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# Nonlinear Engineering

### Modeling and Application

Editor-in-Chief: Nakahie Jazar, Gholamreza

Managing Editor: Skoryna, Juliusz

CiteScore 2018: 1.33

SCImago Journal Rank (SJR) 2018: 0.313
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Volume 6, Issue 2

# Nonlinear Dynamic Characteristics of the Railway Vehicle

Çağlar Uyulan
• Corresponding author
• Istanbul Technical University, Graduate School of Science, Engineering and Technology, Department of Mechatronics Engineering, Istanbul, Turkey
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• Other articles by this author:
/ Metin Gokasan
• Istanbul Technical University, Faculty of Electrical and Electronic Engineering, Control Engineering Department, Istanbul, Turkey
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Published Online: 2017-02-18 | DOI: https://doi.org/10.1515/nleng-2016-0070

## Abstract

The nonlinear dynamic characteristics of a railway vehicle are checked into thoroughly by applying two different wheel-rail contact model: a heuristic nonlinear friction creepage model derived by using Kalker ’s theory and Polach model including dead-zone clearance. This two models are matched with the quasi-static form of the LuGre model to obtain more realistic wheel-rail contact model. LuGre model parameters are determined using nonlinear 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. The symmetric/asymmetric bifurcation attitude and stable/unstable motion of the railway vehicle in the presence of nonlinearities which are yaw damping forces in the longitudinal suspension system are analyzed in great detail by changing the vehicle speed. Phase portraits of the lateral displacement of the leading wheelset of the railway vehicle are drawn below and on the critical speeds, where sub-critical Hopf bifurcation take place, for two wheel-rail contact model. Asymmetric periodic motions have been observed during the simulation in the lateral displacement of the wheelset under different vehicle speed range. The coexistence of multiple steady states cause bounces in the amplitude of vibrations, resulting instability problems of the railway vehicle. By using Lyapunov’s indirect method, the critical hunting speeds are calculated with respect to the radius of the curved track parameter changes. Hunting, which is defined as the oscillation of the lateral displacement of wheelset with a large domain, is described by a limit cycle-type oscillation nature. The evaluated accuracy of the LuGre model adopted from Kalker’s model results for prediction of critical speed is higher than the results of the LuGre model adopted from Polach’s model. From the results of the analysis, the critical hunting speed must be resolved by investigating the track tests under various kind of excitations.

## 1 Introduction

The system dynamic of the railway vehicles have been evolved into much more complex states with the developing high-speed railway technology. From the viewpoint of the performance, it has become very essential to examine the lateral stability features of the railway vehicle [23]. The accurate estimation of the critical speed of a railway vehicle running on a specific track in the design state is also an important issue must be solved [24]. This critical speed directly limit the maximum admissible operating speed of the railway vehicle and is a nonlinear function of various conditions such as wheel-rail contact profile, the adhesion coefficient, suspension dynamics of the bogie, topology of the railway line etc. Above this certain forward speed, an undamped motion of the wheelset restricted by the wheel flange and rail emerges and can cause high lateral creep forces at the wheel-rail contact, that might be concluded with a derailment situation [25].

Many researches have been conducted about the subject of the bifurcation behaviours and chaotic motion in the railway vehicle systems. Kaas-Petersen et al. [21, 22] studied the periodic, biperiodic, and chaotic dynamical behavior of the railway vehicle. The chaotic motions take place when a flange force, that was designed as a spring and flange with a dead band, was used and the whole system jumps into an asymmetric reciprocating motion condition by experiencing a bifurcation, which breaks the symmetric state. The non-smooth contact of the flange-rail induces the quasi-periodic and chaotic motion of the railway vehicle. The Hopf bifurcation and hunting instability of the wheelset has been investigated by using Floquet multiplier in [26]. Gao et al. [27] analyzed the lateral bifurcations of a four-axle railway passenger car within a large speed interval results in the chaotic motions, which express period doubling or quasi-periodic motion characteristic. The wheel-rail contact models, which their contact relations are raw approximation of the realistic wheel-rail contact model, were implemented to examine the wheel-rail contact forces. Hopf bifurcation behavior of a railway wheelset in the presence of dead-zone and yaw damper nonlinearities has been investigated using Bogoliubov-Mitropolsky averaging method [32] and two-dimensinal bifurcation diagrams to determine the influences of the various parameters into critical hunting speed, are depicted in [33]. As a result of this study, it has been found that, yaw stiffness has a major effect on critical hunting speed. Yang and Shen analyzed the Hopf bifurcation and hunting stability of a bogie system with hysteretic and nonlinear suspensions [34]. A symmetric/asymmetric motion differs from chaotic, periodic or quasi-periodic motions. The vehicle models are designed as a multi-body with multi-DOF and set up with left/right or front/back symmetry. The whole system is left or right symmetric under the condition that a left or right symmetric vehicle operates on straight and smooth path. When a left/right symmetric vehicle operates on smooth and straight track, the whole system is also left/right symmetric [35].

Accurate modeling of the wheel-rail contact greatly affects contact forces, vehicle dynamics, contact surfaces wear and vehicle safety. Degraded adhesion has become an obvious problem with wheel-rail contact. Achieving a realistic adhesion model is very difficult due to the complex and highly nonlinear behavior of the adhesion coefficient and the presence of unknown external contaminants. The contact model involving the realistic adhesion model should guarantee good accuracy and high numerical efficiency at the same time so that the tools can be used directly online in the general multi-body model.

Meli and Ridolfi [39], analyzed an adhesion model based on the principal factors that characterize the degraded adhesion (sliding/slipping at the contact interface, high energy dissipation, adhesion recovery, etc.), and this model is confirmed by experimental data. Numerical performances (calculation times and memory consumption) are very important in multi-body applications. For this reason, in practice, it is often impossible to model the wheel-rail contact as an elastic continuous body. Multi-bodied contact models are characterized by three main stages: tangential problem solving, which includes contact point detection, normal problem solving, and the adhesion model. The most important methods include linear Kalker theory saturated according to Johnson-Vermeulen formula [42], the non-linear Kalker theory implemented in the FASTSIM algorithm, and Polach theory [30], which describe the diminishing adhesion coefficient with respect to the growing creepage.

A simple wear model, which was embedded into a friction versus-time model has been investigated in [40]. As a result of this research, kinetic friction behaviors have been observed to change over time under both dry and boundary-lubricated conditions and it has been proven that wear contributes to such behavior. In such cases, wear models can be placed into the friction models.

The presence of contaminants significantly reduces the existing friction between the wheel-rail surfaces, which is a limitation for the tangential forces transmitted in the contact patch. Meli et al. [41], developed a new wheel-rail friction law that takes into account the non-linear behavior of the traction for very large creeps. The proposed friction law has been tested on brake tests with the WSP system (Wheel Slide Protection) under degraded adhesion conditions.

In this paper, complex and realistic quasi-static nonlinear wheel-rail contact models are compared in the investigation of the symmetric/asymmetric bifurcation attitude and stable/unstable motion of the railway vehicle in the presence of nonlinearities which are yaw damping forces in the longitudinal suspension system are analyzed completely by changing the vehicle speed. The main superiority of this paper is that, the more realistic approximation of the wheel-rail contact has been developed to investigate the critical bifurcation speed in a railway vehicle system. An 35-DOF railway vehicle model considering lateral, vertical, rolling, yawing, pitching motions has been used in simulations. The models have been built by considering all of the possible design criteria from the viewpoint of hunting stability in the literature to predict the critical hunting speed more accurately. The results obtained from the simulation will be improved and validated by using more realistic wheel-rail adhesion model in the real-time track tests. It is possible to encounter various kind of chaotic, symmetric/asymmetric periodic solutions, bifurcations and changes to different solution branches by using these two wheel-rail contact model.

## 2 Nonlinear dynamical model

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 ( Ivbx, Ivby, Ivbz, mvb) supported by two bogie frames ( Ibx,1, Ibx,2, Iby,1, Iby,2, Ibz,1, Ibz,2, mb,1, mb,2) with the secondary suspensions ( Ksx, Ksy, Ksz, Csx, Csy, Csz) in longitudinal, lateral and vertical directions, and two wheelsets ( mw, Iwx, Iwy, Iwz), which are mounted with each bogie frame with primary suspensions ( Kpx, Kpy, Kpz, Cpx, Cpy, Cpz) 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.

bc1, bc3 are half of the secondary longitudinal and vertical spring arms, bc2, bc4 are half of the secondary longitudinal and vertical damper arms, bt1, bt3 are half of the primary longitudinal and vertical spring arms, bt2, bt4 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, h0 is the height of the secondary suspension above the bogie frame mass center, hG is the height of the bogie mass center above the wheelset mass center, Lc is the distance between the vehicle body and the bogie frame mass center, Lt1, Lt2 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].

$mvby¨vb=2Fsyvb+mvbgϕse−mvbV2Ry$(1) $mb,iy¨b,i=Fsyb,i+(mb,i+mvb2)V2Ry−gϕse(mb,i+mvb2)$(2) $IvbzΨ¨vb=Mszvb$(3) $Ibz,iΨ¨b,i=Mszb,i$(4) $mvbz¨vb=Fszvb−mvbg−mvbV2Ry$(5) $mb,iz¨b,i=Fszb,i−mb,ig−mb,iV2Ry$(6) $Ivbxϕ¨vb=Msxvb$(7) $Ibx,iϕ¨b,i=Msxb,i$(8) $Ivbyγ¨vb=Msyvb$(9) $Iby,iγ¨b,i=Msyb,i$(10) $mwy¨w,ij=mwV2Ry−mwgϕse+FwLy,ijn+FwRy,ijn+Nwy,ijL+Nwy,ijR+Fsyw,ij−Fwt,ij+V2WextgRy−Wextϕse$(11) $IwzΨ¨w,ij=−Iwy(Vr0+Vsin⁡(ϕse)Ry)ϕ˙w,ij+Rx,ijRFwRy,ijn−Ry,ijRFwRx,ijn+Rx,ijLFwLy,ijn−Ry,ijLFwLx,ijn+Rx,ijRNwy,ijR+Rx,ijLNwy,ijL+MwLz,ijn+MwRz,ijn+Mszw,ij$(12) $mwz¨w,ij=−mwV2ϕseRy−mwg+Fwz,ijL+Fwz,ijR+Nwz,ijR+Nwz,ijL+Fszw,ij$(13) $Iwxϕ¨w,ij=−IwyV(VRy−Ψ˙w,ij)r0+Ry,ijRFwz,ijR−Rz,ijRFwRy,ijn+Ry,ijLFwz,ijL−Rz,ijLFwLy,ijn+Ry,ijLNwz,ijL+Ry,ijRNwz,ijR−Rz,ijRNwy,ijR−Rz,ijLNwy,ijL+Mwx,ijL+Mwx,ijR+Msxw,ij$(14) $Iwyγ¨w,ij=Rz,ijRFwRx,ijn+Rz,ijLFwLx,ijn−Rx,ijRFwz,ijR−Rx,ijRNwz,ijR−Rx,ijLFwz,ijL−Rx,ijLNwz,ijL+Mwy,ijL+Mwy,ijR+Msyw,ij$(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, Ry is the radius of the curved track, r0 is the nominal wheelset rolling radius, yvb, zvb, ψvb, ϕvb, γvb are lateral and vertical displacements, and yaw, roll and pitch angles of the vehicle body, yb,i, zb,i, ψb,i, ϕb,i, γb,i are lateral and vertical displacements, and yaw, roll and pitch angles of the bogies, yw,ij, zw,ij, ψw,ij, ϕw,ij, γw,ij are lateral, vertical displacements and yaw,roll, pitch angles of the wheelsets, ${\stackrel{˙}{y}}_{w,ij},{\stackrel{˙}{z}}_{w,ij},{\stackrel{˙}{\mathrm{\Psi }}}_{w,ij},{\stackrel{˙}{\varphi }}_{w,ij},{\stackrel{˙}{\gamma }}_{w,ij}$ 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, Wext is the external load on the wheelset axles, g is the gravitational constant, ϕse is the cant angle, mvb, mb, 1 mb,2 are vehicle body, first and second bogie frame masses, mw is the wheelset mass, Fwt,ij is the flange contact force, Ivbx, Ib,xi, Ivby, Ib,yi, Ivbz, Ib,zi are roll, pitch, and yaw moments of inertia of the vehicle body and bogies, Iwx, Iwy, Iwz are roll, pitch, and yaw moment of inertia of the wheelsets, Kpx, Kpy, Kpz are primary longitudinal, lateral, and vertical stiffness, Ksx, Ksy, Ksz are secondary longitudinal, lateral, and vertical stiffness, Cpx, Cpy, Cpz are primary longitudinal, lateral, and vertical damping coefficients, Csx, Csy, Csz are secondary longitudinal, lateral, and vertical damping coefficients. Fsyvb, Fszvb, Msxvb, Msyvb, Mszvb are lateral, vertical suspension forces and longitudinal, lateral, vertical suspension moments of the vehicle body, Fsyb,i, Fszb,i, Msxb,i, Msyb,i, Mszb,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, Fsyw,ijFszw,ij are lateral and vertical suspension forces of the wheelset, Msxw,ij, Msyw,ij, Mszw,ij are suspension moments of the wheelset in the longitudinal, lateral, and vertical directions, respectively.

The nonlinear longitudinal yaw damping forces Fd,ij are included in Mszw,ij and are considered to the Yang and Ahmadian [8] conjecture as follows: $Fd,ij={C1Vx,ijψ+C2(Vx,ijψ)2+C3(Vx,ijψ)3+C4(Vx,ijψ)4⇒Vx,ijψ≻0C1Vx,ijψ−C2(Vx,ijψ)2+C3(Vx,ijψ)3−C4(Vx,ijψ)4⇒Vx,ijψ≺0}$(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 ( Fd,ij) are included in vertical suspension moment ( Mszw,ij). These damping forces can be expressed by fourth-order polynomials of ${V}_{x,ij}^{\psi }$. The coefficients C1 to C4 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 $Md,ij=Fd,ijbti$(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 (Kry, Krz). 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: $Fwt,ij=Kry(yw,ij−δ)[12tanh⁡{104(yw,ij−δ)}+12]+Kry(yw,ij+δ)[12tanh⁡{104(−yw,ij−δ)}+12]$(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 $[Fwx,ijL,RFwy,ijL,RMwz,ijL,R]=G.a.b.[−C11000−C22a.b.C230−a.b.C23a.b.C33][ξx,ijL,Rξy,ijL,Rξψ,ijL,R]$(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 Cij 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 $αij=1βij(βij−13βij2+127βij3)[12tanh{104(3−βij)}+12]+1βij[12tanh{104(βij−3)}+12]$(20) $βij=βijR+βijL2$ $βijL,R=(Fwx,ijL,R)2+(Fwy,ijL,R)2μ(Nwy,ijL,R)2+(Nwz,ijL,R)2,$ $(Nwy,ijL,R)2+(Nwz,ijL,R)2=Nw,ijL,R$ where αij ise the saturation constant in the heuristic creep model, βij 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]. $FwLx,ijn=αijFwx,ijL$(21) $FwRx,ijn=αijFwx,ijR$(22) $FwLy,ijn=αijFwy,ijL$(23) $FwRy,ijn=αijFwy,ijR$(24) $MwLz,ijn=αijMwz,ijL$(25) $MwRz,ijn=αijMwz,ijR$(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 $ϵijL,R=2.C.π.a2.b3.ma.g.μfξijL,R$(27) where; ma is the adhesion mass of the wheel, μf is the friction coefficient.

The tangential force is defined as $Fwτ,ijL,R=−2magμfπ(ϵijL,R1+(ϵijL,R)2+arctan(ϵijL,R))$(28)

The creep forces ${F}_{wx,ij}^{L,R},\phantom{\rule{thinmathspace}{0ex}}{F}_{wy,ij}^{L,R}$ are evaluated as [12] $Fwk,ijL,R=Fwτ,ijL,Rξk,ijL,RξijL,R,k=x,y$(29) where $ξijL,R=(ξx,ijL,R)2+(ξy,ijL,R)2$(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 $ξψ,ijL,R=−Ψ˙w,ijV+sinλrw,ijL,R$(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 $Fψwy,ijL,R=−9ma.g16μf|ϵijL,R|((ϵδ,ijL,R)33−(ϵδ,ijL,R)22+16)−13(1−(ϵδ,ijL,R)2)3$(32) where $ϵδ,ijL,R=(ϵijL,R)2−1(ϵijL,R)2+1$ and creepage ${\xi }_{ij}^{L,R}$ defined in Eqn. 27 is converted as ${\xi }_{\psi ,\mathrm{i}\mathrm{j}}^{L,R}×a$. 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 $ξijL,R=(ξx,ijL,R)2+(ξyψ,ijL,R)2$(33) where ${\xi }_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}$ is given as $ξyψ,ijL,R=ξy,ijL,R+ψw,ija⇒|ξy,ijL,R+ψw,ija|≻|ξy,ijL,R|ξyψ,ijL,R=ξy,ijL,R⇒|ξy,ijL,R+ψw,ija|≤|ξy,ijL,R|$(34)

The resulting force effect in lateral direction is demonstrated as the sum of above explained effects respecting the creep saturation as $Fwy,ijL,R=Fψwy,ijL,R+ΔFψwy,ijL,R$(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. $ΔFψwy,ijL,R=−916(ma.g)aμf[|ϵyψ,ijL,R|((ϵyψδ,ijL,R)23−(ϵyψδ,ijL,R)22+16)−13(1−(ϵyψδ,ijL,R)2)3][1+6.3(1−e−ab)]ξψ,ijL,RξijL,R$(36) where $ϵyψδ,ijL,R=(ϵyψ,ijL,R)2−1(ϵyψ,ijL,R)2+1$ ${ϵ}_{\mathrm{y}\psi ,\mathrm{i}\mathrm{j}}^{L,R}$ is actually given as $ϵyψ,ijL,R=23Cπa2bmagμfξyψ,ijL,R1+6.3[1−e−ab]$(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. $C=38GaCjj$(38) where Cjj is the Kalker’s constants ( C11 for longitudinal direction, C22 for lateral direction).

After the substitution of Eqn. 38 in Eqn. 19 the gradient ϵ for tangential stress is $ϵijL,R=14GπabCjjmagμfξijL,R$(39)

Because C11C22, constant Cjj can be obtained as follows $Cjj=(C11ξ¯x,ijL,Rs)2+(C22ξ¯y,ij−L,Rs)2$(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 $Cyψ=4πGba3C23$(41)

After the substitution of Eqn. 41 in Eqn. 37 the gradient of tangential stress ϵyψδ used for calculation of spin influence is $ϵyψ,ijL,R=83babmagμfC23ϵyψ,ijL,R1+6.3(1−e−ab)$(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 $Fw,ijL,R=(σ0pijL,R+σ1dpijL,Rdt+σ2νr,ijL,R)Fz,ijL,R$(43) $dpijL,Rdt=νr,ijL,R−σ0|νr,ijL,R|h(νr,ijL,R)pijL,R$(44) $νr,ijL,R=r0ωw,ijL,R−V$(45) $h(νr,ijL,R)=μc+(μs−μc)e−|νr,ijL,Rvs,ijL,R|γ$(46)

Here, ${p}_{\mathrm{i}\mathrm{j}}^{L,R}$ is the internal friction state, σ0, σ1, σ2, 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, μc and μs 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 δFz. 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. $Fw,ijL,R=sgn(νr,ijL,R)h(νr,ijL,R)Fz,ijL,R(1+(1−σ1|νr,ijL,R|h(νr,ijL,R))h(νr,ijL,R)σ0A|sw,ijL,R|(e−σ0A|sw,ijL,R|h(νr,ijL,R)−1))+Fz,ijL,Rσ2νr,ijL,R$(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.

## 3.1 Hopf Bifurcation

The bifurcation and stability theories related to the nonlinear dynamical system should be applied to examine the railway vehicle behaviour. The dynamic properties such as multiple solution branches, symmetric/asymmetric vibrations, limit cycle behavior are demonstrated in terms of bifurcation diagrams or phase portraits. Two types of bifurcation characteristic, which is observed in railway vehicles, are depicted in Fig. 2a and 2b. The point Q in the Fig. 2a is the super-critical Hopf bifurcation point. The Hopf bifurcation point can be estimated from the eigenvalues of the Jacobian matrix of the system. If the speed exceeds the point Q at critical speed, the railway vehicle becomes unstable, whose amplitude is directly proportional with the increasing speed of the vehicle, reveals. The point Q in the Fig. 2b is the sub-critical Hopf bifurcation point. The unstable periodic solution is recognized at the point Q. The amplitude of the unstable periodic solution increases with decreasing speed from the point of Q to the saddle-node bifurcation point R. After that, the unstable periodic solution evolves into stable condition and its amplitude become larger with increasing vehicle speed. The point R characterizes the critical speed of the railway vehicle [17].

Fig. 2

Hopf Bifurcation forms for railway vehicle a) Super-critical b) Sub-critical

The state vector of the system can be designated by x(t). Then the equations of motion of the railway vehicle from Eqn. 1 to Eqn. 15 is converted to an initial value problem for a system of first order ordinary differential equations with respect to time t (t ≥ 0). $dx¯dt=f(x¯,V)$(48) where f is a vector function of the state vector of the system and V is the railway vehicle speed denoted as the control parameter. The system was analyzed in a numerical way because of the nonlinear and non-smooth wheel-rail interactions and high order.

The non-smoothness is originated from the flange contact when the lateral displacements of the wheelset arrives at the flange clearance. The continuation approximation can be applied to examine the stationary and periodic solutions [14].

The symmetry properties is also essential to be determined, since the wear, which is falling to one side, is most likely to occur and it may cause to chaotic motion at higher speeds, when the asyymetric condition exists.

Oscillation is one of the most important phenomena that occur in dynamical systems. A system vibrates when it has a nontrivial periodic solution. $x(t+T)=x(t),∀t≥0$(49) for some T > 0.

"Nontrivial" term is used to exclude constant solutions corresponding to equilibrium points. The image of a periodic solution in the phase portrait is a closed trajectory, which is usually called a periodic orbit or a closed orbit [8]. The oscillation of nonlinear systems due to a center equilibrium point is not structurally stable. Infinitesmally perturbations may change the type of the equilibrium point to a stable focus (decaying oscillation) or unstable focus (growing oscillation).

The pseudo-arclength continuation and Newton’s iteration can also be applied to track the progression of the stationary and periodic solution [15]. The eigenvalues of the Jacobian matrix and the Floquet multipliers are also evaluated in order to discover the stability of the stationary and periodic solutions. In this paper, the bifurcation diagram were constructed via increasing or decreasing the vehicle speed to investigate the periodic and aperiodic solutions. An implicit and variable-order differential equation solver "ode15s function" in Matlab for the numerical integrations based on the backward difference of the Klopfenstein-Shampine formulas were used to resolve this problem [16]. The absolute and relative errors are selected as 10 –7 for acceptable computational time and accurate results.

## 3.2 Lyapunov’s indirect method to determine the critical speed

The dynamic equations of motion are inherently nonlinear, which avoids applying a variety of linear theories. In order to evaluate the critical hunting velocity, the dynamic equation of motion were transformed into the state space form as $x˙(t)=F[x(t)]$(50) where x(t) denotes 70 dimensional state vector. F[x(t)] is also same dimensional nonlinear vector, that is a function of the state variables [36, 37]. Equilibrium points can be found by solving the nonlinear algebraic from Eqn. 1 to Eqn. 15. The right side of Eqn. 50 can be expanded in a Taylor series at the equilibrium point. As a result the linearized version can be represented by the following matrix/vector forms: $x~˙=Ax~(t)$ where AJ[x(t)]x = xe, J is the Jacobian matrix evaluated at the equilibrium point from the linearized system matrix.

The hunting stability can be evaluated by solving the eigenvalue problem for the autonomous system. If all eigenvalues of A matrix are strictly in the left-half complex plane, then the equilibrium point is asymptotically stable for the actual nonlinear system. If all eigenvalues of A are in the left-half complex plane, but at least one of them is on the imaginary axis, then one cannot conclude anything from the linear approximation, this is the marginally stable case. If at least one eigenvalue of A is strictly in the right-half complex plane, then the equilibrium point is unstable for the nonlinear system [38].

## 4 Results

In this part, the dynamics of a high-speed railway vehicle for the speed range 20 ≤ V ≤ 90 m/s. The forward speed of the railway vehicle was initiated from V = 20 m/s and was increased with small steps of 0.1 m/s. Every simulations at these control speeds were conducted during 20 second. The parameter values were given in Appendix I. The coefficient of adhesion parameter has a great influence on the wheel-rail creep forces, vehicle stability and it can not be controlled as well. The coefficient of adhesion can be estimated through the Kalman filtering algorithm or disturbance observers by using experimental data [18, 19]. In this paper, the value of the coefficient of adhesion parameter was selected as μ = 0.2 and the radius of the curved track parameter was selected as Ry = 6500 m.

There is an asymtotically stable solution until the speed reaches to V = 39.5 m/s for Kalker’s model and V = 41.3 m/s for Polach’s model represented in Fig. 3. When the speeds regularly increase from this speed values, then the trivial solutions lose their stability, and the amplitude of the periodic motions of the leading wheelset increase.

Fig. 3

(a) The phase portrait diagram for the first leading wheelset showing its lateral displacement at V = 39.5 m/s and V >39.6 m/s in the Kalker’s model (b)The phase portrait diagram for the first leading wheelset showing its lateral displacement at V = 41.3 m/s and V > 41.4 m/s in the Polach’s model.

In Fig. 4, the time series of the leading wheelset of the railway vehicle were shown for both adhesion models at V=39.5 m/s – V=39.6 m/s and V=41.3 m/s – V=41.4 m/s, respectively. Time series of the lateral displacements of the leading wheelset of the railway vehicle at V=39.5 m/s – V=39.6 m/s and V=41.3 m/s – V=41.4 m/s are with the same initial conditions. The leading wheelset of the railway vehicle vibrates periodically with the constant amplitude as a function of time, while the lateral displacements of the the wheelsets jump to a higher value at V=39.6 m/s and V=41.4 m/s, respectively. The oscillations are not desired for the passenger comfort and safe operation. Therefore the non-stationary behavior must be suppressed in the normal operation of the railway vehicles.

Fig. 4

The time series of the leading wheelset of the railway vehicle at V=39.5 m/s and V=39.6 m/s in the Kalker’s model (b) the timeseries of the leading wheelset of the railway vehicle at V=41.3 m/s and V=41.4 m/s in the Polach’s model.

There is a transition zone between V=62.2 m/s – V=63.4 m/s and V ≤78.1 m/s – V ≤78.9 m/s speed. The amplitudes of the limit cycle decay continuously between these speed zones, which is shown in Fig. 5.

Fig. 5

(a) The phase portrait diagram for the first leading wheelset showing its lateral displacement at V=62.2 m/s and V≤78.1 m/s in the Kalker’s model (b)The phase portrait diagram for the first leading wheelset showing its lateral displacement at V=63.4 m/s and V≤78.9 m/s in the Polach’s model.

The leading wheelset of the railway vehicle oscillate periodically with the constant amplitude as a function of time, while the lateral displacements of the the wheelsets start decaying to a lower values from V=62.2 m/s and V=63.4 m/s, respectively and this kind of motion was illustrated in Fig. 6.

Fig. 6

(a) The timeseries of the leading wheelset of the railway vehicle at V=62.2 m/s and V=78.1 m/s in the Kalker’s model (b) the timeseries of the leading wheelset of the railway vehicle at V=63.4 m/s and V=78.9 m/s in the Polach’s model

In Fig. 7, the amplitude of the unstable periodic solution of the leading wheelset become larger very quickly with increasing vehicle speed, an aperiodic-chaotic motion progress and periodic solutions of the railway vehicle grow rapidly with increasing speed. When the forward vehicle speed increases from V>84.7 m/s and V>89.4 m/s, the symmetric attractor vanishes and the motion jump to asymmetric chaotic attractor.

Fig. 7

(a) The phase portrait diagram for the first leading wheelset showing its lateral displacement at V=84.6 m/s and V>84.7 m/s in the Kalker’s model (b)The phase portrait diagram for the first leading wheelset showing its lateral displacement at V=89.3 m/s and V>89.4 m/s in the Polach’s model.

From the speeds 84.7 m/s and 89.4 m/s, an unstable periodic solutions, which was shown in Fig. 8, occur. Their amplitudes grow from zero to a certain values and stationary solutions lose their stability in a Hopf bifurcation. The critical speeds of the railway vehicle are V=84.6 m/s and V=89.3 m/s for two adhesion models, respectively. It should be considered as the highest permitted speed for the railway vehicle design criteria. In Fig. 9, the critical hunting speed variations with respect to the radius of the curved track are evaluated in Kalker’s and Polach’s model, respectively.

Fig. 8

(a) The timeseries of the leading wheelset of the railway vehicle at V=84.6 m/s and V>84.7 m/s in the Kalker’s model (b) the timeseries of the leading wheelset of the railway vehicle at V=89.3 m/s and V>89.4 m/s in the Polach’s model.

Fig. 9

Critical hunting speed variations with respect to the radius of the curved track

When the results of the Kalker and Polach models are compared, it can be deduced that the Kalker model is more sensitive in capturing the critical speeds. In addition, It can be seen hat the critical hunting speed is increasing when the radius of the curved track increases.

The wheel-rail contact model with the degraded adhesion model is compatible for multi-body applications in railway vehicle dynamics. High computational performances are required in these studies. The data corresponding to the parameters of CPU and differential equation (ODE) solvers used in the numerical simulations are shown in Table 1. Simulation times for the 3D multi-bodied model and wheel-rail contact model have been measured to verify the computational efficiency of the adhesion model. The calculation times shown in Table 2 belong to the 3D multi-bodied model, the 3D wheel-rail contact model and the full coupled model. The calculation times of two different advanced contact models applied to this paper are compared.

Table 1

CPU data and integration parameters.

Table 2

Computation times of the different wheel-rail contact models.

## 5 Conclusion

Nonlinaer lateral dynamic characteristics of railway vehicle was analyzed by using Kalker’s and Polach’s wheel-rail contact models. The transition zones from low to high amplitude periodic motion and vice versa have been observed in the phase diagrams of two different adhesion model. Flange contact model consist of a very stiff linear spring with a dead band, which leads to a non-smooth behaviour and causes occurrence of chaotic domains. The sub-critical Hopf bifurcations from the stationary solution causes to two different unstable periodic solution at V=84.6 m/s for Kalker’s model and V=89.3 m/s for Polach’s model. The coexistence of multiple steady states generate bounce in the vibration amplitude. Critical hunting speeds are calculated with respect to the different radius of the curved track through Lyapunov’s indirect method and the results are compared with the simulations by using the contribution of the two nonlinear and complex adhesion models. The critical speed in the Polach’s model is higher than that derived from the Kalker’s model. The non-linear dynamic behaviour investigated by modelling the railway vehicle with the Kalker’s method is more complex and accurate than that the Polach’s method. The critical speeds were evaluated by using a comprehensive numerical examinations of the vehicle model and will be examined under more realistic wheel-rail contact model and be validated by on-track tests for future studies.

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## Appendix I

Table 3

Parameters and their numerical values of the railway vehicle [6].

Accepted: 2017-01-14

Published Online: 2017-02-18

Published in Print: 2017-06-27

Citation Information: Nonlinear Engineering, Volume 6, Issue 2, Pages 123–137, ISSN (Online) 2192-8029, ISSN (Print) 2192-8010,

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