A combined experimental and analytical method to determine the EHL friction force distribution between rollers and outer raceway in a cylindrical roller bearing

Friction force is a crucial factor causing power loss and fatigue spalling of rolling element bearings. A combined experimental and analytical method is proposed to quantitatively determine the elastohydrodynamic lubrication (EHL) friction force distribution between rollers and outer raceway in a cylindrical roller bearing (CRB). An experimental system with the instrumented bearing and housing was developed for measuring radial load distribution and friction torque of bearings. A simplified model of friction force expressed by dimensionless speed, load, and material parameters was given. An inequality constrained optimization problem was established and solved by using an experimental data-driven learning algorithm for determining the uncertain parameters in the model. The effect of speed, load, and lubricant property on friction force and friction coefficient was discussed.


Introduction
The wide application of rolling element bearings stems from the basic fact that rolling friction is less than sliding friction [1]. Although the tangential friction force between rollers and raceways is much smaller than the normal contact force, it represents heat generation and power loss of the rolling element bearing [1,2]. Besides, the tangential friction force is one of the key mechanical parameters affecting the fatigue life of the rolling element bearing. On the one hand, local friction force on the raceway surface caused by abnormal sliding will aggravate the wear of the surface [3][4][5]. On the other hand, the tangential friction force induces subsurface cracks of raceways to propagate towards to the surface and finally leads to material peeling [6,7]. In rolling element bearings, the local tangential friction force varies greatly at different azimuths of roller-raceway contact [8], which is called friction force distribution in this paper. Thus, obtaining the uneven friction force distribution between rollers and raceways accurately allows the engineers to correctly determine the heat generation rate and locate the weak area facing fatigue or wear failure in rolling element bearings.
Many models have been developed to understand the friction force produced by roller-raceway contacts of rolling element bearings in the past decades. Most of the models included parameters such as normal load, relative speed, lubricant properties, and material modulus as variables [9][10][11][12][13][14]. For instance, in isoviscousrigid models proposed by Hamrock and Jacobson [11] and Dowson and Higginson [15], the friction force was considered to be affected only by the relative velocity and normal load with almost equal contribution. With the development of elastohydrodynamic lubrication (EHL) theory, lubricant properties were introduced in the calculation of friction force and relative velocity in roller-raceway contact became a dominant factor in the EHL model [12]. Goksem and Hargreaves [10], Zhou and Hoeprich [16], and Pan and Hamrock [17] proposed some simplified EHL models to qualitatively determine the correlation between the friction force and other mechanical parameters such as rotating speed and contact normal load. However, the weights of these mechanical parameters varied a lot among these models and led to very different predictions when scanning on a realistic range of operating conditions [12]. Besides, most of these models were built by numerical methods, and there were not enough experiment results to support the ability of these simplified models in prediction of the bearing friction.
As for the experimental investigation of the friction force, some measurement methods were proposed based on optical, ultrasonic, or electrical technique to study the frictional contact of two bodies [13,18,19]. However, these methods did not apply to measuring the distributions of the internal and tiny friction forces in a rolling bearing component due to the difficulties caused by multi-body contact in the bearing. As another mechanical parameter related to relative motion except for friction force, the friction torque was paid much attention to experimental studies of bearing friction [20][21][22][23]. Compared with the local friction force in the bearing, the friction torque of the bearing can be more easily measured by several methods [20,21,24]. Nevertheless, the friction torque of a bearing is a macro parameter, and it is not enough to determine the local friction force in the bearing only by obtaining the measured friction torque.
Motivated by the significance of determining the friction force distribution upon the power loss and fatigue damage of bearings, this paper presents a new method for obtaining the EHL tangential friction force distribution between rollers and outer raceway in a cylindrical roller bearing (CRB). It is a combined experimental and analytical method, in which a simplified EHL model is adopted, and then optimized based on experimentally measured friction torque, radial load distribution, rotational speed, and properties of lubricants. The distributions of friction force and coefficient within the test CRB are then given by using the experimental data-driven EHL model to demonstrate the improvement in the quantitative prediction of the roller-raceway friction in CRBs.

Combined experimental and analytical method
A flow chart of the proposed method is illustrated in Fig. 1. To accurately obtain the local EHL friction force distributed at different azimuths in a CRB based on an analytical model, first, measured input parameters such as distributed radial load, rotational speed, and lubricant properties are necessary. Then, by introducing the measured parameters to a selected analytical model of friction force, calculated EHL friction forces between the raceway and each roller can be expressed. Further, to assess the accuracy of the calculated result, the analytical friction torque of the bearing calculated M is accordingly calculated and compared with the measured one measured .

M
If the analytical friction torque calculated M agrees well with the measured friction torque measured M , the analytical model is proved to be accurate and can be used to predict the EHL tangential friction force distribution in a bearing. Otherwise, the parameters in the analytical model should be adjusted and ensured by further comparison. Three assumptions are given in this paper as follows.
1) An elastohydrodynamic oil film exists between the rollers and the raceway while rolling contact bearings are rotating, which has been proved practically [25].
2) For EHL, the motion of a CRB is regarded as pure rolling [15,25].
3) The total friction torque of a CRB consists of rolling friction torque, sliding friction torque, friction torque of seals, and drag losses. Rolling friction torque absolutely dominates the total friction torque of a CRB under pure rolling conditions [26].
Based on these assumptions and Refs. [10,11,[15][16][17], for the line contact between the roller and raceway under pure rolling conditions with EHL, a simplified EHL model for the dimensionless friction force f F expressed by dimensionless speed U, dimensionless load W, and material parameter G can be given by Eq. (1): where β represents the specified azimuth position in a CRB, and 1 c -4 c are the undetermined parameters, which should be adjusted if analytical friction torque calculated M does not agree well with the measured one measured M in Fig. 1. U, W, and G can be calculated by Eq. (2): where  is the kinematic viscosity of lubricant, u is the relative rolling speed, i.e., the mean value of the linear speed of the two contact surfaces, E is the equivalent Young's modulus, ( ) R is the reduced radius of curvature in line contact, F is the normal contact load, l is the length of the roller-raceway contact line along the roller axis, and p  is the pressure viscosity index of lubricant. Then, the friction force f , F  in roller-outer raceway contact at the azimuth position β and the calculated friction torque calcualted M can be expressed by Eqs. (3) and (4), respectively: where o ( ) R is the reduced radius of curvature in rollerouter raceway contact, and o R is the radius of outer raceway.
In order to compare the calculated friction torque  5) and (6). This optimization problem can be solved by using statistical machine learning.

Experimental principle
According to the flow chart of the proposed method, the experimentally measured radial load distribution and friction torque within a bearing are necessary for the determination of the simplified EHL friction force model.
A new method based on strain detection was used here to measure the dynamic radial load distribution in CRBs with good accuracy. This measurement method has been successfully developed and implemented on a CRB in a high-speed train gearbox [27,28]. It is significant that instrumentation of the test bearing by strain sensors and calibration of a compliance matrix for strain-load transfer in this method. The detailed measurement principle of the roller-raceway normal loads and calibration of the compliance matrix are illustrated in Ref. [27].
To measure the friction torque of the roller bearing, | https://mc03.manuscriptcentral.com/friction a wire tension method [24] was adopted and developed. The principle of the wire tension method is shown in Fig. 2. The wire rope is wound once around the bearing housing and fixed in place. Two force sensors are connected at the two ends of the wound wire and fixed on the ground. When the bearing is stopping ( Fig. 2(a)), the radial load applied to the bearing can be controlled by adjusting the tension degree of the wire rope, and the total radial load F r applied on the bearing is equal to the sum of the test values F 1 and F 2 from the two force sensors. When the inner ring is rotating in the counterclockwise direction, as shown in Fig. 2(b), the friction force and torque of the roller-raceway rolling contact are generated, which result in a counterclockwise rotation trend of the bearing housing. The left side of the wire is relaxed, and the right side of the wire is tensioned by this rotation trend. The force measured by the left force sensor decreases while it increases on the right side. Thus, according to the principle of torque balance, the measured friction torque M measured generated by the roller-outer raceway rolling contact can be obtained by the product of the total numerical change of the two force sensors and the distance L between the wire and the bearing center in Eq. (7): In this wire tension method, the total applied load and the friction torque are simultaneously measured by the force sensor. Consequently, due to the high technical difficulty and cost of force sensors with both very high accuracy and wide range, the contradiction between measuring range and dividing value of the regular force sensor (the highest accuracy: 0.03% full scale) limits the use of the wire tension method to the condition of small applied radial load of the bearing or small bearing in Refs. [24,29]. In order to accurately measure the friction torque of rolling element bearings under the condition of heavy applied radial load, the wire tension method is developed in this paper by using multiple small range force sensors in parallel, as shown in Fig. 2(c). Multiple wire ropes instead of one are wound around a widened bearing housing while each wire rope connects with two small range force sensors at the two ends. When it is difficult to improve the accuracy of the sensor, the measuring accuracy of the friction torque M can be improved by small dividing value of ΔF 1 + ΔF 2 in Eq. (7) caused by small measuring range of the sensor, and the upper limit of the applied radial load can be increased by increasing the number of the force sensors.

Experimental setup
A bearing of 17 cylindrical rollers SKF NU 214 ECML was instrumented as well as its housing in the experiment. The specifications of the bearing are shown in Table 1. Figure 3(a) shows the test bearing instrumented by several strain gauges. The surface of outer ring corresponding to nine adjacent rollers was evenly divided into nine regions and used as possible load zone during the test. Strain gauges were glued at the center of each region circumferentially. The top test position is set to be 0, and the others are numbered as 1L, 1R, etc., as shown in Fig. 3(a). Accordingly, to provide enough space for the strain gauge, several  Radial clearance (mm) 0.040-0.075 [26] Roller diameter (mm) 15 Roller length (mm) 14 Contact angle (°) 0

Number of rollers 17
notches were introduced evenly distributed on the inner surface of the bearing housing as shown in Fig. 3(b). It has been proved that the radial load distribution within the bearing is not affected by the introduction of notches [27]. Each strain gauge location of the instrumented bearing corresponds to the center of a notch of the instrumented housing during the fit process. Besides, five parallel grooves were machined on the outer surface of the instrumented bearing housing for the application of the developed wire tension method to measure the friction torque. The distance L between the wire and the bearing center is 86.5 mm. Figure 4 provides the assembly diagram and main structure of the test rig. As displayed in Fig. 4(a), the strain gauge cables were gathered and led out through the holes on the housing and end cover after interference fit of the instrumented bearing and its housing. Two oil baffles were fixed on the two ends of the assembly, and the instrumented bearing was lubricated by oil bath in the experiment. Furthermore, the test shafting is shown in Fig. 4(b). The shaft was fitted with the bearing in interference and supported by two support bearings. A belt wheel was connected with the shaft on the end for the power input of the shafting. Figure 4(c) displays the torque measuring and loading structure of test rig. To measure the friction torque, the top of 10 tension sensors was connected with the wire ropes by the thread sleeve, while the bottom was fixed at the base. To apply the external radial load on the test bearing, the constraint on the bearing housing given by wire ropes provided the downward load to the bearing, while the loading platform was lifted by four jacks. The value of the external load of the bearing can be determined by the tension sensors or the pressure sensors located between the platform and jacks.
The photographic view of the experiment setup is shown in Fig. 5. The whole experimental system consists of five modules including transmission, loading, lubrication, temperature monitoring, and mechanical parameter measurement. In the transmission module, a motor connected with the test shafting by timing belt was used to drive the rotation of the test bearing. The rotational speed of the shaft can be set and adjusted in a range from 0 to 5,600 r/min by a motor controller. A hydraulic pump and four jacks were joined by a four-way oil valve and used as the loading module. Ten tension sensors with the full scale range of 0-2,000 N and the accuracy of 0.03% were used here, which decided that the maximum bearing load applied by the loading module was up  to 20 kN. Besides, the lubricating oil was supplied by the oil pump, and the temperatures of the bearing outer ring and lubricating oil were monitored by Temperature sensors 1 and 2, respectively. As for the module of mechanical parameter measurement, an electronic data acquisition (eDAQ) system (serial number 5448, HBM Company) was adopted in the experiment. The strain and tension sensors are connected with the eDAQ by signal cables.

Test conditions
Total 192 operating conditions of test bearing were  www.Springer.com/journal/40544 | Friction conducted. These operating conditions were combined by 12 applied load conditions, 8 shaft speed conditions, and 2 lubrication conditions. The external load was applied on the test bearing increasing from 0 to 12 kN with the increment of 1 kN. The contribution of self-weight was removed here. The shaft speed was set to increase from 500 to 4,000 r/min with the increment of 500 r/min. Two kinds of lubricating oil with different viscosity grades SAE 5W-30 and 75W-90 were used. During the experiment, in order to exclude the influence of temperature variation on the properties of the lubricant, the temperature of lubricating oil was kept at 43±1 °C by controlling the rotation time of the test bearing and using a cooling fan. The properties of the lubricating oil listed in Table 2 were tested according to the international standard ISO 3104:2020 [30]. Besides, a monitoring technique of cage slip based on frequency analysis of strain signals [31] was used to make sure the pure rolling of the test bearing. If the measured frequency of roller passing through the outer ring is equal to the kinematic one, the cage slip ratio is zero, and the bearing operates in pure rolling [31]. Under each combined operating condition, first, the test bearing was loaded after the zero adjustment of the strain signal. Then the rotating and stopping states of the test bearing were controlled to transfer to each other for five times as repeated test of the friction torque. The sampling frequency of both the strain and force signal was set to 10 kHz during the experiment.

Strain-based pure rolling proof and measured radial load distribution
As the key for proofing the pure rolling condition and measuring the radial load distribution within the test bearing, strain signals from all the 9 strain test positions were measured during the experiment. A part of the measured strain signals from once test under the operating conditions of shaft speed 4,000 r/min, applied load 12 kN, and lubricant oil 5W-30 is intercepted as an example and displayed in Fig. 6.
To monitor the cage slip based on frequency analysis of strain signals, the measured frequency of roller passing through the outer ring was obtained by the fast Fourier transform (FFT) of the measured strain signals, while the kinematic one was determined by the bearing size and shaft speed [31]. For example, Fig. 7 shows the FFT frequency spectra of the measured strain signals from the test position No. 0 under the test conditions of shaft speed 4,000 r/min, lubricant oil 5W-30, and several applied loads. The main frequency component of 479.8 Hz was the measured frequency of roller passing through the outer ring and did not change with the applied load. Meanwhile, the kinematic frequency of roller passing through the outer ring was calculated to be equal to 480.3 Hz under the conditions of shaft speed 4,000 r/min. Accordingly, cage slip ratios were almost nil within   the margin of error under these test conditions. Similarly, under other test conditions, the cage slip ratio was found to be nil as well based on the FFT of the measured strain signal. Therefore, the test bearing operated under the pure rolling condition in this work.
Besides, in Fig. 6, it is apparent that the strain signals from seven of the nine test positions (0, 1L, 1R, 2L, 2R, 3L, and 3R) vary sensitively with the applied load during loading process and fluctuate obviously during the rotation of the bearing. It reveals that there are seven adjacent rollers in the load zone under this operating condition. Besides, the variation in the peck values of each single strain signal can be observed in the magnified view in Fig. 6(b), which implies the variation in the roller-raceway contact load during different rollers running through the same test position.
According to the measurement principle of the radial load distribution, the measured radial load distribution under different test conditions can be obtained by the peak values of the measured distributed strain signal. Figure 8  considered to be mainly caused by radial geometrical differences among the rollers. Then, the radial load distribution under different test conditions can be determined by the real-time contact radial load. Figures 8(b), 8(c), and 8(d) present the effect of the applied load, shaft speed, and lubricant on the measured load distribution, respectively. In Figs. 8(b)-8(d), the range and mean were determined by the measured real-time contact radial load, and the distribution curves were fitted according to Harris calculating model about bearing load distribution [2]. As can be seen from Fig. 8(b), when the shaft speed and the lubricant was fixed, the load zone became wider, and the contact load distributed on each test position increased with the applied load increasing. In Fig. 8(c), under the condition of the applied load 12 kN, the measured load distribution under the shaft speed of 500 r/min was almost the same as that under the shaft speed of 2,000 r/min. By contrast, when the shaft speed was 4,000 r/min, the load distributed on each test position increased slightly due to the centrifugal force of rollers. Moreover, under test conditions, no effect of the lubricant property on the radial load distribution existed, as shown in Fig. 8(d).

Measured friction torque
To obtain the measured friction torque of the test bearing, the real-time force difference between the two ends of the wire ropes 1 2 ( ) ( ) F t F t  was recorded based on the tension sensors during the tests. As an example, the black curve in Fig. 9 provides the force difference of the wire ends 1 2 ( ) ( ) F t F t  during 5-time repeated tests under a certain operating condition.
The sum of 1 ( ) F t and 2 ( ) F t , which represents the applied load of the test bearing, is shown in Fig. 9 as well by the red curve. When the bearing switched between rotation and stop states during once test, the variation of the force difference 1 2 ( ) ( ) F t F t  was highlighted and marked as 1 Fig. 10. The point in Fig. 10 represents the mean value of the measured friction torque under each combined test condition, and the bound refers to the range of the minimum to the maximum. It was observed that under the operating conditions of fixed shaft speed and lubricating oil, the measured friction torque increased with the increase of the applied load, and presented an approximate power function correlation with the applied load. Besides, greater friction torque can be found under the test conditions of faster shaft speeds and higher viscosity of the lubricants in Fig. 10. As a result of the combined effect of applied load, shaft speed, and lubricating oil property, the values of the measured torque distributed between 0 and 3 N·m under these test conditions.

Simplified analytical model of EHL friction force
In order to ensure the parameters 1 c -4 c in the Fig. 9 Force sum F 1 (t) + F 2 (t) and difference F 1 (t) − F 2 (t) of the measured total load at both ends of the wire rope (test condition: shaft speed 1,000 r/min, applied load 12 kN, and lubricant oil 75W-90). In this paper, the Fmincon function [33] provided by the optimization toolbox in MATLAB was used to solve the inequality constrained optimization problem described in Eq. (6). Therefore, the parameters 1 c -4 c , which contributed to the global minimum value of the objective function Introducing these parameters into Eqs. (3) and (4) Fig. 11. A calculation error was defined as  to quantify the predictive effect of the simplified EHL model. As can be seen from Fig. 11, the calculated friction torque calculated M agrees well with the measured friction torque measured M and most of the calculation error is about 6 to 7 percent, which indicates that friction force f , F  can be well predicted by the simplified EHL analytical model, as shown in Eq. (9). Compared with those in the models in Refs. [10,16,17], the exponent of G in the simplified EHL model expressed by Eq. (9) is almost the same. However, dimensionless load, rather than speed, dominates the EHL friction force generated from roller-outer raceway contact in Eq. (9).

Distribution of the friction force between rollers and outer raceway
Introducing the measured contact radial load, shaft speed, and lubricant property into Eq. (9), the EHL friction force f , F  generated from different rollers and outer raceway contact can be predicted. Figure 12 presents the predicted EHL friction force distributions under test conditions of different shaft speeds and applied loads. The points in Fig. 12 represent the mean values of the EHL friction force distributed under different test positions, while the range shows the variation in friction force caused by the fluctuation of the radial contact load. Besides, the load zone refers to the region involved in the transfer and distribution of applied radial load. As shown in Fig. 12, under each combined test condition, the friction force within the bearing is in symmetric distribution, which is determined by the symmetric radial load distribution shown in Fig. 8(b). The maximum local friction force appears at the test position 0 and varies with shaft speed and applied load from 1 to no more than 7 N under these combined test conditions. Besides, in load zone, the value and distribution range of the friction force trend to increase with the increase of the applied load and shaft speed. Nevertheless, in unload zone, the friction force varies only with the shaft speed, while the increase of the applied load have no effect on the friction force. Thus, the speed can be found to contribute to the friction force generated from the roller-outer raceway contact by two ways. On the one hand, the friction force is directly influenced by the speed, as illustrated in Eq. (9). On the other hand, the dimensionless load W increases with the roller centrifugal force, which is positively correlated with the shaft speed, leading to the increase of the friction force.
In addition to friction force, the friction coefficient f  is also a significant parameter to characterize the friction performance of the roller-raceway contact [5,13] and can be predicted based on the simplified EHL model by Eq.   | https://mc03.manuscriptcentral.com/friction than 0.1 in unload zone. A symmetric distribution of concave in the middle and convex on both sides can be observed in Fig. 13. This kind distribution of friction coefficient is totally opposite to the distribution of friction force, which depends on the negative power of dimensionless load W in Eq. (11). Besides, the negative power of dimensionless load W contributes to the negative correlation between the friction coefficient f  at a certain position and the applied load. However, differing from the effect of shaft speed on the friction force, the trend of friction coefficient varying with the shaft speed in load zone is not in accord with that in unload zone. For example, as shown in Fig. 13(a), under the conditions of fixed applied load and lubricant, the friction coefficient at a certain position increases with the increase of shaft speed due to the positive power of dimensionless speed U in Eq. (11). In contrast, the friction coefficient at a certain position decreases with the increase of shaft speed in unload zone. It is implied that the roller centrifugal force, which is positively correlated with the shaft speed rather than the speed, dominates the friction coefficient in unload zone.
The distributions of friction force and coefficient under different lubricant conditions are displayed in Fig. 14. According to Eqs. (2), (9), and (11), two important properties of lubricant, kinematic viscosity  and pressure viscosity index p  , contribute to the friction behavior of the bearing by acting on dimensionless speed U and material parameter G, respectively. Furthermore, the friction force and coefficient are both positively related to the viscosity of lubricant and negatively related to the pressure viscosity index of lubricant. Consequently, due to bigger viscosity and smaller pressure viscosity index of lubricant 75W-90 than those of 5W-30 shown in Table 2, local friction force and coefficient at a certain position under the lubricating conditions of 75W-90

Conclusions
A combined experimental and analytical method is presented for the determination of the friction force generated from different rollers and the outer raceway contacts within a CRB. In this method, a simplified model of EHL friction force in pure rolling line contact was given. Its parameters were determined by experimentally measured radial load distribution and friction torque of the CRB. Key findings of this study include: 1) The calculated friction torque calculated M based on the simplified analytical model agrees well with the measured friction torque measured M , which indicates that friction force f , F  can be well predicted by the simplified EHL analytical model.
2) Both the friction force and coefficient within the bearing are in symmetric distribution, which is attributed to the symmetric distribution of the radial load.
3) Under test conditions, the value of friction force was no more than 8 N, while most of the value of friction coefficient was between 0.001 and 0.006 in load zone and no more than 0.1 in unload zone.
Note also that once the rotating speed, radial load distribution, and lubricant properties are obtained, this simplified analytical model driven by experimental data can also be used to predict the friction behavior between rollers and raceway of in-service CRBs in drivetrains.