Measured load spectra of the bearing in high-speed train gearbox under different gear meshing conditions

The load spectrum is a crucial factor for assessing the fatigue reliability of in-service rolling element bearings in transmission systems. For a bearing in a high-speed train gearbox, a measurement technique based on strain detection of bearing outer ring was used to instrument the bearing and determine the time histories of the distributed load in the bearing under different gear meshing conditions. Accordingly, the load spectrum of the total radial load carried by the bearing was compiled. The mean value and class interval of the obtained load spectrum were found to vary non-monotonously with the speed and torque of gear meshing, which was considered to be caused by the vibration of the shaft and the bearing cage. As the realistic service load input of bearing life assessment, the measured load spectrum under different gear meshing conditions can be used to predict gearbox bearing life realistically based on the damage-equivalent principle and actual operating conditions.


Introduction
Over the past decade, the high-speed railway has become an indispensable mode of transportation worldwide, with outstanding features of safety, high efficiency, comfort and reduced-pollution [1]. The development of railway passenger During the operation of trains, the gearbox bearing in highspeed trains experiences complex and various load conditions under the influence of internal excitation from gear meshing [12] and external excitation from wheel-rail contact [13,14]. Therefore, realistic load spectra under different service conditions are required to assess the fatigue reliability and to demonstrate the in-service integrity of gearbox bearings. The indications of bearing health based on monitoring dynamic bearing load can then be used to organize effective maintenance and minimize unplanned downtime.
Both numerical methods and measurement techniques were developed to understand the bearing load in a transmission system under service conditions. Evans et al. [14] presented a multistep dynamic analysis process for linking railcar vibration conditions to bearing component motions and obtained the bearing load numerically. Wang et al. [15,16] developed a comprehensive vehicle-track coupled dynamics model and investigated the influence of wheel-rail excitation on the axle box bearing load. Liu et al. [17,18] discussed the effect of surface waviness and rotor eccentricity on the motor bearing load by using a locomotive-track spatially coupled dynamics model with detailed traction power transmissions. To measure the load of the tapered roller bearing in the wind turbine gearbox, four equally spaced axial grooves were machined in their outer rings and eight strain gauge pairs were used by Keller and Guo et al. [19][20][21][22]. However, this modification of the bearing changed the original load distribution within the bearing, which led to an amount of uncertainty higher than the desired in the estimation of the measured bearing radial load [19,22]. Recently, new ultrasonic reflectometry techniques have been implemented by Chen et al. [23] and Nicholas et al. [24] to measure the contact load in a field operational wind turbine gearbox bearing. Nevertheless, a high sampling frequency of 100 MHz is required for ultrasonic measurement, which results in a large amount of data accumulation and difficulty in long-term monitoring. Overall, although the bearing load conditions have been numerically or experimentally studied, few works applied to the in-situ monitoring of bearing load in high-speed train gearbox and no load spectrum was compiled and presented in the literature due to the limitations of existing methods.
Previously, for in-situ measurement of contact load distribution within rolling element bearings with good accuracy, a new method based on strain measurement has been successfully developed, implemented and applied without any modifications to the bearing itself [25,26]. In the current study, instrumentation based on this approach is added to a bearing in high-speed train gearbox. The load spectra of the gearbox bearing are measured and compiled. The effect of the gear meshing conditions on the load spectra of the bearing is analyzed. Furthermore, the application of the measured load spectrum in bearing life assessment is discussed.

Instrumentation and methodology
This work focuses on a single-stage 650 kW gearbox with a ratio of 1:2.429 from the CRH380B electric multiple units (EMU) as shown in Fig. 1. This configuration of the gearbox represents a majority of real-scale high-speed train gearboxes. The high-speed shaft is supported with two cylindrical roller bearings and one four-point contact ball bearing. The cylindrical roller bearing NU214ECML from the wheel side was selected as the test bearing. Key specifications of the test bearing are shown in Table 1. This cylindrical roller bearing reacts against the radial load whose direction and amplitude depend on and vary with the operational conditions of the gear meshing [20]. Thus, to monitor the dynamic radial load of the bearing, it is necessary to measure the    Fig. 3. During the assembly, the bearing housing is connected with instrumented bearing under an interference fit and then assembled on the gearbox by bolts under a clearance fit. Before the experiment, the original bearing housing was modified; 17 notches were introduced by computerized numerical control (CNC) milling and evenly distributed on the inner surface of the bearing housing [25]. Each notch had the same size with axial length of 19 mm, circumferential length of 11 mm, and radial depth of 2 mm (Fig. 3d). Each strain gauge location of the instrumented bearing corresponds to the center of a notch of the instrumented housing during the fit process.
By instrumenting the bearing and its housing, once a calibration matrix K α_β is obtained and the real-time strain distribution ε α (t) is measured, the real-time load distribution F β (t) within the bearing can be determined as follows [25]: where the subscripts α and β represent the sensing location number (1, 2, …, 17); elements of the calibration matrix K α_β , denoted by k α_β , represent the strains at the sensing location α due to a unit force applied at sensing location β when all other loads are zero. Then, according to the relationship between the total radial load and the load distribution [27], the real-time total radial load F r (t) can be calculated by where ψ β represents the angle between the distributed load F β and the maximum roller-raceway contact load; Z is the number of rollers in the test bearing and equal to 17 in this paper. (1)

Experiment
The experiment was divided into two steps. First, the assembly of instrumented bearing and its housing was loaded in an MTS test machine with a specially designed loading fixture to determine the calibration matrix K α_β . Second, the instrumented bearing and housing were installed in the gearbox and a test bench of high-speed train traction system was used to measure the dynamic distributed strain signals of the test bearing under different gear meshing conditions. An eDAQ (serial number 5448) from HBM Company was adopted as the strain acquisition equipment in the whole experiment. The sampling frequency was, respectively, set to 2.5 kHz in the first step and 20 kHz in the second step.

Determination of the calibration matrix
The experimental setup for determining the calibration matrix of the test bearing is shown in Fig. 4. As shown in Fig. 4a, the whole setup consists of a base, two supports, a shaft, bearings, bearing housing and a polygonal loading ring. A regular 17 polygon fixture was designed and assembled with the instrumented bearing housing by bolts as shown in Fig. 4b. The ball bearing and the instrumented roller bearing were fitted in the housing as usual. A shaft was designed with a concave protrusion to apply the force only on a specific sensing position as shown in Fig. 4c. The setup was installed in an MTS test machine as shown in Fig. 4d.
The calibration matrix components k α_β can be obtained as follows: where ε α_β is the measured strain at the test point No. α when the load P β is applied to the test point No. β. As shown in Fig. 4e, when the load P 1 is applied to the test point No. 1, ε α_1 (α represents 1 to 17) can be measured. The applied load increases from 0 to 9 kN with the increment of 1 kN during a test. With the increase of P 1 , the change of ε α_1 is recorded. Thus, the calibration matrix components k α_1 can be obtained by Eq. (3). All the values of applied loads in this experiment have already removed the part of self-weight. The shaft is kept static to ensure that the loading roller is located at the top and in the line of the loading direction.
Other calibration components k α_β can be obtained similarly by rotating the corresponding sensing location to the top position. Finally, a 17 × 17 calibration matrix K α_β can be determined.

Measurement of dynamic strain distribution under gear meshing
After the calibration, the assembly of instrumented bearing and its housing was removed from the calibration setup and mounted in the test gearbox; 17 small holes were machined on the end cover of the gearbox for leading out the signal cables of strain gauges and sealed with the silicone rubber to prevent the lubricant leakage as shown in Fig. 5a. Then, the test gearbox was installed in the test bench of the highspeed train traction system as shown in Fig. 5b. Figure 5c shows the construction and arrangement of the test bench. The maximum input speed of the test bench is 6500 rpm and the maximum load torque is 7900 N⋅m. The experiment was conducted in two phases and two series of gear meshing conditions were, respectively, examined based on the actual working conditions of the highspeed train traction system. In the first phase, constantspeed operations were conducted throughout a range of input torque levels. The high-speed shaft rotated stably at 1550 rpm (corresponding to the train speed of 100 km/h) in this phase. In the second phase, several field-representative input speeds were conducted while the input torque was fixed at 1000 N⋅m. During the tests, the rotational direction of the high-speed shaft was clockwise (looking from the wheel side) and the high-speed shaft bearing was lubricated with 75 W-90 gear oil. Figure 6 shows the real-time load applied by the MTS load cell and typical results of the strain signals from the calibration test when the test point No. 1 is loaded. The radial load was applied with a rate of 25 N/s and kept at each target value for 10 s. It can be found that ε 1_1 is positive and linearly increases with the load P 1 , while ε 2_1 and ε 17_1 are negative and decrease with the increase of the load P 1 . It indicates that, when the load is applied on the test point No. 1, the outer surface deformation of sensing location No. 1 is in tension while the sensing locations No. 2 and No. 17 are in compression. Among the strains of test points which were farther from the test point No. 1, ε 3_1 , ε 4_1 , ε 5_1 and ε 16_1 , ε 15_1 , ε 14_1 are also positively correlated with the load applied on the test point No. 1, while ε 6_1 and ε 13_1 are nearly zero all the time. The other strain signals are also almost zero since they are far away from the loading point and they are not shown here.

Measured real-time bearing load
A part of the strain signal ε 12 (t) from test position No. 12 under the condition of input speed 1550 rpm and input torque 3000 N⋅m is intercepted as an example and displayed in Fig. 7a. A magnified view of a section of the strain signal shows five roller revolution cycles in Fig. 7b. Corresponding to 17 rollers of test bearing passing this specific strain gauge position in a roller revolution cycle, each cycle consists of 17 strain pulses with a very similar pulse shape but various peak values which are caused by dynamic contact load between the rollers and outer raceway. The data processing procedure from measured real-time strain signals to time series of roller-raceway contact load contains two steps.   (1) based on the strain distribution matrix ε α (t) and the compliance matrix K α_β . Accordingly, the real-time dynamic distributed contact load F 12 (t) can be obtained and the result is shown in Fig. 7c. It should be noted that the compliance matrix in Eq. (1) is determined when the bearing is stationary and the roller is in the center of each notched region as shown in Fig. 4e, which corresponds to the peak position in cyclic strain response when the bearing rotates. Therefore, the time series of peak values of the strain pulse were used in Eq. (1) to calculate the roller-raceway contact load. In Fig. 7c, the dynamic contact load distributed on this outer raceway position is observed to irregularly fluctuate around the mean value of 4.73 kN with a range about 0.6 kN. In Fig. 7d, a magnified view of the real-time contact load F 12 (t) in five roller revolution cycles indicates the variation of the contact load among rollers and cycles. This variation of roller-raceway contact load is considered as a comprehensive result of roller size deviation [24], gear meshing excitation and structural vibration [14]. Figure 8a shows the time histories of all measured nonzero contact loads under the gear meshing condition of input speed 1550 rpm and input torque 3000 N⋅m. Accordingly, the load distribution within the instrumented bearing is determined and shown in Fig. 8b and c. The points and bounds in Fig. 8b represent, respectively, the mean value and minimum to maximum range of dynamic roller-raceway contact loads shown in Fig. 8a. Besides, the load distribution  Fig. 8b is fitted based on the points according to the Harris theory of bearing radial load distribution [27]. Under this specified condition, the load zone contains seven test positions and the maximum contact load is F 12 (t). Moreover, the maximum contact load F 12 (t) is located in the center of the loaded zone while contact loads on either side of the load F 12 (t) show an axisymmetric distribution with respect to the direction of the load F 12 (t). Thus, according to Harris theory of bearing radial load distribution [27], the direction of the total radial load F r (t) of the test bearing is precisely the direction of F 12 (t), while the real-time amplitude of the total radial load can be obtained by Eq. (2). Figure 8d shows a section of the measured real-time total radial load F r (t) with a mean value about 18.19 kN and a variation range about 2 kN under this specified condition. For other conditions of gear meshing, the measured load time histories of the instrumented bearing can be determined similarly and then the load spectra of the bearing load under these experimental conditions can be compiled.

Measured load spectra of the gearbox bearing under different conditions
Frequency distribution histograms of the measured bearing load under the two series of gear meshing conditions are, respectively, shown in Figs. 9 and 10. Load spectra of 32-step classification are compiled here according to the Sturges formula [28]. Furthermore, to compare the characteristics of the load spectra under different conditions with the same sample size and classification number of the histogram, the statistical parameter, spatial frequency, is used here, which refers to the count of a certain range of load carried by the gearbox bearing for each kilometer of the train running. The spatial frequency is determined by the input speed, transmission ratio and wheel diameter which is assumed to be 830 mm in this paper. As shown in Figs. 9 and 10, the load spectra under most of the gear meshing conditions approximately conform to the normal distribution. However, under some conditions, such as the condition of input speed 1550 rpm and input torque 1000 N⋅m, the frequency distribution of the load spectra is different from the others. Two peaks of spatial frequency appear under these conditions. To characterize the influence of gear meshing conditions on the load spectra of the gearbox bearing, the mean value and class interval of the frequency load spectra under different conditions are, respectively, presented in Figs. 11 and 12. Due to the fixed classification of load spectra under different gear meshing conditions, the class interval represents the discrete degree of the dynamic load of the gearbox bearing.
The mean values of the measured bearing load varying with the test input torque are shown in Fig. 11a. Under conditions of a fixed input speed, the bearing loads increase linearly with input torque as the gear mesh forces increase linearly with torque. In most previous experimental studies, the load of a shaft bearing in a gear transmission system without external excitation was believed to be decided by the shaft bending moments and gear meshing torque as the   effect of the rotational speed on the bearing load was usually small enough to be ignored [19,21,24]. Some modeling or simulating research considered the rotational speed as a factor of the bearing load, which affected the gear mesh force and further the bearing load with a monotonous slight trend under fixed torque conditions [20]. Interestingly, differing from previous results about the effect of speed on the bearing load in the literature, for the test gearbox bearing of high-speed train, a fluctuation of the average load with rotational speed is observed in Fig. 11b. When the input torque is fixed at 1000 N⋅m, the mean value of the measured bearing load increases initially, followed by a decrease, but then again increases with the increase of the rotational speed. The same trend is reflected in the variation of load spectrum class interval with input speed in Fig. 12b. Besides, as shown in Fig. 12a, the influence of torque on the class interval of the measured bearing load spectra shows a nonmonotonic trend as well. Overall, the characteristics of the measured load spectra of high-speed train gearbox bearing  for test under different conditions of gear meshing can be drawn as follows: (1) The average level of the bearing load is mainly determined by torque, and affected by rotational speed under the condition of a specific torque. (2) The class interval of the measured bearing load spectrum varies non-monotonously with torque and speed, which means that the torque and speed of gear meshing are not the direct cause of the dispersion of the measured bearing load spectrum.
To clarify the mechanism of the non-monotonic variation of the measured load spectrum characteristics with the torque and speed of gear meshing, the frequency spectra of the measured time history of bearing load under different conditions are shown in Fig. 13. In Fig. 13a, when the input speed is fixed at 1550 rpm, there are several obvious frequency components under the torque of 500, 1000, 1500 and 2000 N⋅m. The power of these frequency components varies with torque following the same trend of the class interval with torque presented in Fig. 12a. Under the conditions of constant-torque, the consistency of the various trends of frequency component power and load spectrum characteristics with speed is indicated in the comparison between Figs. 11b, 12b and Fig. 13b. Table 2 provides the rotational frequency of the high-speed shaft and cage under different input speeds. The shaft frequency is calculated by the input speed while the cage frequency is obtained by using a method based on the analysis of measured strain signals [29]. Comparing the frequency components of the measured bearing load in Fig. 13 with the characteristic rotational frequencies of the bearing listed in Table 1, it can be found that these obvious frequency components of the measured bearing load mostly correspond to the rotation frequency of the high-speed shaft or 2 times the harmonic frequency of the cage rotation. For example, under the speed of 1550 rpm and torques of 1000, 1500 and 2000 N⋅m, the frequency components about 22.2 Hz shown in Fig. 13a are close to 2 times the harmonic frequency (i.e., 21.8 Hz) of the cage rotation. In Fig. 13b, under the speed of 2325 rpm, the frequency 32.3 Hz is close to 2 times the harmonic frequency (i.e., 32.6 Hz) of the cage rotation while 37.6 Hz is close to the rotational frequency 38.7 Hz of the shaft. It is difficult to say with certainty what causes these obvious frequency components. However, the following evidence from Fig. 13 strongly suggests that it is caused by the in-situ modal vibration of the high-speed shaft or the cage in the bearing: (1) The obvious frequency component only appears under some combined conditions of specific speed and torque. (2) The vibration of the rotor and the resonance effect induced by the characteristic vibration of cage will arise under specific critical conditions and increase the dynamic response of the bearing [30].
In summary, the non-monotonic variation of the measured load spectrum characteristics is believed to be caused by the vibration of the high-speed shaft or the bearing cage under some combined conditions of specific speed and torque.

Bearing life assessment based on measured load spectra
According to some theories of bearing life [10,31,32], an accurate prediction of bearing life needs the input of realistic service bearing loads. In previous research, due to the difficulty of measuring the actual bearing loads and their variations, the load of a shaft bearing in a gearbox was frequently assumed to be a constant load under a specified condition and calculated by using analytical models [19][20][21]33]. Compared with the analytical bearing load, the measured load spectrum in this paper can be used to predict gearbox bearing life more realistically.
The gearbox bearing maintenance duration in China is 1.2 million km operation of the train and the scrapping duration is 2.4 million km. Thus, to assess the bearing life using the measured load spectrum, two L-P curves, respectively, corresponding to 1.2 and 2.4 million km operating gearbox bearings are obtained according to Lundberg-Palmgren theory [10] as illustrated in Eq. (5) and given in Fig. 14. where C is the basic dynamic load rating of the bearing and equal to 13.7 kN for the test bearing NU214 [34]; P is the bearing load acting under the specified condition and p = 10/3 for roller bearings; L is the life in millions of revolutions and has been counted by cumulative spatial frequency in Fig. 14. Besides, the measured cumulative spatial frequency spectrum of the bearing load under different gear meshing conditions is shown in Fig. 14. In Fig. 14a, all of the cumulative frequency-amplitude spectra are below the two L-P curves under the gear meshing conditions of 1000 N·m input torque and different speeds. It means that gearbox bearings operating continuously under such conditions can meet the safe operating requirements of 2.4 million km and perform even better. When the gear meshing speed is fixed, as shown in Fig. 14b, the load spectrum gradually intersects the L-P curve at a smaller cumulative spatial frequency with the increase of gear meshing torque. The damage-equivalent bearing load F e defined by the Palmgren-Miner rule [33] is used here to assess the bearing life. In general, for k loads each operating for N k revolutions, the damage-equivalent load F e is given by where F k is the kth load component in the measured load spectrum (k = 1, 2, …, 32 for the measured load spectra in this paper) and N is the total number of revolutions under one condition of gear meshing. As an example, under the condition of 3000 N·m, the damage-equivalent load of the measured load spectrum is calculated to be 18.1 kN and highlighted by a horizontal dotted line in Fig. 14b. Then, two intersection points A (395.1, 18.1) and B (821.7, 18.1) of the horizontal dotted line and the L-P curves are determined.
According to the damage-equivalent principle [33], intersection points A or B, respectively, represent that to ensure the safe operation of 2.4 or 1.2 million km of the gearbox bearing, the operation time of gearbox under the condition of 3000 N·m input torque should not exceed 5.94% (equal to 395.1/6651.1) or 12.35% (equal to 821.7/6651.1) of the total operation time. Similarly, the limited time of safe operation under other conditions can be obtained. Furthermore, once the gearbox operating conditions in the actual operation process of high-speed train are recorded, the actual life and damage of the gearbox bearing can be accurately predicted and evaluated according to the combination of the measured load spectrum under each operating condition.

Implications and limitations
The implication of the proposed approach in measuring the load spectrum of gearbox bearing has a number of assumptions and limitations.
To measure the load distribution of the gearbox bearing, the same number of notches as the number of rollers were introduced to the test bearing housing circumferentially. It has been verified that the number and geometry of notches on the housing could change the local stiffness of bearings and further the load distribution [35]. To obtain the real load distribution, the introduction of notches must obey the following rules [25]: (1) For a given bearing under an applied load with an unknown direction, the number of notches must be the same as the number of rollers. (2) All the notches should have the same dimension and be evenly distributed on the housing according to the relative position of rollers so that the removal of the material has the same effect for each test position and subsequently no effect on the load distribution.
There are some complexities associated with the installation of the measurement system. The strain gauges are connected to the acquisition equipment by cables. The existence of signal cables increases the difficulty of gearbox assembly. Besides, the presence of an end cover and limited space within the gearbox restrict available regions for cable routing. As a result, some holes were machined on the end cover for leading out the signal cables, which is not allowed for gearboxes in the field. Wireless data acquisition modules based on flexible sensors and integrated circuits are expected to drive the application of this measurement method on gearboxes in the field besides in the test bench.
Notwithstanding the above, this measurement method of bearing load has been successfully applied to load spectrum measurement of bearing in high-speed train gearbox. The measured load spectra of the gearbox bearing under different conditions of gear meshing were compiled for the first time. Furthermore, the measured load spectra of the bearing in high-speed train gearbox can reflect the actual load conditions and their variations applied on the bearing under different gear meshing conditions, which can be used to monitor the gearbox bearing damage and evaluate the gearbox bearing fatigue reliability.

Conclusions
In this paper, the load spectra of the bearing in high-speed train gearbox under different conditions of gear meshing were measured and compiled by instrumenting the bearing. Major conclusions are given below: 1. The time histories of the dynamic radial load carried by an instrumented gearbox bearing were deduced from the measured strain signals and a calibrated compliance matrix. 2. Non-monotonic trends of the measured load spectrum characteristics (mean value and class interval) varying with gear meshing speed and torque were found and believed to be caused by the vibration of the high-speed shaft or the bearing cage under some combined conditions of specific speed and torque. 3. Combined with the actual service conditions of the gearbox, the measured load spectrum under different gear meshing conditions can be used to predict gearbox bearing life more realistically.
It should be noted that, besides the gear meshing excitation, the wheel-rail excitation is another significant factor contributing to the load variation and fatigue life of the drivetrain bearing. In future research, the influence of wheel-rail excitation on the gearbox bearing load will be evaluated experimentally. Furthermore, the load spectra of the gearbox bearing under the combined effect of internal gear meshing excitation and external wheel-rail contact excitation will be measured and complied. In addition, the measurement technique of dynamic roller-raceway contact load will be further applied to other drivetrain bearings in high-speed train such as axle box bearings or motor bearings.