An advanced and effective dynamical model of the railway vehicle, which provides a detailed analysis of the lateral dynamic features were build in this section. The dynamic model constitutes a 35-DOF equations of motion considering lateral, vertical, rolling, yawing, pitching motions, respectively related to a vehicle body ( *I*_{vbx}, I_{vby}, I_{vbz}, m_{vb}) supported by two bogie frames ( *I*_{bx,1}, *I*_{bx,2}, *I*_{by,1}, *I*_{by,2}, *I*_{bz,1}, *I*_{bz,2}, *m*_{b,1}, *m*_{b,2}) with the secondary suspensions ( *K*_{sx}, K_{sy}, K_{sz}, C_{sx}, C_{sy}, C_{sz}) in longitudinal, lateral and vertical directions, and two wheelsets ( *m*_{w}, I_{wx}, I_{wy}, I_{wz}), which are mounted with each bogie frame with primary suspensions ( *K*_{px}, K_{py}, K_{pz}, C_{px}, C_{py}, C_{pz}) also in three directions. The states of the railway vehicle model are lateral, vertical displacement and velocities of the vehicle body, bogies, wheelsets, and yaw, roll, pitch angles and angular velocities of the vehicle body, bogies, wheelsets, respectively. All of these possible motions are connected with primary and secondary suspensions, whose springs and dampers exhibit linear elastic characteristics [1]. Track irregularities are disregarded, the wheels and rails are always in contact, the wheels move on a smooth plane, and radius of the curved track is infinite. Railway vehicle diagram is presented in Figure 1.

Fig. 1 Dynamic model of the railway vehicle.

Nonlinearities of the first and second point of the contact of the wheel-rail and the creep forces and moments are also included in the model and will be explained later.

*b*_{c1}, *b*_{c3} are half of the secondary longitudinal and vertical spring arms, *b*_{c2}, *b*_{c4} are half of the secondary longitudinal and vertical damper arms, *b*_{t1}, *b*_{t3} are half of the primary longitudinal and vertical spring arms, *b*_{t2}, *b*_{t4} are half of the primary longitudinal and vertical damper arms, *h* is the height of the vehicle body mass center above the wheelset mass center, *h*_{0} is the height of the secondary suspension above the bogie frame mass center, *h*_{G} is the height of the bogie mass center above the wheelset mass center, *L*_{c} is the distance between the vehicle body and the bogie frame mass center, *L*_{t1}, *L*_{t2} are half of the primary lateral spring and damper arms. The vehicle body, the bogie frames, and wheelsets dynamics in lateral, vertical, roll,yaw, and pitch directions are evaluated in the following forms by using Newton’s second law [2].

$${m}_{vb}{\ddot{y}}_{vb}=2{F}_{syvb}+{m}_{vb}{g\varphi}_{se}-\frac{{m}_{vb}{V}^{2}}{{R}_{y}}$$(1)
$${m}_{b,i}{\ddot{y}}_{b,i}={F}_{syb,i}+\frac{({m}_{b,i}+\frac{{m}_{vb}}{2}){V}^{2}}{{R}_{y}}-{g\varphi}_{se}({m}_{b,i}+\frac{{m}_{vb}}{2})$$(2)
$${I}_{vbz}{\ddot{\mathit{\Psi}}}_{vb}={M}_{szvb}$$(3)
$${I}_{bz,i}{\ddot{\mathit{\Psi}}}_{b,i}={M}_{szb,i}$$(4)
$${m}_{vb}{\ddot{z}}_{vb}={F}_{szvb}-{m}_{vb}g-\frac{{m}_{vb}{V}^{2}}{{R}_{y}}$$(5)
$${m}_{b,i}{\ddot{z}}_{b,i}={F}_{szb,i}-{m}_{b,i}g-\frac{{m}_{b,i}{V}^{2}}{{R}_{y}}$$(6)
$${I}_{vbx}{\ddot{\varphi}}_{vb}={M}_{sxvb}$$(7)
$${I}_{bx,i}{\ddot{\varphi}}_{b,i}={M}_{sxb,i}$$(8)
$${I}_{vby}{\ddot{\gamma}}_{vb}={M}_{syvb}$$(9)
$${I}_{by,i}{\ddot{\gamma}}_{b,i}={M}_{syb,i}$$(10)
$$\begin{array}{rl}{m}_{w}{\ddot{y}}_{w,ij}=& \frac{{m}_{w}{V}^{2}}{{R}_{y}}-{m}_{w}{g\varphi}_{se}+{F}_{wLy,ij}^{n}+{F}_{wRy,ij}^{n}+{N}_{wy,ij}^{L}\\ & +{N}_{wy,ij}^{R}+{F}_{syw,ij}-{F}_{wt,ij}+\frac{{V}^{2}{W}_{ext}}{{gR}_{y}}-{W}_{ext}{\varphi}_{se}\end{array}$$(11)
$$\begin{array}{rl}{I}_{wz}{\ddot{\mathit{\Psi}}}_{w,ij}=& -{I}_{wy}\left(\frac{V}{{r}_{0}}+\frac{V\mathrm{sin}({\varphi}_{se})}{{R}_{y}}\right){\dot{\varphi}}_{w,ij}+{R}_{x,ij}^{R}{F}_{wRy,ij}^{n}\\ & -{R}_{y,ij}^{R}{F}_{wRx,ij}^{n}+{R}_{x,ij}^{L}{F}_{wLy,ij}^{n}-{R}_{y,ij}^{L}{F}_{wLx,ij}^{n}\\ & +{R}_{x,ij}^{R}{N}_{wy,ij}^{R}+{R}_{x,ij}^{L}{N}_{wy,ij}^{L}+{M}_{wLz,ij}^{n}+{M}_{wRz,ij}^{n}\\ & +{M}_{szw,ij}\end{array}$$(12)
$$\begin{array}{rl}{m}_{w}{\ddot{z}}_{w,ij}=& -\frac{{m}_{w}{V}^{2}{\varphi}_{se}}{{R}_{y}}-{m}_{w}g+{F}_{wz,ij}^{L}+{F}_{wz,ij}^{R}+{N}_{wz,ij}^{R}\\ & +{N}_{wz,ij}^{L}+{F}_{szw,ij}\end{array}$$(13)
$$\begin{array}{rl}{I}_{wx}{\ddot{\varphi}}_{w,ij}=& -\frac{{I}_{wy}V(\frac{V}{{R}_{y}}-{\dot{\mathit{\Psi}}}_{w,ij})}{{r}_{0}}+{R}_{y,ij}^{R}{F}_{wz,ij}^{R}-{R}_{z,ij}^{R}{F}_{wRy,ij}^{n}\\ & +{R}_{y,ij}^{L}{F}_{wz,ij}^{L}-{R}_{z,ij}^{L}{F}_{wLy,ij}^{n}+{R}_{y,ij}^{L}{N}_{wz,ij}^{L}\\ & +{R}_{y,ij}^{R}{N}_{wz,ij}^{R}-{R}_{z,ij}^{R}{N}_{wy,ij}^{R}-{R}_{z,ij}^{L}{N}_{wy,ij}^{L}\\ & +{M}_{wx,ij}^{L}+{M}_{wx,ij}^{R}+{M}_{sxw,ij}\end{array}$$(14)
$$\begin{array}{rl}{I}_{wy}{\ddot{\gamma}}_{w,ij}=& {R}_{z,ij}^{R}{F}_{wRx,ij}^{n}+{R}_{z,ij}^{L}{F}_{wLx,ij}^{n}-{R}_{x,ij}^{R}{F}_{wz,ij}^{R}-{R}_{x,ij}^{R}{N}_{wz,ij}^{R}\\ & -{R}_{x,ij}^{L}{F}_{wz,ij}^{L}-{R}_{x,ij}^{L}{N}_{wz,ij}^{L}+{M}_{wy,ij}^{L}+{M}_{wy,ij}^{R}\\ & +{M}_{syw,ij}\end{array}$$(15)
where subscript "*i*" denotes the position of bogies: *i* = 1 (front bogie), *i* = 2 (rear bogie), and subscript "*j*" denotes the position of wheelsets: *j* = 1 (leading wheelset), *j* = 2 (trailing wheelset).

*V* ise the forward speed of the vehicle, *R*_{y} is the radius of the curved track, *r*_{0} is the nominal wheelset rolling radius, *y*_{vb}, z_{vb}, ψ_{vb}, ϕ_{vb}, γ_{vb} are lateral and vertical displacements, and yaw, roll and pitch angles of the vehicle body, *y*_{b,i}, z_{b,i}, ψ_{b,i}, ϕ_{b,i}, γ_{b,i} are lateral and vertical displacements, and yaw, roll and pitch angles of the bogies, *y*_{w,ij}, z_{w,ij}, ψ_{w,ij}, ϕ_{w,ij}, γ_{w,ij} are lateral, vertical displacements and yaw,roll, pitch angles of the wheelsets, ${\dot{y}}_{w,ij},{\dot{z}}_{w,ij},{\dot{\mathrm{\Psi}}}_{w,ij},{\dot{\varphi}}_{w,ij},{\dot{\gamma}}_{w,ij}$ are lateral, vertical velocities and yaw, roll, pitch angular velocities of the wheelsets, ${N}_{wy,ij}^{L}{N}_{wz,ij}^{L}{N}_{wy,ij}^{R}{N}_{wz,ij}^{R}$ are normal forces on the left and right wheels in the lateral and vertical directions, *W*_{ext} is the external load on the wheelset axles, *g* is the gravitational constant, *ϕ*_{se} is the cant angle, *m*_{vb}, *m*_{b, 1} *m*_{b,2} are vehicle body, first and second bogie frame masses, *m*_{w} is the wheelset mass, *F*_{wt,ij} is the flange contact force, *I*_{vbx}, I_{b,xi}, I_{vby}, I_{b,yi}, I_{vbz}, I_{b,zi} are roll, pitch, and yaw moments of inertia of the vehicle body and bogies, *I*_{wx}, I_{wy}, I_{wz} are roll, pitch, and yaw moment of inertia of the wheelsets, *K*_{px}, K_{py}, K_{pz} are primary longitudinal, lateral, and vertical stiffness, *K*_{sx}, K_{sy}, K_{sz} are secondary longitudinal, lateral, and vertical stiffness, *C*_{px}, C_{py}, C_{pz} are primary longitudinal, lateral, and vertical damping coefficients, *C*_{sx}, C_{sy}, C_{sz} are secondary longitudinal, lateral, and vertical damping coefficients. *F*_{syvb}, F_{szvb}, M_{sxvb}, M_{syvb}, M_{szvb} are lateral, vertical suspension forces and longitudinal, lateral, vertical suspension moments of the vehicle body, *F*_{syb,i}, F_{szb,i}, M_{sxb,i}, M_{syb,i}, M_{szb,i} are lateral, vertical suspension forces and longitudinal, lateral, vertical suspension moments of the bogies, ${F}_{wx,ij}^{L},{F}_{wy,ij}^{L},{F}_{wz,ij}^{L}$ are linear creep forces at the equilibrium coordinate system of the left wheel in the longitudinal, lateral, vertical directions, ${F}_{wx,ij}^{R},{F}_{wy,ij}^{R},{F}_{wz,ij}^{R}$ are linear creep forces at the equilibrium coordinate system of the right wheel in the longitudinal, lateral, vertical directions, ${M}_{wx,ij}^{L},{M}_{wy,ij}^{L},{M}_{wz,ij}^{L}$ are linear creep moments at the equilibrium coordinate system of the left wheel in the longitudinal, lateral, and vertical directions, ${M}_{wx,ij}^{R},{M}_{wy,ij}^{R},{M}_{wz,ij}^{R}$ are linear creep moments at the equilibrium coordinate system of the right wheel in the longitudinal, lateral, and vertical directions, ${F}_{wLx,ij}^{n},{F}_{wLy,ij}^{n},{F}_{wRx,ij}^{n},{F}_{wRy,ij}^{n}$ are nonlinear heuristic creep forces of the left and right wheels in the longitudinal and lateral directions, ${M}_{wLz,ij}^{n},{M}_{wRz,ij}^{n}$ are nonlinear heuristic creep moment of the left and right wheels in the vertical direction, ${R}_{x,ij}^{L},{R}_{y,ij}^{L},{R}_{z,ij}^{L},{R}_{x,ij}^{R},{R}_{y,ij}^{R},{R}_{z,ij}^{R}$ are *xyz* components of the contact position vector on the left and right wheels, *F*_{syw,ij}F_{szw,ij} are lateral and vertical suspension forces of the wheelset, *M*_{sxw,ij}, M_{syw,ij}, M_{szw,ij} are suspension moments of the wheelset in the longitudinal, lateral, and vertical directions, respectively.

The nonlinear longitudinal yaw damping forces *F*_{d,ij} are included in *M*_{szw,ij} and are considered to the Yang and Ahmadian [8] conjecture as follows:
$$\begin{array}{rl}& {F}_{d,ij}=\\ & \left\{\begin{array}{c}{C}_{1}{V}_{x,ij}^{\psi}+{C}_{2}{({V}_{x,ij}^{\psi})}^{2}+{C}_{3}{({V}_{x,ij}^{\psi})}^{3}+{C}_{4}{({V}_{x,ij}^{\psi})}^{4}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\Rightarrow {V}_{x,ij}^{\psi}\succ 0\\ {C}_{1}{V}_{x,ij}^{\psi}-{C}_{2}{({V}_{x,ij}^{\psi})}^{2}+{C}_{3}{({V}_{x,ij}^{\psi})}^{3}-{C}_{4}{({V}_{x,ij}^{\psi})}^{4}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\Rightarrow {V}_{x,ij}^{\psi}\prec 0\end{array}\right\}\end{array}$$(16)
where ${V}_{x,ij}^{\psi}={b}_{t,i}{\mathrm{\Psi}}_{w,ij}$ is the relative longitudinal speed between the bogies and their wheelsets, which cause the yaw movement. The moments produced by the nonlinear longitudinal damping forces ( *F*_{d,ij}) are included in vertical suspension moment ( *M*_{szw,ij}). These damping forces can be expressed by fourth-order polynomials of ${V}_{x,ij}^{\psi}$. The coefficients *C*_{1} to *C*_{4} are obtained from the experimental tests on the actual dampers by Yang and Ahmadian [8].The additive moments generated by these forces are represented as
$${M}_{d,ij}={F}_{d,ij}{b}_{ti}$$(17)

The contact relation between the wheel and rail interface exhibits highly non-linear characteristic and is responsible for the occurrence of the limit-cycle dynamic behavior and hunting phenomenon. The contacts between the wheel and rail can be categorized into primary and secondary contacts, which takes place at the interface surface and the wheel flange-rail side, respectively. As the primary contact causes to happen creep forces in the elliptical contact area, the secondary contact collides the rail side when the clearance between the wheel flange and the rail is zero.

In this paper, the wheel is considered to be conical, and the rail is designed as having a knife-edge spring element with a lateral and vertical stiffnesses (*K*_{ry}, K_{rz}). The axle load, suspension, and inertial forces of the wheelset are implicitly contained in this spring force [6].

The flange contact forces at the secondary contact were designed as follows:
$$\begin{array}{rl}{F}_{wt,ij}=& {K}_{ry}({y}_{w,ij}-\delta )\left[\frac{1}{2}\mathrm{tanh}\{{10}^{4}({y}_{w,ij}-\delta )\}+\frac{1}{2}\right]\\ & +{K}_{ry}({y}_{w,ij}+\delta )\left[\frac{1}{2}\mathrm{tanh}\{{10}^{4}(-{y}_{w,ij}-\delta )\}+\frac{1}{2}\right]\end{array}$$(18)

The function used in Eqn. 18 are used to characterize the nonlinear dead-band dynamics of the vehicle, which leads to the saturation of the creep constant and the flange contact. The flange contact force are approached by embedding hyperbolic tangents into the function by the assistance of numerical treatment and the actual contact mechanism.

Two wheel-rail contact models, a heuristic nonlinear friction creepage model derived by using Kalker’s theory [3, 5] and Polach model [4] including dead-zone clearance are applied to inspect the lateral dynamic features of the railway vehicle.

## Kalker’s Model

Kalker’s model has been used when the combinations of the spin and creepages to model three dimensional rolling contact mechanical model were considered. Kalker has found an exact analytical method named as Kalker linear-theory to evaluate the contact forces in the linear part of the creep force-creepage curve. In this model a simple linear relationship between the surface deformation and traction is considered. In the case without slipping, the tangential strain is constant on the contact patch. This indicates that the tractions are finite except for the trailing edge of the contact patch. In fact, the unloaded material flows into the contact patch from the leading edge, where there is no traction, and as it goes towards the trailing edge the traction is built up to infinity. Therefore, there is always a slip zone adjacent to this edge.

There exists a linear relationship between the creepages, and creep forces, creep moments. Therefore, the creep forces and moment can be evaluated for an elliptic patch using
$$\begin{array}{rl}& \left[\begin{array}{c}{F}_{wx,ij}^{L,R}\\ {F}_{wy,ij}^{L,R}\\ {M}_{wz,ij}^{L,R}\end{array}\right]=\\ & G.a.b.\left[\begin{array}{ccc}{-C}_{11}& 0& 0\\ 0& -{C}_{22}& \sqrt{a.b}.{C}_{23}\\ 0& -\sqrt{a.b}.{C}_{23}& a.b.{C}_{33}\end{array}\right]\left[\begin{array}{c}{\xi}_{x,ij}^{L,R}\\ {\xi}_{y,ij}^{L,R}\\ {\xi}_{\psi ,ij}^{L,R}\end{array}\right]\end{array}$$(19)
where ${\xi}_{x,ij}^{L,R},{\xi}_{y,ij}^{L,R},{\xi}_{\psi ,ij}^{L,R}$ are longitudinal, lateral, spin creepages of the left and right wheels, *a* and *b* are the half distance of the elliptical contact plane, *G* is the tangential elasticity modulus and *C*_{ij} coefficients are known as Kalker’s coefficients and can be evaluated as functions of the *a/b* ratio and of the Poisson’s module ( *ν*). The coefficients are available in tabulated form in [10].

To account for the creep mechanism that takes place at the primary contact point, a heuristic nonlinear approach which contains the nonlinear effect of the adhesion limit is utilized. Arising kinematic creepages from the primary contact point and a heuristic nonlinear model that consists of the nonlinear effect of the adhesion boundary was evaluated. By using Johnson’s approach [7], the saturation constant *α* can be evaluated from the unlimited resultant creep force such that
$$\begin{array}{rl}{\alpha}_{ij}=& \frac{1}{{\beta}_{ij}}\left({\beta}_{ij}-\frac{1}{3}{\beta}_{ij}^{2}+\frac{1}{27}{\beta}_{ij}^{3}\right)\left[\frac{1}{2}tanh\{{10}^{4}(3-{\beta}_{ij})\}+\frac{1}{2}\right]\\ & +\frac{1}{{\beta}_{ij}}\left[\frac{1}{2}tanh\{{10}^{4}({\beta}_{ij}-3)\}+\frac{1}{2}\right]\end{array}$$(20)
$${\beta}_{ij}=\frac{{\beta}_{ij}^{R}+{\beta}_{ij}^{L}}{2}$$
$${\beta}_{ij}^{L,R}=\frac{\sqrt{{\left({F}_{wx,ij}^{L,R}\right)}^{2}+{\left({F}_{wy,ij}^{L,R}\right)}^{2}}}{\mu \sqrt{{\left({N}_{wy,ij}^{L,R}\right)}^{2}+{\left({N}_{wz,ij}^{L,R}\right)}^{2}}},$$
$$\sqrt{{\left({N}_{wy,ij}^{L,R}\right)}^{2}+{\left({N}_{wz,ij}^{L,R}\right)}^{2}}={N}_{w,ij}^{L,R}$$
where *α*_{ij} ise the saturation constant in the heuristic creep model, *β*_{ij} is the nonlinearity in the heuristic creep model, ${\beta}_{ij}^{L},{\beta}_{ij}^{R}$ are the nonlinearity of the left wheel and the right wheel in the heuristic creep model, respectively. The nonlinear heuristic creep forces and moments can be introduced by their corresponding linear creep force multiplied by the related saturation factor as [9].
$${F}_{wLx,ij}^{n}={\alpha}_{ij}{F}_{wx,ij}^{L}$$(21)
$${F}_{wRx,ij}^{n}={\alpha}_{ij}{F}_{wx,ij}^{R}$$(22)
$${F}_{wLy,ij}^{n}={\alpha}_{ij}{F}_{wy,ij}^{L}$$(23)
$${F}_{wRy,ij}^{n}={\alpha}_{ij}{F}_{wy,ij}^{R}$$(24)
$${M}_{wLz,ij}^{n}={\alpha}_{ij}{M}_{wz,ij}^{L}$$(25)
$${M}_{wRz,ij}^{n}={\alpha}_{ij}{M}_{wz,ij}^{R}$$(26)

## Polach’s Model

Polach’s method [11], has been widely used in commercial codes, because its computation time is relatively short and its accuracy is satisfactory as well as compared with other methods. This model grants evaluation of full non-linear creep forces and taking spin into account. Tangential stress is proportional to slip ratio and the distance from the leading edge of the contact area with a constant "*C”* as tangential contact stiffness. The gradient of tangential stress distribution in the adhesion patch is
$${\u03f5}_{ij}^{L,R}=\frac{2.C.\pi .{a}^{2}.b}{3.{m}_{a}.g.{\mu}_{f}}{\xi}_{ij}^{L,R}$$(27)
where; *m*_{a} is the adhesion mass of the wheel, *μ*_{f} is the friction coefficient.

The tangential force is defined as
$${F}_{w\tau ,ij}^{L,R}=\frac{-2{m}_{a}{g\mu}_{f}}{\pi}\left(\frac{{\u03f5}_{\mathrm{i}\mathrm{j}}^{L,R}}{1+{{(\u03f5}_{\mathrm{i}\mathrm{j}}^{L,R})}^{2}}+\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}({\u03f5}_{\mathrm{i}\mathrm{j}}^{L,R})\right)$$(28)

The creep forces ${F}_{wx,ij}^{L,R},\phantom{\rule{thinmathspace}{0ex}}{F}_{wy,ij}^{L,R}$ are evaluated as [12]
$${F}_{wk,ij}^{L,R}=\frac{{F}_{w\tau ,ij}^{L,R}{\xi}_{k,ij}^{L,R}}{{\xi}_{ij}^{L,R}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}k=x,y$$(29)
where
$${\xi}_{ij}^{L,R}=\sqrt{{({\xi}_{x,ij}^{L,R})}^{2}+{({\xi}_{y,ij}^{L,R})}^{2}}$$(30)

The spin effect, which is a rotation about the vertical axis *z* due to the wheel conicity, is also important to solve the wheel-rail contact problem.

The relative spin ${\xi}_{\psi ,\mathrm{i}\mathrm{j}}^{L,R}$ can be stated as
$${\xi}_{\psi ,\mathrm{i}\mathrm{j}}^{L,R}=\frac{{-\dot{\mathit{\Psi}}}_{w,\mathrm{i}\mathrm{j}}}{V}+\frac{sin\lambda}{{r}_{w,\mathrm{i}\mathrm{j}}^{L,R}}$$(31)
where *λ* is the wheel conicity, and ${r}_{w,\mathrm{i}\mathrm{j}}^{L,R}$ are the rolling radius of the left and right wheel, respectively.

The center of rotation is located on the longitudinal axis of the contact patch, but its position relies upon the equilibrium of the forces and is initally unknown. If the longitudinal semiaxis is too small, the center of spin rotation is approaching the origin of the coordinate system. Using the conversion of the tangential stress distribution ellipsoid to a hemisphere, the lateral tangential force generated by pure spin can be obtained as
$$\begin{array}{rl}{\mathrm{F}}_{\psi \mathrm{w}\mathrm{y},\mathrm{i}\mathrm{j}}^{\mathrm{L},\mathrm{R}}& =\frac{-9{\mathrm{m}}_{\mathrm{a}}.\mathrm{g}}{16}{\mu}_{\mathrm{f}}|{\u03f5}_{\mathrm{i}\mathrm{j}}^{\mathrm{L},\mathrm{R}}|\left(\frac{{({\u03f5}_{\delta ,\mathrm{i}\mathrm{j}}^{\mathrm{L},\mathrm{R}})}^{3}}{3}-\frac{{({\u03f5}_{\delta ,\mathrm{i}\mathrm{j}}^{\mathrm{L},\mathrm{R}})}^{2}}{2}+\frac{1}{6}\right)\\ & -\frac{1}{3}{\sqrt{(1-{({\u03f5}_{\delta ,\mathrm{i}\mathrm{j}}^{\mathrm{L},\mathrm{R}})}^{2})}}^{3}\end{array}$$(32)
where
$${\u03f5}_{\delta ,ij}^{L,R}=\frac{{{(\u03f5}_{ij}^{L,R})}^{2}-1}{{{(\u03f5}_{ij}^{L,R})}^{2}+1}$$
and creepage ${\xi}_{ij}^{L,R}$ defined in Eqn. 27 is converted as ${\xi}_{\psi ,\mathrm{i}\mathrm{j}}^{L,R}\times a$. However this solution is valid only for *a*→0.

The detailed solution for different relations *a/b* given by Kalker [13] represented that with an increasing relation *a/b* the force effect of the spin increases. The forces originated from the longitudinal and lateral creepages and the lateral force caused by spin creepage are evaluated seperately.

The resulting creepage is
$${\xi}_{\mathrm{i}\mathrm{j}}^{L,R}=\sqrt{{({\xi}_{x,\mathrm{i}\mathrm{j}}^{L,R})}^{2}+{({\xi}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R})}^{2}}$$(33)
where ${\xi}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}$ is given as
$$\begin{array}{rl}{\xi}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}& ={\xi}_{y,\mathrm{i}\mathrm{j}}^{L,R}+{\psi}_{w,\mathrm{i}\mathrm{j}}a\Rightarrow |{\xi}_{y,\mathrm{i}\mathrm{j}}^{L,R}+{\psi}_{w,\mathrm{i}\mathrm{j}}a|\succ |{\xi}_{y,\mathrm{i}\mathrm{j}}^{L,R}|\\ {\xi}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}& ={\xi}_{y,\mathrm{i}\mathrm{j}}^{L,R}\Rightarrow |{\xi}_{y,\mathrm{i}\mathrm{j}}^{L,R}+{\psi}_{w,\mathrm{i}\mathrm{j}}a|\le |{\xi}_{y,\mathrm{i}\mathrm{j}}^{L,R}|\end{array}$$(34)

The resulting force effect in lateral direction is demonstrated as the sum of above explained effects respecting the creep saturation as
$${F}_{\mathrm{w}\mathrm{y},\mathrm{i}\mathrm{j}}^{L,R}={F}_{\psi \mathrm{w}\mathrm{y},\mathrm{i}\mathrm{j}}^{L,R}+{\mathrm{\Delta}\mathrm{F}}_{\psi \mathrm{w}\mathrm{y},\mathrm{i}\mathrm{j}}^{L,R}$$(35)
where ${\mathrm{\Delta}\mathrm{F}}_{\psi \mathrm{w}\mathrm{y},\mathrm{i}\mathrm{j}}^{L,R}$ is the growth of the tangential force originated from the spin.
$$\begin{array}{rl}{\mathrm{\Delta}F}_{\psi wy,ij}^{L,R}& =\frac{-9}{16}\left({m}_{a}.g\right){a\mu}_{f}\\ & [|{\u03f5}_{y\psi ,ij}^{L,R}|\left(\frac{{\left({\u03f5}_{y\psi \delta ,ij}^{L,R}\right)}^{2}}{3}-\frac{{\left({\u03f5}_{y\psi \delta ,ij}^{L,R}\right)}^{2}}{2}+\frac{1}{6}\right)\\ & -\frac{1}{3}\sqrt{{\left(1-{\left({\u03f5}_{y\psi \delta ,ij}^{L,R}\right)}^{2}\right)}^{3}}]\left[1+6.3\left(1-{e}^{\frac{-a}{b}}\right)\right]\frac{{\xi}_{\psi ,ij}^{L,R}}{{\xi}_{ij}^{L,R}}\end{array}$$(36)
where
$${\u03f5}_{y\psi \delta ,ij}^{L,R}=\frac{{{(\u03f5}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R})}^{2}-1}{{{(\u03f5}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R})}^{2}+1}$$
${\u03f5}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}$ is actually given as
$${\u03f5}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}=\frac{2}{3}\frac{{C\pi a}^{2}b}{{m}_{a}{g\mu}_{f}}\frac{{\xi}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}}{1+6.3[1-{e}^{\frac{-a}{b}}]}$$(37)

The contact stiffness in Eqn. 19 and Eqn. 27 can be obtained by experiments or can be derived from Kalker’s constants [13]. The amount of the tangential contact stiffness *C* is obtained by considering a linear part of the creep-force law according to Kalker theory without spin effect.
$$C=\frac{3}{8}\frac{G}{a}{C}_{jj}$$(38)
where *C*_{jj} is the Kalker’s constants ( *C*_{11} for longitudinal direction, *C*_{22} for lateral direction).

After the substitution of Eqn. 38 in Eqn. 19 the gradient *ϵ* for tangential stress is
$${\u03f5}_{\mathrm{i}\mathrm{j}}^{L,R}=\frac{1}{4}\frac{{G\pi abC}_{jj}}{{m}_{a}{g\mu}_{f}}{\xi}_{ij}^{L,R}$$(39)

Because *C*_{11}≠*C*_{22}, constant *C*_{jj} can be obtained as follows
$${C}_{jj}=\sqrt{{\left(\frac{{C}_{11}{\overline{\xi}}_{x,ij}^{L,R}}{s}\right)}^{2}+{\left(\frac{{C}_{22}{\overline{\xi}}_{y,ij}^{-L,R}}{s}\right)}^{2}}$$(40)
where ${\overline{\xi}}_{x,ij}^{L,R}$ and ${\overline{\xi}}_{y,ij}^{L,R}$ are the mean value of the corresponding creepages.

The tangential contact stiffness related with the lateral force caused by the spin can be derived as
$${C}_{y\psi}=\frac{4}{\pi}\frac{G\sqrt{b}}{{\sqrt{a}}^{3}}{C}_{23}$$(41)

After the substitution of Eqn. 41 in Eqn. 37 the gradient of tangential stress *ϵ*_{yψδ} used for calculation of spin influence is
$${\u03f5}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}=\frac{8}{3}\frac{b\sqrt{ab}}{{m}_{a}{g\mu}_{f}}\frac{{C}_{23}{\u03f5}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}}{1+6.3(1-{e}^{\frac{-a}{b}})}$$(42)

## LuGre Model

Tangential adhesion force formulation is a complex nonlinear problem. There are different friction models in the literature, many of them are insufficient to model realistic dynamic features of the adhesion phenomenon. LuGre model [28] gives a better information of this natural phenomenon by combining experimentally detected different behaviors including stiction,that needs to be overcome to enable relative motion of stationary wheel-rail in contact, Dahl effect and Stribeck effect [29].

This model is given by the following equations
$${F}_{w,\mathrm{i}\mathrm{j}}^{L,R}=\left({\sigma}_{0}{p}_{\mathrm{i}\mathrm{j}}^{L,R}+{\sigma}_{1}\frac{d{p}_{\mathrm{i}\mathrm{j}}^{L,R}}{dt}+{\sigma}_{2}{\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}\right){F}_{z,\mathrm{i}\mathrm{j}}^{L,R}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}$$(43)
$$\frac{\mathrm{d}{p}_{\mathrm{i}\mathrm{j}}^{L,R}}{\mathrm{d}\mathrm{t}}={\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}-\frac{{\sigma}_{0}|{\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}|}{\mathrm{h}({\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R})}{p}_{\mathrm{i}\mathrm{j}}^{L,R}$$(44)
$${\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}={r}_{0}{\omega}_{w,\mathrm{i}\mathrm{j}}^{L,R}-V$$(45)
$$\mathrm{h}({\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R})={\mu}_{\mathrm{c}}+({\mu}_{\mathrm{s}}-{\mu}_{\mathrm{c}}){\mathrm{e}}^{-{|\frac{{\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}}{{v}_{s,\mathrm{i}\mathrm{j}}^{L,R}}|}^{\gamma}}$$(46)

Here, ${p}_{\mathrm{i}\mathrm{j}}^{L,R}$ is the internal friction state, *σ*_{0}, *σ*_{1}, *σ*_{2}, *γ* are model parameters, ${\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}$ is the relative velocity between peripheral wheel velocity and railway vehicle velocity, ${F}_{z,\mathrm{i}\mathrm{j}}^{L,R}$ is the normal force acting on each wheelset, ${v}_{s,\mathrm{i}\mathrm{j}}^{L,R}$ is the Stribeck velocity, *μ*_{c} and *μ*_{s} are normalized Coulomb and normalized static friction coefficients, respectively.

## Deriving the dynamic friction model parameters

Despite of providing better description of adhesion phenomenon, LuGre approach is hard to be applied since it requires comprehensive experiments to determine model parameters. There are simpler static friction models in the literature which give steady-state characteristics of adhesion successfully. Polach Model [30] is one of these railway specific models, which has been verified with thorough experiments. Moreover, parameters of Polach Model can be easily determined using geometrical and mass properties of the railway vehicle. By reason of quasi–static form of LuGre model contains all parameters of the dynamic form, nonlinear optimization techniques to match Polach and Kalker model with the quasi–static form of the LuGre model can be used.

The distributed approach for LuGre model is necessitating to obtain quasi-static equation. Distributed model seperates the contact patch between wheel and rail into slices. Each of these slices is defined by their specific differential internal friction states *δp* and differential normal forces *δF*_{z}. Assuming that normal force is uniformly distributed and relative velocities of all slices are equal, tangential adhesion force can be given by the following equation 31.
$$\begin{array}{rl}& {F}_{w,\mathrm{i}\mathrm{j}}^{L,R}=sgn\left({\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}\right)h\left({\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}\right)\\ & {F}_{z,\mathrm{i}\mathrm{j}}^{L,R}(1+(1-\frac{{\sigma}_{1}|{\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}|}{h({\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R})})\frac{h({\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R})}{{\sigma}_{0}A|{s}_{w,\mathrm{i}\mathrm{j}}^{L,R}|}({e}^{-\frac{{\sigma}_{0}A|{s}_{w,\mathrm{i}\mathrm{j}}^{L,R}|}{h({\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R})}}-1))+{F}_{z,\mathrm{i}\mathrm{j}}^{L,R}{\sigma}_{2}{\nu}_{r,\mathrm{i}\mathrm{j}}^{L,R}\end{array}$$(47)
where ${s}_{w,\mathrm{i}\mathrm{j}}^{L,R}$ is the slip ratio and *A* is the length of the contact patch.

Since all four axles of the railway vehicle have a traction motor, it is not possible to measure velocity of the vehicle using wheel velocities of the trailing axles. Parameter determination procedure can be summarized as

1) Parameters of Polach and Kalker model are evaluated using geometrical and mass properties of the railway vehicle

2) LuGre model parameters are determined using genetic optimization method, which it’s objective is to minimize the error between the output of the Polach and Kalker model and quasi-static LuGre model for specific operating conditions.

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