with the theoretical value. By using the dynamic mathematical model, the influence of load on vibration acceleration of bearing is analyzed. It is concluded that the vibration response of the bearing is different under the same rotating speed and different load, and the effect of load on vibration acceleration is non-monotonic. Based on the mentioned model, the influence of rotating speed on vibration acceleration of bearing is analyzed. The results show that the vibration response of the bearing is different under the same load and different rotating speed. And the vibration acceleration of bearing increases with the increase of rotating speed and it is consistent with people's intuitive understanding. Acknowledgement Thanks for all authors of the references. Special thanks for the members of the research team and senior engineer Mr. Wang Jialiang who helps to develop the vibration measurement system of rolling bearings. References [1] C. Tao, P. Zhong, R. Wang, Failure Analysis and Prevention for Rotor in Aero-engine, National Defend Industry Press, Beijing, 2000. [2] P. Wang, Vibro-acoustic Coupling Algorithm of Bearing and Simulation Research on Vibration and Acoustic Features with Bearing Waviness, Master Dissertation, Chongqing University, Chongqing, 2012. [3] J. Liu, Y. Shao, Dynamic Modeling on Localized Defect of Cylindrical Roller Bearing Based on Non-Hertz Line Contact Characteristics, Journal of Mechanical Engineering, 50(2014) 91-97. [4] P. Zhou, MATLAB: Application of numerical analysis, Publishing House of Electronics Industry, Beijing, 2014. [5] F. Chen, Dynamic Characteristics of Ball Bearing Based on Explicit Dynamics, Master Dissertation, Shanghai University, Shanghai, 2015. [6] GB/T 24610-2009/ ISO 15242-2004 Rolling bearings-Measuring methods for vibration (acceleration).