Assessment of Bearing Dynamic Characteristics by Numerical Modeling with Effects of Oil Film and Centrifugal Deformation

This paper developed a modified quasi-static model (MQSM), considering the oil film thickness between the bearing parts and the centrifugal deformation of the inner ring, and contrasting with traditional quasi-static model (TQSM), to analyze the dynamic characteristics of spindle bearing.Themodel was verified with the experimental results. A systematic parametric analysis wasmade to investigate the influence of applied load, inner ring rotation speed (ni), and the radius coefficient of groove curvature (RCR) on the contact load, contact angle, and heat generating rate. The results show that there is a smaller influence on the contact load, contact angle, and heat generation of bearing with the changes of ni and axial load (Fa) of bearing in the case of MQSM and TQSM. But the radial load (Fr) and RCR have great influence on this.


Introduction
Kinematic accuracy of the main shafts commonly determines the machining quality, performance, service life, and reliability of high-speed machine tool, in which the supporting spindle bearings play a very important role [1,2].The higher speed and reliability of spindle bearings subjected to the continuously large thrust loading are demanded during high rotational speed machining.Therefore, it is necessary to create an accurate model to reflect the dynamic characteristics of spindle bearings, including rotational speed, contact temperature, and dynamic loading [3][4][5][6].Especially, the oil film thickness and inner ring centrifugal deformation should be taken into account for the assessments of the spindle bearings' dynamic characteristics, which may significantly affect the geometry relationship of the bearing internal channel curvature center and the ball center obviously.
It was found that the centrifugal expansion deformation of the bearing inner ring is much larger than the deformation of the shaft with the increase of rotational speed.Based on a dynamic model of spindle bearings, the research of Cao et al. [7] indicated that the bearing contact angle decreases with the increase of contact load by introducing the centrifugal expansion deformation of the bearing inner ring.The general spindle bearing FEM [8] showed that the impact factors of high-speed system, such as bearing radial stiffness, centrifugal force, and gyroscopic moments, have a significant effect on dynamic characteristics of spindle system in high speeds.A coupled spindle bearing model was established considering the centrifugal deformation by Hong et al. [9], and the simulation and experimental results indicated that there is a correlation between the natural frequency and the number of bearings.The Jones' bearing model was updated by Guo et al. [10], by considering of radial centrifugal expansion and thermal deformations on the geometric displacement in the bearings, which can predict the contact angle, deformation, and load between rolling elements and bearing raceways more accurately.
During the high rotating process of the spindle bearings, lubrication oil film can also bear a part of multiple orientation loading, which plays significantly important roles in the dynamic characteristics and the distance between the bearing internal channel curvature center and the ball center [11][12][13][14].The shear of oil film will cause remarkable heat of hydrostatic 2 Mathematical Problems in Engineering hydrodynamic hybrid bearings, which will lead to decreasing of bearing clearance and even seizure especially at high rotational speed [15].An integrated thermal model [16] was numerically established to calculate the heat generation of spindle bearings and temperature distribution of the spindle system, considering the rotation speed, preload, and oil film thickness.Therefore, the machine performance and dynamic design of machine spindle can be improved by taking into account the impact of oil film thickness.
To analyze the bearing dynamic characteristics, the typical quasi-statics modeling of spindle bearings was widely utilized.A five degrees of freedom quasi-static model of ball bearing under nonuniform preload was established by Li et al. [17], and the results indicated that proper nonuniform preload can improve the contact status between balls and rings and can also reduce the heat generation rate and the excessive local heat in ball bearing under practical working conditions.Kim et al. [18] employed the quasi-static analysis model by taking dynamic effects into account to optimize the nonstandard angular contact ball bearing for the main shaft of a grinder.The fatigue life calculations in rolling bearing simulations were evaluated by a new quasi-static multidegree of freedom tapered roller bearing model by considering non-Hertz contact pressures in time-domain simulations [19].However, the assessment of the dynamic characteristics of the contact load, contact angle, and bearing heat with the increase of rotational speed, axial load, radial load, and coefficient of raceway curvature radius (RCR) had not taken into account both the factors of film oil thickness and centrifugal deformation.Therefore, the establishment of considering both the deformation of film thickness and centrifugal bearing mechanics model is extraordinarily important for a more accurate analysis of the dynamic characteristics of the spindle bearings.
In this paper, a modified quasi-statics model (MQSM) was established by considering the influence of film thickness and centrifugal deformation, which is expected to make the bearing dynamic performance analysis more effective.Section 2 showed the calculation algorithms of the deformation of inner rings and rotors with the effect of film thickness and centrifugal expansion deformation.The MQSM was established in Section 3 by considering film thickness and centrifugal expansion deformation to reflect spindle bearings' dynamic characteristics.The spindle bearing heat was applied as the comparison parameter with the experimental results, the calculation of which was shown in Section 4. The experimental and MQSM results in Section 5 were given and discussed, to reveal the changing rule of the dynamic characteristic parameters of the spindle bearings.Section 6 presented the concluding remarks.

Deformation of Inner Ring and Rotor
. .Introducing Centrifugal Expansion Deformation.The centrifugal force of bearing inner rings can hardly be ignored and will result in centrifugal expansion deformation, particularly in high rotational speed.On the basis of the elastic mechanics theory, the bearing inner rings and the cooperating rotating shafts are simplified in advance as the thin-walled rings and the thick walled cylinder, respectively.Thereafter, centrifugal expansion deformation calculation formula can be deduced at the coordination position of the shaft and the inner ring by ( 1) and (2) [7]: where   is the centrifugal expansion deformation of shafts The mutual interference fit coupling and constraints among the inner ring and shaft have frequently occurred, whereas the centrifugal expansion of the inner ring diameter and rotor outer diameter can reduce the amount of interference on fit surfaces, when subjected to centrifugal force.The magnitude of interference caused by centrifugal expansion of the shaft and the inner ring can thus be determined as As known, centrifugal force deformation is proportional to the square of the rotating speed, and the centrifugal force deformation mainly occurred on the inner ring on the mating surfaces of the shaft and the inner ring with the rotational speed increase.The shaft centrifugal expansion deformation has less effect on the change of bearing radial internal clearance.Thereby, the factor of the centrifugal expansion deformation of the inner rings will be merely taken into account to research the impact of the centrifugal expansion deformation of the inner rings and shafts on the bearing internal geometry relationship.
. .Introducing Oil Film ickness.Lubrication condition plays remarkable role in spindle bearings; therefore, certain thickness of oil film between ring channels should be recommended.Based on the theory of elastic hydrodynamic lubrication (EHL), oil film stiffness is determined by the minimum oil film thickness.Under the condition of isothermal and adequate oil supply, dimensionless center oil film thickness   and dimensionless minimum oil film thickness   between rolling body and ring channel can be determined as [11]  0 = 2.69 0.67  0.53  −0.067 (1 − 0.61 −0.73 )  min = 3.63 0.68  0.49  −0.073 (1 −  −0.68 ) (4) where  is the dimensionless velocity parameters;  is the dimensionless material parameters;  is the dimensionless load parameters; k is the ellipticity.Thereafter, center oil film thickness ℎ 0 and minimum oil film thickness   between the rolling body and ring channel are then determined as ℎ 0 =    0 = 2.69 0.67  0.53  −0.067 (1 − 0.61 −0.73 )   ℎ min =    min = 3.63 0.68  0.49  −0.073 (1 −  −0.68 )   (5) where   is the rolling element along the movement direction of the equivalent radius of curvature.
Figure 1 shows the distribution of film thickness and pressure under EHL and nonlinear Hertz contact condition, where the difference is whether considering the existence of lubricating oil film.The existence of the lubricating oil film will reduce the elastic deformation between the roller and the ring channel.Consequently, to calculate the actual contact stiffness, the amount of the radial tendency can be regarded as the radial elastic deformation between the roller and ring channel by considering the oil film thickness, as follows: where Δ is the elastic deformation under EHL condition;  is the elastic deformation at Hertz contact position; ℎ 0 is the central oil film thickness at EHL contact position.

The Spindle Bearing Mechanical Characteristics Analysis
. .e Ball Motion Analysis.For easy analysis, coordinate system of spindle bearing is established as shown in Figure 2. The parameters   ,   , and   are the rotation angular velocity corresponding to the ball 3-D coordinate components of x, y, and z, respectively.The parameters  and   are the ball's spiral angle and yaw angle, respectively.Assuming   = 0 in this paper, the rotation angular velocity in the coordinate component of  will be   = 0.
The outer ring of spindle bearing is set as the fixed part, and constant angular velocity of inner ring rotates is denoted by   .Consequently, the angular velocity of the ball   can be determined as [3] where   =   /  ,   is the ball diameter,   is the pitch diameter of the bearing, and   and   are the contact angles between the ball and the inner and outer ring, respectively.The ball's rotation angular velocity   is then calculated by [3] Given that the spindle bearings are a kind of high-speed bearings, on the basis of the control theory of ring and assuming the outer raceway is controlled, the spiral angle   can be determined as Then the spin angular velocity of bearing inner ring can be calculated by . .Displacement Coordinate Equation.The outer ring and bearing seat are fixed in advance; therefore, the outer ring curvature center will be always in the same position.In our dynamic characteristics analysis model, the relative radial, axial, and angular displacements of bearing inner and outer rings are denoted as   ,   , and , respectively, caused by the combined action of the radial force   , axial force   , and moment .Meanwhile, the geometric positions of the bearing components with the effect of   ,   , and M are shown in Figure 3, in which (i) the axial and radial distances between the inner and outer channel curvature centers are denoted as   and   , respectively, (ii) the distances between the ball center and the inner and outer channel curvature centers are denoted as  ij and   , respectively, and (iii) the axial and radial changed distances of the center of channel curvature are denoted as   and   , respectively.U aj , U rj Figure 3 shows the geometric position relationship between the jth ball centers with angle position and the inner and outer channel curvature centers.
Without the effect of loading, the distance between the inner and outer channel curvature centers can be determined as  = (  +   − 1)    (11) where   and   are the inner and outer raceway groove curvature radius ratios, respectively.   is the ball diameter after considering deformation.
The ball locations of the jth ball are obtained by where  0 is the initial contact angle.As shown in Figure 3, according to the relative position of curvature center, the displacement coordinate equations of the ball center change: Assessments of the spindle bearings' dynamic characteristics by traditional quasi-statics model (TQSM) used to pay no attention to the effect of the oil film thickness and centrifugal force, which will affect the geometry relationship of the bearing internal channel curvature center and the ball center obviously.In this paper, to make the bearing dynamic performance analysis more accurate, a modified quasi-statics model (MQSM) was established by considering the influence of film thickness and centrifugal deformation.Therefore, the modified parameters of   ,   ,  ij , and   , as shown in Figure 3, are redefined considering the oil film thickness as follows: where   is the circular radius of inner ring channel curvature center,   is the inner ring centrifugal force deformation,   and   are the Hertz elastic deformations at the jth ball and the inner and outer contact points, respectively, and ℎ  and ℎ  are the center oil film thickness at the jth ball and inner and outer ring contact point, respectively.
. .Force Balance Equation of Ball and Inner Ring.When the bearing is at high speed, the ball and the inner ring stresses can be shown as in Figure 2. Force equilibrium equations of each ball can be established as follows: where   and   are the contact stiffness coefficients between ball and inner and outer rings, respectively;   is the centrifugal force of the jth ball;   and   are the friction force between the ball and inner and outer rings, respectively.
The force balance equations of the inner ring in the horizontal and vertical direction, and moment equilibrium conditions can be written as where   is the curvature radius of the inner ring channel.

The Spindle Bearing Heat
Based on the calculation theory of Harris high-speed bearing frictional heat, the generated heat of the spindle bearing mainly includes the following several parts.
(1) Power loss of the mating pairs of ball and raceway during relative sliding process can be determined as where   is traction force between the jth ball and its ring in the direction of the ellipse long axis and V  is relative sliding velocity between the jth ball and its ring in the direction of the ellipse long axis.
(2) Spin movement between the ball with the inside and outside raceway is an important factor to the power loss [6] where   is the jth ball spin torque,  is the friction coefficient,   is the contact load between the jth ball and its ring, Σ  is the complete elliptic integral of the second kind between the jth ball and its ring,   is the contact ellipse semimajor axis, and   is the spin angular velocity of the jth ball.
(3) Because of power loss caused by the gyroscopic motion, the jth ball can be determined as where   is the gyroscopic moment of the jth ball, J is the inertia moment,   is the orbital angular velocity of the jth ball, and   is the rotation angular velocity of the jth ball.(4) With the ball revolution, the power loss caused by viscous damping effect of the mixture of oil and gas can be determined as where   is the resistance force of the jth ball caused by the mixture of oil and gas,   is the density of the mixture of oil and gas, and   is the resistance coefficient.Finally, the total power loss of the bearing can be determined as where  is the number of the ball.

The MQSM Model Results and Discussion
. .Experimental Assessment.The MQSM of the spindle bearing was established by MATLAB tool, and the flow chart is shown in Figure 4. To evaluate the MQSM availability, experiments were conducted on a self-designed motorized spindle bearing rig (Figure 5) in Shanghai Intelligent Manufacturing and Robot Key Laboratory.The required equipment and test instruments include test spindle (model 80GS60A), electric spindle support, a pair of B7006C bearing, motorized spindle overhang shaft, bearing inner and outer baffle, sleeve, compressive bar, waveform, Jordan weight, piezoelectric pressure sensor, temperature sensor and related Labview signal acquisition device, and KOLLMORGEN transducer (model: ACO5000D), as shown in Figure 5.
The test principle is shown in Figure 6.The tested bearings installed back-to-back are installed on the outer end of the rotor of the driving electric spindle, which makes the bearings under test suspended.The driving electric spindle drives the inner ring of the bearings rotating.Because of the friction between the bearing inner and outer ring and balls, the bearing overhanging outer ring and the outer sleeve with its interference fit tend to rotate together with the inner ring without additional resistance.A pressure lever is installed on the sleeve that is fitted with the bearing outer ring, so that the upper end of the sleeve is attached to the piezoelectric pressure sensor fixed on the support.When the outer sleeve rotates with the inner ring, the pressure sensor will be pressed and the pressure   will be measured.
The voltage signal measured by the piezoelectric pressure sensor is inputted into the computer through the acquisition card and analyzed and processed by Labview.The value of the friction moment of a single bearing and the actual friction moment of the bearing can be obtained: where   is friction moment of the bearing,   is the pressure value measured, and   is the pressure test point with a distance of 55 mm from the axle center.
where   is the heat generation power of a single bearing and   is angular velocity of bearing inner ring.
In the test, the change of electric spindle speed is adjusted by frequency changer, and the height difference of inner and outer ring spacer is controlled by waveform ring, so as to adjust the size of bearing axial load.The magnitude of the radial load is changed by lifting the weight on the bolt directly below the bearing outer sleeve.
In the tests, the bearing heat production rates were obtained by the monitored bearing friction torque.Compared with the MQSM results, (i) the experimental results of the bearing heat production rate with various rotation speeds under the conditions of the axial load   = 130 N and radial load   = 0 N were shown in Figure 7; (ii) the experimental results of the bearing heat production rate with various axial loads under the conditions of inner ring speed   = 5000 r/min and radial load   = 0N were shown in Figure 8; (iii) the experimental results of the bearing heat production rate with various radial loads under the conditions of inner ring speed   = 20000 r/min, axial load   = 130 N, were shown in Figure 9.
The results indicate that the change of bearing heat production rates with the increase of rotational speed and axial load, and radial load has similar tendency in experiments and MQSM.Therefore, it can be concluded that the high-speed angular contact ball bearing mechanics model of MQSM in this paper has excellent performance.
. .Results of Dynamic Characteristics.The changes of   ,   , ni, and RCR have significant influence on the dynamic characteristics of bearing.Based on the variation of these four conditions (Table 1), the variation rule of bearing dynamic parameters is analyzed.
The results of bearing dynamic characteristics in TQSM and MQSM are compared in Figures 10-25.The parameters in this mathematical model of bearing are in ADOSE of P.K. Gupta bearing dynamics program [10] was tabulated in Table 2.The 10 spindle oil was chosen with the lubrication oil dynamic viscosity of  0 =0.027Pa•s and viscosity coefficient =2.3e-8Pa−1 [11].The environment temperature of 25 ∘ C was set.Ring and ball materials are bearing steel.
During the calculation process, it is found that the bearing sizes are very small comparing the whole bearing system, and the small centrifugal deformation has then little influence on the bearing dynamic characteristics.Therefore, the effect of the centrifugal deformation on the dynamic characteristics can be ignored, and then MQSM just considers the influence of oil film thickness.

. . . Influence of Working Condition and RCR on Oil Film
ickness.  ,  r ni and RCR have significant influence on the oil film thickness between the bearing inner and outer ring and the ball.The changes of oil film thickness were analyzed in four cases listed in Table 1.
Figures 10 and 11 show the relation curve between the bearing inner and outer rings and the ball with the bearing speed.The figures show that the oil film thickness increases obviously with the increase of ni.The oil film thickness between the outer ring and the ball is larger than that of the inner ring.
Figures 12 and 13 show the relation curve of   between bearing inner and outer ring and ball.As shown in the figures, as   increases, oil film thickness decreases.The reason for this phenomenon is that the increase of   reduces the axial clearance inside the bearing.
Figures 14 and 15 show the relation curve of   between bearing inner and outer rings and ball.As shown in the figure, with the increase of   , the oil film thickness between the bearing inner and outer ring and the ball decreases with   the increase of   in the loading area.In the nonloaded area, it increased with the increase of   .The reason for this phenomenon is that the increase of   compacts the ring and ball at the bearing loading area and reduces the radial clearance.In the nonloaded area, the ring and the ball are relaxed and the radial clearance increases.
Figures 16 and 17 show the variation curve of RCR between the bearing inner and outer ring and the ball.As shown in the figure, oil film thickness increases with the increase of RCR.The reason for this phenomenon is that the increase of RCR reduces curvature tolerance between the ring and the ball, which is conducive to the production of oil film.However, if RCR is too large, bearing capacity of bearing will be reduced.

. . . Influence of Working Condition on Dynamic Parameters.
Figures 18-20 indicate that with the axial load of 3000 N, the dynamic characteristics of the contact load and bearing heat in TQSM and MQSM have little difference with the increase of the spindle speed from 15000 r/min to 35000 r/min.However, the contact angle of inner rings in MQSM is slightly less than that in TQSM as shown in Figure 10.Meanwhile, with the spindle speed of 30000 r/min, the similar results of    Figures 24-26 show that with the spindle speed of 30000 r/min and the thrust load of 3000 N, (i) the result values of dynamic characteristics of the contact load and bearing heat in MQSM are larger than that in TQSM when the radial load is beyond 1500 N, and (ii) the contact angle in MQSM is less than that in TQSM under the radial load of 2500 N. Therefore, the change of inner ring contact angle in MQSM is more stable than that in TQSM, which proves that the results of MQSM are more credible.
With the spindle speed of 30000 r/min and the axial load of 3000 N, Figures 27-29 show the dynamic characteristics with various RCR coefficients from 0.52 to 0.55 in MQSM and TQSM.Results indicate that all the values of the dynamic characteristics of the contact load, contact angle, and bearing heat in MQSM are less than those in TQSM.When RCR is more than 0.54, TQMS calculation results have a relatively  obvious mutation (except the inner ring contact angle).This is caused by the fact that the influence of oil film is not taken into account when calculating with TQMS.When RCR is greater than 0.54, the contact load and contact angle will increase, which will lead to the increase of heat generation.When calculating with MQMS, the lubricating oil film will form a layer of oil film between ball and raceway due to the consideration of lubrication, which will weaken the influence of the change of the curvature coefficient of the groove and will not cause mutation.Center oil film thickness : Magnitude of interference caused by centrifugal expansion : Inertia moment : Stiffness coefficients : Distances between the ball center and the ring channel curvature centers   : Gyroscopic moment   : Spin torque : Contactload

Figure 1 :
Figure 1: The distribution of film thickness and pressure under EHL and Hertz contact condition.

Figure 2 :
Figure 2: Movement and force of the inner ring and ball in the bearing coordinate system.

Figure 3 :
Figure 3: The position relationship between the jth ball center and the inner and outer channel curvature centers.

Figure 4 :
Figure 4: Flow chart of the MQSM establishment.

Figure 6 :
Figure 6: Configuration of the bearing test system.

Figure 7 :
Figure 7: Results of the bearing heat production rate with various rotation speeds.

Figure 8 :
Figure 8: Results of the bearing heat production rate with various axial loads.

Figure 9 :
Figure 9: Results of the bearing heat production rate with various radial load.

Figure 10 :
Figure 10: The influence of ni on oil film thickness between ball and outer ring.

4 Figure 11 : 3 Figure 12 :
Figure 11: The influence of ni on oil film thickness between ball and inner ring.

4 Figure 13 :
Figure 13: The influence of   on oil film thickness between ball and inner ring.

Figure 14 :
Figure 14: The influence of   on oil film thickness between ball and outer ring.

4 Figure 15 :
Figure 15: The influence of   on oil film thickness between ball and inner ring.

Figure 16 : 4 Figure 17 :
Figure 16: The influence of RCR on oil film thickness between ball and outer ring.

Figure 18 :
Figure 18: Contact load with various ni.

Figure 19 :
Figure 19: Contact angle with various ni.

Figure 20 :
Figure 20: Heat generation rate with various ni.

Figure 29 :
Figure 29: Heat generation rate with various RCR.

;
, ]  , and   are the density, poison ratio, and elasticity modulus of the shaft, respectively;   is the inner diameter of the spindle;   is the centrifugal expansion deformation of the inner ring;   , ]  , and   are the density, poison ratio, and elasticity modulus of inner ring, respectively;   is the angular speed of inner ring; d and   are the inner diameters of the inner ring and its channel, respectively.

Table 1 :
Four different conditions.

Table 2 :
The bearing parameters in MQSM.
: Resistance coefficient   : Ball diameter    : Ball diameter after considering deformation   : Bearing pitch diameter   : Inner diameters   : Inner diameter of the spindle   : Elasticity modulus of inner ring   : Elasticity modulus of shaft   : Inner raceway groove curvature radius ratios   : Outer raceway groove curvature radius ratios : Centrifugal force