A coupled dynamics model for studying the outer ring fault characteristics and mechanisms of axle-box bearings in urban rail vehicles

2.1. Dynamic modelling of DTRB with the outer ring raceway fault

A DTRB using a back-to-back arrangement is adopted for axle-box bearing, as shown in Fig. 2(a) and (b). It consists of two inner rings, an outer ring, multiple rollers, two cages, and a spacer. In this paper, we assume that its outer ring and inner ring are fixed in the axle-box housing and axle end of the wheelset respectively. During operation, it undertakes the longitudinal load, the lateral load, and the vertical load generated by the secondary mass (including the bogie frame and car body), the traction (braking) action, and wheel-rail contact. Its loading process and contact force analysis are depicted in Fig. 2(c) and (d).

The force analysis of DTRB and the detailed derivation of motion equations have been provided in [16], [17]. Based on these developed DTRB models, we need to couple it with the urban rail vehicle dynamic model. Namely, the dynamic response of the wheelset and axle-box, as the external excitations, should be considered in the DTRB model. As seen in Fig. 2(c), under the vertical load from the axle-box, the relative displacement will occur between the outer ring and the inner ring, the roller is squeezed by the raceway to generate the internal contact load to balance the external load of the axle-box. The contact loads are related to elastic deformations induced by relative displacements between rollers and raceways. Therefore, to calculate the contact load, it is necessary to first analyse the relative displacement. In Fig. 2(c), the relative radial displacement of the $k$-th roller at the azimuth angle ${\phi }_{r\left(L,R\right)ijk}$, between the rollers and outer ring raceways, as well as between the rollers and inner ring raceways can be determined by Eq. (1) [18]:

where ${\delta }_{r\left(L,R\right)ijk}^{out}$ and ${\delta }_{r\left(L,R\right)ijk}^{in}$ represent the radial relative displacement between the roller and outer ring, as well as between the roller and inner ring respectively. $X_{ab(L,R)i}$, $X_{r(L,R)ijk}$, and $X_{wi}$ are the longitudinal displacements of the axle-box, the roller, and the wheelset respectively. $Z_{ab(L,R)i}$, $Z_{r(L,R)ijk}$, and $Z_{wi}$ are the vertical displacements of the axle-box, the roller, and the wheelset respectively. ${\alpha }_{wi}$ and ${\gamma }_{wi}$ are the roll and yaw angles of the wheelset. $d_w$ is the lateral semi-span of primary suspensions. $h_{\phi r\left(L,R\right)ijk}$ is the initial clearance [19], $h_{\phi r\left(L,R\right)ijk}=0.5u_r\left(1-cos{\phi }_{r\left(L,R\right)ijk}\right)$. $u_r$ is the bearing radial clearance.

Fig. 2Structural composition and schematics of the dynamic model for DTRB

a) Schematics of axle-box bearing

b) Structural composition of the DTRB.

c) Schematic of loading process for DTRB

d) Contact force analysis of DTRB

Likewise, the relative axial displacement of the $k$-th roller, between the rollers and outer rings and between the rollers and inner rings can be expressed as Eq. (2):

where ${\delta }_{a\left(L,R\right)ijk}^{out}$ and ${\delta }_{a\left(L,R\right)ijk}^{in}$ represent the axial relative displacement between the roller and outer ring, as well as between the roller and inner ring respectively. $Y_{ab(L,R)i}$, $Y_{r(L,R)ijk}$, and $Y_{wi}$ are the lateral displacements of the axle-box, the roller, and the wheelset respectively. $u_a$ is the bearing axial clearance.

Combining Eq. (1) and Eq. (2), the elastic deformations between the roller and the outer ring raceways, as well as between the roller and the inner ring raceways can be obtained by Eq. (3):

where ${\alpha }_o$ and ${\alpha }_i$ represent the contact angles between the roller and the outer ring raceway, as well as the roller and the inner ring raceway respectively, as seen in Fig. 2(d).

Likewise, the elastic deformation generated by the contact between the roller and the guiding flange of the inner ring can be expressed as Eq. (4):

where ${\alpha }_f$ represents the contact angle between the roller and the guiding flange.

Consequently, the contact load between the roller and raceways, as well as between the roller and the guiding flange in Fig. 2(d) can be obtained according to the load-deformation relationship based on the Hertzian line contact theory and the Hertzian point contact theory, they are given in Eq. (5):

where $Q_{\left(L,R\right)ijk}^{o\left(i\right)}$ represent the contact load between the roller and raceways. $Q_{f\left(L,R\right)ijk}$ represents the contact load between the roller and the guiding flange. $K_{o\left(i\right)}$ and $K_f$ are the contact equivalent stiffness coefficient of the roller-raceways and the roller-guiding flange respectively, they can be calculated by [19], [20].

Furthermore, the corresponding frictional forces and additional moments caused by the above contact loads are calculated according to [16]. Simultaneously, the motion differential equations of DTRB in the proposed coupling model can be obtained, they are given in [16]. Then, the forces and moments acting on the inner ring (i.e., the wheelset) and outer ring (i.e., the axle-box) can be continuously solved at each time instant.

Fig. 3Mathematical characterisation diagram of outer ring fault for DTRB

a) Fault on the outer ring raceway

b) Local expansion of outer ring fault

In order to study the outer ring fault characteristics and mechanisms of axle-box bearing, the localised surface fault needs to be set in the outer ring raceway of above DTRB dynamic model to simulate the bearing fault. Typically, the localised faults are characterised as additional deformations in the contact pair of bearing elements [21],[22]. As shown in Fig. 3, when the $k$-th roller at the azimuth angle ${\phi }_{r\left(L,R\right)ijk}$ passes through the fault of length $L$, width $W$, and depth $H$, the additional clearance caused by the fault can be expressed as [22]:

where $A$ is the maximum value of clearance variation, $A=r_b-\sqrt{{r_b}^2-{\left(W/2\right)}^2}$. $r_b$ is the radius of the roller, $r_b=0.5D_b$. ${\theta }_o$ is the initial offset angle between the outer ring raceway fault and the first roller, it is set to 0 in this paper. ${\tau }_o$ is the angle corresponding to the outer ring raceway fault, ${\tau }_o=2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(W/D_o\right)$. $D_o$ is the diameter of the outer ring raceway (seen in Fig. 2(d)). The contact azimuth angle of the $k$-th roller can be obtained by Eq. (7):

where ${\omega }_c$ is the rotation angular velocity of the cage, ${\omega }_c=\frac{1}{2}\left(1-\frac{D_b}{D_m}\right){\omega }_w$. ${\omega }_w$ is the rotation angular velocity of the wheelset. $D_m$ is the pitch diameter, as seen in Fig. 2(d).

Thus, in the faulty DTRB dynamic model, the elastic deformations between the roller and the outer ring raceways in Eq. (3) is redefined as follows:

Based on Eq. (8), the contact deformation and contact load between the roller and raceway will be calculated in the DTRB model with outer ring fault. And the vibration response of the faulty DTRB model can be solved.

2.2. DTRB-vehicle-track coupling dynamic model considering the flexibility of vehicle components

Based on the above theoretical foundations and vehicle-track coupled dynamics theory [23], a comprehensive simulation model of the DTRB-vehicle-track coupling dynamic model is formulated, as illustrated in Fig. 4. The simulation model integrates an urban rail vehicle-track coupling dynamic model and the faulty DTRB dynamic model. The interactions between the vehicle and the DTRB are described through the dynamic contact forces and moments of the DTRB acting on the axle-box and wheelset. Thereinto, the vehicle-track model considering the flexibilities of the bogie frame, the axle-box, and the wheelset is modelled in the SIMPACK platform. The flexibilities of these three components of the front bogie are considered only to improve computational efficiency. Their dynamic responses are obtained by modal superposition.

Fig. 4Proposed DTRB-vehicle-track coupling dynamic model considering the flexibility of vehicle components and bearing outer ring fault

They are coupled in the dynamic model by the Guyan substructure modal reduction method [24]. The FE models of the wheelset and axle-box in the modal analysis are discretised using the Solid 185 element, and the bogie frame is mainly discretised using the Shell 181 element. Vibration modes of approximately 300 Hz, 1000 Hz, and 3000 Hz for the bogie frame, wheelset, and axle-box are considered. The profiles of the wheel tread and rail are the LM and the CHN 60. The FASTSIM algorithm is used to solve wheel-rail contact relationships. The typical America 5-class track spectrum is employed to express the characteristics of track random irregularity excitation. The wheelset and axle-box are connected through the DTRB model, which is modelled as two dummy bodies, two time-excitation force elements (i.e., No. 93 force element in SIMPACK), and a revolute joint. Dummy body 2, as an intermediate component, is used to transmit the contact forces and moments from the SIMAT co-simulation interface to the axle-box and the wheelset via the No. 93 force element. Dummy body 1 is employed to convert motion forms through the rotation joint. Correspondingly, the dynamic model of the DTRB is established and integrated into the MATLAB/Simulink platform. The dynamic responses of the DTRB elements, and the force and moment acting on the wheelset and axle-box are calculated and solved according to the preliminaries in Subsection 2.1 and the developed DTRB dynamic model in [16], respectively. The solution results are subsequently applied to the axle-box and the wheelset of the vehicle dynamic model via the SIMAT co-simulation interface to investigate the dynamic responses of the proposed coupling dynamic system. Simultaneously, the vibration responses of the axle-box and wheelset are fed back to the DTRB model through the co-simulation interface. This process is iterated at the next time step until the simulation is complete.

3.1. Theoretical verification of the proposed model

The axle-box bearings at wheelset 2 on the right side of the front bogie are selected (as seen in Fig. 1 and Fig. 4). The analysis of contact loads under normal and faulty conditions, as well as the slip characteristics under normal conditions, are depicted in Fig. 5. A localised fault, with the size of length 50.4 mm, width 2 mm and depth 2 mm, is set on the outer ring raceway of row 2 in DTRB. The vehicle running speed is set to 60 km/h, and without considering the track irregularity.

Fig. 5Theoretical verification and analysis of the correctness and rationality for the proposed model

a) Contact load history of the normal DTRB

b) Instantaneous contact load of two time-instants

c) Theoretical verification of contact load

d) Contact load history with fault and its spectrum

e) Slip analysis of the cage

f) Slip analysis of the roller

As shown in Fig. 5(a), the contact load time-history curve of the roller-outer ring raceway for the normal DTRB appears a periodic fluctuation feature with constant amplitude, and the amplitudes of the two rows are consistent. Then, we choose the contact load time-history curve of single contact cycles. On the one hand, the contact load distribution of all rollers at two time-instant from the loaded zone to the unloaded zone is analysed, as seen in Fig. 5(b). On the other hand, the contact load at single contact cycles is compared with the theoretical value, as seen in Fig. 5(c). From Fig. 5(b) and (c), we can see that the contact load curves are basically consistent with the theoretical values, and the instantaneous contact loads from A time-instant to B time-instant are reasonable in accordance with [8]. Thereafter, the contact load history and its load spectrum of faulty DTRB are discussed in Fig. 5(d). There are significant periodic impulses on the contact load history. The reciprocal of the period is just equal to the rotation frequency of cage ($f_c$= 2.745 Hz), which corresponds to the fault characteristics of the outer ring. Therefore, it is feasible to use the faulty DTRB model established in this paper to study the fault characteristics of the outer ring of the axle-box bearing. Finally, the slip rate of the roller and cage are analysed to further demonstrate the rationality of the proposed model. As shown in Fig. 5(e) and (f), the actual rotation speed of the cage is consistent with its theoretical rotation speed, but the consistency between the actual rotation speed of the roller and its theoretical rotation speed is better in the loaded zone than in the unloaded zone. This is because the cage drives the roller movement in the unloaded zone, while the roller drives the cage movement in the load zone. The former is more likely to cause the roller slippage. The maximum slip rates of the cage and roller are 3.1 % and 4.3 % respectively. In summary, the coupled dynamic model proposed in this paper is reasonable and feasible.

3.2. Field test verification of the proposed model

As shown in Fig. 6, the real urban rail train test is conducted to verify the correctness of the proposed model. In Fig. 6(a), the vibration response of the axle-box bearing with outer ring fault is measured through the acceleration sensor fixed in the axle-box housing. The rotational speed of the wheelset is measured through a rotating-speed sensor installed at the axle-end. In Fig. 6(b), the data acquisition system is depicted, it includes the vehicle-mounted monitoring computer and two pre-processors. Based on the rotating-speed input signal from the rotating-speed sensor in Fig. 6(a), the vehicle-mounted monitoring computer controls the pre-processor to complete the even-angle resampling of axle-box vibration signals, i.e., the order tracking. The resampled angular domain vibration signal is transmitted to the vehicle-mounted monitoring computer for storage via the data bus. As shown in Fig. 6(c), we choose the acceleration process and constant speed process between two stations to validate the proposed model. In the proposed simulation model, the size of the outer ring fault and the position of the tested axle-box are the same as those in section 3.1. The typical America 5-class track spectrum is employed. The simulation results of the proposed model are compared with on-site test data. Their vibration amplitude in angle domain and order spectrum under variable and constant speed conditions are given in Fig. 7, and the contact load characteristics from the simulation model are also shown and discussed.

Fig. 6Field tests and verification of the proposed model with outer ring raceway fault on the axle-box bearing

a) Selection of measurement positions

b) Data acquisition

c) Speed-distance curve of testing vehicle between two stations

In Fig. 7(a) and (c), the resampled vibration signal in the angle domain exhibits periodic characteristics with the same intervals. And the periodic interval of the simulation and the field test is generally consistent. At the variable speed stage, the amplitude variation tendencies of the simulation and the field test are also consistent, both increasing with the increase of speed. However, owing to the difference between track irregularities in the field tests and simulations, linear representation of the rail support, and consideration of the limited number of vibration modes of the flexible body in subsection 2.2, the axle-box acceleration calculated by the proposed model is smaller than that of the test measurement in the angle domain. In Fig. 7(b) and (d), significant FCOs of the outer ring listed in Table 2 and their harmonics can be found. Under variable and constant speed conditions, the first-order FCOs of simulation and field test are 8.663, 8.662, 8.957, and 8.939 respectively. The corresponding errors with theoretical FCO are 0.36 %, 0.37 %, 3.03 % and 2.82 %. The FCO deviation of the field test is greater than that of the simulation. The reason is that the rolling elements of the axle-box bearing are slipping more severely in the field test, and the packet loss of data transmission may also cause deviation in the FCO. Furthermore, the periodic impulse can be seen in the roller-raceway contact force curve. The interval period gradually decreases as the speed increases under variable speed operating conditions. Compared to not setting up the track irregularity in Fig. 5(a) and (d), the amplitudes of the contact load will fluctuate and are not equal between two rows in actual operation simulation. It is also sufficient to demonstrate the necessity of considering vehicle-track coupling factors.

Fig. 7Verification and comparison with field test for the proposed model with the faulty DTRB

a) Comparisons of vibration signals in angle domain

b) Comparisons of order spectrum, and contact loads

c) Comparisons of vibration signals in angle domain

d) Comparisons of order spectrum, and contact loads

In summary, the characteristics and mechanism of the outer ring fault of the axle-box bearing are correct. By modifying various fault sizes and the vehicle operating conditions, the outer ring fault characteristics of the axle-box bearing in real scenarios can be simulated and obtained. Based on these fault features, fault diagnosis algorithms can be improved for the condition monitoring of axle-box bearings. Furthermore, the remaining life and instantaneous wear features of the faulty axle-box bearing can be evaluated though the contact loads and rolling-sliding results output by the DTRB dynamic model [14]. Therefore, the proposed coupled model can be used to analyse and reveal the fault characteristics of axle-box bearings in urban rail vehicles. It can provide a feasible theoretical basis for the research of fault diagnosis algorithms and life assessment.