Abstract
Spindle bearing with high rotational speed is the kernel of many rotating machines which plays massive important role in the performance of the whole machine. There are also some deficiencies of the present quasi-dynamic mathematical models of the rolling bearing in analyzing the bearing dynamic parameters. This paper develops the 3D quasi-dynamic model with ball’s 6-DOF, cage’s 6-DOF, and driving ring’s 5-DOF under steady working stage, based on the analysis of the relative position of each component to determine the normal forces between each component and the analysis of the relative velocity of the contacting point to determine the tangential force between each component. This model can analyze the dynamic parameters, such as the spin-to-roll ratio of inner and outer rings, the angular velocity of rolling elements in y-axis of the location coordinate, the normal contact force of cage pocket and ball, the cage angular velocity and ball revolution velocity, and the gyroscopic moment under different working condition. Compared with the numerical calculation results of ADORE program, the results of the proposed model are able to accurately describe the change laws of the spindle bearing parameters.
1. Introduction
Development of a spindle bearing system can significantly improve the efficiency and accuracy of machine tools. The quasi-dynamic models of the spindle bearing are not only better capable of analyzing bearing dynamic parameters than quasi-static models and could also need less numerical calculation time than the dynamic model. Generally, the steady working duration of ball bearings would last 90 percent of the whole service life under normal condition. Therefore, establishment of perfect bearing quasi-dynamic model plays massive significant role on the analysis of the bearing operational performance [1–3], such as cage slipping, cage impacts, roller skidding, roller-flange contact, and roller skewing.
Many contributions for spindle bearing system were conducted based on the bearing quasi-dynamic modeling, such as Harris [4] and Yuan [5], who established the equation groups about 6 degrees of rolling elements, 3 degrees of cages, and 5 degrees of rings by dropping the hypothesis of the ring control theory and considering the oil film drags of the contacting ellipse between rolling elements and raceways with oil film lubrication. Wang et al. [6] analyzed the boundary conditions of aeroengine bearing by the quasi-dynamic method and then discussed the relationship between status parameters associated with Failure modes (such as lubricating oil film thickness, etc.) and working parameters (such as rotation speed and external load). Ren et al. [7] established a new quasi-dynamic model of the counter-rotating cylindrical roller bearing based on elastohydrodynamic lubrication theory, hydrodynamic lubrication theory, and the quasi-dynamic analysis method to dynamic behaviors of counter-rotating cylindrical roller bearing. However, those researches have not considered the complete degree of freedom of the rolling bearing components which is very significant to determine their dynamic characteristics. Therefore, it is badly necessary to establish the bearing quasi-dynamic model for the analysis of the component complete degree of freedom.
In order to reflect the real operation of high speed angular contact ball bearings, a quasi-dynamic calculation model for the high speed angular contact ball bearing was built by embedded Broyden-Fletcher-GoldFarb-Shanno mutative scale optimization in quasi-dynamic calculation program [8]. For the spindle bearing with a certain contact angle, gyroscopic moment will inevitably induct large sliding between the balls and raceways and increase the bearing friction moment, which are the main reasons for the bearing heat generation and wear [9–11]. Therefore, accurately analyzing the spin angular velocity between ball and raceway is benefit for the correct calculation of bearing power dissipation, assessment of the rate of heat generation, and appropriate decision of lubrication flow rate, etc. Meanwhile, the bearing cage is a critical component which is often the most Failure part for high speed bearings, typically. With the increase of bearing rotation speed, the analytical precision of the contact force between the cage and the balls or the guide rings is dramatically important for the force characteristics of bearing research [12–14].
To analyze the operating characteristics of rolling bearings at steady working stage, the bearing quasi-dynamic model is developed in this paper. In the optimal model, (i) by the analysis of the relative positions of each bearing component, the normal forces between the bearing components are determined and (ii) the tangential forces between the bearing components are obtained according the study of the directions and magnitudes of the relative velocities of the contact points and then (iii) the force mathematical models of balls’ 6-DOF, cage’s 6-DOF, and the driving ring’s 5-DOF. On the base of those models, the gyroscopic moment model is established to analyze the pitching angles and yaw angles, and the cage forces and operating condition are studied by the contacting force models between the cage and the ball and guide ring. The results of the improved quasi-dynamic model are furthermore compared with the results of the typical rolling bearing force calculation example of Gupta [15]. Studies show that the model is able to accurately describe the dynamic variations of the parameter characteristics of spindle bearings.
2. Mechanics Modeling of Spindle Bearing
2.1. Techniques of Mechanics Modeling
Figure 1 shows the interaction model of bearing components. The interaction forces between these parts are mainly the normal contact forces and the tangential forces. First, by means of determining the position vectors of part B and part A in space, judging whether part B in contact with part A and calculating the normal contact force, the normal force of part B relative to part A can be obtained. Second, according to the velocity vectors of part A and part B in space, the velocity direction of part B relative to part A at the contact point can be determined, and on the basic of Coulomb’s law, the tangential force of part B relative to part A is obtained. Finally, the influence of the lubricant on the force will be considered to solve the force of part B relative to part A. Equilibrium equations of part A and part B will be established and the numerical solution can be done by considering these three factors.
2.2. Interaction Model of Each Parts of Bearing
2.2.1. Interaction of Ball and Ring
The vector position relationship of ball and ring in space is shown in Figure 2. The points of Oi and Oe are the centers of inner and outer raceway groove curvatures, respectively. The position vectors of and are from the center of the ball to the centers of groove curvature of the outer ring and the inner ring, respectively.
In the vector triangle of ΔOOaOe (seeing Figure 2), the vector of is the position vector from the ball center to the outer raceway groove curvature center and can be expressed byLikewise, the vector of in the vector triangle of ΔOrOaOi is the position vector from the ball center to the inner raceway groove curvature center and can be described asThe elastic approaches of ball and ring raceways are expressed bywhere k = i, e denote the inner and outer ring, respectively. When δk ≤ 0, the ball and raceway have no contact with each other and contact deformation will not occur, and the normal force is zero, whereas when δk > 0, the contact deformation occurred, and the normal contact load can be obtained by (4) using Hertz theory of contact stress.where k is the stiffness coefficient of Hertz contact.
As illustrated in Figure 3, the local coordinate of o_xy is established to analyze the contact ellipse between ball and inner raceway. Relative linear velocity can be described in the direction of x-axis and y-axis as (5) and (6), respectively.
The angle between the direction of relative linear velocity and the direction of semimajor axis of the contact ellipse is
According to Coulomb’s law, tangential friction with respect to semimajor axis and semiminor axis of the contact ellipse is stated as
The force of contact point between ball and inner raceway is then obtained as
As illustrated by Figure 4, the vectors from the ball center to the contact ellipse centers of inner and outer raceways are and , respectively. The friction torque between the ball and the inner raceway is as follows:and the friction torque of the inner raceway iswhere is the vector from the ball center to inner ring coordinate center. Similarly, the contact force between ball and outer raceway can be given as
2.2.2. Interaction of Ball and Cage
As illustrated in Figure 5, the vectors of and are the first and the second components of the vectors from the ball center to the cage pocket center in the pocket coordinates op_xpypzp, respectively. The contact point between the ball and the pocket wall and the ball center and the pocket center is aligned in line. The minimum distance from pocket wall to the ball surFace can be expressed asWhen δbp ≤ 0, the elastohydrodynamic lubrication state will occur between the ball and cage pocket, and the normal force component between the ball and cage in the contact coordinate system can be obtained based on Brewe formula by [16]where , , and . And the corresponding tangential force component can be given using Coulomb Law as follows:where Rx is the equivalent radius along semimajor axis of the contact area in the contact coordinate, Rz is the equivalent radius along semiminor axis of the contact area in the contact coordinate, η0 is the dynamic viscosity of lubricating oil, u is the relative sliding speed of the ball and the pocket wall, is the equivalent elastic modulus of the ball and the cage, and μbp of 0.02 ~ 0.08 is the friction coefficient between the ball and the pocket wall.
When δbp < 0, contact deformation will occur between the ball and the pocket wall, and the normal contact load can be obtained by using Hertz theory asThe relative line velocity at the contact point of the ball and the pocket wall isIn contact coordinate _, the angle between νbp with axis isThe tangential friction forces along the directions of and are expressed as follows, respectively:
The force and the moment subjected to the ball by cage pocket can be stated as follows, respectively:
According to Newton’s third law, the force and the moment subjected to the cage by the ball can be obtained as follows, respectively:where is the position vector from the contact point of ball and pocket wall to the pocket center and is the position vector from the pocket center to the cage coordinate center.
2.2.3. Interaction of Cage and Guide Ring
Because of the effect of inertial force during steadily working process, the cage and guide ring exist potential collision, which can be assessed by the minimum clearance between the cage and the guide ring. As shown in Figure 6, the minimum clearance in the vector triangle ΔOr is given aswhere the symbols “+” and “-” are for outer and inner guide rings, respectively. and are the second and third components of , respectively, which is the vector from the guide ring center to contact point of cage and ring in ring coordinate system or_xryrzr.
When δrc ≤ 0, hydrodynamic lubrication will occur between the cage and the guide ring, and the interaction condition can be regarded as short sliding bearing. In the contact coordinate system _ (seeing in Figure 5), the interaction forces of each direction are calculated by the following formula [17, 18]:where is the relative speed of the cage with respect to the guide ring at the minimum clearance in the direction of axis, Bc is the width of the cage, u is drag speed of lubricating oil, u=(ωr+ωc), is the cage guide clearance, and ε is relative offset of the cage center.
The friction moment of the moving cage caused by distribution pressure of fluid pressure oil film can be expressed aswhereVcr is the relative sliding speed of the cage rail with respect to the ring land and can be stated as follows:When δrc > 0, contact deformation will occur between the cage and the guide ring, and the normal contact load can be obtained by using Hertz theory as where Kl is the Hertz contact stiffness coefficient with finite length.
The relative sliding speed of the ring with respect to the cage at contact point iswhere is the velocity at the contact point of the ring and is the velocity at the contact point of the cage. and are the first and second components of the tangential relative velocity, respectively, and the angel along axis is
Applying Coulomb’s law, the tangential force at the contact point iswhere is the friction coefficient between the cage and the guide ring and is set 0.02 ~ 0.08.
The tangential forces in the direction of and axis of the contact coordinate _ are as follows, respectively:
According to Newton’s third law, the force and the moment of the cage caused by the guide ring arewhere is the vector distance from the center of cage coordinate system to the contact point between the cage and the ring.
2.2.4. Drag of Ball from Oil-Gas Mixture
The balls should overcome the viscous friction drag caused by the bearing internal lubrication during orbital motion. The viscous friction drag of a ball shown in [19] can be approximately expressed bywhere ρef is the lubricant density in the bearing free space divided by the volume of the free space and the drag coefficients CD can be determined from [20].
2.3. The Establishment of Equilibrium Equations
The equilibrium equations of the ball are established in the azimuth coordinate system of the ball, in which there are six unknown parameters that need to be calculated, and they are xb, rb, ωm, , ωby, and ωbz.
The inertial forces on a ball areand the inertial moments on a ball arewhere Ib is the inertia moment and can be calculated by .
The equilibrium equations of ball’s force and moment are
The force equilibrium equations of the cage are established in the inertial coordinate system of the bearing, and the moment equilibrium equation are established in the fixed coordinate system of the cage. There are the six parameters need to be determined as , , , ωcx θcy, and θcz.
The force and moment equilibrium equations of the cage are
The force equilibrium equations of the guide ring are established in the inertial coordinate system of the bearing, and the moment equilibrium equations are established in the ring fixed coordinate system. Similarly, there are also six parameters need to be solved , , , , . The force and moment equilibrium equations of the guide ring are
2.4. Numerical Solution
In this paper, six equations of the ball and cage and five equations of the guide ring were solved numerically by Matlab tools. The magnitude order of the displacement and angular velocity is significantly different; therefore, it is easy to get a nonconvergent solution by combining them to solve. In order to solve this problem, the method from first partial to whole is adopted in this paper. The displacement variables are determined first, and then the angular velocity variables are calculated. Thereafter, the initial value, the displacement, and angular velocity variables will be solved together. The solution process is shown in Figure 7.
3. Validation
In order to verify the accuracy of the updated mathematical model of the angular contact ball bearing, the results of a typical example is illustrated to compare the results with ADORE calculation. The values of bearing parameters are listed in Table 1.
Bearings working condition is set as Fa of 2000 N, of 400 N, ni of 120000 r/min with the fixed outer ring. Comparing the results of this paper’s program with Gupta’s ADORE dynamics program at 12.87 ms, the results are compared in Table 2.
As listed in Table 2, the results of this paper and Gupta’s show that the maximum error is about 10%. Such phenomenon is due to the fact that (i) the interference fit generated by bearing installed on the spindle was not considered, which would lead the initial contact angle to decrease, (ii) Gupta’s ADORE dynamics program considered the acceleration influence, but the proposed mechanical model of ball bearing in this paper did not considered such factor.
4. Dynamic Analysis of Spindle Bearing
4.1. Influence of Working Condition Parameters
The changes of Fa, Fr, and ni have significant influence on the dynamic characteristics of bearing. Based on the variation of these three working conditions (Table 3), the variation rule of bearing dynamic parameters is analyzed.
4.1.1. Influence of Working Conditions on the Spin-To-Roll Ratio
As an important parameter of the angular contact bearing at high speed, spin-to-roll ratio represents the state of the ball rolling in the raceway. Friction and heating are produced during the spinning of the raceway relative to the ball, and the rotation sliding is more serious with large spin-to-roll ratio, which will lead to the severe heat and wear. Therefore, the spin-to-roll ratio should be reduced as much as possible during operation. The change rule of spin-to-roll ratio is analyzed under three working conditions as tabulated in Table 3.
Figures 8 and 9 show that the variations of the spin-to-roll ratio of ball- inner and ball-outer raceway with the change of Fa. It can be illustrated that (i) the spin-to-roll ratio of the ball-inner raceway is much larger than that of the ball-outer raceway under the constant Fr and ni, (ii) the ball-raceway spin-to-roll ratio decreases with the increase of Fa, and such phenomena are due to the fact that Fa can reduce axial clearance and the spin angular velocity of the ball-inner and ball-outer raceways and increase the spindle bearing axial preload, which will result in spin-to-roll ratio lower, (iii) the ball spin-to-roll ratio is different at any ball azimuth location with the influence of Fr, (iv) there exists a certain spin-to-roll ratio between the ball and the outer raceway, which indicates that the high speed spindle bearing of oil lubrication does not conform to the hypothesis of outer raceway control, and (v) when Fa=1000 N and Fa=1500 N, the ball at the location angle of 120° has the maximum spin-to-roll ratio; with the increase of the load, the spin-to-roll ratio of the ball in this position gradually becomes the minimum. The reason for this phenomenon is that when bearing axial load is relatively small, there is axial clearance between ball and outer ring, which cannot make the ball press outer ring tightly that results in high spin angular velocity.
Figures 10 and 11 show the variation curves of the spin-to-roll ratio of ball-inner and ball-outer raceways with respect to the radial load. Results show that, with constant Fa and ni, the spin-to-roll ratio of the ball-inner raceway decreases in the loading area and increases in the nonloading area, because the radial load can decrease the radial clearance in the loading area and then the ball and the raceway are tightly pressed, which resulted in the spin-to-roll ratio reducing. On the contrary, the spin-to-roll ratio will increase in the nonloading area. It also can be found that the radial load has less effect on the spin-to-roll ratio of the ball-outer raceway than of the ball-inner raceway.
As shown in Figures 12 and 13, under the constant loads Fa and Fr, the spin-to-roll ratio of the ball-inner raceway increases obviously with ni increase; comparatively, the spin-to-roll ratio of the ball-outer raceway increases gradually. Such phenomena are due to the fact that the centrifugal force increased with ni increasing, which leads to the decrease of the contact load of the ball-inner raceway and the increase of the sliding of the ball-inner raceway and the spin-to-roll ratio obviously. The increase of the centrifugal force can also increase the contact load of the ball and the raceway, restrain the sliding between the ball and the raceway, and slow down the spin-to-roll ratio increase. The ball at location angle of 120° has the maximum spin-to roll ratio with the inner ring and the minimum spin-to-roll ratio with the outer ring; the reason for this phenomenon is that the ball at location angle of 120° is affected by the maximum centrifugal force.
4.1.2. Influence of Working Conditions on
Figure 14 shows the curves in the direction of y-axis in the azimuth coordinate system with the change of Fa. Under the condition load Fr and the constant rotational speed ni, gradually increases with Fa increasing. When Fa=3000 N, is basically stable at any ball azimuth location. Such phenomena suggest that Fa can reduce the axial clearance and decrease the influence of Fr on the ball, and the load condition of the ball tends to be steady.
The changing curves of along y-axis in the azimuth coordinate system with the change of Fr are shown in Figure 15. It can be illustrated that when Fa and ni are constant, gradually decreases with the increase of Fr, which is larger in the loading area. Such phenomena are due to the fact that Fr can reduce the radial clearance and inhibits the gyroscopic motion between the ball and the raceway and typically lead to decreasing significantly in loading area.
As shown in the Figure 16 of changing curves with the change of ni, under the constant load Fa and Fr, the value of gradually increases with ni increasing, and the of the ball varies obviously with different ball azimuth location. increases slowly with the increase of ni in the loading area, whereas the increase is obvious in nonloading areas. Such phenomena are due to the fact that the radial load holds the ball and the raceway tight in the loading area and inhibits the changes of the ball in the space position, which causes increasing slowly. When ni is greater than 110000r/min, at location angle of 120°, the value of the ball reaches the minimum. The reason for this phenomenon is that the ball at this position bear the maximum load because of radial load , and the increased centrifugal force compacts the ball and the raceway, resulting in the minimum here.
4.2. Influence of Working Conditions on Cage
4.2.1. Cage Forces
There exits three types of interaction conditions between the ball and the pocket of cage, such as (i) the pocket colliding the balls, (ii) the balls colliding the pocket, and (iii) no contact between the ball and the pocket. In order to study the relative motions between the ball and the pocket, the three working condition as tabulated in Table 2 were analyzed.
Figures 17–19 illustrate the curves between the ball and the pocket of cage with different Fa, Fr, and ni. In these Figures, the negative value represents the pocket colliding the ball and conversely the positive is the ball colliding the pocket. (i) It can be illustrated that the normal load between the ball and the pocket of cage is very small during the steady-state working period; (ii) the ball and the pocket collide with each other. It will last until the dynamic equilibrium state is achieved during the bearing steady working state. (iii) Figure 19 shows that as ni increases, the normal load between the cage and the ball gradually decreases in the loading area. The reason for this phenomenon is that the centrifugal force increases with ni increasing in the loading area; it leads to the fact that the ball and the raceway are compressed; and the result is that the ball rotation angular velocity decreases, making the contact force between the ball and the cage decrease. In no-load areas, the opposite is true.
4.2.2. Cage Rotation Speed
In this paper, the relationship between the cage rotational angular speed and the ball revolution angular speed is studied from the three working conditions listed in Table 2.
and curves of with different Fa, , and ni are shown in Figures 20–22, which suggest that and decrease with the increase of Fa, and increase with the increase of ni, while the change of Fr has little effect on and . The value of lies in the middle position of the values of . Therefore, it is reasonable to take the average speed value of the balls as the cage rotational angular speed.
4.3. Influence of Working Conditions on Gyroscopic Moment
The gyroscopic moment is a dynamic inertial force produced during the process of bearing operation, which is harmful for the bearing operation condition. Especially, under the condition of high speed and light load, it causes the sliding, which will lead to serious friction, makes the bearing internal fever, heats up, and accelerates the bearing degeneration.
Figures 23 and 24 show the curves of My and Mz with the change of Fa. Under the constant values of Fr and ni, (i) My and Mz increase obviously with the increase of Fa, (ii) when Fa=3000 N, Mz tends to be stable at each ball azimuth location. Such phenomena are due to the fact that Fa reduces the axial clearance and the impact of Fr on balls and results in the same load condition for balls.
Figures 25 and 26 show the curves of My and Mz with the change of Fr. Under the constant values of Fa and ni, My and Mz gradually decrease with the increase of Fr. It is because that the increase of Fr will decrease the radial clearance of the bearing; therefore, the rolling body and the ring are pressed tightly, and the gyroscopic moment is suppressed and reduced.
Figures 27 and 28 show the curves of My and Mz with the change of ni in the direction of y and z of the azimuth coordinate system of the ball. It is illustrated that My and Mz increase significantly with the increase of ni under the constant load of Fa and Fr.
5. Conclusions
(1)The change of Fa, Fr, and ni can affect the spin-to-roll ratio of ball-inner raceway and ball-outer raceway. In different working conditions, the spin-to-roll ratio of the ball-inner raceway is greater than that of the ball-outer raceway. In high speed, the outer raceway control hypothesis is only a special case. In general, there is a spin angular velocity between the ball and the outer raceway.(2)The increase of Fa and ni of spindle bearing can cause increase; however, the increase of Fr will cause decrease. A specific Fa can reduce the axial clearance of the bearing, which will make the azimuth of the ball tend to be stable in the space.(3)The interaction between the ball and the pocket of cage is that not only does the ball collide with the pocket but also the pocket collides with the ball, which can achieve the dynamic equilibrium state of the bearing steady-state work.(4)It is reasonable to take the average speed value of the balls as the cage rotational angular speed.(5)Gyroscope moment increases significantly with the increase of Fa and ni. The increase of Fr can suppress the gyroscopic movement.
Nomenclature
| O_XYZ: | Inertial coordinate system |
| O_xyz: | Coordinate system |
| : | Contact angle (°) |
| : | Semimajor axis of the contact area (mm) |
| b: | Semiminor axis of the contact area (mm) |
| : | Spiral angle (°) |
| : | Yaw angle (°) |
| : | Pitch diameter (mm) |
| F: | Hertz contact force (N) |
| : | Thrust load (N) |
| : | Radial load (N) |
| : | Force act on cage by ball (N) |
| f: | Groove curvature radius coefficient |
| G: | Dimensionless material parameters |
| h: | Center oil film thickness (mm) |
| : | Central oil film thickness (mm) |
| : | The inertia moment of ball |
| J: | Inertia moment |
| K: | Hertz contact stiffness coefficients |
| m: | Quality (kg) |
| : | The gyroscopic moment along z-axis in the azimuth coordinate system of ball (N.mm) |
| : | The gyroscopic moment along y-axis in the azimuth coordinate system of ball (N.mm) |
| ni: | The inner ring rotational speed (r/min) |
| : | Displacement vector |
| : | Radius (mm) |
| : | Density (g/cm3) |
| : | Density of the mixture gas (g/cm3) |
| T: | Tangential force (N) |
| : | Angular speed (rad/s) |
| : | The cage rotational angular velocity (rad/s) |
| : | The ball revolution angular velocity (rad/s) |
| : | The ball rotation angular velocity along y-axis in the azimuth coordinate system of ball(rad/s) |
| : | Relative sliding velocity (m/s) |
| δ: | Elastic approach (mm) |
| : | Relative angular displacement (°) |
| μ: | Friction coefficient. |
| a: | The azimuth coordinates of the ball |
| r: | The fixed body coordinate system of inner ring |
| i: | Inertial coordinate |
| c: | The fixed body coordinate system of the cage |
| p: | The pocket coordinate |
| g: | Contact coordinate. |
| b: | Ball |
| i: | Inner ring |
| e: | Outer ring |
| c: | Cage |
| p: | Cage pocket |
| g: | Contact point |
| j: | The jth ball |
| x, y, z: | The coordinate components. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors confirm that article content has no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 51805299), Shandong High School Research Project (J18KA004), and the subtopic of the program in the field of Intelligent Manufacturing and Advanced Materials, Scientific and Technological Innovation Plan (no. 13521103002) in Shanghai University.