Dynamic responses of locomotive axle box bearings under varying operating conditions

Abstract

Axle box bearings (ABBs) are a vital component of a railway locomotive, ensuring their reliable performance requires an in-depth understanding of their dynamic behaviour. This work theoretically derives a locomotive–track longitudinal–vertical coupled dynamics model that takes detailed dynamics of the ABBs into account. The model considers a number of complex interactions, such as non-linear contact and friction forces within the ABB, wheel–rail interactions, and track irregularities. The motions of ABB are coupled with the wheelset and bogie frame, allowing detailed investigation of the ABB dynamic behaviour during operation. The correctness of the proposed model is verified in both the time and frequency domains by comparing with field test. The dynamic characteristics of ABBs under varying operating conditions were comprehensively investigated. The results demonstrate that track irregularities have a non-negligible influence on the dynamic forces of the locomotive ABB. Furthermore, the axle load transfer occurs during loading conditions, which leads to significant differences in vertical and longitudinal dynamic interactions within the ABBs across wheelsets. It is clear that the operating conditions of the locomotive should be suitably considered in the dynamic assessment, operation, and maintenance of ABBs.

Keywords

locomotive–track coupled dynamics model dynamics interactions axle box bearing dynamic responses loading process

1. Introduction

Heavy-haul trains play a pivotal role in freight transportation, significantly affecting the distribution of essential commodities and bolstering the national economies. While much scholarly attention has been directed towards the longitudinal impulse of trains, but the study of the dynamic behaviour of locomotive and their key components remains relatively underserved. Given that the locomotive is the primary source of traction for heavy-haul trains, its behaviour is crucial to the overall dynamic performance of the train (Zhang et al., 2022). A critical part of the locomotive, the axle box bearings (ABBs), endures various dynamic impacts during operation, such as car body loading, traction, and braking forces (Wang et al., 2019a). Recently, with the increased capacity and running speed of heavy-haul trains, ABBs are typically subjected to cyclic alternating loads in a harsher working environment. This exacerbates the deterioration of ABBs’ dynamic performance, increases the probability of failure, and potentially threatens the safety of train operations (Cheng et al., 2019). Therefore, it is particularly essential to investigate the dynamic properties of locomotive ABBs to help ensure the safety and stability of heavy-haul trains.

Various studies have analysed the dynamic response of vehicle systems. Researchers (Garg and Dukkipati, 1984; Wickens, 2003) introduced the classical theory and fundamental analysis of rail vehicle dynamics. Zhai (2020); Zhai and Cai (1997) proposed a novel concept for coupling independent vibration systems into an overall system via detailed mathematical modelling of the wheel–rail dynamic relationship. The overall vehicle–track vertical coupled dynamics model (CDM) was developed (Zhai and Sun, 1994), and extended this to include a locomotive–track spatial CDM to study the dynamic responses of the forces between the wheel and rail (Zhai et al., 1996). Aggestam et al. (2018); Aggestam and Nielsen (2020) created a vehicle–track CDM using two-dimensional and three-dimensional slab track models, enabling analysis of track dynamic responses and the exploration of vehicle–track vertical interactions. Tang et al. (2023) developed a flexible wheelset-damped track-tunnel-soil CDM that considered the interaction between the flexible wheelset and track, demonstrated the adequacy of CDM for analysing vibration and transmission of track and vehicle systems.

The more detailed vehicle–track CDMs have been further developed by adopting coupled dynamics theory to consider key components, such as the traction system, gear transmission (GT) system, and flexible wheelset. These models facilitate a more accurate exploration of the dynamic responses between components. A locomotive–track spatial CDM that accounts for the GT system is presented (Chen et al., 2018; Zhang et al., 2019), the dynamic response of locomotives under various excitations and external forces were analysed, and elucidating the interaction of the motor gear and the locomotive. Numerous scholars applied the vehicle–track vertical mathematical model to investigate dynamic responses of the vehicle system and the wheelset with the presence of wheel flats (Baeza et al., 2008; Nielsen and Igeland, 1995; Uzzal et al., 2013). Their findings highlight the significant impact of wheelset defects on wheelset’s force, displacement, and acceleration responses. Liang et al. (2021) developed a locomotive–track CDM with wheel polygonal wear and studied the wheel–rail dynamic interaction, and Spangenberg (2020) analysed the vibration theory of polygonal wheels. Their collective research verified that the polygonal wheel is the primary reason for the periodic wear of wheelsets and dynamic fluctuations in coupler force, and proved that wheel polygonal wear leads to severe vibration under loading and deteriorates the traction performance of the locomotive.

Most recently, some researchers have investigated the dynamic response of rolling element bearings (Harsha et al., 2004; Kalker, 2013) and ABBs of the vehicle (Wang et al., 2019b). A vehicle–track three-dimensional CDM (Wang et al., 2020) is developed to investigate the dynamic performance of ABBs with wheel polygonal wear, proving that wheel wear defects significantly increase ABB roller contact forces. Li et al. (2019) developed a vehicle spatially-CDM that considers the time-varying non-linear contact load of ABBs. They investigated the load distribution characteristics of ABBs excited by track irregularity and found that not only the contact forces of ABBs but also the lateral and vertical vibration of vehicle components increases. Yu et al. (2023b) established an urban rail vehicle–track CDM considering ABBs, studied the influence of raceway surface waviness on the skidding characteristics of ABBs, and revealed the cage and roller slip ratio law of different amplitudes and wavenumbers of raceway surface waviness. In general, the extant research on the dynamic responses of ABBs of high-speed train and urban rail vehicle, finding that the dynamic behaviour of ABBs is significantly influenced by vehicle vibration. However, research on the dynamic characteristics of locomotive ABBs under heavy load conditions remains limited. Given that heavy-haul trains typically involve significant axle loads and long formations, this gap underscores a critical need for further investigation to ensure the dynamic performance and operational reliability of ABBs.

Consequently, this paper proposes a novel locomotive–track longitudinal–vertical CDM with ABBs. This model specifically accounts for the non-linear interaction between the inner and outer rings within the ABB subsystem, including contact and friction forces. By incorporating these elements, the model enhances our understanding of the simulation techniques and dynamics, particularly when considering locomotive ABBs in a coupled mechanical system under varying operating conditions.

The structure of this paper is organised below: In Section 2, the dynamic forces between different components are derived. The CDM is established consisting of a locomotive, an ABB subsystem, and a track dynamics subsystem. The accuracy of the locomotive–track CDM with ABBs is verified by comparing simulated results with experimental tests in Section 3. Section 4 presents the analysis of the dynamic radial contact forces and stress, longitudinal/vertical relative displacement, and the loaded regions of ABB rings. Finally, conclusions are drawn in Section 5.

2. Dynamic model

To analyse the dynamic response of ABBs, a locomotive–track longitudinal–vertical CDM was established. The ABB subsystem is integrated into the locomotive–track longitudinal–vertical CDM to more accurately reflect the interactions within locomotive ABBs under realistic operating conditions. This model implements the force transmission process from wheelset to axle box and bogie frame, using ABBs. As shown in Figure 1, the dynamic model includes the locomotive structure, the axle box (shown in gray), and the track subsystem. The bounce, longitudinal, and pitch motions of these components are considered in detail.

In the locomotive dynamics model, two bogie frames supported the car body, the secondary suspension between them is regarded as a spring-damping unit ( $K_{s z}, C_{s z}$ ), the traction rods are considered as a spring-damping unit ( $K_{s x}, C_{s x}$ ) to provide a longitudinal suspension force. Each bogie frame is supported by two axle boxes connected to the wheelset. Similarly, the vertical structures of primary suspension and the traction rods are considered to be spring-damping units ( $K_{p z}, C_{p z}, K_{p x}, C_{p x}$ ).

Each axle box and wheelset are connected through the ABB system, as shown in Figure 2. Note that in this ABB system, the outer ring is integrated within the axle box, and the inner ring is installed on the wheel-axle.

For the track system, only the vertical motion of the track structure is under consideration. The track system is a damping spring–mass system. The stiffness and damping between rail, sleeper, ballast, and subgrade are taken into account ( $K_{p i}, C_{p i}, K_{b i}, C_{b i}, K_{f i}, C_{f i}$ ), and the system also accounts for the shear elasticity and damping ( $K_{b i}, C_{b i}$ ) effect between adjacent ballasts.

2.1. Internal forces of the locomotive system

In this model, the force between each component is thought of as a spring-damping unit. The secondary suspension forces of the locomotive are given as follows.

The secondary suspension vertical force is:

F_{s z i} = K_{s z} ({Z_{c} - Z_{t i} - (- 1)}^{i} β_{c} l_{c}) + C_{s z} ({{\dot{Z}}_{c} - {\dot{Z}}_{t i} - (- 1)}^{i} {\dot{β}}_{c} l_{c}); (i = 1, 2)

(1)

The longitudinal force of the traction rod linking the car body and the bogie frame is:

F_{s x i} = K_{s x} (l_{s e} - \sqrt{{l_{s e z i}}^{2} + {l_{s e x i}}^{2}}) - C_{s x} (\frac{{\dot{l}}_{s e z i} l_{s e z i} + {\dot{l}}_{s e x i} l_{s e x i}}{\sqrt{{l_{s e z i}}^{2} + {l_{s e x i}}^{2}}}); (i = 1, 2)

(2)

where

\begin{array}{l} l_{s e z i} = l_{s e} \sin α_{s r i} + Z_{t i} - Z_{c} + {(- 1)}^{i} β_{t i} l_{p s x} + {(- 1)}^{i} β_{c} l_{s x} \\ l_{s e x i} = l_{s e} \cos α_{s r i} - {(- 1)}^{i} X_{t i} + {(- 1)}^{i} X_{c} + {(- 1)}^{i} β_{t i} l_{p s z} - {(- 1)}^{i} β_{c} l_{s z} \end{array}

(3)

In the equations above, $X, Z$ and $β$ are the longitudinal, vertical displacements and angular displacement of pitch motion, respectively, the subscripts $c$ and $t i$ denote the car body and the $i$ th bogie frame, respectively. $K$ and $C$ represent the stiffness and damping, subscripts $s$ stands for the secondary suspension, $x$ and $z$ mean the directions of longitudinal and vertical, respectively. $l_{c}$ is the half-distance from the centres of two frames, $l_{s e}$ denotes the length of the traction rod between the car body and bogie frame, $l_{s x}$ and $l_{s z}$ are the longitudinal and vertical distance between the secondary suspension traction rod and the car body centroid, respectively, $l_{p s x}$ and $l_{p s z}$ are the longitudinal and vertical distances between the secondary suspension traction rod and the bogie frame centroid, respectively, and $α_{s r}$ is the tilt angle of the traction rod.

The primary suspension forces between the bogie frame and axle box are illustrated below. The vertical forces can be calculated by:

F_{p z i} = K_{p z} ({Z_{t r o u n d (i / 2)} - Z_{w i} - (- 1)}^{i} β_{t o u n d (i / 2)} l_{t}) + C_{p z} ({{\dot{Z}}_{t r o u n d (i / 2)} - {\dot{Z}}_{w i} - (- 1)}^{i} {\dot{β}}_{t o u n d (i / 2)} l_{t}); (i = 1, 2, 3, 4)

(4)

The longitudinal forces can be calculated by:

F_{p x i} = K_{p x} (l_{p e} - \sqrt{{l_{p e z i}}^{2} + {l_{p e x i}}^{2}}) - C_{p x} (\frac{{\dot{l}}_{p e z i} l_{p e z i} + {\dot{l}}_{p e x i} l_{p e x i}}{\sqrt{{l_{p e z i}}^{2} + {l_{p e x i}}^{2}}}); (i = 1, 2, 3, 4)

(5)

\begin{array}{l} l_{p e z i} = l_{p e} \sin α_{p r i} + Z_{w i} - Z_{t r o u n d (i / 2)} + {(- 1)}^{i} β_{t r o u n d (i / 2)} l_{p x} \\ l_{p e x i} = l_{p e} \cos α_{p r i} - {(- 1)}^{i} X_{w i} + {(- 1)}^{i} X_{t r o u n d (i / 2)} - {(- 1)}^{i} β_{t r o u n d (i / 2)} l_{p z} \end{array}

(6)

where the

r o u n d

function is used to round numbers to integers. The subscript

p

means the primary suspension and

w i

denotes the

i

th wheelset.

l_{t}

denotes the half-distance between the centres of the two wheelsets,

l_{p e}

is the length of the traction rod between the bogie frame and the axle box,

l_{p x}

and

l_{p z}

are the longitudinal and vertical distance between the traction rod of the primary suspension and the bogie frame centroid, respectively, and

α_{pr}

is the tilt angle of the traction rod.

2.2. Non-linear forces of the ABB

A double-row roller dynamic ABB model was established because the locomotive dynamic model takes no account of lateral motion, and the motion of two-row rollers can be regarded as identical. The radial relative displacement between the roller and the inner ring is ignored, rollers only rotate along the inner ring, and the motion between the roller and the bearing rings is kept to pure rolling. The influences of lubrication are ignored. The interaction between the inner and outer rings is performed as a non-linear spring, every roller is regarded as a spring unit ( $K_{r}$ ) that can react to the non-linear contact between the outer and inner rings, as illustrated in Figure 3.

When the ABB bears the external load, the inner and outer rings and rollers will interact. The compressive displacement between the inner and outer rings are identified as follows:

δ_{b i j} = {\begin{cases} (X_{a i} - X_{w i}) \cos θ_{i j} + (Z_{a i} - Z_{w i}) \sin θ_{i j}; (δ_{b i j} > 0) \\ 0; (δ_{b i j} \leq 0); (i = 1 - 4; j = 1 - 17) \end{cases}

(7)

where

X_{a}

Z_{a}

represent the longitudinal/vertical displacements of the axle box. The subscript

j

means the

j

th roller. And the angular distribution of the ABB roller,

θ_{i j}

can be calculated by:

θ_{i j} = \frac{2 π}{N_{j}} (j - 1) + \sum_{Δ t = 0}^{T e n d} Δ t \cdot ω_{c i} (t)

(8)

where

N_{j}

denotes the total number of rollers.

ω_{c}

is the angular velocity of the roller along the raceway, which can be calculated by:

ω_{c i} = \frac{(d_{m} - d_{r}) ω_{r i}}{2 \cdot d_{m}}

(9)

where

ω_{r}

denotes the rotation speed of the inner ring.

The force of roller along the radial direction can be expressed as follows (Yang et al., 2018):

F_{N i j} = K_{r} {(δ_{b i j})}^{1.11}

(10)

The contact stiffness can be determined by (Liu et al., 2022):

K_{r} = {(\frac{{l_{r}}^{0.8} \cdot π E_{0}}{3.81 \cdot 2 \cdot {(1 - u)}^{2}})}^{\frac{10}{9}}

(11)

where

l_{r}

denotes the effective contact length.

E_{0}

and

u

denote the elastic modulus and Poisson’s ratio, respectively.

According to the classical sliding friction theory, the friction force exerted by the rollers on the outer ring can be derived as follows:

F_{s i j} = ξ_{i j} F_{N i j}

(12)

where

ξ

is the dynamic friction coefficient, which is determined by the material of ABB and the relative slip velocity, details can be found in reference (Yu et al., 2023a).

The total longitudinal and vertical forces of the rollers exerted on the outer ring can be expressed as follows:

{\begin{cases} F_{r x i} = \sum_{j = 1}^{N_{j}} (F_{N i j} \sin θ_{i j} + F_{s i j} \cos θ_{i j}) \\ F_{r z i} = \sum_{j = 1}^{N_{j}} (F_{N i j} \cos θ_{i j} - F_{s i j} \sin θ_{i j}) \end{cases}; (i = 1, 2, 3, 4)

(13)

The vertical and longitudinal forces between the axle box and the wheelset are:

{\begin{cases} F_{a w z i} = 2 \cdot F_{r z i} \\ F_{a w x i} = 2 \cdot F_{r x i} \end{cases}; (i = 1, 2, 3, 4)

(14)

2.3. Wheel–rail forces

Wheel–rail contact relationship is one of the crucial aspects that affects the dynamic performance of a vehicle, which determines the forces between wheels and rails, and results in normal, traction, braking, or vibration-induced forces. The wheel–rail contact relationship consists predominantly of the normal and tangential forces. In this study, the wheel–rail normal contact force can be calculated by (Zhai and Sun, 1994):

P_{i} = {(\frac{1}{G} δ Z_{i})}^{\frac{3}{2}}; (i = 1, 2, 3, 4)

(15)

where

G

denotes the Hertz wheel–rail contact constant.

Considering the rail vertical displacement $Z_{r}$ and the track irregularity $Z_{i r r}$ , the compression of the wheel and rail can be derived as follows:

δ Z_{i} (t) = Z_{w i} (t) - Z_{r} (x_{w i}, t) - Z_{i r r} (t)

(16)

The wheel–rail creep force can be determined by (Chen et al., 2017):

F_{c r e e p i} = μ_{i} P_{i}; (i = 1, 2, 3, 4)

(17)

where

μ

denotes the adhesion coefficient, as defined in Ishikawa and Kawamura (1997):

μ_{i} = c \cdot \exp (- a \cdot v_{s l i p i}) - d \cdot \exp (- b \cdot v_{s l i p i})

(18)

According to the dry rail surface condition of Chinese railway lines studied in Chen et al. (2017), the parameters in the formula are $a = 0.12$ , $b = 0.24$ , $c = 0.53$ , and $d = 0.53$ . The wheel–rail relative velocity is:

v_{s l i p} = {\dot{θ}}_{w} R_{0} - v_{w}

(19)

where

θ_{w}

is the angular displacement of the wheelset rotation,

R_{0}

denotes the radius of the wheelset, and

v_{w}

is the longitudinal speed of the wheel.

2.4. Coupler-induced resistance forces

The coupler-induced resistance force $F_{l o a d}$ , is the drag force imposed on the locomotive car body and applied at a height $l_{f}$ from the centre of the car body, and can be modelled as the force of the locomotive to drag the trailers through the coupler in heavy-haul trains. This resistance force can be calculated by the product of the load weight and the unit weight resistance coefficient $w r$ , with reference to Chen et al. (2018):

w r = a_{0} + a_{1} \cdot v + a_{2} \cdot v^{2} + i

(20)

where

a_{0}

a_{1}

, and

a_{2}

represent the resistance coefficients,

v

is the locomotive speed, and

i

is the ramp slope. The parameters employed in the literature (Chen et al., 2018) are adopted here.

2.5. Motion equations of the locomotive

The motion equations of the locomotive comprise relationships for a car body, two bogie frames, eight axle boxes, and four wheelsets. Based on the previously calculated forces, the locomotive system motion equations can be obtained using the D'Alembert principle.

Bounce, longitudinal, and pitch motions of the car body are:

{\begin{cases} M_{c} {\ddot{Z}}_{c} = M_{c} g - 2 F_{s x 1} \sin α_{s r 1} - 2 F_{s x 2} \sin α_{s r 2} - 2 F_{s z 1} - 2 F_{s z 2} \\ M_{c} {\ddot{X}}_{c} = 2 F_{s x 2} \cos α_{s r 2} - 2 F_{s x 1} \cos α_{s r 1} - F_{l o a d} \\ J_{c} {\ddot{β}}_{c} = 2 F_{s z 2} l_{c} - 2 F_{s z 1} l_{c} - 2 F_{s x 1} (l_{s x} \sin α_{s r 1} - l_{s z} \cos α_{s r 1}) \\ + 2 F_{s x 2} (l_{s x} \sin α_{s r 2} - l_{s z} \cos α_{s r 2}) + F_{l o a d} l_{f} \end{cases}

(21)

Bounce, longitudinal, and pitch motions of the bogie frame are:

\begin{array}{l} {\begin{cases} M_{t} {\ddot{Z}}_{t i} = M_{t} g - 2 {F_{p z}}_{(2 i - 1)} - 2 F_{p x (2 i - 1)} \sin α_{pr (2 i - 1)} - 2 F_{p z (2 i)} \\ - 2 F_{p x (2 i)} \sin α_{pr (2 i)} + 2 F_{s z i} + 2 F_{s x i} \sin α_{s r i} \\ M_{t} {\ddot{X}}_{t i} = {(- 1)}^{i + 1} \cdot 2 F_{s x i} \cos α_{s r i} - 2 F_{p x (2 i - 1)} \cos α_{pr (2 i - 1)} + 2 F_{p x (2 i)} \cos α_{pr (2 i)} \\ J_{t} {\ddot{β}}_{t i} = 2 F_{p z (2 i)} l_{t} - 2 F_{p z (2 i - 1)} l_{t} + {(- 1)}^{i} \cdot 2 F_{s x i} (l_{p s x} \sin α_{s r i} + l_{p s z} \cos α_{s r i}) \end{cases}; \\ - 2 F_{p x (2 i - 1)} (l_{p x} \sin α_{pr (2 i - 1)} - l_{p z} \cos α_{pr (2 i - 1)}) \\ + 2 F_{p x (2 i)} (l_{p x} \sin α_{p r (2 i)} - l_{p z} \cos α_{p r (2 i)}) \end{array} (i = 1, 2)

(22)

Bounce and longitudinal motions of the axle box are:

{\begin{cases} M_{a} {\ddot{Z}}_{a i} = M_{a} g + F_{p x i} \sin α_{p r i} + F_{p z i} - F_{a w z i} \\ M_{a} {\ddot{X}}_{a i} = {(- 1)}^{i + 1} F_{p x i} \cos α_{p r i} - F_{a w x i} \end{cases}; (i = 1, 2, 3, 4)

(23)

Bounce, longitudinal, and pitch motions of the wheelset are:

{\begin{cases} M_{w} {\ddot{Z}}_{w i} = M_{w} g + 2 F_{a w z i} - 2 P_{i} \\ M_{w} {\ddot{X}}_{w i} = 2 F_{a w x i} + 2 F_{c r e e p i} & ; (i = 1, 2, 3, 4) \\ J_{w} {\ddot{θ}}_{w i} = T_{w i} - 2 F_{c r e e p i} R_{0} \end{cases}

(24)

where

T_{w i}

is the tractive torque applied to the wheelset,

g

is the gravity acceleration,

M

and

J

denote the mass and moment of inertia of different component, respectively.

2.6. Equations of the track system

A rail track is a damped elastic structure that bears longitudinal, lateral, and vertical loads. The track model of this paper refers to Zhai and Cai (1997), the details will not be repeated here. The track model is regarded as an Euler beam supported by continuous elastic discrete points. The foundation supporting the rail is discrete along the longitudinal direction and the discrete position is determined by each sleeper fulcrum. The motion equation of rail can be expressed as follows:

E I \frac{\partial^{4} Z_{r} (x, t)}{\partial x^{4}} + m_{r} \frac{\partial^{2} Z_{r} (x, t)}{\partial t^{2}} = - \sum_{i = 1}^{N_{r}} F_{r s i} (t) δ (x - x_{i}) + \sum_{j = 1}^{4} P_{j} (t) δ (x - x_{w j})

(25)

where

E I

represents the rail bending stiffness,

Z_{r} (x, t)

represents the vertical displacement of the rail,

m_{r}

is the track mass per unit length, and

N_{r}

is the number of sleepers (rail support points).

δ (x)

denotes the Dirac delta function,

x_{i}

is the horizontal coordinate of the

i

th sleeper, and

x_{w j}

is the coordinate of the

j

th wheel.

F_{r s i} (t)

is the

i

th contact force between the rail and the sleeper, which can be calculated by:

F_{r s i} (t) = K_{p i} (Z_{r} (x_{i}, t) - Z_{s i} (t)) + C_{p i} ({\dot{Z}}_{r} (x_{i}, t) - {\dot{Z}}_{s i} (t))

(26)

The fourth-order partial differential equation (25) can be transformed into a series of second-order ordinary differential equations by the Ritz method. The details of the solution can be found in the authors’ previous work (Zhai and Cai, 1997) and are not repeated here.

For the structure shown in Figure 1, the motion equations of the sleeper and the ballast are (Chen et al., 2017):

\begin{array}{l} M_{s i} {\ddot{Z}}_{s i} (t) + (C_{p i} + C_{b i}) {\dot{Z}}_{s i} (t) + (K_{p i} + K_{b i}) Z_{s i} (t) - C_{b i} {\dot{Z}}_{b i} (t) - K_{b i} Z_{b i} (t) \\ - C_{p i} \sum_{k = 1}^{N m} Y_{k} (x_{i}) {\dot{q}}_{k} (t) - K_{p i} \sum_{k = 1}^{N m} Y_{k} (x_{i}) q_{k} (t) = 0; (i = 1, 2, . . ., N_{r}) \end{array}

(27)

\begin{array}{l} M_{b i} {\ddot{Z}}_{b i} (t) + (C_{p i} + C_{f i} + 2 C_{b i}) {\dot{Z}}_{b i} (t) + (K_{b i} + K_{f i} + 2 K_{b i}) Z_{b i} (t) - C_{b i} {\dot{Z}}_{s i} (t) - K_{b i} Z_{s i} (t) \\ - C_{b i} {\dot{Z}}_{b (i + 1)} (t) - K_{b i} Z_{b (i + 1)} (t) - C_{b i} {\dot{Z}}_{b (i - 1)} (t) - K_{b i} Z_{b (i - 1)} (t) = 0; (i = 1, 2, . . ., N_{r}) \end{array}

(28)

where

Y_{k} (x)

is the

k

th orbital mode shape function and

q_{k} (t)

is the second-order ordinary differential equation obtained using the rail motion equation. Details can be found in the literature (Chen et al., 2017; Zhai and Cai, 1997) and will not be repeated here.

3. Simulation method and model validation

A locomotive–track longitudinal–vertical CDM considering the ABB subsystem was established. On the basis of this model, the dynamic response of the system was investigated.

The simulation was carried out for a locomotive of a type that is widely used across China, and relevant vehicle dynamics data is shown in Table 1. The structure and material parameters of the ABB are listed in Table 2. The Association of American Railroads (AAR) Class 4 track irregularity data was adopted to accurately reflect the influence of track excitation on the locomotive dynamic model. Taking into account the accuracy and stability of the dynamic equations, a fourth-order Runge–Kutta integration algorithm was used to solve the established vehicle system and bearing dynamic equations.

Table 1.

Main parameters of the locomotive model.

Notation	Specification	Value
$M_{c}$	Car body mass (kg)	6.26 × 10⁴
$M_{t}$	Bogie frame mass (kg)	6.275 × 10³
$M_{w}$	Wheelset mass (kg)	2.77 × 10³
$M_{a}$	Axle-box mass (kg)	368
$J_{c}$	Moment of inertia of car body (kg∙m2)	1.434 × 10⁶
$J_{t}$	Moment of inertia of bogie frame (kg∙m2)	1.311 × 10⁴
$J_{w}$	Moment of inertia of wheelset (kg∙m2)	1.081 × 10³
$K_{s x}$ / $K_{s z}$	Stiffness of secondary suspension along x/z (N∙m-1)	1 × 10⁸/2.14 × 10⁶
$K_{p x}$ / $K_{p z}$	Stiffness of primary suspension along x/z (N∙m-1)	1.645 × 10⁶/3.14 × 10⁷
$C_{s z}$	Damping of secondary suspension along z (N∙s∙m-1)	5 × 10⁴
$C_{p z}$	Damping of primary suspension along z (N∙s∙m-1)	9 × 10⁴
$l_{c}$	Semi-longitudinal distance between bogies (m)	5.03
$l_{t}$	Semi-longitudinal distance between wheelsets in bogie (m)	1.3

Table 2.

Basic parameters of axle box bearing.

Parameters	Value
Number of rollers in each row ( $j$ )	17
The average diameter of roller ( $d_{r}$ /mm)	27
The pitch diameter of the bearing ( $d_{m}$ /mm)	213
Modulus of elasticity ( $E_{0}$ /MPa)	2.07 × 10⁵
Poisson’s ratio ( $u$ )	0.29

Data from field experiments (a straight line in China) were collected to verify and validate the dynamic model. The vibration behaviour of the axle box was acquired through installation of accelerometers to measure the wheelset’s vertical acceleration. The actual vibration acquisition position is shown in Figure 4. The established coupled dynamics model was run with the same operating conditions as the field experiment, and simulation results were compared with experimental data.

Figure 5 compares the field test data and the simulation results of the wheelset vertical acceleration with a speed of 80 km/h in the time and frequency domains. As seen in Figure 5(a), the peak acceleration values of the field test and simulation are 52.2 m/s² and 45.8 m/s², respectively. Considering the random error of vibration caused by the track irregularity, the root mean squares (RMS) of the accelerations are calculated. It is shown that the RMS of the acceleration is 9.4 m/s² in the field test, and 8.7 m/s² in the simulation. In addition, the vibration acceleration frequency spectrum shows that the field test and simulation signals are basically consistent. Therefore, the locomotive–track longitudinal–vertical CDM with ABBs developed in this paper can be seen to accurately reflect the dynamic responses of the whole system and each component, and is available for further analysis of the dynamic characteristics of locomotive ABB.

4. Simulation and result analysis

To obtain the dynamic response of the ABB, simulations were performed under different conditions, such as with or without track irregularities, with or without loading, and with varying loading weights. The contact force and stress within the ABB and the relative displacement of bearing rings and their loaded regions were used to evaluate the bearings dynamic characteristics under varying operating conditions.

4.1. Dynamic responses of the ABB excited by track irregularities

The dynamic interaction between the wheels and the rails on locomotives is non-negligible. As an external disturbance of the locomotive system, track irregularity is one of the primary input disturbances of the dynamic system. This section investigates the effect of track irregularities on the dynamic properties of ABBs. The AAR Class 4 track irregularities are adopted during simulations.

Figure 6 shows the radial contact force and contact stress between the roller and outer ring within the ABB of the No.1 wheelset (the first position wheelset in the forward direction is No.1, the second is No.2, and so on) with and without the track irregularity at a speed of 80 km/h. The roller does not consistently bear contact forces but experiences a periodically loaded, and Figure 6(b) also reveals that the roller is only in partial contact. Track irregularity results in higher frequency vibration forces and larger peaks. Compared to the results acquired without track irregularities, the amplitude of the contact force and contact stress of the roller–outer ring increases by 7.2% and 4.6%, respectively.

Figure 7 illustrates distribution of the radial contact forces between the rollers and the outer ring of the ABB at a particular time. About half of the rollers bear the pressure during locomotive operation. The number of load-bearing rollers did not change compared to the condition without track irregularities, but the maximum amplitude of radial contact force increased by 15%.

Figure 8 shows the longitudinal/vertical relative displacement and radial compression between the outer and inner rings, with or without track irregularity. The vibration intensity and loaded region of the ABB roller are analysed.

Owing to the weight of the locomotive car body and bogie frame, the roller is compressed and bears the load in the upper part of the ABB. In this case, the vertical displacement of the outer ring of the ABB is greater than that of the inner ring, which is why the vertical relative displacement is always negative. The longitudinal relative displacement is effectively maintained at the original centroid position, the vibration in this direction is extremely low. Moreover, the vibrations of the roller under the track irregularity are more random and severe than without irregularity excitation.

4.2. Dynamic responses of the ABB with or without loading

To simulate the loading conditions of heavy-haul train operation, a coupler-induced load resistance is exerted on the locomotive car body, as the force from the wagons passing through the coupler. The simulation cases are without loading (the coupler-induced load resistance is 0) and loading 5000 t weight by the wagons. Note that the coupler-induced load resistance and a traction force are applied to maintain the locomotive dynamics model at a speed of 80 km/h. The traction characteristics applied here are referred to in Chen et al. (2018). The dynamic response of ABBs with or without loading is compared and discussed.

Figures 9(a) and (b) show the longitudinal/vertical relative displacements and radical compression between the ABB outer and inner rings of different wheelsets when the locomotive without loading. Because of the weight of the locomotive, there is a vertical relative displacement between the rings of the ABB. However, without loading, the longitudinal relative displacement remains approximately zero. It is revealed that the distributions of the loaded region of the ABB rollers in different wheelsets virtually coincide, the trajectories of the rollers of different wheelset are essentially the same.

Figure 10(a) and (b) show the longitudinal/vertical relative displacement and radical compression between the outer and inner rings of different wheelsets ABB with a loading of 5000 t weight. The figures show that the rollers of the upper and front parts of the ABB are compressed and bearing under load. Due to the weight of the locomotive system, the rollers are subjected to pressure in the upper part. As the load resistance is applied, the car drags the axle, the longitudinal relative displacements increase, thus the front part of the ABB rollers is compressed.

It is worth noting that in Figure 10(a), from the No.1 to the No.4 wheelset, the longitudinal relative displacements become smaller while the vertical relative displacements become gradually greater. Meanwhile, as shown in Figure 10(b), the compression displacement and the load force have obvious changes, the wheelsets on the front bogie decreases while then the rear bogie increases, compared with those without loading over the same wheelsets, indicating that significant axle load transfer occurred.

Axle load transfer occurs when the locomotive undergoes tractive, braking, and loading processes and, to a certain extent, affects the wheel–rail dynamic interaction. Figure 11 compares normal contact forces between the wheel and the rail with the 5000 t weight loading and the average axle load without loading. As seen, the axle loads of the front bogie frame wheelsets decrease while the loads of the rear bogie frame increase. Which means that there is an axle load transfer, with the wheelset of front bogie frame being unloaded and the rear ones being loaded. Additionally, the No.1 and No.4 wheelsets (wheelsets farther from the centre of the locomotive) experience a more significant change. The axle load transfer changes the wheel–rail contact force and further affects the force between the wheel and the axle box, resulting in a corresponding change within the ABB. For example, when the wheel–rail normal contact force of the No.4 wheelset becomes larger and is delivered to the axle box, the bearing rollers of the No.4 wheelset ABB carry a larger contact force accordingly. Therefore, the ABBs of the No.4 wheelsets with the greatest increase axle loads were selected for subsequent investigations.

4.3. Dynamic responses of the ABB with different loading conditions

To further compare and analyse the dynamic response of the locomotive ABB under different loading conditions. Similarly, the coupler-induced load resistance is applied to simulate the different loading conditions, three weights of the wagons were considered in this case: 0 t (without loading), 2500 t, and 5000 t. The speed of the locomotive is still maintained at 80 km/h.

Figure 12 represents the radial contact force and contact stress of the No.4 wheelset ABB rollers when the locomotive was under different load conditions. It shows that as the loading weight applied by the coupler increases from 0 t to 2500 t and then to 5000 t, the radial contact force between the roller and the outer ring of ABB on axle No.4 increases by 6.64% and 18.20%, and the contact stress increased by 7.69% and 11.01%, respectively.

Figure 13 shows the longitudinal/vertical relative displacement and radical compression between the outer and inner rings of the No.4 wheelset ABB with different loading conditions. The greater the loading weight, the greater the load increase of the wheelset, so the vertical displacement of the roller becomes greater. This change in vertical displacement is caused by the axle load transfer when the coupler imparted load resistance is applied. Meanwhile, the drag effect of the locomotive system on the axle becomes more obvious with increased loading weight, that is, the longitudinal relative displacement between the outer and inner rings of the ABB increases. The change of loaded region is also demonstrated within in Figure 13(b).

5. Conclusions

This study establishes a locomotive–track longitudinal–vertical CDM taking ABBs into account. This model integrated the dynamic subsystem of ABB rollers, considers the realistic force transmission process of the primary suspension when the wheel–rail force is transmitted to the bogie frame through the axle box, and explores the dynamic interaction of ABBs fitted to the locomotive. Furthermore, the model was contrasted with the field test, and the dynamic behaviours of ABBs were demonstrated and analysed undergoing a range of different operating conditions.

By comparing the model with the field test, it was confirmed that this model accurately reflects and effectively simulates the vibration properties of locomotive components. The simulation results show that track irregularities have a highly noticeable impact on the ABB of the railway locomotive. The ABBs generate more intense vibration and larger contact when there is excitation of track irregularity. Moreover, when the coupler imparted load resistance is applied, the rollers in the front part of the ABB rollers are subjected to more dynamic forces. This proves that the longitudinal dynamic behaviour of rollers is affected to some extent, the loading region of the ABB rollers and the primary suspension force are also significantly affected during the loading process. The greater the coupler-induced load resistance, the worse the working environment of the ABB and the running safety of the locomotive. Moreover, axle load transfer occurs in the presence of the coupler-induced resistance force, and the degree of transfer becomes larger as the load resistance increase.

In conclusion, understanding the dynamic behaviour of locomotives and ABBs has profound theoretical and practical significance for ensuring the safety of heavy-haul train operations. In addition, this model can also be used for dynamic response analysis, component and system fault diagnosis, and safety assessment of ABBs under different operating conditions. However, this model does not consider the dynamic interactions with other components, such as motors, so there is scope for further model refinement.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 52205217), National Key Research and Development Program of China (Nos. 2021YFB3400703 and 2021YFB3400704), and the Natural Science Foundation of Sichuan (No. 2022NSFSC1964).

ORCID iD

Yukun Wang

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