Simulation of a freight wagon bogie in double roller test rig

Abstract

Test benches are useful tools for testing bogies in laboratory conditions, most of which consist of large rollers to move the wheels of the bogies, either at a reduced scale or at full size. The full-scale test rig located at the BOGLAB laboratory uses two small rollers to move each wheel of one of the bogie’s wheelsets. This paper presents a 21-DOF dynamic model of a two-axle bogie running on this double roller test rig that considers the motion of the two wheelsets, the bogie frame and the four axle boxes, which is not usually implemented in the models of the literature. The model also computes the creepages, creep and normal forces originating from the contact of the two rollers on each wheel. The simulation results show the movement transmission from the front wheelset to the rear wheelset due to the specific configuration of this test bench. A sensitivity analysis is performed to study the influence of bogie load and speed on the vertical acceleration of the axle boxes. Data show that increasing the bogie load leads to higher vertical acceleration in the axleboxes. The effect of speed is nonlinear, revealing a threshold speed beyond which vertical accelerations significantly decrease.

Introduction

In 2015, the United Nations adopted the 2030 Agenda for Sustainable Development and its 17 Sustainable Development Goals (SDGs) to promise a healthier planet and secure the rights and well-being of everyone on Earth. Railways can play a relevant role in achieving the SDGs, as it is a very efficient transport method that can help reduce pollution and energy consumption in the movement of people and goods.

Maintenance is a key aspect of keeping the efficiency and safety standards of railways and thereby contributing to the achievement of the SDGs. However, predictive maintenance requires extensive knowledge of the rolling stock to determine the best time to perform the maintenance tasks. In this sense, mathematical models of rolling stock offer a good approach, as they allow studying the performance of rolling stock in different conditions and at lower costs than experimental testing with real vehicles.

Lots of models can be found in the scientific literature,^1–9 most of them focused on the stability of the railway vehicle. For example, Lee and Cheng use a linearised bogie model to investigate the influence of several parameters on the bogie’s performance in straight ¹⁰ and curved tracks. ¹¹ Song et al. ¹² introduce a distributed masses model to control the stability of a high-speed train. This is achieved by moving two actuators placed transversely in both bogie ends. The system is tested in a 1:5 scale bogie test rig. Bosso et al. ¹³ present a bi-dimensional model of the friction dampers used in the Y-25 freight bogie. Then, stability analyses are performed. Molatefi et al. ⁸ proposes a mathematical model for studying the stability of a freight wagon. This model includes several non-linear phenomena such as non-linear springs and clearances. Shin et al. ¹⁴ use a dynamic model implemented in Simpack to develop an algorithm able to detect hunting instability from acceleration signals. All these works focus on the stability of rail vehicles but don’t take into account the effect of external excitations like, for example, track irregularities.

Among the models that also consider the influence of vibrations in the dynamic behaviour of the rolling stock, the following can be highlighted. Demir ¹⁵ develops a multibody model of a rapid transit vehicle with several degrees of freedom. Then, track irregularities are simulated according to the standards and the suspension parameters are characterised so that the measured vibration is within the standard margins. True et al. ¹⁶ use the model developed in previous works ¹⁷ to attempt to detect the track geometry from the dynamic reactions measured in the wheelset. Lebel et al. ¹⁸ use Bayesian methods for monitoring the suspension health condition based on acceleration data. Dumitriu ¹⁹ uses rail vehicle models that combine rigid and flexible solids to establish the operating condition of primary suspension dampers. Simulation results are combined with vibration data measured in the actual vehicle. Bernal et al. ²⁰ present a bogie model to detect the wheel flat from the accelerations measured in the wheelset, the bogie and the wagon. The model is implemented in GENSYS and compared to previous works, obtaining similar results. These authors also ²¹ propose a quarter-vehicle model to estimate the characteristics of a low-cost system that detects wheel flats. This system is subsequently tested on a 1:4 scale test bench.

Nevertheless, the number of works that focus on simulating the behaviour of a bogie in a roller test rig is limited. Iwnicki and Wickens ²² proposed the dynamic model of a 1:5 scale bogie running on a test rig. Each wheel of the bogie is moved by a large roller (in relation to the wheel radius). Later, Liang et al. ²³ developed a two-degree-of-freedom model of this scaled bogie to study wheel flats and rail defects. The acceleration measured in the axle box of a scale bogie is the input of the mathematical model. Liu et al. ²⁴ propose a dynamic model of a test rig. The system consists of two rollers of large diameter, the wheelset and the primary suspension of the bogie. The model is used to investigate the relationship between defects in the wheel and/or rail and the vibrations registered in the gearbox attached to the axle. Shrestha et al. ²⁵ compare the performance of a scaled bogie roller test rig with de numerical model developed in GENSYS.

As Myamlin et al. ²⁶ expound, the majority of roller test rigs use a large roller to induce the motion of the wheelset (or wheel in some cases), which is the case of the aforementioned studies. However, BOGLAB has a full-scale bogie test bench that uses two small rollers to move one of the wheelsets of the bogie. A system that is not found in the examined scientific literature. This kind of test bench derives from the underfloor wheel lathes used for wheel reprofiling and has the advantage of not requiring an auxiliary structure for keeping the bogie or rail vehicle correctly positioned on the test bench. For geometric and dynamic reasons, placing two cylinders one on top of the other leads to an unstable position, as the upper cylinder will tend to roll over the lower one and fall. Therefore, single-roller test stands have an auxiliary structure to hold the bogie or wheelset in its proper position. This is unnecessary on two-roller test benches, as the geometry of the test bench allows the wheelsets to be supported without external support. However, two-roller test benches have two major disadvantages. By using small-diameter rollers, the contact pressures are higher, and there are two points of contact instead of just one. In addition, this configuration completely changes the running behaviour of the wheelset.

In essence, the reviewed scientific literature can be classified into three groups according to the main topic of the works:

1. Development of dynamic models for studying the stability of rail vehicles on tracks without including irregularities or defects.^1–14

2. Dynamic models that include any kind of defects or irregularities to analyse their impact on the vehicle’s behaviour.^15–21

3. Simulation of the performance of rail vehicles on a one-roller test rig, usually at scales 1:4 and 1:5,^22–25 and occasionally study of defects.

It is extremely rare to find dynamic models that consider axle boxes, though this is the typical place to install the accelerometers when measuring the performance of actual rail vehicles.

The authors propose a bogie model to simulate the performance of a freight bogie tested in a double roller test rig. This model considers the bogie frame, the wheelsets and the axle boxes, as well as the contact between the two rollers of the test rig and the wheel. Therefore, this paper introduces two key innovations to the scientific community. Firstly, it considers the axle boxes in the model, which are not usually taken into account in scientific literature, and paves the way for roller bearing models to be included in future studies. Secondly, it calculates the contact forces due to the action of two small rollers on a wheel.

The structure of the paper is as follows: the second section describes the bogie under study. The third section expounds the mathematical model of the bogie. The results obtained from the simulation are discussed in the fourth section. Finally, the conclusions are described in the fifth section.

System description

Bogie description

The bogie under study in this work is a freight bogie of type Y-21 Cse, which is based on the Y-25 bogie widely used in Europe for freight wagons. This type of bogie has two wheelsets that can load up to 20 tonnes per axle and is adapted to run on the Spanish rail network, so its track gauge is wider than the standard. The main characteristics of the Y-21 Cse bogie are summarised in Table 1.

Table 1.

Bogie y21 CSE characteristics.

Parameter	Value
Maximum axle load	20 t
Brake	Pneumatic
Maximum speed	100 km/h
Width	2.5 m
Length	3.5 m
Wheelbase	2 m
Track gauge	1668 mm
Wheel diameter (new)	920 mm

The bogie is modelled in CREO Parametric from technical documentation to obtain certain characteristics that would be difficult to get in other ways. Figure 1 shows this 3D CAD model of the bogie.

Figure 1.

CAD model of the Y-21 Cse bogie and the double roller test rig.

Test bench description

The BOGLAB laboratory, placed in one of the Renfe main workshops located in the South of Madrid (Spain), has a bogie test rig specifically designed and manufactured by Dannobat Railway Systems to test bogies under different conditions (see Figure 2). The DTR-25 test rig derives from wheel lathes used for wheel reprofiling and is formed by a bench and a drive system that transmits movement to the wheels of the wheelset by means of two small rollers. These rollers have a diameter of 350 mm, and the contact geometry is the same as the profile of UIC-54 rails. One wheelset (rear) stands over the fixed bed, and the other (front wheelset) is driven by the rollers. The system that drives the rollers allows the simulation of the operating conditions of the front wheelset of the bogie for speeds between 0 and 80 km/h. It is also possible to change the rotating direction of the wheelset, simulating the movement of the bogie forward or backwards. Two of the four rollers are moved by electric traction motors, while the other two rollers are moved by wheel/roller friction. Traction motors are coupled following a master-slave logic.

Figure 2.

Installed bogie on the test bench.

The load is applied vertically thanks to a hydraulic system. This system consists of a couple of hydraulic cylinders that pull a horizontal beam placed above the front axle by using two chains. The maximum bogie load provided by the hydraulic loading system is 20 tonnes.

The geometry of the bogie test rig can support the bogie without the need for external structures, which allows the free movement of the bogie in the vertical and lateral directions, and also some movement in the longitudinal direction. This is an advantage over one roller test rigs, which require a structural support or coupler to maintain the wheel/bogie/vehicle in position.

Having two small rollers, the wheel has at least two contact points at all times, which is a disadvantage if the research aims to study the contact between the wheel and rollers in similar conditions to a tangent track. This configuration results in reduced contact areas and elevated contact pressures, which may lead to accelerated wheel wear if this device were employed in wear simulation applications.

The test rig allows for experiments to be carried out, among others, to analyse the vertical dynamics of the bogie, the performance of the suspension elements or the effect of the presence of defects in components such as rolling bearings, axles, gearboxes, etc. Although the contact conditions differ from those found on track, this type of test rig offers a platform for fatigue testing, the simulation of axle cracks ²⁷ and for studying the behaviour of the bogie on wheel lathes. To extrapolate the results of the test bench to reality, a correlation should be established between the data, which is beyond the scope of this work.

Mathematical model

A 21-degree-of-freedom model is proposed for simulating the behaviour of the Y-21 bogie over the double roller test bench. The model comprises the bogie frame, the two wheelsets and the four axle boxes. The scheme of this model is summarised in Figure 3. As can be seen, the inputs are the bogie load, the speed and the kinematic conditions of the wheels due to the roller and/or wheel irregularities. The outputs are the displacements, velocities and accelerations of the wheelsets, the bogie frame and the axle boxes. The model parameters are the mass and moments of inertia of the bodies, the distances between them and the stiffness and damping coefficients linking all the bodies.

Figure 3.

Scheme of model representation.

Five coordinate systems are used to establish the dynamics of the wheelset, as can be seen in Figure 4. Coordinate system number 4 is the equilibrium reference frame, which is common to the axle boxes and the bogie frame. The other coordinate systems are obtained by rotating a finite angle about one axis of the previous reference frame. In addition, coordinate systems 0 and 1 are particularised for the front (subscript f) and rear (subscript r) rollers and the left (subscript i) and right (subscript d) wheels, respectively. The coordinate transformation matrices are given by equations (1)–(6), where the left superscript means the destination coordinate system and the right subscript means the origin coordinate system.

$${}^1{R}_{0f}=\left(\begin{array}{ccc}\cos \alpha \hfill & -\sin \alpha \hfill & 0\hfill \\ \multicolumn{1}{c}{\sin \alpha }& \cos \alpha \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right)$$

(1)

$${}^1{R}_{0r}=\left(\begin{array}{ccc}\cos \alpha \hfill & \sin \alpha \hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin \alpha }& \cos \alpha \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right)$$

(2)

$${}^2{R}_{1i}=\left(\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \cos {\delta }_L\hfill & -\sin {\delta }_L\hfill \\ \multicolumn{1}{c}{0}& \sin {\delta }_L\hfill & \cos {\delta }_L\hfill \end{array}\right)$$

(3)

$${}^2{R}_{1d}=\left(\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \cos {\delta }_R\hfill & \sin {\delta }_R\hfill \\ \multicolumn{1}{c}{0}& -\sin {\delta }_R\hfill & \cos {\delta }_R\hfill \end{array}\right)$$

(4)

$${}^3{R}_2=\left(\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \cos \varphi \hfill & -\sin \varphi \hfill \\ \multicolumn{1}{c}{0}& \sin \varphi \hfill & \cos \varphi \hfill \end{array}\right)$$

(5)

$${}^4{R}_3=\left(\begin{array}{ccc}\cos \psi \hfill & \sin \psi \hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin \psi }& \cos \psi \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right)$$

(6)

Figure 4.

Coordinate systems transformation.

Coordinate systems 0 and 1 are related to the contact planes that contain the contact points. ${\delta }_R$ and ${\delta }_L$ are the right and left contact angles (see Figure 5), and $\alpha$ is the angle between the normal contact force and the wheelset load (see Figure 6), which is given by the geometry of the test bench and the wheel radius as

$$\alpha ={\sin }^{-1}\left(\frac{L_{\mathrm{roller}}/2}{R_w+R_r}\right)$$

(7)

The multibody model developed is shown in Figure 7, where all the bodies and their connections can be observed. Reference frames are placed at the centre of mass of each body. The wheelsets are modelled with four degrees of freedom and can move in the vertical and lateral direction and rotate around the longitudinal (roll) and vertical (yaw) axes. The axle boxes can move in the vertical and axial direction. The bogie frame is allowed to move vertically and laterally and to rotate around the longitudinal (roll), lateral (pitch) and vertical (yaw) axes.

Figure 5.

Front view of contact plane axes.

Figure 6.

Lateral view of contact plane axes and normal forces.

Figure 7.

Two-axle bogie model in front (top) and lateral (bottom) views. For clarity, the rollers have not been drawn in the front view.

The model is developed based on seven subsystems composed of bogie frame (subscript b), forward wheelset (subscript w1), rear wheelset (subscript w2), front left axle box (subscript ab1L), front right axle box (subscript ab1R), rear left axle box (subscript ab2L) and rear right axle box (subscript ab2R). For each subsystem, the classical dynamic equations (Newton’s second law and conservation of angular momentum) are applied in matrix form, as stated in (8) and (9).

$$\mathit{M}·{\stackrel{··}{\mathit{q}}}_{\mathit{L}}+\mathit{C}·{\stackrel{·}{\mathit{q}}}_{\mathit{L}}+\mathit{K}·\mathit{q}=\mathit{F}$$

(8)

$$\mathit{I}·{\stackrel{··}{\mathit{q}}}_{\mathbf{\alpha }}+{\mathit{C}}_{\mathbf{\alpha }}·{\stackrel{·}{\mathit{q}}}_{\mathbf{\alpha }}+{\mathit{K}}_{\mathbf{\alpha }}·{\mathit{q}}_{\mathbf{\alpha }}=\mathit{T}$$

(9)

Where $\mathit{M}$ is the mass matrix, $\mathit{I}$ is the inertia matrix, $\mathit{K}$ and ${\mathit{K}}_{\mathbf{\alpha }}$ are the stiffness matrices and $\mathit{C}$ and ${\mathit{C}}_{\mathbf{\alpha }}$ are the damping matrices of the seven subsystems. $\mathit{F}$ and $\mathit{T}$ are the input forces and moments. The coordinate vectors are:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{q}}_{\mathit{L}}=z_{w1},z_{\mathrm{ab}1L},z_{\mathrm{ab}1R},z_{w2},z_{\mathrm{ab}2L},z_{\mathrm{ab}2R},z_b,y_{w1},y_{w2},y_{\mathrm{ab}1L},y_{\mathrm{ab}1R},y_{\mathrm{ab}2L},y_{\mathrm{ab}2R},y_b\}}\\ \multicolumn{1}{c}{{\mathit{q}}_{\mathbf{\alpha }}=\{{\varphi }_{w1},{\varphi }_{w2},{\theta }_b,{\varphi }_b,{\psi }_{w1},{\psi }_{w2},{\psi }_b\}}\end{array}$$

In (8) and (9), dots over the vector and variables indicate time differentiation. Variables $y_i$ and $z_i$ correspond to the axial and vertical movements, respectively, and ${\varphi }_i,{\theta }_i$ and ${\psi }_i$ are the roll, pitch and yaw movements, respectively.

Normal contact forces

Hertz’s contact model is used to calculate the shape of the contact area and pressure. The theory predicts the shape of the contact area, its size as a function of the applied normal load N, and the magnitude and distribution of the forces that appear in the contact. The hypotheses on which Hertz’s theory is based are the following^28,29:

1. The surfaces are continuous and non-conformal.

2. The deformations are small.

3. Each solid can be considered as an elastic half-space.

4. There is no friction between the surfaces.

Hertz showed that the contact surface is flat and elliptical, with a major semi-axis $a$ and a minor semi-axis $b$ . The most notable results of Hertz’s theory are the following:

1. The ratio $a/b$ only depends on the curvatures of both bodies.

2. The two semi-axes $a$ and $b$ are proportional to $N^{1/3}$ and their product is proportional to $N^{2/3}$ , independently of the distribution of tangential stresses.

3. The penetration ${\delta }_{\mathrm{indent}}$ is proportional to $N^{2/3}$ .

The dimensions of the semi-axes are given by equations (10) and (11).

$$a=m{\left[\frac{3\pi N}{4}\frac{k_1+k_2}{k_3}\right]}^{\frac{1}{3}}$$

(10)

$$b=n{\left[\frac{3\pi N}{4}\frac{k_1+k_2}{k_3}\right]}^{\frac{1}{3}}$$

(11)

$$k_1=\frac{1-{\nu }_W^2}{\pi E_W}$$

(12)

$$k_2=\frac{1-{\nu }_R^2}{\pi E_R}$$

(13)

$$k_3=\frac{1}{2}(\frac{1}{R_{W1}}+\frac{1}{R_{W2}}+\frac{1}{R_{R1}}+\frac{1}{R_{R2}})$$

(14)

Where:

• $N$ is the normal force in the contact.

• $v_W$ and ${\nu }_R$ are the Poisson coefficients of the wheel and the roller/rail, respectively.

• $E_W$ and $E_R$ are the Young’s modulus of the wheel and the roller/rail, respectively.

• $R_{W1}$ is the principal curvature radius of the wheel at the contact point.

• $R_{W2}$ is the tangential curvature radius of the wheel at the contact point.

• $R_{R1}$ is the principal curvature radius of the roller/rail at the contact point.

• $R_{R2}$ is the tangential curvature radius of the roller/rail at the contact point.

Parameters $m$ and $n$ depend on the angle $\psi$ formed by the normal planes to the tangent plane that contains the principal or tangential curvatures of the bodies. The values of $m$ and $n$ are given in Table 2.

$$\psi ={\cos }^{-1}\left(\frac{k_4}{k_3}\right)$$

(15)

$$k_4=\frac{1}{2}\sqrt{{(\frac{1}{R_{W1}}-\frac{1}{R_{W2}})}^2+{(\frac{1}{R_{R1}}-\frac{1}{R_{R2}})}^2+2(\frac{1}{R_{W1}}-\frac{1}{R_{W2}})(\frac{1}{R_{R1}}-\frac{1}{R_{R2}})\cos 2\psi }$$

(16)

Table 2.

Parameters $m$ and $n$ as function of angle $\psi$ .

$\psi {(}^{°})$	0	10	20	30	40	50	60	70	80	90
m	∞	6.6120	3.7779	2.7307	2.1357	1.7542	1.4858	1.2835	1.1278	1
n	0.0000	0.3186	0.4079	0.4930	0.5673	0.6407	0.7171	0.8017	0.8927	1

The contact pressure distribution on the contact surface, $P(x,y)$ , is given by (17)

$$P(x,y)=P_0\sqrt{1-{\left(\frac{x}{a}\right)}^2-{\left(\frac{y}{b}\right)}^2}$$

(17)

$$P_0=\frac{3N}{2\pi \mathrm{ab}}$$

(18)

Tangential forces

The first step to calculate the tangential forces is to compute the creepages, which are, by definition ²⁹ :

$${\xi }_x=\frac{\mathrm{proj}./\mathrm{x}({\mathit{V}}_{\mathit{W}}-{\mathit{V}}_{\mathit{R}})}{\frac{1}{2}\Vert {\mathit{V}}_{\mathit{W}}+{\mathit{V}}_{\mathit{R}}\Vert }$$

(19)

$${\xi }_y=\frac{\mathrm{proj}./\mathrm{y}({\mathit{V}}_{\mathit{W}}-{\mathit{V}}_{\mathit{R}})}{\frac{1}{2}\Vert {\mathit{V}}_{\mathit{W}}+{\mathit{V}}_{\mathit{R}}\Vert }$$

(20)

$${\xi }_{\mathrm{sp}}=\frac{\mathrm{proj}./\mathrm{z}(\mathbf{\omega }-{\mathbf{\omega }}_{\mathit{R}})}{\frac{1}{2}\Vert {\mathit{V}}_{\mathit{W}}+{\mathit{V}}_{\mathit{R}}\Vert }$$

(21)

${\mathit{V}}_{\mathit{W}}$ and ${\mathit{V}}_{\mathit{R}}$ are the absolute speeds of the wheel and the roller/rail at the contact. $\mathbf{\omega }$ and ${\mathbf{\omega }}_{\mathit{R}}$ are the angular speeds of the wheel and the roller/rail.

The angular velocity of the wheelset in the coordinates of the wheelset body (CS2) is

$$\mathbf{\omega }=\stackrel{·}{\varphi }{\mathit{a}}_{\mathit{x}2}+(\stackrel{·}{\psi }\sin \varphi +\Omega ){\mathit{a}}_{\mathit{y}2}+\stackrel{·}{\psi }\cos \varphi {\mathit{a}}_{\mathit{z}2}$$

(22)

The position vectors of the four contact points in terms of the wheelset coordinate system (CS2) are given by equations (23)–(26).

$${}^2{\mathit{R}}_{\mathit{L}1}=r_L\sin (\alpha ){\mathit{a}}_{\mathit{x}2}+(e-{\Delta }_L){\mathit{a}}_{\mathit{y}2}-r_L\cos (\alpha ){\mathit{a}}_{\mathit{z}2}$$

(23)

$${}^2{\mathit{R}}_{\mathit{L}2}=-r_L\sin (\alpha ){\mathit{a}}_{\mathit{x}2}+(e-{\Delta }_L){\mathit{a}}_{\mathit{y}2}-r_L\cos (\alpha ){\mathit{a}}_{\mathit{z}2}$$

(24)

$${}^2{\mathit{R}}_{\mathit{R}1}=r_R\sin (\alpha ){\mathit{a}}_{\mathit{x}2}+(-e-{\Delta }_R){\mathit{a}}_{\mathit{y}2}-r_R\cos (\alpha ){\mathit{a}}_{\mathit{z}2}$$

(25)

$${}^2{\mathit{R}}_{\mathit{L}2}=-r_R\sin (\alpha ){\mathit{a}}_{\mathit{x}2}+(-e-{\Delta }_R){\mathit{a}}_{\mathit{y}2}-r_R\cos (\alpha ){\mathit{a}}_{\mathit{z}2}$$

(26)

The left superscript refers to the coordinate system, as well as the subscript of the unitary vectors. The subscripts (L, R)(1, 2) refer to the left or right wheel and the front (1) or rear (2) rollers. $r_L$ and $r_R$ are the rolling radii at the contact point. Considering conical profiles, ${\Delta }_L$ and ${\Delta }_R$ are the lateral displacements of the wheelset from its equilibrium position. $e$ is half of the track gauge.

In terms of the equilibrium reference frame and using equations (5) and (6), the position vectors are

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{R}}_{\mathit{L}1}=[x+\cos \left(\varphi \right)\sin \psi ({\Delta }_L-e)+r_L\cos \left(\psi \right)\sin (\alpha )-r_L\sin \left(\varphi \right)\sin \left(\psi \right)\cos \left(\alpha \right)]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[y-\cos \left(\varphi \right)\cos \left(\psi \right)({\Delta }_L-e)+r_L\sin \left(\psi \right)\sin \left(\alpha \right)+r_L\cos \left(\psi \right)\sin \left(\varphi \right)\cos \left(\alpha \right)]{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+[z-\sin \left(\varphi \right)({\Delta }_L-e)-r_L\cos \left(\varphi \right)\cos \left(\alpha \right)]{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(27)

$$\begin{array}{l}{\mathit{R}}_{\mathit{L}\mathbf{2}}=\left[x+\cos \left(\varphi \right)\sin \psi \left({\Delta }_L-e\right)-r_L\cos \left(\psi \right)\sin \left(\alpha \right)-r_L\sin \left(\varphi \right)\sin \left(\psi \right)\cos \left(\alpha \right)\right]{\mathit{a}}_{\mathit{x}\mathbf{4}}\\ +\left[y-\cos \left(\varphi \right)\cos \left(\psi \right)\left({\Delta }_L-e\right)-r_L\sin \left(\psi \right)\sin \left(\alpha \right)+r_L\cos \left(\psi \right)\sin \left(\varphi \right)\cos \left(\alpha \right)\right]{\mathit{a}}_{\mathit{y}\mathbf{4}}\\ +\left[z-\sin \left(\varphi \right)\left({\Delta }_L-e\right)-r_L\cos \left(\varphi \right)\cos \left(\alpha \right)\right]{\mathit{a}}_{\mathit{z}\mathbf{4}}\end{array}$$

(28)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{R}}_{\mathit{R}1}=[x+\cos \left(\varphi \right)\sin \psi ({\Delta }_R-e)+r_R\cos \left(\psi \right)\sin (\alpha )-r_R\sin \left(\varphi \right)\sin \left(\psi \right)\cos \left(\alpha \right)]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[y-\cos \left(\varphi \right)\cos \left(\psi \right)({\Delta }_R-e)+r_R\sin \left(\psi \right)\sin \left(\alpha \right)+r_L\cos \left(\psi \right)\sin \left(\varphi \right)\cos \left(\alpha \right)]{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+[z-\sin \left(\varphi \right)({\Delta }_R-e)-r_R\cos \left(\varphi \right)\cos \left(\alpha \right)]{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(29)

$$\begin{array}{l}{\mathit{R}}_{\mathit{R}\mathbf{2}}=\left[x+\cos \left(\varphi \right)\sin \psi \left({\Delta }_R-e\right)-r_R\cos \left(\psi \right)\sin (\alpha )-r_R\sin \left(\varphi \right)\sin \left(\psi \right)\cos \left(\alpha \right)\right]{\mathit{a}}_{\mathit{x}\mathbf{4}}\\ +\left[y-\cos \left(\varphi \right)\cos \left(\psi \right)\left({\Delta }_R-e\right)-r_R\sin \left(\psi \right)\sin \left(\alpha \right)+r_L\cos \left(\psi \right)\sin \left(\varphi \right)\cos \left(\alpha \right)\right]{\mathit{a}}_{\mathit{y}\mathbf{4}}\\ +\left[z-\sin \left(\varphi \right)\left({\Delta }_R-e\right)-r_R\cos \left(\varphi \right)\cos \left(\alpha \right)\right]{\mathit{a}}_{\mathit{z}\mathbf{4}}\end{array}$$

(30)

The velocities of the wheels at the contact points are obtained by time-differentiating equations (27)–(30).

The unitary vectors of the contact plane at the contact point between the left wheel and the front left roller in coordinates of the equilibrium reference frame are given by equations (31)–(33).

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_{\mathit{L}1\mathit{x}}=[\cos (\psi )\cos (\alpha )+\sin ({\delta }_L+\varphi )\sin (\psi )\sin (\alpha )]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[\sin (\psi )\cos \left(\alpha \right)-\sin ({\delta }_L+\varphi )\cos (\psi )\sin (\alpha )]{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+\cos ({\delta }_L+\varphi )\sin (\alpha ){\mathit{a}}_{\mathit{z}4}}\end{array}$$

(31)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_{\mathit{L}1\mathit{y}}=-\cos ({\delta }_L+\varphi )\sin (\psi ){\mathit{a}}_{\mathit{x}4}+\cos ({\delta }_L+\varphi )\cos (\psi ){\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+\sin ({\delta }_L+\varphi ){\mathit{a}}_{\mathit{z}4}}\end{array}$$

(32)

$${\mathit{e}}_{\mathit{L}1\mathit{z}}=-\sin \left(\alpha \right){\mathit{a}}_{\mathit{x}4}-\cos \left(\alpha \right)\sin ({\delta }_L){\mathit{a}}_{\mathit{y}4}+\cos (\alpha )\cos ({\delta }_L){\mathit{a}}_{\mathit{z}4}$$

(33)

Similarly, the unitary vectors of the contact plane at the contact point between the left wheel and the rear left roller in coordinates of the equilibrium coordinate system can be obtained using the equations (2), (3), (5) and (6) to get the expressions given by equations (34)–(36).

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_{\mathit{L}2\mathit{x}}=[\cos (\psi )\cos (\alpha )-\sin ({\delta }_L+\varphi )\sin \left(\psi \right)\sin \left(\alpha \right)]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[\sin \left(\psi \right)\cos \left(\alpha \right)+\sin ({\delta }_L+\varphi )\cos \left(\psi \right)\sin \left(\alpha \right)]{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{-\cos ({\delta }_L+\varphi )\sin (\alpha ){\mathit{a}}_{\mathit{z}4}}\end{array}$$

(34)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_{\mathit{L}2\mathit{y}}=-\cos ({\delta }_L+\varphi )\sin (\psi ){\mathit{a}}_{\mathit{x}4}+\cos ({\delta }_L+\varphi )\cos (\psi ){\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+\sin ({\delta }_L+\varphi ){\mathit{a}}_{\mathit{z}4}}\end{array}$$

(35)

$${\mathit{e}}_{\mathit{L}2\mathit{z}}=\sin \left(\alpha \right){\mathit{a}}_{\mathit{x}4}-\cos \left(\alpha \right)\sin ({\delta }_L){\mathit{a}}_{\mathit{y}4}+\cos (\alpha )\cos ({\delta }_L){\mathit{a}}_{\mathit{z}4}$$

(36)

Equations (37)–(39) define the unitary vectors of the contact plane at the contact point between the right wheel and the front right roller in the coordinates of the equilibrium reference frame.

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_{\mathit{R}1\mathit{x}}=[\cos \left(\psi \right)\cos \left(\alpha \right)-\sin ({\delta }_R-\varphi )\sin (\psi )\sin (\alpha )]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[\sin (\psi )\cos \left(\alpha \right)+\sin ({\delta }_R-\varphi )\cos (\psi )\sin (\alpha )]{\mathit{a}}_{\mathit{y}4}}\end{array}+\cos ({\delta }_R-\varphi )\sin (\alpha ){\mathit{a}}_{\mathit{z}4}$$

(37)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_{\mathit{R}1\mathit{y}}=-\cos ({\delta }_R-\varphi )\sin \left(\psi \right){\mathit{a}}_{\mathit{x}4}+\cos ({\delta }_R-\varphi )\cos \left(\psi \right){\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{-\sin ({\delta }_R-\varphi ){\mathit{a}}_{\mathit{z}4}}\end{array}$$

(38)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_{\mathit{R}1\mathit{z}}=-\sin \left(\alpha \right){\mathit{a}}_{\mathit{x}4}+\cos \left(\alpha \right)\sin ({\delta }_R){\mathit{a}}_{\mathit{y}4}+\cos (\alpha )\cos ({\delta }_R){\mathit{a}}_{\mathit{z}4}}\end{array}$$

(39)

The unitary vectors of the contact plane at the contact point between the right wheel and the right rear roller in coordinates of the equilibrium coordinate system are given by equations (40)–(42).

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_{\mathit{R}2\mathit{x}}=[\cos \left(\psi \right)\cos \left(\alpha \right)+\sin ({\delta }_R-\varphi )\sin (\psi )\sin (\alpha )]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[\sin (\psi )\cos \left(\alpha \right)-\sin ({\delta }_R-\varphi )\cos (\psi )\sin (\alpha )]{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{-\cos ({\delta }_R-\varphi )\sin (\alpha ){\mathit{a}}_{\mathit{z}4}}\end{array}$$

(40)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_{\mathit{R}2\mathit{y}}=-\cos ({\delta }_R-\varphi )\sin \left(\psi \right){\mathit{a}}_{\mathit{x}4}+\cos ({\delta }_R-\varphi )\cos \left(\psi \right){\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{-\sin ({\delta }_R-\varphi ){\mathit{a}}_{\mathit{z}4}}\end{array}$$

(41)

$${\mathit{e}}_{\mathit{R}2\mathit{z}}=\sin \left(\alpha \right){\mathit{a}}_{\mathit{x}4}+\cos \left(\alpha \right)\sin ({\delta }_R){\mathit{a}}_{\mathit{y}4}+\cos (\alpha )\cos ({\delta }_R){\mathit{a}}_{\mathit{z}4}$$

(42)

The velocities of the rollers at the contact points in the coordinates of the equilibrium reference frame are

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{V}}_{\mathit{rL}1}=\frac{V{r}_L\cos (\psi )\cos (\alpha )}{r_0}{\mathit{a}}_{\mathit{x}4}+\frac{V{r}_L\sin (\psi )\cos (\alpha )}{r_0}{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+\frac{V{r}_L\sin (\alpha )}{r_0}{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(43)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{V}}_{\mathit{rL}2}=\frac{V{r}_L\cos (\psi )\cos (\alpha )}{r_0}{\mathit{a}}_{\mathit{x}4}+\frac{V{r}_L\sin (\psi )\cos (\alpha )}{r_0}{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{-\frac{V{r}_L\sin (\alpha )}{r_0}{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(44)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{V}}_{\mathit{rR}1}=\frac{V{r}_R\cos (\psi )\cos (\alpha )}{r_0}{\mathit{a}}_{\mathit{x}4}+\frac{V{r}_R\sin (\psi )\cos (\alpha )}{r_0}{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+\frac{V{r}_R\sin (\alpha )}{r_0}{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(45)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{V}}_{\mathit{rR}2}=\frac{V{r}_R\cos (\psi )\cos (\alpha )}{r_0}{\mathit{a}}_{\mathit{x}4}+\frac{V{r}_R\sin (\psi )\cos (\alpha )}{r_0}{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{-\frac{V{r}_R\sin (\alpha )}{r_0}{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(46)

Applying equations (19)–(21) and considering that the projections of the wheel speed to the X and Y axes are ${\stackrel{·}{\mathit{R}}}_{(\mathit{L},\mathit{R})(1,2)\mathit{x}}·{\mathit{e}}_{(\mathit{L},\mathit{R})(1,2)\mathit{x}}$ and ${\stackrel{·}{\mathit{R}}}_{(\mathit{L},\mathit{R})(1,2)\mathit{y}}·{\mathit{e}}_{(\mathit{L},\mathit{R})(1,2)\mathit{y}}$ we can obtain the creepages. After some algebra and assuming small angles (except for alpha), the creepages are given by equations (47)–(58).

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_{L1x}=-\frac{1}{V}({\Delta }_L\sin \left(\alpha \right)\stackrel{·}{\varphi }-\sin \left(\alpha \right)\stackrel{·}{z}-y\cos \left(\alpha \right)\stackrel{·}{\psi }-V\cos \left(\alpha \right)}\\ +e\cos \left(\alpha \right)\stackrel{·}{\psi }-e\sin \left(\alpha \right)\stackrel{·}{\varphi }+{\delta }_L\sin \left(\alpha \right){\stackrel{·}{y}}_{w1}+r_L\varphi \stackrel{·}{\psi }\hfill \\ -\cos \left(\alpha \right)\psi \stackrel{·}{y}+\sin \left(\alpha \right)\varphi \stackrel{·}{y}+{\delta }_L{r}_L\stackrel{·}{\psi }-{\delta }_L{r}_L{\cos }^2\left(\alpha \right)\stackrel{·}{\psi }\hfill \\ +\frac{{\delta }_L{r}_L\sin \left(2\alpha \right)\stackrel{·}{\varphi }}{2}+{\delta }_L\sin \left(\alpha \right)\varphi \stackrel{·}{z}-V\sin \left(\alpha \right)\varphi \psi \hfill \\ +\frac{V{r}_L\cos \left(\alpha \right)}{r_0}-V{\delta }_L\sin \left(\alpha \right)\psi )\hfill \end{array}$$

(47)

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_{L1y}=\frac{1}{V}(\varphi \stackrel{·}{z}+{\delta }_L\stackrel{·}{z}-V\psi +\stackrel{·}{\mathrm{y}}+r_L\cos \left(\alpha \right)\stackrel{·}{\varphi }+r_L\sin \left(\alpha \right)\stackrel{·}{\psi }-{\delta }_L\varphi \stackrel{·}{y}-{\Delta }_L{\delta }_L\stackrel{·}{\varphi }+{\delta }_{L}e\stackrel{·}{\varphi }+V{\delta }_L\varphi \psi -\frac{V{r}_L\cos \left(\alpha \right)\psi }{r_0}-{\delta }_L{r}_L,\sin \left(\alpha \right)\varphi \stackrel{·}{\psi })}\end{array}$$

(48)

$${\xi }_{L1\mathrm{sp}}=\frac{1}{V}(\sin \left(\alpha \right)\stackrel{·}{\varphi }-\cos \left(\alpha \right)\stackrel{·}{\psi }+{\delta }_L\cos \left(\alpha \right)(\varphi \stackrel{·}{\psi }+\Omega )+\frac{V{r}_L\sin \left(\alpha \right)}{r_0})$$

(49)

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_{L2x}=\frac{1}{V}(\cos \left(\alpha \right)V-\sin \left(\alpha \right)\stackrel{·}{z}+\mathrm{ycos}\left(\alpha \right)\stackrel{·}{\psi }+{\Delta }_l\sin \left(\alpha \right)\stackrel{·}{\varphi }-}\\ e\cos \left(\alpha \right)\stackrel{·}{\psi }-e\sin \left(\alpha \right)\stackrel{·}{\varphi }+{\delta }_L\sin \left(\alpha \right)\stackrel{·}{y}-r_L\varphi \stackrel{·}{\psi }\hfill \\ +\cos \left(\alpha \right)\psi \stackrel{·}{y}+\sin \left(\alpha \right)\varphi \stackrel{·}{y}-{\delta }_L{r}_L\stackrel{·}{\psi }\hfill \\ +{\delta }_L{r}_L{\cos }^2\left(\alpha \right)\stackrel{·}{\psi }+\frac{{\delta }_L{r}_L\sin \left(2\alpha \right)\stackrel{·}{\varphi }}{2}-{\delta }_L\sin \left(\alpha \right)\psi \mathrm{V}+{\delta }_L\sin \left(\alpha \right)\varphi \left(t\right)\stackrel{·}{z}\hfill \\ -\frac{V{r}_L\cos \left(\alpha \right)}{r_0}-\sin \left(\alpha \right)\varphi \psi V)\hfill \end{array}$$

(50)

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_{L2y}=\frac{1}{V}({\delta }_l\stackrel{·}{z}-\psi \left(t\right)\mathrm{V}+\stackrel{·}{\mathrm{y}}+r_L\cos \left(\alpha \right)\stackrel{·}{\varphi }-r_L\sin \left(\alpha \right)\stackrel{·}{\psi }-{\delta }_L\varphi \stackrel{·}{y}-y{\delta }_L\stackrel{·}{\varphi }+\varphi \stackrel{·}{z}+{\delta }_{L}e\stackrel{·}{\varphi }+{\delta }_L\varphi \psi V-\frac{V{r}_L\cos \left(\alpha \right)\psi }{r_0}+{\delta }_L{r}_L\sin \left(\alpha \right)\varphi \stackrel{·}{\psi })}\end{array}$$

(51)

$${\xi }_{L2\mathrm{sp}}=\frac{1}{V}(\cos \left(\alpha \right)\stackrel{·}{\psi }+\sin \left(\alpha \right)\stackrel{·}{\varphi }-{\delta }_L\cos \left(\alpha \right)(\varphi \stackrel{·}{\psi }+\Omega )+\frac{V{r}_L\sin \left(\alpha \right)}{r_0})$$

(52)

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_{R1x}=\frac{1}{V}(\cos \left(\alpha \right)V+\sin \left(\alpha \right)\stackrel{·}{z}+{\Delta }_R\cos \left(\alpha \right)\stackrel{·}{\psi }-{\Delta }_R\sin \left(\alpha \right)\stackrel{·}{\varphi }}\\ +e\cos \left(\alpha \right)\stackrel{·}{\psi }-e\sin \left(\alpha \right)\stackrel{·}{\varphi }+{\delta }_R\sin \left(\alpha \right)\stackrel{·}{y}-r_R\varphi \stackrel{·}{\psi }\hfill \\ +\cos \left(\alpha \right)\psi \stackrel{·}{y}-\sin \left(\alpha \right)\varphi \stackrel{·}{y}+{\delta }_R{r}_R\stackrel{·}{\psi }-{\delta }_R{r}_R{\cos }^2\left(\alpha \right)\stackrel{·}{\psi }\hfill \\ +\frac{{\delta }_R{r}_R\sin \left(2\alpha \right)\stackrel{·}{\varphi }}{2}-{\delta }_R\sin \left(\alpha \right)\psi V+{\delta }_R\sin \left(\alpha \right)\varphi \stackrel{·}{z}\hfill \\ -\frac{V{r}_R\cos \left(\alpha \right)}{r_0}+\sin \left(\alpha \right)\varphi \psi V)\hfill \end{array}$$

(53)

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_{R1y}=\frac{1}{V}(\varphi \stackrel{·}{z}-\psi V-{\delta }_R\stackrel{·}{z}+\stackrel{·}{y}+r_R\cos \left(\alpha \right)\stackrel{·}{\varphi }+r_R\sin \left(\alpha \right)\stackrel{·}{\psi }+{\delta }_R\varphi \stackrel{·}{y}+y{\delta }_R\stackrel{·}{\varphi }+{\delta }_{R}e\stackrel{·}{\varphi }-{\delta }_R\varphi \psi V-\frac{V{r}_R\cos \left(\alpha \right)\psi }{r_0}+{\delta }_R{r}_R\sin \left(\alpha \right)\varphi \stackrel{·}{\psi })}\end{array}$$

(54)

$${\xi }_{R1\mathrm{sp}}=\frac{1}{V}(\cos \left(\alpha \right)\stackrel{·}{\psi }-\sin \left(\alpha \right)\stackrel{·}{\varphi }+{\delta }_R\cos \left(\alpha \right)(\varphi \stackrel{·}{\psi }+\Omega )-\frac{V{r}_R\sin \left(\alpha \right)}{r_0})$$

(55)

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_{R2x}=\frac{1}{V}(\cos \left(\alpha \right)\mathrm{V}-\sin \left(\alpha \right)\stackrel{·}{z}+y\cos \left(\alpha \right)\stackrel{·}{\psi }+{\Delta }_R\sin \left(\alpha \right)\stackrel{·}{\varphi }+e\cos \left(\alpha \right)\stackrel{·}{\psi }+e\sin \left(\alpha \right)\stackrel{·}{\varphi }-{\delta }_R\sin \left(\alpha \right)\stackrel{·}{y}-r_R\varphi \stackrel{·}{\psi }}\\ +\cos \left(\alpha \right)\psi \stackrel{·}{y}\hfill \\ +\sin \left(\alpha \right)\varphi \stackrel{·}{y}+{\delta }_R{r}_R\stackrel{·}{\psi }-{\delta }_R{r}_R{\cos }^2\left(\alpha \right)\stackrel{·}{\psi }-\frac{{\delta }_R{r}_R\sin \left(2\alpha \right)\stackrel{·}{\varphi }}{2}\hfill \\ +{\delta }_R\sin \left(\alpha \right)\psi V-{\delta }_R\sin \left(\alpha \right)\varphi \stackrel{·}{z}-\frac{V{r}_R\cos \left(\alpha \right)}{r_0}-\sin \left(\alpha \right)\varphi \psi V)\hfill \end{array}$$

(56)

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_{R2y}=-\frac{1}{V}(\psi \left(t\right)\mathrm{V}-\varphi \stackrel{·}{z}+{\delta }_R\stackrel{·}{z}-\stackrel{·}{y}-r_R\cos \left(\alpha \right)\stackrel{·}{\varphi }+r_R\sin \left(\alpha \right)\stackrel{·}{\psi }-{\delta }_R\varphi \stackrel{·}{y}-y{\delta }_R\stackrel{·}{\varphi }-{\delta }_{R}e\stackrel{·}{\varphi }+{\delta }_R\varphi \psi \mathrm{V}+\frac{V{r}_R\cos \left(\alpha \right)\psi }{r_0}+{\delta }_R{r}_R\sin \left(\alpha \right)\varphi \stackrel{·}{\psi })}\end{array}$$

(57)

$${\xi }_{R2\mathrm{sp}}=\frac{1}{V}(\cos \left(\alpha \right)\stackrel{·}{\psi }+\sin \left(\alpha \right)\stackrel{·}{\varphi }+{\delta }_R\cos \left(\alpha \right)(\varphi \stackrel{·}{\psi }+\Omega )+\frac{V{r}_R\sin \left(\alpha \right)}{r_0})$$

(58)

Next, the tangential contact forces are computed using the Fastsim algorithm ³⁰ in the contact plane reference systems. After applying equations (1)–(6) and assuming small angles, the forces are expressed in the equilibrium coordinate system as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{F}}_{\mathit{cL}1}=[T_{\mathrm{XL}1}(\cos \left(\alpha \right)+\sin \left(\alpha \right)\psi ({\delta }_L+\varphi ))-T_{\mathrm{YL}1}\psi ]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[T_{\mathrm{YL}1}+T_{X1L}(\cos \left(\alpha \right)\psi -\sin \left(\alpha \right)({\delta }_L+\varphi ))]{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+[T_{\mathrm{YL}1}({\delta }_L+\varphi )+T_{\mathrm{XL}1}\sin \left(\alpha \right)]{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(59)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{F}}_{\mathit{cL}2}=[T_{\mathrm{XL}2}(\cos \left(\alpha \right)-\sin \left(\alpha \right)\psi ({\delta }_L+\varphi ))-T_{\mathrm{YL}2}\psi ]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[T_{\mathrm{YL}2}+T_{\mathrm{XL}2}(\cos \left(\alpha \right)\psi +\sin \left(\alpha \right)({\delta }_L+\varphi ))]{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+[T_{\mathrm{YL}2}({\delta }_L+\varphi )-T_{\mathrm{XL}2}\sin \left(\alpha \right)]{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(60)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{F}}_{\mathit{cR}1}=[T_{\mathrm{XR}1}(\cos \left(\alpha \right)-\sin \left(\alpha \right)\psi ({\delta }_R-\varphi ))-T_{\mathrm{YR}1}\psi ]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[T_{\mathrm{YR}1}+T_{\mathrm{XR}1}(\cos \left(\alpha \right)\psi +\sin \left(\alpha \right)({\delta }_R-\varphi ))]{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+[T_{\mathrm{XR}1}\sin \left(\alpha \right)-T_{\mathrm{YR}1}({\delta }_R-\varphi )]{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(61)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{F}}_{\mathit{cR}2}=[T_{\mathrm{XR}2}(\cos \left(\alpha \right)+\sin \left(\alpha \right)\psi ({\delta }_R-\varphi ))-T_{\mathrm{YR}2}\psi ]{\mathit{a}}_{\mathit{x}4}}\\ \multicolumn{1}{c}{+[T_{\mathrm{YR}2}+T_{\mathrm{XR}2}(\cos \left(\alpha \right)\psi -\sin \left(\alpha \right)({\delta }_R-\varphi ))]{\mathit{a}}_{\mathit{y}4}}\\ \multicolumn{1}{c}{+[-T_{\mathrm{XR}2}\sin \left(\alpha \right)-T_{\mathrm{YR}2}({\delta }_R-\varphi )]{\mathit{a}}_{\mathit{z}4}}\end{array}$$

(62)

Where $T_{X(L,R)(1,2)}$ and $T_{Y(L,R)(1,2)}$ are the creep forces computed from Fastsim.

By assuming that the displacements ${\Delta }_L$ and ${\Delta }_R$ are the lateral movements of the wheelset ( $y$ ), and that ${\delta }_L={\delta }_L=\lambda$ , being $\lambda$ the wheel conicity, the radii of the wheels at the contact points can be computed as

$$\begin{array}{c}{r}_L=r_0+y·\lambda \hfill \\ r_R=r_0-y·\lambda \hfill \end{array}$$

(63)

Motion equations

The expressions that describe the suspension forces of the front and rear wheelsets are given by equations (64)–(71). The flexibility of the supports is given by $K_t$ and $C_t$ .

$$\begin{array}{c}\multicolumn{1}{c}{F_{w1z}=-2K_{a{b}_{\mathrm{xz}}}{z}_{w1}+K_{a{b}_{\mathrm{xz}}}{z}_{\mathrm{ab}1L}+K_{a{b}_{\mathrm{xz}}}{z}_{\mathrm{ab}1R}-2C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{w1}}\\ \multicolumn{1}{c}{+C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{\mathrm{ab}1L}+C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{\mathrm{ab}1R}-2K_t{z}_{w1}-2C_t{\stackrel{·}{z}}_{w1}}\end{array}$$

(64)

$$\begin{array}{c}\multicolumn{1}{c}{F_{w1y}=-2K_{a{b}_y}{y}_{w1}+K_{a{b}_y}{y}_{\mathrm{ab}1L}+K_{a{b}_y}{y}_{\mathrm{ab}1R}-2C_{a{b}_y}{\stackrel{·}{y}}_{w1}}\\ \multicolumn{1}{c}{+C_{\mathrm{aby}}{\stackrel{·}{y}}_{\mathrm{ab}1L}+C_{a{b}_y}{\stackrel{·}{y}}_{\mathrm{ab}1R}}\end{array}$$

(65)

$$\begin{array}{c}\multicolumn{1}{c}{M_{w1x}=-2K_{a{b}_{\mathrm{xz}}}{b}_1^2{\varphi }_{w1}-2K_{a{b}_{\mathrm{xz}}}{y}_{w1}\lambda e-2C_{a{b}_{\mathrm{xz}}}{b}_1^2{\stackrel{·}{\varphi }}_{w1}}\\ \multicolumn{1}{c}{-2C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{y}}_{w1}\lambda e)}\end{array}$$

(66)

$$M_{w1z}=-2K_{a{b}_{\mathrm{xz}}}{b}_1^2{\psi }_{w1}-2C_{a{b}_{\mathrm{xz}}}{b}_1^2{\stackrel{·}{\psi }}_{w1}$$

(67)

$$\begin{array}{c}\multicolumn{1}{c}{F_{w2z}=-2K_{a{b}_{\mathrm{xz}}}{z}_{w2}+K_{a{b}_{\mathrm{xz}}}{z}_{\mathrm{ab}2L}+K_{a{b}_{\mathrm{xz}}}{z}_{\mathrm{ab}2R}-2C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{w2}}\\ \multicolumn{1}{c}{+C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{\mathrm{ab}2L}+C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{\mathrm{ab}2R}-2K_t{z}_{w2}-2C_t{\stackrel{·}{z}}_{w2}}\end{array}$$

(68)

$$\begin{array}{c}\multicolumn{1}{c}{F_{w2y}=-2K_{a{b}_y}{y}_{w2}+K_{a{b}_y}{y}_{\mathrm{ab}2L}+K_{a{b}_y}{y}_{\mathrm{ab}2R}-2C_{a{b}_y}{\stackrel{·}{y}}_{w2}}\\ \multicolumn{1}{c}{+C_{a{b}_y}{\stackrel{·}{y}}_{\mathrm{ab}2L}+C_{a{b}_y}{\stackrel{·}{y}}_{\mathrm{ab}2R}}\end{array}$$

(69)

$$\begin{array}{c}\multicolumn{1}{c}{M_{w2x}=-2K_{a{b}_{\mathrm{xz}}}{b}_1^2{\varphi }_{w2}-2K_{a{b}_{\mathrm{xz}}}{y}_{w2}\lambda e-2C_{a{b}_{\mathrm{xz}}}{b}_1^2{\stackrel{·}{\varphi }}_{w2}}\\ \multicolumn{1}{c}{-2C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{y}}_{w2}\lambda e)}\end{array}$$

(70)

$$M_{w2z}=-2K_{a{b}_{\mathrm{xz}}}{b}_1^2{\psi }_{w2}-2C_{a{b}_{\mathrm{xz}}}{b}_1^2{\stackrel{·}{\psi }}_{w2}$$

(71)

The flange contact force is given by Ahmadian and Yang ³¹ as

$$F_{\mathrm{ti}}=\{\begin{array}{c}\multicolumn{1}{c}{K_r(y_i-\delta ),y_i\ge \delta }\\ \multicolumn{1}{c}{0,-\delta <y_i<\delta }\\ \multicolumn{1}{c}{K_r(y_i+\delta )y_i\le -\delta }\end{array}$$

(72)

By combining the suspension forces with the normal and tangential contact forces, the equations of motion of the front wheelset are obtained.

$$\begin{array}{c}\multicolumn{1}{c}{m_{w1}{\stackrel{··}{z}}_{w1}=-2K_{a{b}_{\mathrm{xz}}}{z}_{w1}+K_{a{b}_{\mathrm{xz}}}{z}_{\mathrm{ab}1L}+K_{a{b}_{\mathrm{xz}}}{z}_{\mathrm{ab}1R}-2C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{w1}}\\ \multicolumn{1}{c}{+C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{\mathrm{ab}1L}+C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{\mathrm{ab}1R}-2K_t{z}_{w1}-2C_t{\stackrel{·}{z}}_{w1}}\\ \multicolumn{1}{c}{+\frac{F_{n1L}2\cos \left(\alpha \right)}{\cos \left(\lambda \right)}+\frac{F_{n1R}2\cos \left(\alpha \right)}{\cos \left(\lambda \right)}-W_{e1}+F_{c1\mathrm{Lz}}+F_{c2\mathrm{Lz}}}\\ \multicolumn{1}{c}{+F_{c1\mathrm{Rz}}+F_{c2\mathrm{Rz}}+F_{\mathrm{in}1L}+F_{\mathrm{in}1R}}\end{array}$$

(73)

$$\begin{array}{c}\multicolumn{1}{c}{m_{w1}{\stackrel{··}{y}}_{w1}=-2K_{a{b}_y}{y}_{w1}+K_{a{b}_y}{y}_{\mathrm{ab}1L}+K_{a{b}_y}{y}_{\mathrm{ab}1R}-2C_{a{b}_y}{\stackrel{·}{y}}_{w1}}\\ \multicolumn{1}{c}{+C_{a{b}_y}{\stackrel{··}{y}}_{\mathrm{ab}1L}+C_{a{b}_y}{\stackrel{··}{y}}_{\mathrm{ab}1L}-F_{t1}-W_{e1}{\varphi }_{w1}+F_{c1\mathrm{Ly}}}\\ \multicolumn{1}{c}{+F_{c2\mathrm{Ly}}+F_{c1\mathrm{Ry}}+F_{c2\mathrm{Ry}}}\end{array}$$

(74)

$$\begin{array}{l}{I}_{wx}{\stackrel{··}{\varphi }}_{w1}=I_{wy}\frac{V}{r_0}{\stackrel{·}{\psi }}_{w1}-2K_{a{b}_{xz}}{b}_1^2{\varphi }_{w1}-2K_{a{b}_{xz}}{y}_{w1}\lambda e-2C_{a{b}_{xz}}{b}_1^2{\stackrel{·}{\varphi }}_{w1}-2C_{a{b}_{xz}}{\stackrel{·}{y}}_{w1}\lambda e\\ +\left[T_{Y1L}\left({\delta }_L+{\varphi }_{w1}\right)+F_{n1L}\cos \left(\alpha \right)+T_{X1L}\sin \left(\alpha \right)\right]·\left[e-y_{w1}+r_L\cos \left(\alpha \right){\varphi }_{w1}+r_L\sin \left(\alpha \right){\psi }_{w1}\right]\\ +\left[r_L\cos \left(\alpha \right)+{\varphi }_{w1}\left(y_{w1}-e\right)\right]·\left[T_{Y1L}-F_{n1L}\left(\sin \left(\alpha \right){\psi }_{w1}+\cos \left(\alpha \right)\left({\delta }_L+{\varphi }_{w1}\right)\right)+T_{X1L}\left(\cos \left(\alpha \right){\psi }_{w1}-\sin \left(\alpha \right)\left({\delta }_L+{\varphi }_{w1}\right)\right)\right]\\ +(\left[r_L\cos \left(\alpha \right)+{\varphi }_{w1}\left(y_{w1}-e\right)\right]·\left[T_{Y2L}+F_{n1L}\left(\sin \left(\alpha \right){\psi }_{w1}-\cos \left(\alpha \right)\left({\delta }_L+{\varphi }_{w1}\right)\right)+T_{X2L}\left(\cos \left(\alpha \right){\psi }_{w1}+\sin \left(\alpha \right)\left({\delta }_L+{\varphi }_{w1}\right)\right)\right]\\ -\left[T_{Y2L}\left({\delta }_L+{\varphi }_{w1}\right)+F_{n1L}\cos \left(\alpha \right)-T_{X2L}\sin \left(\alpha \right)\right]·\left[y_{w1}-e-r_L\cos \left(\alpha \right){\varphi }_{\mathrm{w}1}+r_L\sin \left(\alpha \right){\psi }_{w1}\right]\\ +\left[{\varphi }_{w1}\left(y_{w1}+e\right)+r_R\cos \left(\alpha \right)\right]·\left[T_{Y1R}-F_{n1R}\left(\sin \left(\alpha \right){\psi }_{w1}-\cos \left(\alpha \right)\left({\delta }_R-{\varphi }_{w1}\right)\right)+T_{X1R}\left(\cos \left(\alpha \right){\psi }_{w1}+\sin \left(\alpha \right)\left({\delta }_R-{\varphi }_{w1}\right)\right)\right]\\ -\left[F_{n1R}\cos \left(\alpha \right)+T_{X1R}\sin \left(\alpha \right)-T_{Y1R}\left({\delta }_R-{\varphi }_{w1}\right)\right]·\left[{\psi }_{w1}+e-r_R\cos \left(\alpha \right){\varphi }_{w1}-r_R\sin \left(\alpha \right){\psi }_{w1}\right]\\ +\left[T_{X2R}\sin \left(\alpha \right)-F_{n1R}\cos \left(\alpha \right)+T_{Y2R}\left({\delta }_R-{\varphi }_{w1}\right)\right]·\left[y_{w1}+e-r_R\cos \left(\alpha \right){\varphi }_{w1}+r_R\sin \left(\alpha \right){\psi }_{w1}\right]\\ +\left[{\varphi }_{w1}\left(y_{w1}+e\right)+r_R\cos \left(\alpha \right)\right]·\left[T_{Y2R}+F_{n1R}\left(\sin \left(\alpha \right){\psi }_{w1}+\cos \left(\alpha \right)\left({\delta }_R-{\varphi }_{w1}\right)\right)+T_{X2R}\left(\cos \left(\alpha \right){\psi }_{w1}-\sin \left(\alpha \right)\left({\delta }_R-{\varphi }_{w1}\right)\right)\right]\end{array}$$

(75)

$$\begin{array}{c}\multicolumn{1}{c}{I_{\mathrm{wz}}{\stackrel{··}{\psi }}_{w1}=-\frac{I_{\mathrm{wy}}V}{r_0}{\stackrel{·}{\varphi }}_{w1}-2K_{a{b}_{\mathrm{xz}}}{b}_1^2{\psi }_{w1}-2C_{a{b}_{\mathrm{xz}}}{b}_1^2{\stackrel{·}{\psi }}_{w1}}\\ \multicolumn{1}{c}{+(1-{\psi }_{w1}^2)·[T_{X1L}{r}_L{\varphi }_{w1}+T_{X1L}{\delta }_L{r}_L-y_{w1}{T}_{X1L}\cos \left(\alpha \right)+y_{w1}{F}_{n1L}\sin \left(\alpha \right)+T_{X1L}e\cos \left(\alpha \right)-F_{n1L}e\sin \left(\alpha \right)-T_{Y1L}{r}_L\sin \left(\alpha \right)-T_{X1L}{\delta }_L{r}_L{\cos }^2\left(\alpha \right)+F_{n1L}{\delta }_L{r}_L\cos \left(\alpha \right)\sin \left(\alpha \right)]}\\ +(1-{\psi }_{w1}^2)·[T_{X2L}{r}_L{\varphi }_{w1}+T_{X2L{\delta }_L{r}_L}-y_{w1}{T}_{X2L}\cos \left(\alpha \right)-y_{w1}{F}_{n1L}\sin \left(\alpha \right)+T_{X2L}e\cos \left(\alpha \right)+F_{n1L}e\sin \left(\alpha \right)+T_{Y2L}{r}_L\sin \left(\alpha \right)-T_{X2L}{\delta }_L{r}_L{\cos }^2\left(\alpha \right)-F_{n1L}{\delta }_L{r}_L\cos \left(\alpha \right)\sin \left(\alpha \right)]\hfill \\ +({\psi }_{w1}^2+1)·[T_{X1R}{\delta }_R{r}_R-T_{X1R}{r}_R{\psi }_{w1}+y_{w1}{T}_{X1R}\cos \left(\alpha \right)-y_{w1}{F}_{n1R}\sin \left(\alpha \right)+T_{X1R}e\cos \left(\alpha \right)-F_{n1R}e\sin \left(\alpha \right)+T_{Y1R}{r}_R\sin \left(\alpha \right)-T_{X1R}{\delta }_R{r}_R{\cos }^2\left(\alpha \right)+F_{n1L}{\delta }_R{r}_R\cos \left(\alpha \right)\sin \left(\alpha \right)]\hfill \end{array}$$

(76)

Where $m_{w1}$ is the wheelset mass and $I_{\mathrm{wx}},I_{\mathrm{wy}}\mathrm{and}{I}_{\mathrm{wz}}$ are the inertia of the wheelset in the X, Y and Z-directions. $K_{a{b}_{\mathrm{xz}}}$ , $K_{a{b}_y}$ , $C_{a{b}_{\mathrm{xz}}}$ and $C_{a{b}_y}$ are the stiffness and damping coefficients of the rolling bearing in the radial (suffix $\mathrm{xz}$ ) and axial (suffix $y$ ) directions; $a$ and $b$ are constant lengths. $T_{(X,Y)(1,2)(L,R)}$ are the creep forces and $F_{n(1,2)(L,R)}$ are the normal forces at the contact points. $F_{t1}$ is the reaction of the rollers due to flange contact. $W_{e1}$ is the load on the front wheelset, which includes the weight of the wheelsets, the axle boxes, the half bogie and the applied load. $F_{\mathrm{in}1L}$ and $F_{\mathrm{in}1R}$ are external input forces derived from roller irregularities or other external phenomena.

Similarly, the expressions for the motion of the rear wheelset have been defined in (77)–(80), being $\mu$ the friction coefficient between the wheels and the rails.

$$\begin{array}{c}\multicolumn{1}{c}{m_{w2}{\stackrel{··}{z}}_{w2}=-2K_{a{b}_{\mathrm{xz}}}{z}_{w2}+K_{a{b}_{\mathrm{xz}}}{z}_{\mathrm{ab}2L}+K_{a{b}_{\mathrm{xz}}}{z}_{\mathrm{ab}2R}-2C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{w2}}\\ \multicolumn{1}{c}{+C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{\mathrm{ab}2L}+C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{z}}_{\mathrm{ab}2R}-2K_t{z}_{w2}-2C_t{\stackrel{·}{z}}_{w2}+F_{n2L}}\\ \multicolumn{1}{c}{+F_{n2R}-W_{e2}}\end{array}$$

(77)

$$\begin{array}{c}\multicolumn{1}{c}{m_{w2}{\stackrel{··}{y}}_{w2}=-2K_{a{b}_y}{y}_{w2}+K_{a{b}_y}{y}_{\mathrm{ab}2L}+K_{a{b}_y}{y}_{\mathrm{ab}2R}-2C_{a{b}_y}{\stackrel{·}{y}}_{w2}}\\ \multicolumn{1}{c}{+C_{a{b}_y}{\stackrel{··}{y}}_{\mathrm{ab}2R}+C_{a{b}_y}{\stackrel{··}{y}}_{\mathrm{ab}2R}-F_{t2}-W_{e2}{\varphi }_{w2}}\\ \multicolumn{1}{c}{-\mu W_{e2}g\mathrm{sign}({\stackrel{·}{y}}_{w2})}\end{array}$$

(78)

$$\begin{array}{c}\multicolumn{1}{c}{I_{\mathrm{wx}}{\stackrel{··}{\varphi }}_{w2}=-2K_{a{b}_{\mathrm{xz}}}{b}_1^2{\varphi }_{w2}-2K_{a{b}_{\mathrm{xz}}}{y}_{w2}\lambda e-2C_{a{b}_{\mathrm{xz}}}{b}_1^2{\stackrel{·}{\varphi }}_{w2}}\\ \multicolumn{1}{c}{-2C_{a{b}_{\mathrm{xz}}}{\stackrel{·}{y}}_{w2}\lambda e}\end{array}$$

(79)

$$I_{\mathrm{wz}}{\stackrel{··}{\psi }}_{w2}=-2K_{a{b}_{\mathrm{xz}}}{b}_1^2{\psi }_{w2}-2C_{a{b}_{\mathrm{xz}}}{b}_1^2{\stackrel{·}{\psi }}_{w2}$$

(80)

The equations that define the motion of the axle boxes are derived from the analysis of the actuating forces on them.

$$\begin{array}{c}\multicolumn{1}{c}{m_{\mathrm{ab}}{\stackrel{··}{z}}_{\mathrm{ab}1L}=-K_{a{b}_{\mathrm{xz}}}(z_{\mathrm{ab}1L}-z_{w1}-b_1{\phi }_{w1})}\\ \multicolumn{1}{c}{-K_{\mathrm{pz}}(z_{\mathrm{ab}1L}-z_b-L{\theta }_b+b_1{\phi }_b)}\\ \multicolumn{1}{c}{-C_{a{b}_{\mathrm{xz}}}({\stackrel{·}{z}}_{\mathrm{ab}1L}-{\stackrel{·}{z}}_{w1}-b_1{\stackrel{·}{\phi }}_{w1})}\\ \multicolumn{1}{c}{-C_{\mathrm{pz}}({\stackrel{·}{z}}_{\mathrm{ab}1L}-{\stackrel{·}{z}}_b-L{\stackrel{·}{\theta }}_b+b_1{\stackrel{·}{\phi }}_b)}\end{array}$$

(81)

$$\begin{array}{c}\multicolumn{1}{c}{m_{\mathrm{ab}}{\stackrel{··}{z}}_{\mathrm{ab}1R}=-K_{a{b}_{\mathrm{xz}}}(z_{\mathrm{ab}1R}-z_{w1}+b_1{\phi }_{w1})}\\ \multicolumn{1}{c}{-K_{\mathrm{pz}}(z_{\mathrm{ab}1R}-z_b-L{\theta }_b-b_1{\phi }_b)}\\ \multicolumn{1}{c}{-C_{a{b}_{\mathrm{xz}}}({\stackrel{·}{z}}_{\mathrm{ab}1R}-{\stackrel{·}{z}}_{w1}+b_1{\stackrel{·}{\phi }}_{w1})}\\ \multicolumn{1}{c}{-C_{\mathrm{pz}}({\stackrel{·}{z}}_{\mathrm{ab}1L}-{\stackrel{·}{z}}_b-L{\stackrel{·}{\theta }}_b-b_1{\stackrel{·}{\phi }}_b)}\end{array}$$

(82)

$$\begin{array}{c}\multicolumn{1}{c}{m_{\mathrm{ab}}{\stackrel{··}{z}}_{\mathrm{ab}2L}=-K_{a{b}_{\mathrm{xz}}}(z_{\mathrm{ab}2L}-z_{w2}-b_1{\phi }_{w2})}\\ \multicolumn{1}{c}{-K_{\mathrm{pz}}(z_{\mathrm{ab}2L}-z_b+L{\theta }_b+b_1{\phi }_b)}\\ \multicolumn{1}{c}{-C_{a{b}_{\mathrm{xz}}}({\stackrel{·}{z}}_{\mathrm{ab}2L}-{\stackrel{·}{z}}_{w2}-b_1{\stackrel{·}{\phi }}_{w2})}\\ \multicolumn{1}{c}{-C_{\mathrm{pz}}({\stackrel{·}{z}}_{\mathrm{ab}2L}-{\stackrel{·}{z}}_b+L{\stackrel{·}{\theta }}_b+b_1{\stackrel{·}{\phi }}_b)}\end{array}$$

(83)

$$\begin{array}{c}\multicolumn{1}{c}{m_{\mathrm{ab}}{\stackrel{··}{z}}_{\mathrm{ab}2R}=-K_{a{b}_{\mathrm{xz}}}(z_{\mathrm{ab}2R}-z_{w2}+b_1{\phi }_{w2})}\\ \multicolumn{1}{c}{-K_{\mathrm{pz}}(z_{\mathrm{ab}2R}-z_b+L{\theta }_b-b_1{\phi }_b)}\\ \multicolumn{1}{c}{-C_{a{b}_{\mathrm{xz}}}({\stackrel{·}{z}}_{\mathrm{ab}2R}-{\stackrel{·}{z}}_{w2}+b_1{\stackrel{·}{\phi }}_{w2})}\\ \multicolumn{1}{c}{-C_{\mathrm{pz}}({\stackrel{·}{z}}_{\mathrm{ab}2R}-{\stackrel{·}{z}}_b+L{\stackrel{·}{\theta }}_b-b_1{\stackrel{·}{\phi }}_b)}\end{array}$$

(84)

$$\begin{array}{c}\multicolumn{1}{c}{m_{\mathrm{ab}}{\stackrel{··}{y}}_{\mathrm{ab}1L}=-K_{a{b}_y}(y_{\mathrm{ab}1L}-y_{w1}-b_1{\psi }_{w1})}\\ \multicolumn{1}{c}{-K_{\mathrm{py}}(y_{\mathrm{ab}1L}-y_b-L{\psi }_b)}\\ \multicolumn{1}{c}{-C_{a{b}_y}({\stackrel{·}{y}}_{\mathrm{ab}1L}-{\stackrel{·}{y}}_{w1}-b_1{\stackrel{·}{\psi }}_{w1})}\\ \multicolumn{1}{c}{-C_{\mathrm{py}}({\stackrel{·}{y}}_{\mathrm{ab}1L}-{\stackrel{·}{y}}_b-L{\stackrel{·}{\psi }}_b)}\end{array}$$

(85)

$$\begin{array}{c}\multicolumn{1}{c}{m_{\mathrm{ab}}{\stackrel{··}{y}}_{\mathrm{ab}1R}=-K_{a{b}_y}(y_{\mathrm{ab}1R}-y_{w1}+b_1{\psi }_{w1})}\\ \multicolumn{1}{c}{-K_{\mathrm{py}}(y_{\mathrm{ab}1R}-y_b+L{\psi }_b)}\\ \multicolumn{1}{c}{-C_{a{b}_y}({\stackrel{·}{y}}_{\mathrm{ab}1R}-{\stackrel{·}{y}}_{w1}+b_1{\stackrel{·}{\psi }}_{w1})}\\ \multicolumn{1}{c}{-C_{\mathrm{py}}({\stackrel{·}{y}}_{\mathrm{ab}1R}-{\stackrel{·}{y}}_b+L{\stackrel{·}{\psi }}_b)}\end{array}$$

(86)

$$\begin{array}{c}\multicolumn{1}{c}{m_{\mathrm{ab}}{\stackrel{··}{y}}_{\mathrm{ab}2L}=-K_{a{b}_y}(y_{\mathrm{ab}2L}-y_{w2}-b_1{\psi }_{w2})}\\ \multicolumn{1}{c}{-K_{\mathrm{py}}(y_{\mathrm{ab}2L}-y_b+L{\psi }_b)}\\ \multicolumn{1}{c}{-C_{a{b}_y}({\stackrel{·}{y}}_{\mathrm{ab}2L}-{\stackrel{·}{y}}_{w2}-b_1{\stackrel{·}{\psi }}_{w2})}\\ \multicolumn{1}{c}{-C_{\mathrm{py}}({\stackrel{·}{y}}_{\mathrm{ab}2L}-{\stackrel{·}{y}}_b+L{\stackrel{·}{\psi }}_b)}\end{array}$$

(87)

$$\begin{array}{c}\multicolumn{1}{c}{m_{\mathrm{ab}}{\stackrel{··}{y}}_{\mathrm{ab}2R}=-K_{a{b}_y}(y_{\mathrm{ab}2R}-y_{w2}+b_1{\psi }_{w2})}\\ \multicolumn{1}{c}{-K_{\mathrm{py}}(y_{\mathrm{ab}2R}-y_b-L{\psi }_b)}\\ \multicolumn{1}{c}{-C_{a{b}_y}({\stackrel{·}{y}}_{\mathrm{ab}2R}-{\stackrel{·}{y}}_{w2}+b_1{\stackrel{·}{\psi }}_{w2})}\\ \multicolumn{1}{c}{-C_{\mathrm{py}}({\stackrel{·}{y}}_{\mathrm{ab}2R}-{\stackrel{·}{y}}_b-L{\stackrel{·}{\psi }}_b)}\end{array}$$

(88)

Where $m_{\mathrm{ab}}$ is the mass of the axle box; $K_{a{b}_{\mathrm{xz}}},C_{a{b}_{\mathrm{xz}}},K_{\mathrm{pz}}$ and $C_{\mathrm{pz}}$ are the stiffness and damping coefficients of the rolling bearing and the primary suspension, respectively. $L$ and $b_1$ are constant lengths given by the geometry of the bogie.

The four equations that define the motion of the bogie frame in the vertical direction and pitch, roll and yaw angles are the following:

$$\begin{array}{c}\multicolumn{1}{c}{m_b{\stackrel{··}{z}}_b=-4K_{\mathrm{pz}}{z}_b+K_{\mathrm{pz}}{z}_{\mathrm{ab}1L}+K_{\mathrm{pz}}{z}_{\mathrm{ab}1R}+K_{\mathrm{pz}}{z}_{\mathrm{ab}2L}}\\ \multicolumn{1}{c}{+K_{\mathrm{pz}}{z}_{\mathrm{ab}2R}-4C_{\mathrm{pz}}{\stackrel{·}{z}}_b+C_{\mathrm{pz}}{\stackrel{·}{z}}_{\mathrm{ab}1L}+C_{\mathrm{pz}}{\stackrel{·}{z}}_{\mathrm{ab}1R}}\\ \multicolumn{1}{c}{+C_{\mathrm{pz}}{\stackrel{·}{z}}_{\mathrm{ab}2L}+C_{\mathrm{pz}}{\stackrel{·}{z}}_{\mathrm{ab}2R}-2K_{\mathrm{sz}}{z}_b-2C_{\mathrm{sz}}{\stackrel{·}{z}}_b}\end{array}$$

(89)

$$\begin{array}{c}\multicolumn{1}{c}{m_b{\stackrel{··}{y}}_b=-4K_{\mathrm{py}}{y}_b+K_{\mathrm{py}}(y_{\mathrm{ab}1L}+y_{\mathrm{ab}1R}+y_{\mathrm{ab}2L}+y_{\mathrm{ab}2R})}\\ \multicolumn{1}{c}{-4C_{\mathrm{py}}{\stackrel{·}{y}}_b+K_{\mathrm{py}}({\stackrel{·}{y}}_{\mathrm{ab}1L}+{\stackrel{·}{y}}_{\mathrm{ab}1R}+{\stackrel{·}{y}}_{\mathrm{ab}2L}+{\stackrel{·}{y}}_{\mathrm{ab}2R})}\\ \multicolumn{1}{c}{-2K_{\mathrm{sy}}{y}_b-2C_{\mathrm{sy}}{\stackrel{·}{y}}_b}\end{array}$$

(90)

$$\begin{array}{c}\multicolumn{1}{c}{I_{\mathrm{by}}{\stackrel{··}{\theta }}_b=-4K_{\mathrm{pz}}{L}^2{\theta }_b-4C_{\mathrm{pz}}{L}^2{\stackrel{·}{\theta }}_b+K_{\mathrm{pz}}L{z}_{\mathrm{ab}2L}}\\ \multicolumn{1}{c}{+K_{\mathrm{pz}}L{z}_{\mathrm{ab}2R}-K_{\mathrm{pz}}L{z}_{\mathrm{ab}1L}-K_{\mathrm{pz}}L{z}_{\mathrm{ab}1R}}\\ \multicolumn{1}{c}{+C_{\mathrm{pz}}L{\stackrel{·}{z}}_{\mathrm{ab}2L}+C_{\mathrm{pz}}L{\stackrel{·}{z}}_{\mathrm{ab}2R}-C_{\mathrm{pz}}L{\stackrel{·}{z}}_{\mathrm{ab}1L}}\\ \multicolumn{1}{c}{-C_{\mathrm{pz}}L{\stackrel{·}{z}}_{\mathrm{ab}1R}}\end{array}$$

(91)

$$\begin{array}{c}\multicolumn{1}{c}{I_{\mathrm{bx}}{\stackrel{··}{\phi }}_b=-4K_{\mathrm{pz}}{b}_1^2{\phi }_b-4C_{\mathrm{pz}}{b}_1^2{\stackrel{·}{\phi }}_b+K_{\mathrm{pz}}{b}_1{z}_{\mathrm{ab}2L}}\\ \multicolumn{1}{c}{-K_{\mathrm{pz}}{b}_1{z}_{\mathrm{ab}2R}+K_{\mathrm{pz}}{b}_1{z}_{\mathrm{ab}1L}-K_{\mathrm{pz}}{b}_1{z}_{\mathrm{ab}1R}}\\ \multicolumn{1}{c}{+C_{\mathrm{pz}}{b}_1{\stackrel{·}{z}}_{\mathrm{ab}2L}-C_{\mathrm{pz}}{b}_1{\stackrel{·}{z}}_{\mathrm{ab}2R}+C_{\mathrm{pz}}{b}_1{\stackrel{·}{z}}_{\mathrm{ab}1L}}\\ \multicolumn{1}{c}{-C_{\mathrm{pz}}{b}_1{\stackrel{·}{z}}_{\mathrm{ab}1R}-K_{\mathrm{rot}}{\phi }_b-2K_{\mathrm{sz}}{b}_2^2{\phi }_b-2C_{\mathrm{sz}}{b}_2^2{\stackrel{·}{\phi }}_b}\end{array}$$

(92)

$$\begin{array}{c}\multicolumn{1}{c}{I_{\mathrm{bz}}{\stackrel{··}{\psi }}_b=K_{\mathrm{py}}L(y_b-y_{\mathrm{ab}1L})-K_{\mathrm{py}}L(y_b-y_{\mathrm{ab}1R})}\\ \multicolumn{1}{c}{-K_{\mathrm{py}}L(y_b-y_{\mathrm{ab}2L})+K_{\mathrm{py}}L(y_b-y_{\mathrm{ab}2R})}\\ \multicolumn{1}{c}{+C_{\mathrm{py}}L({\stackrel{·}{y}}_b-{\stackrel{·}{y}}_{\mathrm{ab}1L})-C_{\mathrm{py}}L({\stackrel{·}{y}}_b-{\stackrel{·}{y}}_{\mathrm{ab}1R})}\\ \multicolumn{1}{c}{-C_{\mathrm{py}}L({\stackrel{·}{y}}_b-{\stackrel{·}{y}}_{\mathrm{ab}2L})+C_{\mathrm{py}}L({\stackrel{·}{y}}_b-{\stackrel{·}{y}}_{\mathrm{ab}2R})}\end{array}$$

(93)

Where $m_b$ is the mass of the bogie frame, $I_{\mathrm{bx}},I_{\mathrm{by}}$ and $I_{\mathrm{bz}}$ are the inertia of the bogie frame in the $X,Y$ and $Z$ directions. $K_{\mathrm{pz}},C_{\mathrm{pz}},K_{\mathrm{sz}}$ and $C_{\mathrm{sz}}$ are the stiffness and damping coefficients of the primary and secondary suspensions, respectively. $L$ and $b$ are constant lengths given by the geometry of the bogie.

This model is developed for a two-roller test rig, but it can be easily modified to simulate one-roller test rigs by setting the angle $\alpha =0$ and removing the contact calculations of the rear rollers.

The parameters of the model are listed in Table 3. Mass and inertia data and dimensions are obtained from the 3D CAD model. The stiffnesses and dampings are obtained from technical documentation and other research works ²⁴ or calculated from measurements of the actual bogie.

Table 3.

Model parameters.

Parameter	Symbol	Value
Bogie frame mass	m b	1.424 × 103 kg
Bogie frame moment of inertia around the roll axis	I bx	1.179 × 103 kgm2
Bogie frame moment of inertia around the pitch axis	I by	1.193 × 103 kgm2
Bogie frame moment of inertia around the yaw axis	I bz	2.326 × 103 kgm2
Wheelset mass	m w	1.099 × 103 kg
Wheelset moment of inertia around the roll axis	I wx	694.09 kgm2
Wheelset moment of inertia around the pitch axis	I wy	1141.82 kgm2
Wheelset moment of inertia around the yaw axis	I wz	694.09 kgm2
Axle box mass	m ab	158.42 kg
Half of the track gauge	e	0.834 m
Wheel conicity	$\lambda$	0.05
Wheel radius	r 0	0.46 m
Flange clearance	δ	0.01 m
Flange contact stiffness	K r	109 N/m
Total supporting stiffness	$K_t$	7.6 × 108 N/m
Supporting damping	$C_t$	4 × 105 N s/m
Half the distance between the centres of axle boxes along the pitch axis	b 1	1.070 m
Half the distance between wagon supports	b 2	0.175 m
Half the wheelbase	L	1 m
Bogie load	W	10 t
Bearing stiffness in the radial direction	$K_{a{b}_{\mathrm{xz}}}$	3.87 × 108 N/m
Bearing stiffness in the axial direction	$K_{a{b}_y}$	1 × 107 N/m
Bearing damping in the radial direction	$C_{a{b}_{\mathrm{xz}}}$	5 × 104 N s/m
Bearing damping in the axial direction	$C_{a{b}_y}$	1.4 × 104 N s/m
Vertical stiffness of the primary suspension	$K_{\mathrm{pz}}$	1.016 × 106 N/m
Lateral stiffness of the primary suspension	$K_{\mathrm{py}}$	5.6 × 104 N/m
Vertical damping of the primary suspension	C pz	105 N s/m
Lateral stiffness of the primary suspension	$C_{\mathrm{py}}$	4.9 × 103 N s/m
Vertical stiffness of the secondary suspension	K sz	7.55 × 105 N/m
Lateral stiffness of the secondary suspension	$K_{\mathrm{sy}}$	7.55 × 105 N/m
Vertical damping of the secondary suspension	C sz	0 N s/m
Lateral damping of the secondary suspension	$C_{\mathrm{sy}}$	0 N s/m
Distance between roller centres	L roller	0.7 m
Roller radius	r roller	0.175 m

Damping on Y-25/Y-21 type bogies is provided by a dry friction-based Lenoir link, which is highly non-linear. Several authors^13,32 have proposed different models to simulate the performance of this type of damping. However, computation times become extremely long. This work proposes a linear approach for the damping by setting an equivalent damping based on the research developed by Mazilu and Dinu. ³² The model proposed in this latter work is implemented using our values for the Y-21 bogie and a bogie load of 10 tonnes. The resulting friction force at each time step is divided by the velocity at that time step, giving an equivalent damping as a function of time. Based on these results, the equivalent damping value is chosen, which is represented by the black line of Figure 8.

Figure 8.

Friction force computed using Mazilu and Dinu ³² (top) and equivalent damping (bottom).

Simulation results

First, the contact area and pressure distribution are computed, as these data are part of the input to compute the contact forces. Although this work focuses on the behaviour of the bogie on a two-roller test rig, it is interesting to perform a small comparison between the contact parameters in the wheel/rail and wheel/rollers cases. The contact patches are shown in Figure 9 for a bogie load of 10 tonnes (in addition to the weight of the bogie).

Figure 9.

Contact patches in wheel/rail (left) and wheel/rollers (right) cases. The bogie load is 10 t.

The first feature to mention is the aspect ratio of the contact ellipse: in the wheel/rail case, the major semi-axis of the ellipse is in the rolling direction, which results in an $a/b$ ratio of 1.42, while in the wheel/rollers case, the major semi-axis is the transversal direction, resulting in an $a/b$ ratio of 0.51. That is, the small curvature radius of the roller considerably changes the shape of the contact ellipse. The contact area is also smaller in the wheel/rollers case: 41.46 mm² per roller versus 89.73 mm² in the wheel/rail case. Obviously, this is also reflected in the contact pressure, which is higher in the wheel/rollers contact: 1288 versus 993 MPa. These results are in accordance with previous works, ³³ and will imply an accelerated wheel wear.

Simulation at 50 km/h

The differential equations introduced in the previous section are implemented in MATLAB and solved using the Dormand-Prince pair, which is based on the Runge-Kutta formula. ³⁴ A first set of simulations is performed at 50 km/h and a bogie load of 10 tonnes to study the performance of the bogie in nominal conditions. Simulation time is set to 10 s in order to obtain a stationary response of the system after the initial transient. No external forces due to roller irregularities or other phenomena are considered. Therefore, the model variables $F_{\mathrm{in}1L}$ and $F_{\mathrm{in}1R}$ are set to zero. As initial conditions, all the variables are set to zero except the vertical displacement of both wheelsets, which is set to 9.8 mm, and the lateral displacement of the wheelsets. The initial lateral displacement of the front wheelset is set to 5 mm, while the initial position of the rear wheelset is set to 0.5 mm.

The vertical displacements (and the velocities) of the front and rear wheelsets quickly reach a stable vertical position around zero after the initial conditions to excite the system. The front wheelset presents a small oscillation of amplitude about 0.6 μm, and the rear wheelset presents an oscillation of amplitude about 0.025 μm. Therefore, these small oscillations are negligible. The same behaviour is observed in the vertical motion of the bogie frame.

The angular displacements of the front wheelset are shown in Figure 10. The left-hand plots illustrate the evolution of the roll and yaw angle of the first wheelset throughout the complete simulation. The right-hand plots provide a zoomed view to facilitate a more comprehensive understanding of the movement. It is evident that, owing to the initial conditions, the roll angle of the front wheelset manifests an initial transient, which is subsequently followed by a trend towards zero. Within this trend (Figure 10, upper right), an oscillation about 170 Hz can be observed.

Figure 10.

Time evolution of angular variables of wheelset 1. The right plots are zoomed regions of the left ones. V = 50 km/h, W = 10 t.

On the other hand, the yaw angle starts at zero and develops a sustained oscillation of amplitude about 8.4 × 10⁻⁵ rad. The frequency of this oscillation is about 170 Hz, too.

Figure 11 shows the axial and lateral displacements of the wheelsets and bogie. The front wheelset initially experiences a transient period due to the simulation’s initial conditions, after which it begins to trend smoothly towards zero. In contrast, the displacement of the rear wheelset can be considered negligible. Due to lateral friction between the wheel treads and supporting rails, total motion amounts to just 15 μm. The lateral motion of the bogie frame exhibits damped oscillations due to the initial conditions. This oscillation is the result of the model’s internal flexibilities.

Figure 11.

Time evolution of the axial displacement of the front wheelset (up), rear wheelset (middle) and bogie frame (bottom). V = 50 km/h, W = 10 t.

Figure 12 illustrates the behaviour of the angular variables of the bogie frame. After the initial transient, the pitch of the bogie frame stabilises around 1.52 × 10⁻⁶ rad. Closer inspection of its behaviour reveals a sustained oscillation with an amplitude of 0.075 μrad, which could be considered negligible. However, it is interesting that a frequency of approximately 170 Hz can also be identified, as well as its second harmonic. The roll angle of the bogie shows a trend towards zero. Again, a closer look at the plot reveals oscillations of frequency about 170 Hz. The yaw angle of the bogie exhibits a very low frequency oscillation with a peak-to-peak amplitude of 3 × 10⁻⁴ rad. However, what appears to be a simple line turns out to be a higher frequency oscillation on closer inspection. Again, the frequency of around 170 Hz observed earlier appears.

Figure 12.

Time evolution of angular variables of the bogie frame. V = 50 km/h, W = 10 t.

As could be expected from the configuration of the test bench, the behaviour of the front and rear axle boxes differs notably in terms of kinematics. These displacements are shown in Figure 13. The front axle boxes follow the same trend as the front wheelset, moving towards zero after the initial transient due to the initial conditions of the simulation. Conversely, the rear axle boxes exhibit damped oscillations for approximately 0.7 s. After this, there is almost no lateral movement of the rear axle boxes: they move laterally by about 12 μm in 9 s.

Figure 13.

Time evolution of the axial displacement of the axle boxes. V = 50 km/h, W = 10 t.

As occurs with the wheelsets and the bogie frame, the vertical displacement of the axle boxes is almost imperceptible. However, studying the vertical accelerations reveals interesting features. Due to the initial conditions, the four axle boxes experience vertical accelerations reaching 1.5 × 10⁴ m/s², which are damped rapidly. Figure 14 shows a zoomed region of the vertical acceleration between 0.3 and 0.5 s. As can be observed from these plots, the vertical acceleration of the front axle boxes oscillates at a frequency of around 170 Hz with an amplitude of around 5 m/s². As no external periodic forces are applied to the bogie, this phenomenon is due to the model’s internal flexibilities and the contact forces between the wheels and rollers. A similar pattern is evident in the rear axle boxes, though the amplitude of the accelerations is only approximately 0.25 m/s². As the rear wheelset and axle boxes are only subject to vertical forces from the bogie itself, the vertical movement of the axle boxes results from the transmission of movement from the front wheelset. According to these results, the vertical accelerations registered in the rear axle boxes are about 20 times lower than those registered in the front axle boxes.

Figure 14.

Vertical vibrations of the axle boxes (zoomed between 0.3 and 0.5 s). V = 50 km/h, W = 10 t.

To get a comprehensive view of the vertical acceleration of the axle boxes, a frequency analysis is performed. The PSDs of the accelerations are shown in Figure 15. The axle boxes of the front wheelset display a frequency component of high amplitude located at 168.81 Hz, which matches the high-frequency oscillations observed in the previous time plots. The second multiplier of this frequency is also visible at 337.52 Hz. The spectra also show two frequency components whose shape suggests that they are natural frequencies of the system. The first one is located at 162.6 Hz and is almost hidden by the component located at 168.81 Hz. The second frequency is located at around 284 Hz.

Figure 15.

PSD of vertical acceleration of the axle boxes. V = 50 km/h, W = 10 t.

On the other hand, the frequency component of 168.81 Hz and its second harmonic are barely visible in the spectra of the rear axle boxes. The acceleration of these mechanical components is clearly dominated by the frequency around 284 Hz.

Sensitivity analysis

We will focus only on the two parameters that are easily modifiable in the actual test bench: the speed and the bogie load. The bogie load is scanned from 0 tonnes (just the weight of the bogie) to 20 tonnes in steps of 2 tonnes. For each bogie load case, the speed is increased in steps of 5 km/h, starting at 10 km/h up to 130 km/h. As in the previous simulations, no forces from external phenomena are applied.

As sensors for monitoring rolling stock dynamics are typically located in the axle boxes, the sensitivity analysis will focus on the evolution of the RMS acceleration in these four boxes. This parameter can be computed quickly and provides a good indication of the level of vibration experienced by the axle boxes. The results are plotted in Figures 16 and 17.

Figure 16.

RMS acceleration of the front axle boxes as a function of speed and bogie load.

Figure 17.

RMS acceleration of the rear axle boxes as a function of speed and bogie load.

The bogie load clearly influences the vertical acceleration of the axle boxes: increasing the bogie load will result in an increase in vertical acceleration at a given speed. However, the effect of speed is more interesting. Increasing speed increases the level of axlebox accelerations, but there is a speed above which vertical accelerations are significantly reduced to the point of being almost imperceptible. The speed at which the trend change occurs depends on the bogie load. For a bogie load of 4 tonnes, this speed is 50 km/h. For 10 and 20 tonnes, this speed increases to 80 and 120 km/h, respectively.

As far as maximum acceleration amplitudes are concerned, the front wheelset axleboxes are in the region of 11 m/s². The maximum RMS values of the rear axle boxes are around 0.65 m/s². Therefore, the ratio between the accelerations is somewhat lower than the value of 20 mentioned in the previous section. It should also be noted that the RMS values are slightly higher on the left axle box than on the right axle box in the front wheelset. The opposite occurs in the rear wheelset, where the RMS values are slightly higher on the right axle box than on the left.

Conclusions

A 21-DOF dynamic model of a two-axle bogie is presented in this work. The model considers the vertical and lateral displacements and the roll and yaw angles of the wheelsets, the vertical and lateral displacements of the four axle boxes and the vertical and lateral displacements and pitch, roll and yaw angles of the bogie frame. The inclusion of the axle boxes in the model allows us to simulate the behaviour of this element, which is where vibrations in the rolling stock are usually measured. This element is not usually simulated in the scientific literature and is one of the novelties of this research.

The model is customised to simulate the performance of a Y-21 freight bogie in a two-roller test rig. This type of test rig device rotates one of the wheelsets while the other rests on a fixed rail section. This configuration is typical of wheel lathes and has not been studied in the scientific literature reviewed. The bogie has also been modelled in 3D, so key parameters to characterise the model can easily be extracted. The creepages, creep and normal forces have been calculated taking into account this configuration and introduced into the dynamic model of the bogie. Although this work simulates that specific configuration, any other bogie type can be simulated by just changing the model parameters. In the same way, a test bench that moves the two wheelsets can be simulated by adding the creepages, creep and normal forces to the rear wheelset. One-roller test rigs can be simulated by setting the parameter $\alpha$ to 0 and removing the contact computations of the rear rollers.

A small comparison between the contact surfaces and pressures of the typical wheel/rail contact and this specific wheel/rollers contact case is performed. The results show that the major semi-axis of the contact ellipse changes its direction when the two small rollers are considered. In addition, the contact area is reduced, and the contact pressure is increased. Therefore, the wheels should be subjected to accelerated wear, opening the door to using this type of test rig for studying wheel wear.

The performance of the model is analysed by simulating the behaviour of the bogie at a speed of 50 km/h under a bogie load of 10 tonnes. The lateral friction between the rear wheels and the supporting rail limits the axial movement of the rear wheelset, which induces a yaw movement on the bogie frame as the front wheelset moves laterally from the initial position to centre itself in the longitudinal axis of the bogie. Contact forces between wheels and rollers and internal elements of the system induce vertical vibrations in the system. Front axle boxes are subject to vertical accelerations of amplitude 5 m/s² that reach the rear axle boxes attenuated around 20 times. The frequency analysis of the vertical acceleration of the axle boxes reveals the existence of a component located at 168.81 Hz and its second multiplier, which explains the high-frequency signals observed in the time domain of certain variables.

A sensitivity analysis is carried out by modifying the two basic parameters that can be changed in the test rig: the speed and the bogie load. According to the results from the simulations, the bogie load has a direct influence on the vertical acceleration of the axle boxes. However, the relationship between the speed and the vertical acceleration of axle boxes is more complicated. The results show that the acceleration increases with speed up to a speed at which the acceleration drops abruptly. In addition, this speed also depends on the bogie load applied.

Future research will focus on experimental validation of the model and simulation of other test setups, including roller bearing models in the axle boxes, as a way to develop a Digital Twin that allows analysing and understanding bogie performance on a test bench for predictive maintenance tasks. However, there is still a long way to go.