A trailer car dynamics model considering brake rigging of a high-speed train and its application

Brake systems are essential for the speed regulation or braking of a high-speed train. The vehicle dynamic performance under braking condition is complex and directly affects the reliability and running safety. To reveal the vehicle dynamic behaviour in braking process, a comprehensive trailer car dynamics model (TCDM) considering brake systems is established in this paper. The dynamic interactions between the brake system and the other connected components are achieved using the brake disc–pad frictions, brake suspension systems, and wheel–rail interactions. The force and motion transmission from the brake system to the wheel–rail interface is performed by the proposed TCDM excited by track irregularity. In addition, the validity of TCDM is verified by experimental test results. On this basis, the dynamic behaviour of the coupled system is simulated and discussed. The findings indicate that the braking force significantly affects vehicle dynamic behaviour including the wheel–rail forces, suspension forces, wheelset torsional vibration, etc. The dynamic interactions within the brake system are also significantly affected by the vehicle vibration due to track irregularity. Besides, the developed TCDM can be further employed to the dynamic assessment of such a coupled mechanical system under different braking conditions.


Introduction
In recent years, high-speed railways have developed rapidly in China [1][2][3]. The high-speed trains usually adopt the disc brake systems, which are mainly composed of brake disc, pad, and clipper. The braking performance is crucial to the vehicle running safety. During braking process, the friction forces generated within the brake disc-pad interface through sliding friction are firstly delivered to the wheel-rail interface. Then, the longitudinal wheel-rail creep forces opposite to the running speed prevent the vehicle moving forward. Thus, the brake system is excited by the internal disc-pad friction and external wheel-rail excitations during braking process. The continuous improvement of train operating speed and the increasing complexity of operation environments bring a great challenge to the reliability of the braking performance [4]. Particularly, interactions between the wheel-rail excitation and the disc-pad friction increase the vibration intensity of the brake system, deteriorate the braking performance of the train, and aggravate the fatigue damage of components [5,6]. Therefore, it is critical to ascertain the vehicle system dynamic behaviour when braking and accurately assess the brake system dynamic performance.
Track irregularity is the main excitation for the railway vehicles and causes the impact load between the wheel and the rail, which determines the dynamic behaviour of a vehicle system [7][8][9]. In recent years, the dynamic characteristics of a vehicle system and its key components with the consideration of track irregularity excitation have been studied by many scholars [10][11][12]. To understand the dynamic interactions within the vehicle-track systems, Zhai et al. [13] suggested a coupled dynamic model regarding the vehicle and track as an integrated system. Thereafter, the coupled dynamics model was further improved and extensively adopted to study railway vehicle dynamics. Naeimi et al. [14] investigated the effect of left and right uneven track irregularities on the dynamic response of a track 1 3 Railway Engineering Science (2023) 31(3):269-280 using a three-dimensional vehicle-track coupled dynamics model. Xu et al. [15] studied the wheel-rail contact force using a dynamics model with the consideration of track geometric random irregularity. Song et al. [16] proposed a high-speed train vehicle dynamics model considering the dynamic effects of the vehicle-track system on the pantograph-catenary system. Chen et al. [17] built a locomotive-track vertical coupled dynamics model to analyse the dynamic behaviour of the system excited by both of the track irregularity and gear mesh force. Liu et al. [18] explored the effect of wheel polygonal wear and track irregularity on vibration characteristics of the gear transmission system through full-scale experiment and numerical simulation. The authors developed a motor car-track coupled dynamics model of a high-speed train containing the traction transmission systems [10] and axle-box bearing [19] and found that the gear meshing and track irregularity excitations significantly impacted the dynamic performance and stability of these key components. These research achievements help to understand the dynamic behaviours of the key components of a train, including the wheelset, slab, pantograph-catenary, gearbox, axle box, traction motor, etc., and the results indicated that the vehicle-track coupled system and the track irregularity had a significant impact on the system components. However, few studies addressed the brake system, which as an important sub-system of a train [20]. Therefore, a vehicle dynamics model containing brake systems need to be established with full consideration of the track irregularity to precisely assess its dynamic performance during operation. For the brake system dynamics, the friction-induced vibration and noise [21,22] and the complex nonlinear dynamic behaviour [23,24] under different working conditions have been widely investigated. Sinou et al. [25] investigated the mechanism of TGV brake squeal by experiments and developed a finite element model of the brake system. Then, the stability and transient dynamic responses of the brake system were simulated to understand and predict the TGV brake squeal. Chen et al. [26] proposed a railway vehicle brake dynamics model considering the brake disc-pad friction to explore the mode-coupling instability of the brake system under different braking pressures. Lorang et al. [27] studied the nonlinear behaviour of a TGV brake system via experiments and numerical simulation. Wu et al. [28] analysed the effect of a disc brake system on wheel polygonisation using a dynamics model of brake system with wheelset and rail. Zeng et al. [29] evaluated the nonlinear vibration characteristics of the brake system based on a railway vehicle dynamics model with brake system. Recently, the authors proposed a trailer bogie dynamics model with brake systems, which considered the disc-block nonlinear friction and the torsional vibration between disc and wheel [30]. Based on the model, the effects of suspension stiffness of pad and parameters of primary suspension system on the stability of the trailer bogie system were comprehensively analysed. All the above research can provide references for understanding the vibration characteristics and the parametric design of brake systems. However, the dynamic interaction between the brake systems and the vehicle was not fully considered in the established brake system dynamics models.
In summary, the dynamic coupling between the brake system and the vehicle system under the combined action of multiple nonlinear excitations including the track geometric irregularity and the disc-pad interface friction is rarely investigated [31,32]. Particularly, there is little research on the spatial coupling with the aspects of motions and forces between the mechanical brake system and vehicle system, although the brake system and vehicle system interact with each other during braking process. Thus, the dynamic behaviour of the vehicle and brake system in real operation conditions with track irregularity cannot be accurately reflected and predicted using the current methods during braking process.
To meet this gap, adopting the classical vehicle dynamical and interface tribological theories, a spatial trailer car dynamics model (TCDM) of a high-speed train considering brake systems is established in this paper. The coupling effects between the brake system and vehicle system are comprehensively considered using the disc-pad interface friction, wheel-rail relationships, and system suspensions. Besides, the effectiveness of the proposed model is validated by field tests. Then, the dynamic response of the vehicle excited by both of the track irregularity and braking interface friction is further simulated and analysed.

Trailer car dynamics model with brake system
Using the vehicle-track coupled dynamics [13], a highspeed train TCDM with brake system is proposed. The mechanical structure of the vehicle and detailed components of the brake system are shown in Fig. 1. The whole trailer car consists of carbody, bogie frame, wheelsets, axle box, and brake systems. The carbody is supported on the bogie frame using secondary suspension systems, and the bogie frame is supported on the wheelset by the primary suspension systems. The clapper of the brake system is hung on the bogie frame, and the brake disc is fixed on the wheelset axle. During a braking process, the braking forces are applied on the points shown in Fig. 1c. Then, the side structure of the clipper will rotate around the z-axle, which makes the brake pad contact the brake disc and generate frictions. The direction of the friction force between the brake disc and pad is determined by their relative velocity. The friction forces within the brake disc-pad interface generated by sliding friction will be delivered to the wheel-rail interface through wheelset axle. Therefore, the dynamic force within the disc-pad interface affects the force within the wheel-rail interface, and vice versa. For the brake system, the disc is mounted on the wheelset axle, and it rotates synchronously with the axle. The clipper is hung on the bogie frame via rubber nodes, which allows the rotation around the y-axis. Besides, the end of clipper close to the pad is also suspended on the bogie frame using two brackets. The structures on both sides of the clipper can rotate around the x-axis. During the braking process, the forces generated by the cylinder will be applied on the side structures of the clipper, and the side structures rotate to make the brake pad contact the brake disc. Finally, the friction forces are generated with the relative motions of disc and pad under the normal contact force. The motions and frictions of the brake system are comprehensively considered in the proposed model. Figure 2 shows the end view of the TCDM with disc brake system.

Dynamics model of brake system
The brake disc-pad contact and friction are essential to the brake system. In the braking process, the pad and disc should be in contact and generate the normal force and friction force (Fig. 3). The disc-pad normal force is performed by a linear unilateral spring-damper combination force element and is calculated by where r ijy is the lateral relative displacement between points i and j located on the pad and disc, respectively; k y and c y are the disc-pad contact stiffness and damping, respectively; and v ijy is the lateral relative velocity between points i and j. The lateral displacement between points i and j is calculated by where y D is the lateral displacement of disc; y i is the lateral displacement of point i located on the pad; x and y are the rotation angles of brake disc around x-and y-axes, respectively; and r jx and r jy are the longitudinal and lateral displacement of point j relative to the disc centre, respectively. The relative velocity between points i and j is calculated by Using Eqs. (1)-(3), the disc-pad normal forces can be calculated. Clipper Clipper The Stribeck model is widely employed to describe the friction characteristics between the brake disc and pad [20], whereby the dynamic friction coefficient can be computed as where k and s are the dynamic and static friction coefficients of the brake system, respectively; v r is the tangential relative velocity between the pad and disc and can be calculated by where v px and v py are the longitudinal and vertical velocities of the brake pad, respectively; v dx and v dy are the longitudinal and vertical velocities of the brake disc, respectively. When the vehicle is braking at high-speed range, the disc and pad are regarded as in sliding friction conditions. The friction force F f can be calculated by The longitudinal friction force of brake pad F dpx is calculated by The vertical friction force of brake pad F dpz is calculated by The friction torque of brake pad T dp is calculated by In Eqs. (7)- (9), V x and V z represent the velocity of the brake pad relative to the disc in x and z directions, respectively, which are given as where v px and v pz are the longitudinal and vertical velocities of the brake pad, respectively; v dx and v dz are the longitudinal and vertical velocities of the brake disc, respectively.

Flexible brake clipper and disc
The finite element method (FEM) is employed to simulate the flexible deformation of brake clipper and disc, which can reveal the dynamic responses of the brake system more realistically [10,33]. The displacement of a flexible body can be calculated by [10] where u is the Eulerian coordinate vector, and w represents the flexible displacement in the spatial point u at instant t. Using the modal superposition method, the flexible deformation can be calculated by where q(t) is the generalised coordinate vector, and (u) is the mode shape matrix. In this study, the finite element models of brake clipper and disc shown in Fig. 4 are established in the ANSYS environment. The model details and the main parameters of the brake clipper and disc are illustrated in Table 1. Then, the modal matrix (u) can be calculated by modal analysis, which is applied to calculate the flexible deformation of the brake system.

Wheel-rail interactions
Because of the nonlinear geometric relations between wheel tread and rail profile, the wheel-rail interactions are extremely complex. In the model, a method in the literature [13] is adopted to obtain the wheel-rail contact parameters, and the wheel-rail normal force F N can be calculated by [34]

Fig. 3 Contact and friction within the brake disc-pad interface
where G is the wheel-rail contact constant, and Z(t) is the wheel-rail normal compression.
The longitudinal creep force, lateral creep force, and creep moment can be calculated by Kalker's linear creep theory [35,36]. To simulate the case of any creep rate and small spin values, the corrected longitudinal creep force, lateral creep force, and creep moment can be obtained by Shen-Hedrick-Elkins model [37]. Figure 5 shows the full-scale test bench of a train brake system adopted to grasp the friction coefficient within the disc-pad interface. It comprises a rotational brake disc derived by motor and an inertia wheel group, a brake pad, and a brake calliper which provides the clamping force to perform the braking states. Based on the test bench, a large number of repeated braking experiments within the speed range of 0-350 km/h were conducted to acquire the tribological characteristics of the brake systems. The braking normal force applied on the system in the test is consistent with that during vehicle operation. Figure 6 shows the friction coefficient of the disc-pad contact interface at different speeds. The sampling frequency and filtering techniques are not listed here due to the confidentiality limitations. It can be Fig. 4 Finite element models of the a brake hanger, b side structure, and c disc  . 6 Tested disc-pad friction coefficient concluded that the friction coefficient is changing with the variation of speed, and the maximum and minimum amplitudes of friction coefficient are 0.45 and 0.33, respectively. The mean value of the friction coefficients is 0.37, which is adopted in the brake dynamics model.

Field experimental test and model validation
During the high-speed train operation, vibration signals of the bogie frame and the axle box are collected to confirm the suggested model. The corresponding monitoring points on the bogie frame and axle box are shown in Fig. 7. The vibration signals are selected from the vehicle running along a straight line at 300 km/h. For comparison, simulations based on the proposed TCDM are implemented in the same conditions by employing the main parameters in author's previous work [33] and adopting the tested track geometric irregularity of a high-speed railway line in China [35]. Figure 8 depicts the vertical vibration acceleration signals of the bogie frame and axle box in time and frequency domains. The maximum amplitudes of the bogie frame (axle box) vibration acceleration from field test and simulation are 0.85 (16.7) and 0.74 (15.8) m/s 2 , respectively. Compared to the tested results, the vibration acceleration errors of the bogie frame and axle box in simulation are 12.9% and 5.4%, respectively. In addition, it can be found that the simulation results have similar frequency components to the test results. Generally, the dynamics model can mainly grasp the vibration characteristics of vehicles during operation, and its effectiveness is verified by the experimental test results.

Vehicle dynamic performance during braking process
Using the proposed TCDM with brake system, the dynamic behaviour of the system is investigated and discussed in this section. During simulation, the track is considered as a straight line, the speed is set to be a constant 300 km/h, and a normal braking force of 13 kN is applied on the side structure of the brake clipper. Figure 9 depicts the vertical vibration acceleration of the bogie frame in time and frequency domains. The maximum amplitude of vibration acceleration reaches 3.6 m/s 2 under braking condition, with an increment of 6.3% compared to the result without braking force. It can be seen from the frequency spectrum that the vibration of bogie frame with and without braking force has little difference. Figure 10 depicts the pitch angle of the bogie frame in time and frequency domains. The maximum amplitude and RMS value of the bogie frame pitch angle are 11.33 × 10 -3 rad and 10.8 × 10 -3 rad under the braking condition, and the corresponding maximum amplitude and RMS value without braking are 5.8 × 10 -4 rad and 2.3 × 10 -4 rad, respectively. The frequency spectrum indicates that the pitch angle of the bogie frame has higher amplitude, especially when the frequency is over 50 Hz. It can be concluded that the braking force enhances the pitch motion of the bogie frame. Figure 11 shows the vertical vibration acceleration of the axle box in time and frequency domains. The maximum amplitude and RMS value of the vibration acceleration reach 37.6 m/s 2 and 10.0 m/s 2 , with an increment of 4.5% and 2.9% compared to those without braking force, respectively. As shown in Fig. 11b, the frequency spectrum obtained with braking force is more complex than that without braking force. Figure 12 depicts the angular acceleration of wheelset in time and frequency domains. It shows that the angular acceleration of wheelset with braking force is much higher than that without braking force. In addition, the frequency results also show the similar phenomenon that the frequency amplitude with braking force is clearly higher than that without braking force. Therefore, the braking force enhances the torsional vibration of wheelset, which is determined by the structure characteristics of the brake system.
To make a clear and better comparison of the vibration responses of the vehicle system with and without braking, the corresponding statistic indexes are listed in Table 2.

Bogie frame
High-speed train Trailer bogie Fig. 7 Field experimental test Figure 13 shows the suspension forces of the brake system in time domain and frequency domains. It indicates that the suspension force is almost vibrating about 0 N without braking force. The suspension forces of the brake system for the first and second wheelsets are rapidly increased due to the braking force. Besides, the suspension force of the brake system located on the first wheelset is higher than that of the second wheelset. The frequency spectrum shows that the amplitude of suspension forces with braking force is higher than that without braking force, especially at high-frequency range. Besides, the frequency amplitude of suspension force for the first wheelset is also higher than that of second wheelset. In the braking process, the main frequencies include the 29, 57, 86 and 173 Hz, namely the rotation frequency of wheelset and its multiplications at 300 km/h. Figure 14 shows the braking friction forces within the disc-pad interface of the brake systems located on the first and second wheelsets. During braking process, the RMS values of the friction force of the first and second wheelset brake systems are 6.1 and 6 kN, respectively. The vibration  amplitude of the friction force of the second wheelset brake system is higher than that of the first wheelset. The frequencies of the friction forces for different brake systems are almost the same, mainly comprising the excitation frequency due to track irregularity, the wheelset rotation frequency, and its multiplications at the running speed of 300 km/h. It can be concluded that the friction force of the first wheelset brake system is higher than that of second wheelset. This difference causes the suspension force of the brake system located on the first wheelset higher than that of the second wheelset. Figure 15 depicts the dynamic force of the primary suspension system in time and frequency domains. During the braking process, the RMS value of the suspension force for the first and second wheelsets is 42.5 and 66.3 kN, respectively. The RMS value for the first wheelset without braking force is 54.2 kN. It can be concluded that the primary suspension forces are significantly affected by the braking force. It causes the primary suspension force of the first wheelset smaller than that without braking force and causes the primary suspension fore of the second wheelset higher than that without braking force. This phenomenon arises from that the friction force for the first wheelset decreases the loads of the primary suspension system, and the friction force for the second wheelset increases the loads of the primary suspension system. Figure  2.1% (5.8%) for the second wheelset. It can be seen from Fig. 16b that the frequency of vertical wheel-rail forces with and without braking force is mainly determined by the track irregularity.
The above analysis indicates that the vehicle dynamic performance during braking conditions is different from that without braking in terms of motions, dynamic forces, etc. In the braking process, the friction force within disc-pad interface is transmitted to the wheel-rail interface and bogie frame via wheelset axle and brake suspensions, respectively. The forces transmitted to the wheel-rail interface change the wheel-rail interactions and result in the variation of vertical and longitudinal wheel-rail forces. The longitudinal wheel-rail force resists the vehicle running forward; then, it causes the load transfer phenomenon in the wheel-rail vertical force (Fig. 16) and enhances the torsional vibration of wheelset (Fig. 12). The friction force transmitted to the bogie frame changes the load of the primary suspension system, causing the suspension forces of the first wheelset smaller than those of the second wheelset (Fig. 15). For the brake system, the disc-pad interaction forces and suspension forces are mainly influenced by the vehicle vibrations due to track irregularity. The effects of wheel-rail excitations should be considered in the dynamic assessment of brake system. However, the dynamic interactions between the brake system and the vehicle system are usually ignored in the traditional vehicle-track coupled dynamics model. This shortage can be mitigated in the developed TCDM by analysing the dynamic behaviour of the brake system in a vehicle vibration environment.
To make a clear comparison of the dynamic forces of the vehicle system in the braking process, the corresponding statistic indexes are listed in Table 3.

Conclusion
A trailer car dynamics model (TCDM) considering the brake system is proposed to reveal the dynamic interactions between the brake system and vehicle system. Using the brake-disc frictions, wheel-rail interactions, and suspension systems, the brake systems are integrated into the classical vehicle dynamics model. In addition, the accuracy of the TCDM is validated by field experimental test results. Compared to the traditional vehicle dynamics model, the developed TCDM has the advantage to reveal the dynamic behaviour of the vehicle-track coupled system more accuracy during braking process.
The results indicate that the vertical vibrations of the bogie frame and the axle box are hardly influenced by braking force. However, the angular displacement of the bogie frame with braking force is significantly higher than that without braking force. In the parametric design of bogies, the effects of braking force can be analysed using this model. In addition, the torsional vibration of wheelset is enhanced by the braking force. The load transfer can be observed in the brake suspension force, primary suspension force, and wheel-rail vertical force under braking conditions, which is caused by the friction force generated within the brake disc-pad interface. It can be used to guide the wheel-rail force control in braking process. The fluctuation of the braking friction force, which determines the braking performance, is mainly dominated by the track irregularity. Therefore, the coupling effects between the brake system and vehicle system require to be considered in the dynamic evaluation of vehicle-track system, especially in braking process.
Further, the proposed TCDM can be employed to investigate the dynamic performance of the vehicle-track coupled system under different braking conditions. In addition, the complex wheel-rail excitations, including wheel polygonal wear and wheel flats, are also worth being involved in the dynamic assessment of such systems. In our future work, the vehicle ride index and the fatigue damage of components will be studied; the friction coefficients will be defined under different conditions to perform the disc-pad tribological characteristics.
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