Nonlinear DynDOI 10.1007/s11071-012-0460-9

O R I G I NA L PA P E R

The “resultant bifurcation diagram” method and itsapplication to bifurcation behaviors of a symmetric railwaybogie system

Xue-Jun Gao · Ying-Hui Li · Yuan Yue

Received: 24 November 2011 / Accepted: 9 May 2012© Springer Science+Business Media B.V. 2012

Abstract The concept of symmetric bifurcation fora symmetric wheel-rail system is defined. After that,the time response of the system can be achieved bythe numerical integration method, and an unfixed anddynamic Poincaré section and its symmetric sectionfor the symmetric wheel-rail system are established.Then the ‘resultant bifurcation diagram’ method isconstructed. The method is used to study the symmet-ric/asymmetric bifurcation behaviors and chaotic mo-tions of a two-axle railway bogie running on an idealstraight and perfect track, and a variety of character-istics and dynamic processes can be obtained in theresults. It is indicated that, for the possible sub-criticalHopf bifurcation in the railway bogie system, the sta-ble stationary solutions and the stable periodic solu-tions coexist. When the speed is in the speed rangeof Hopf bifurcation point and saddle-node bifurcationpoint, the coexistence of multiple solutions can causethe oscillating amplitude change for different kindsof disturbance. Furthermore, it is found that there aresymmetric motions for lower speeds, and then the sys-tem passes to the asymmetric ones for wide ranges

X.-J. Gao (�)College of Environment and Civil Engineering, ChengduUniversity of Technology, Chengdu 610059, Sichuan,Chinae-mail: gaoxj3000@sina.com.cn

Y.-H. Li · Y. YueSchool of Mechanics and Engineering, Southwest JiaotongUniversity, Chengdu 610031, Sichuan, China

of the speed, and returns again to the symmetric mo-tions with narrow speed ranges. The rule of symmetrybreaking in the system is through a blue sky catastro-phe in the beginning.

Keywords Railway bogie · The ‘resultant bifurcationdiagram’ method · Symmetry/asymmetry ·Bifurcation

1 Introduction

With the gradual increasing of the forward speed, rail-way vehicle dynamics become more and more impor-tant to vehicle design, experiment and operation, etc.The lateral stability [1–4] is one of the very impor-tant aspects of the vehicle performance since it in-volves the determination of the critical speed, whichdirectly affects the maximum allowed running speedof a train. High-speed trains need to have a higher crit-ical speed. If the train loses its stability, the huntingmotion will appear in the operation. The severe hunt-ing motion will deteriorate the running quality of thetrain, reduce ride comfort, lead to strong interactionsbetween wheels and rails, and even cause derailmentin a major accident. From the point of view of dy-namics, the hunting movement of the vehicle may boildown to the motion stability and bifurcation problemof dynamical systems. The hunting phenomenon of avehicle is actually a stable periodic solution in its cor-responding dynamical system, while the computation

X.-J. Gao et al.

of the critical speed of a vehicle is a determination ofthe lowest speed where the stable periodic motion ex-ists, or a determination of the highest speed where thestationary solution is a unique and asymptotically sta-ble steady-state solution. In other words, it is a processof the determination of the bifurcation point. In ev-eryday operation, passenger cars usually do not run atspeeds which are higher than the critical speed. It may,however, happen that the critical speed has decreasedbelow the operating speed due to heavily worn wheelprofiles or other reasons, so it is of great importanceto know what can happen at speeds which are higherthan the critical speed.

Many studies have been carried out at home andabroad in the research of bifurcation behaviors andchaotic motions in railway vehicle system. Cooper-rider [5] first formulated the dynamical system wherethe revised longitudinal and lateral creep forces weretaken into account but spin creep forces were not con-sidered to describe the lateral dynamics of a bogie of arailway passenger car running on an ideal straight andperfect track. He also studied the hunting behaviorsof the bogie and described the effects of flange con-tact, wheel slip and coulomb friction with nonlinearexpressions. Kaas-Petersen et al. [6] studied the lat-eral motions of the Cooperrider bogie with and with-out flange forces, they found that the system turns intoasymmetric periodic motion states through symmetry-breaking bifurcations when the wheels and rails haveno flange contact, while the chaotic motions also ex-ist in vehicle systems when the wheels and rails haveflange contacts. On the basis of the Cooperrider bogie,True et al. [7, 8] established their own research modeland studied its bifurcation behaviors. They found thatthere are some asymmetric motions at some velocitiesin the symmetric railway bogie system. They also an-alyzed the Neimark bifurcations, saddle-node bifurca-tions and pitchfork bifurcations appearing in the sys-tem and discussed the mode interactions near a degen-erate bifurcation. Later, Zeng [9] presented a combi-nation of QR algorithm and golden section methodto evaluate the Hopf bifurcation point, and then ap-plied the shooting method to determine the limit cyclesof a high-speed railway passenger car system. Ahma-dian et al. [10, 11] investigated the hunting stabilityof a wheel axle and locomotive bogie with the asymp-totic method and analyzed many factors on the stabil-ity. Yang et al. [12] investigated the Hopf bifurcationand hunting stability of the bogie and locomotive sys-tem with hysteretic and nonlinear suspensions. Ding

et al. [13] established a mathematical model of non-linear hunting vibrations for the bogie of a railwayfreight car with 3 degrees of freedom, where dry fric-tion and wheel-rail impact were taken into considera-tion. They studied the Hopf bifurcation and the criti-cal speed, and found that there were two invariant cir-cles from Hopf bifurcation, one being stable and theother unstable. In recent years, Mark Hoffmann [14]studied the fundamental dynamic behaviors of Euro-pean two-axle railway wagons with special attentionto their unwanted hunting motion on straight track.True [15] illustrated the inherent meanings and manyrelated factors of the critical speed from the point ofview of nonlinear dynamics. He clarified that the valueof the critical speed must be found as the highest speedin the parameter vector for which the stationary so-lution is a unique steady-state solution of the theo-retical vehicle dynamic problem. Therefore the cal-culation of the critical speed must be formulated asan existence problem but not a stability analysis inmathematics. Gao et al. [16] discussed the continu-ation method and its application in hunting motionsand bifurcation behaviors of a six-axle locomotive.They [17] also investigated lateral bifurcation behav-iors of a four-axle railway passenger car and foundthat symmetric/asymmetric periodic and chaotic mo-tions exist in large speed ranges. They also found thatthe attractors underwent processes of symmetry break-ing and symmetry restoring repeatedly and went intoasymmetric chaotic motions through a series of the pe-riodic solutions doublings or quasi-periodic motions.

There is more work on the bifurcation behaviorsand chaotic motions of vehicle system in these studies.However, only little is published about the asymmet-ric motions and their inherent mechanism appearingin the symmetric railway vehicle system. Moreover,there isn’t a better method to describe the asymmet-ric motions in symmetric vehicle system. In the paper,the concept of symmetric bifurcation for symmetricwheel-rail system is defined. After that the ‘resultantbifurcation diagram’ method which can describe thesymmetric/asymmetric motions in a symmetric wheel-rail system is constructed. Then a two-axle railwaybogie is taken as the analysis object and the methodis used to study its symmetric/asymmetric bifurcationbehaviors and chaotic motions in large speed ranges.To illustrate a variety of possible motions in the sys-tem, the results in the forms of bifurcation diagrams,phase trajectories, Poincaré maps, power spectra and

The “resultant bifurcation diagram” method and its application to bifurcation behaviors

Fig. 1 The sketch map of a single wheel axle

Lyapunov exponents [18], etc. are presented. Manynonlinear dynamical phenomena are observed and dis-cussed in great detail from the point of the mathemat-ics or mechanics.

2 The dynamic model

2.1 Wheel-rail contact geometry relation

In a given cross-section parameters of wheels andrails, the rolling radius (rl, rr ) and the contact angle(δl, δr ) for the left or the right wheel, and the roll angleof the wheel axle (φw) and other wheel-rail contact ge-ometry parameters are approximated as the functionsof the lateral displacements of the wheel axle (yw)showed in Fig. 1 and can be expressed as

rl, rr = r0 ± λyw

δl, δr = δ0 ± ε0yw/a0

φw = σyw/a0

⎫⎪⎪⎬

⎪⎪⎭

(1)

where r0 is the nominal rolling radius of the wheelsand λ is the equivalent conicity of the wheel tread. Theδ0 is the contact angle when the wheel axle is situatedin the track centerline and ε0 is a variable parameterof the contact angle due to the lateral displacements ofthe wheel axle. The a0 is the half distance in lateraldirection between the contact points of the left and theright wheel and σ is the parameter of the roll angle.

In Fig. 1, the Fl and Fr are the tangential con-tact forces which are tangent to the wheels’ surfacesin their corresponding contact points for the left andright wheels, respectively. Meanwhile, the Nl and Nr

are the normal forces which are perpendicular to the

wheels’ surfaces in their contact points, also for theleft and right wheels. These forces will be illustratedin detail in Sects. 2.2 and 2.3.

2.2 Wheel-rail creep forces or creep moments

There is an extremely complex physical phenomenonwhen the stiff wheels with elasticity move forward onthe stiff rails with elasticity at a certain speed. The stiffwheels slide relatively to the stiff rails and thereforethere is some velocity differences in the contact areabetween the wheels and the rails. The slip includeselastic deformation and rigid slip. It should be under-stood as micro-slip and indicates the local relative ve-locity in the given point of the contact area betweentwo rolling bodies. The relative velocity or relative an-gular velocity in wheel-rail contact point normalizedby the forward speed is denoted as the creepage. Thelongitudinal creepage ξx , the lateral creepage ξy andthe spin creepage ξsp of the wheel axles are [19]

ξx(l,r) = V + ψwyw + r(l,r)(ψwφw − Ω) ∓ a0ψw

V

ξy(l,r) = −V ψw + yw + r(l,r)φw

V cos(δ(l,r))

ξsp(l,r) = ∓(Ω − ψwφw) sin(δ(l,r)) + ψw cos(δ(l,r))

V

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(2)

where the subscript l and r in the physical quantitiesin the paper represent the left and right wheel of thewheel axle. The signs ± and ∓ are taken upper signsif it is the left wheel, lower signs if the right wheel. TheV is the forward speed of the bogie and Ω = V/r0 isthe nominal angular velocity of the wheel. The ψw isthe yaw angle of the wheel axle.

The creep forces and creep moments are producedby the relative slip of wheels and rails in the contactarea and depended on the creepage. According to theKalker linear creep theory [20], the creep forces andcreep moments between the wheels and rails in a linearregion can be expressed as

Fx = −f11ξx

Fy = −f22ξy − f23ξsp

Mz = f23ξy − f33ξsp

⎫⎪⎪⎬

⎪⎪⎭

(3)

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where Fx and Fy are the longitudinal and lateralcreep forces, and Mz is the spin creep moment. Thef11, f22, f23 and f33 are the longitudinal, lateral, lat-eral/spin, spin creep coefficients that are relevant tothe wheel-rail contact geometry parameters and can befound in Kalker’s contact table [21] and computed eas-ily.

The Kalker’s linear creep theory is only availablefor small creep and small spin conditions. For the pos-sible cases of large creep, the creep theory by Shen etal. [22] can be used to revise the linear results so thatthe obtained creep forces and creep moments can bewidely applied to arbitrary values of the creepage. Itis also applicable to the wheel-rail interaction simula-tion in actual working conditions. Moreover, the the-oretical results of Shen’s creep theory are explicit andcorrespond well to the experimental results. So it issuitable for a dynamical simulation. The revision co-efficient ε is

ε ={

(1/β)[β − β2/3 + β3/27] β ≤ 31/β β > 3

(4)

where β is the normalized resultant creep force. If μ

is the friction coefficient between the wheels and railsand N is the normal force in the contact area, then β

can be given by

β =√

F 2x + F 2

y /(μN) (5)

So the revised creep forces and creep moments are

F ′x = εFx, F ′

y = εFy, M ′z = εMz (6)

Last the coordinate transformations are used toswitch the revised creep forces and creep moments inthe contact patch to the track coordinates. After thatthe equations of the motion for the bogie system canbe established. The detailed transformations are

F(l,r)x = F ′(l,r)x cos(ψw)

− F ′(l,r)y cos(δ(l,r) ± φw) sin(ψw)

F(l,r)y = F ′(l,r)x sin(ψw)

+ F ′(l,r)y cos(δ(l,r) ± φw) cos(ψw)

M(l,r)z = M ′(l,r)z cos(δ(l,r) ± φw)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(7)

2.3 Normal contact forces and flange forces

The normal contact forces in the contact patch for theleft or the right wheels are denoted as N(l,r) and alsodepicted in Fig. 1. The vertical component N(l,r)z ofthe normal contact forces can be obtained from thestatic equilibrium equation in vertical direction of thewheel axle [23]. In the analysis of straight track, it issupposed that N(l,r)z is equal to the half of the axleload and can be given by

N(l,r)z = 0.5W (8)

where W is the axle load.Then the lateral component N(l,r)y of the normal

contact forces can be expressed as

N(l,r)y = ∓N(l,r)z tan(δ(l,r) ± φw) (9)

During the operation of the vehicle, if the lateraldeviation of the wheel axle exceeds the flange clear-ance, there is also a lateral contact between the wheelflange and the rail, which results in a sudden restor-ing force called the flange force. The flange force ismodeled here as a stiff spring with a dead band and nodamping [24], given by a piecewise linear function asfollows:

Ft(yw) =⎧⎨

⎩

k0(yw − η) yw > η

0 |yw| ≤ η

k0(yw + η) yw < −η

(10)

where k0 is the flange contact stiffness and η is theflange clearance.

2.4 Dynamical equations of bogie system

A dynamical model of a two-axle railway bogie sys-tem is investigated and the schematic diagram isshown in Fig. 2. It is a multibody dynamical systemwhich is composed of a bogie frame (Mt, Itx, Itz), twowheel axles (Mw, Iwx, Iwz), the primary suspensions(Kpx,Kpy,Kpz,Cpx,Cpy,Cpz) between the bogieframe and the wheel axles, and the secondary sus-pensions (Ksx,Ksy,Ksz,Csx,Csy,Csz) between thebogie frame and the car body (we suppose that the carbody moves along the track center line with constant

The “resultant bifurcation diagram” method and its application to bifurcation behaviors

Fig. 2 Dynamical model of a two-axle railway bogie system

speed V ). It is assumed that all parts are rigid bod-ies except that the suspension elements all have lin-ear elastic characteristics. The simulations performedneglect any track irregularities and assume that thewheels and rails remain in contact all along and thatthe wheels roll on a smooth, level, and perfect track.The vertical and pitch motions of the bogie are ne-glected and only the lateral, roll and yaw motions aretaken into consideration. Moreover, it is assumed thatthe vertical displacements are so small that the dynam-ical equations for the vertical and horizontal motions,respectively, are uncoupled except for the roll motionsof the wheel axles.

There are seven degrees of freedom in the bogiesystem. They include the lateral motion ywi (i = 1–2)and yaw angle ψwi of the two wheel axles, and thelateral motion yt , roll angle φt and yaw angle ψt ofthe bogie frame. Considering the left-right symmetryand before-after symmetry of the bogie system, thedynamical equations of the bogie system can be ex-

pressed as follows:

mwywi = −Fyf i + Flyi + Fryi + Nlyi + Nryi − Fti

Iwzψwi = −dwFxf i + a0(Frxi − Flxi) + a0ψwi (Fryi

+ Nryi − Flyi − Nlyi) + Mlzi

+ Mrzi − IwyφwiΩ

mt yt = Fyf 1 + Fyf 2 − Fyt

Itzψt = lt [Fyf 1 − Fyf 2] + dw[Fxf 1 + Fxf 2] − dsFxt

Itxφt = htw[Fyf 1 + Fyf 2] + dw[Fzf 1 + Fzf 2]+ hbtFyt − dsFzt

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11)

where Fxf i,Fyf i,Fzf i are the primary suspensionforces in longitudinal, lateral and vertical directions,while Fxt ,Fyt ,Fzt are the secondary suspension forcesin longitudinal, lateral and vertical directions. The sus-

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pension forces can be given by

Fxf i = 2Kpxdw(ψwi − ψt) + 2Cpxdw(ψwi − ψt )

Fyf i = 2Kpy(ywi − yt ∓ ltψt − htwφt )

+ 2Cpy(ywi − yt ∓ lt ψt − htwφt )

Fzf i = 2Kpzdw(φwi − φt ) + 2Cpzdw(φwi − φt )

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(12)

Fxt = 2Ksxdsψt + 2Csxdsψt

Fyt = 2Ksy(yt − hbtφt ) + 2Csy(yt − hbt φt )

Fzt = 2Kszdsφt + 2Cszdsφt

⎫⎪⎪⎬

⎪⎪⎭

(13)

where the subscript i = 1,2 in the physical quantitiesin the paper represents the leading and trailing wheelaxles of the bogie. The signs ± and ∓ are taken uppersigns if i = 1, lower signs if i = 2.

The state vector can be defined as

yT = {yw1, yw2, ψw1, ψw2, yt , ψt , φt , yw1,

yw2,ψw1,ψw2, yt ,ψt , φt } (14)

Then the dynamical problem of the bogie system (11)can be formulated as an initial value problem of an au-tonomous system of ordinary differential equations offirst order with the time t (t ≥ 0) as the single inde-pendent variable.

dy

dt= f (y,V ) (15)

where f is a function of the state vector and V ∈ R+is the system control parameter, which indicates theforward speed of the bogie.

Given a set of system parameters and initial condi-tions, the solutions of the system can be achieved byintegrating equation (15) and they are in the form asfollows:

y = y(t,V ,y0) (16)

The steady-state solutions are relevant to the time t ,the forward speed V and the initial conditions y0.For some given speeds, there are stationary and peri-odic solutions, which correspond to the stationary andhunting motions of the bogie system. In other words,there are stable stationary solutions whose time re-sponse curve is decaying to the track centerline when

the speed is small enough. Then the stationary solu-tions lose their stabilities and the stable periodic so-lutions whose time response curve possess certain am-plitude and phase emerge when the speed is larger thancertain speed. At some speed intervals, aperiodic solu-tions whose time response curve is non-periodic andsometimes seems like erratic may also exist in the sys-tem. Therefore, we can deeply understand the dynam-ical features of the railway bogie system by analyzingthe solutions of the ordinary differential equations.

3 The ‘resultant bifurcation diagram’ forsymmetric wheel-rail system

3.1 Definitions of symmetric bifurcation

Assuming that all springs and damp in the railway bo-gie are laid out symmetrically around a vertical planethrough the track centerline and a vertical plane per-pendicular to the first one through the pivot of the bo-gie, and symmetric nonlinear wheel-rail contact forcesbetween wheels and rails, then the whole bogie systemis symmetric around the track centerline when the bo-gie runs on an ideal straight and perfect track. More-over, the solution that follows the track centerline isthe stationary solution (or say, the zero solution). Thestate space of the bogie system can be defined by

R14 = {y|y ∈ R14} (17)

It is well known that the Poincaré map is an effec-tive tool for the investigation of dynamics of a dynam-ical system. However, the definition of a Poincaré mapneeds to acknowledge the geometric structure of phasespace of the dynamical system. So there actually isn’ta general method for the definition of Poincaré map.For example, the instant of collision is often defined asthe Poincaré section and then the Poincaré map can beestablished in an impact vibration system [25]. For adeterministic and smooth dynamical system, there aremany different choices of the Poincaré sections and theexpressions of results are different from each other.

It is very difficult to know about the geometricstructure of the state space of the whole system forthe high-dimensional and self-excited railway vehiclesystem, and so it brings some trouble for the choices ofthe Poincaré section. In the bifurcation analysis of thebogie system, the choice of a Poincaré section shouldbe made on the one hand to obtain the characteristics

The “resultant bifurcation diagram” method and its application to bifurcation behaviors

of the vibration and achieve the most essential fea-tures of the system, and on the other hand to considerthe demands of engineering practice. In the paper, thePoincaré section is made as the instant where the lat-eral velocity vanishes and the displacement of the bo-gie frame is non-negative. The section is an unfixedsection and can be defined by∏

1

= {y ∈ R14|yt = 0, yt ≥ 0

}(18)

At the same time, a symmetric transformation inphase space R14 is

R : y �→ −y (19)

At a given speed, if y is a solution vector, then−y is another solution vector of the system. In otherwords, all solution sets appear in pairs. If the trans-formed solution is invariable under above transforma-tion, it is a symmetric solution set and corresponds tothe symmetric motions, while if the transformed solu-tion is not invariable, it is an asymmetric solution setand corresponds to the asymmetric motions in the rail-way vehicle system [26].

From the analysis, we see that there is another sym-metric Poincaré section with the original Poincaré sec-tion

∏1 around the track centerline. The symmetric

Poincaré section is made as the instant where the lat-eral velocity vanishes and the displacement of the bo-gie frame is non-positive. It can be given by∏

2

= {y ∈ R14|yt = 0, yt ≤ 0

}(20)

In some circumstances, symmetric as well as asym-metric attractors exist in the system and correspond tothe symmetric and asymmetric motions in the vehi-cle system, respectively. If the attractors through thePoincaré section

∏1 and the attractors through the

Poincaré section∏

2 coalesce, then they are symmetricattractors.

3.2 The method of ‘resultant bifurcation diagram’

The symmetric motions certainly do exist while theasymmetric motions may exist in the system in spiteof the dynamical system being symmetric around thetrack centerline. However, the asymmetric motionsare also very important in vehicle system dynamicsbecause they may cause lopsided wear and violent

wheel-rail interaction and then produce a differencesbetween the profiles of the wheels and the rails [8].The asymmetric wheel-rail contact forces may aggra-vate the lopsided wear and make the tread of wheelsand the profile of rails bruise or crack. This will affectthe normal use of the vehicle and may lead to somepotential dangers for safe operation. Therefore, the bi-furcation analysis of symmetric wheel-rail system isnot only to determine the periodic motions or chaoticmotions of the system, but also to determine the sym-metry of motions around the track centerline.

The ‘resultant bifurcation diagram’ method forsymmetric wheel-rail system is constructed to judgea solution to be symmetric or asymmetric around thetrack centerline. The basic idea is that first the con-cept of symmetric bifurcation for symmetric wheel-rail system is defined, then the time response of thesystem can be achieved by the numerical integrationmethod and an unfixed and dynamic Poincaré sec-tion

∏1 (e.g., defined by the expression (18)) and its

symmetric section∏

2 (e.g., defined by the expression(20)) for symmetric wheel-rail system are established.Using the defined Poincaré section and its symmet-ric Poincaré section, two bifurcation diagrams can beconstructed from which a solution can be judged to beperiodic or aperiodic. But the symmetry of the motionsaround the track centerline cannot be determined fromany one of the diagrams. Then the ‘resultant bifurca-tion diagram’ is constructed. If a solution is symmet-ric, the branches on the ‘resultant bifurcation diagram’are identical. If a solution is asymmetric, the solutionbranches are not identical. In this way, whether a solu-tion is symmetric or not can be determined.

If there are asymmetric motions in the ‘resultant bi-furcation diagram’, for example, two asymmetric pe-riod 1 motions, then there are two branches in two dif-ferent colors that are projected by the Poincaré sec-tion and its symmetric Poincaré section in the ‘resul-tant bifurcation diagram’. They indicate two asymmet-ric period 1 motions but not a period 2 motion. There-fore, the number of solution branches should be distin-guished prudently between the ‘resultant bifurcationdiagram’ and the normal bifurcation diagram, and itcan only be determined by any one of the colors in thediagram.

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4 Numerical results and discussion

In this section, a bogie of a high-speed railway pas-senger car made in Changchun factory is taken asthe research object and its bifurcation behaviors andchaotic motions are investigated and discussed in greatdetail. The values of the mass of rigid bodies, thestiffness coefficients, the damping coefficients, lengthand distance and other parameters can be found inthe literature [27]. The wheel-rail contact geometryparameters are: the equivalent conicity of the wheeltread, λ = 0.056, the contact angle when the wheelaxle is situated in the track centerline, δ0 = 0.056,the variable parameter of the contact angle due tothe lateral displacements of the wheel axle, ε0 = 0.0and the parameter of the roll angle, σ = 0.056 [2].The wheel-rail contact interaction parameters are: theflange contact stiffness, k0 = 14.6 × 107 N/m and theflange clearance, η = 9.1 mm. The creep coefficients,f11 = 6325.463 × 103 N, f22 = 5707.858 × 103 N,f23 = 14.420 × 103 N·m, f33 = 25.338 N·m2. Thefriction coefficient is μ = 0.15 as was done in the pa-pers by Kaas-Petersen [6] and True [7] etc.

A continuation algorithm that can compute the sta-tionary and periodic solution branches step by step ispresented in [28–30] and here it is used to calculateand follow the solutions in the parameter-state space.In the analysis, the forward speed V of the bogie ischosen as the system control parameter. The strategyis to start with a stationary solution (the zero solu-tion) that is known to be unique and asymptoticallystable at a sufficiently low speed. Then, the speed isincreased with small steps and the eigenvalues of theJacobi matrix by numerical techniques for each valueof the speed are calculated and followed. When a bi-furcation point is reached, it may continue along orig-inal solution path, or may follow the new path of thebifurcated solution. If the increments in the control pa-rameter are adequate, the known solution will providean excellent initial value for the determination of thenext solution on the branches. That is, it uses a tangentpredictor and a Newton iteration corrector to followthe family of the stationary solutions and periodic so-lutions, and calculates the eigenvalues of the Jacobimatrix and estimates the Floquet multipliers for de-termining the stability of the stationary and periodicsolutions.

At the same time, the ‘resultant bifurcation di-agram’ method is applied to analyze the symmet-ric/asymmetric motions of the bogie system in large

Fig. 3 Bifurcation diagram showing the lateral deviation of theleading wheel axle of the railway bogie versus the speed

speed ranges. Many nonlinear dynamical behaviorsare illustrated in great detail for a deeper understand-ing of the dynamics of the railway bogie system.

4.1 Stationary and symmetric periodic motions

Some results of the bifurcation analysis computed bythe continuation algorithm are illustrated in Fig. 3. Thedynamic long-time behaviors of the railway bogie de-pending on the forward speed are represented in the bi-furcation diagram by the lateral deviation of the lead-ing wheel axle of the bogie relative to the track center-line. In the diagram, the stable solutions are denoted bysolid lines and the unstable solutions are dotted ones.It can be described from the following aspects.

First, the stationary solutions of the bogie sys-tem are illustrated. The stationary solutions are sta-ble when the speed is small enough. From the solu-tions at this speed and increasing the speed with smallsteps, the stationary solutions will follow the solutionbranches OAB. The stability of the stationary solu-tions can be determined by the positive and negativereal part of the eigenvalues of the Jacobi matrix ofthe system. In the diagram, branches OA are stablestationary solutions, while branches AB are unstablestationary solutions. The point A (VA = 90.065 m/s,α1,2 = 7.4921 × 10−11 ± 16.5807i, αi here indicatesthe eigenvalues of the Jacobi matrix of the bogie sys-tem) is the Hopf bifurcation point where a pair of com-plex conjugate eigenvalues just cross the imaginaryaxis into the positive real half-plane and the previousstable stationary solutions become unstable.

Figure 4 shows the migration process of a pair ofcomplex conjugate eigenvalues of the Jacobi matrix of

The “resultant bifurcation diagram” method and its application to bifurcation behaviors

Fig. 4 The migration process of a pair of complex conjugateeigenvalues of the Jacobi matrix of the system with the speed

the system with the speed. The abscissa and ordinateare real and imaginary parts of the complex conjugateeigenvalues. It can be seen from the diagram that theHopf bifurcation indeed occurs at VA where the stabil-ity of the stationary solutions changes. Moreover, boththe real and imaginary parts of the eigenvalues of theJacobi matrix are increasing with the increasing speed.But the imaginary parts change much more slowly. Itindicates that the frequency of the system grows. Inthis respect, the vibration of the system is more andmore ‘violent’.

Then the periodic motions of the bogie system arediscussed. According to the Hopf bifurcation theory,a periodic solution bifurcates from the Hopf bifurca-tion point. So the Hopf bifurcation point A is selectedas the initial point and the continuation algorithm isused to follow the periodic motions. It can be foundthat an unstable periodic solution AC bifurcates to theleft of point A and the bifurcation is therefore sub-critical. The amplitude of the unstable periodic oscilla-tion grows with the decreasing speed until the point C(VC = 74.811 m/s, max |yC| = 9.1001 mm is the max-imum lateral amplitude of the periodic motion for theleading wheel axle of the bogie in the point C, TC =0.4517 s is the period of the system at the speed ofpoint C) is reached. The bifurcating unstable periodicsolution gains stability in the saddle-node bifurcationpoint C where the amplitude of the periodic oscilla-tion is so large that the flange contact occurs (flangeclearance η = 9.1 mm). After that, the stable peri-odic solution CD (corresponding to the stable huntingmotion of the bogie) grows in amplitude with the in-creasing speed until the point D (VD = 125.03 m/s,

Fig. 5 Unstable limit cycles in solution branches AC at fivedifferent speeds. The speeds from outer to inner in the diagramare 75.0 m/s, 77.0 m/s, 79.0 m/s, 81.0 m/s and 83.0 m/s

max |yD| = 9.992 mm, TD = 0.2564 s). The wholesystem oscillates symmetrically. When the speed islarger than VD, the whole system changes into asym-metric chaotic motions and the detailed analysis willbe conducted later.

In the above analysis, the speed of point A at whichthe stationary solution loses its stability in a sub-critical Hopf bifurcation is the Hopf bifurcation speed.At the same time the speed of point C is the low-est speed for which a periodic motion (a hunting mo-tion) exists and below which the stationary solutionis unique and asymptotically stable, it is the criticalspeed of the vehicle and can be used as the limited de-sign speed.

The coexistence of many nonlinear dynamical phe-nomena in railway vehicles has been discussed bymany outstanding scientists, such as True [31, 32],Hoffmann [33], Schupp [34], Zboinski and Dusza [35]etc. Here it also occurs. It can be seen from the bi-furcation diagram in Fig. 3 that the stable stationarymotions, unstable periodic motions and stable periodicmotions coexist in the speed range VC < V < VA. Towhat motion the system eventually tends is greatly re-lated to the initial disturbances of the system. Figures 5and 6 show the projection of unstable limit cycles inthe solution branches AC and the stable limit cycles inthe solution branches CA’ both at five different speedsin this speed range, respectively. The abscissa is thelateral displacements while the ordinate is the lateralvelocities of the leading wheel axle of the bogie. Itcan be seen from Fig. 5 that the amplitudes of un-stable periodic solutions decrease with the increasingspeed and they are all smaller than the flange clearance

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Fig. 6 Stable limit cycles in solution branches CA’ at five dif-ferent speeds. The speeds from inner to outer in the diagram are75.0 m/s, 77.0 m/s, 79.0 m/s, 81.0 m/s and 83.0 m/s

(η = 9.1 mm). In contrast to Fig. 5, the amplitudes ofthe stable periodic solutions increase with the increas-ing speed and they are all larger than the flange clear-ance in Fig. 6. That is to say, the flange forces have asignificant influence on the periodic motions and theirstability of the system due to the constraint of flange.

4.2 Symmetric/asymmetric bifurcation behaviors atsuper-high speeds

In everyday operation of a vehicle, the forward speedsare usually not larger than the critical speed of the ve-hicle. But from the point of view of mathematics ormechanics, it is also attractive to explore whether thereare asymmetric motions and their inherent mechanismat some ‘extreme’ speeds.

In order to construct the bifurcation diagram to il-lustrate the periodic and aperiodic motions appearingin the system, the integration in time continues un-til the small transients have died out. Then the lat-eral amplitudes of the bogie frame when the bogiemoves to the left side of the track centerline is plot-ted with slowly increasing speed. That is, the pointsin the Poincaré section

∏1 are taken and the bifurca-

tion diagram is depicted in Fig. 7. In a similar way,the lateral amplitudes of the bogie frame when the bo-gie moves to the right side of the track centerline isalso plotted with slowly increasing speed. The pointsin the Poincaré section

∏2 are taken and the bifurca-

tion diagram is depicted in Fig. 8. The initial valuesof the numerical integration for the next step are theend values of the previously computed speed after the

Fig. 7 Bifurcation diagram of the bogie frame (left side)

Fig. 8 Bifurcation diagram of the bogie frame (right side)

transient is negligible in case the speed increments areadequate.

However, neither the bifurcation diagram of Fig. 7nor Fig. 8 can be used to describe the symmetry ofthe motions of the system around the track centerline,namely whether a motion is symmetric or not cannotbe judged from a single bifurcation diagram. So the‘resultant bifurcation diagram’ method is used here tosolve this problem. The bifurcation diagram for theright side amplitudes of the bogie frame with respectto the track centerline are transformed to the left sideby the formula (19), and then superimposed with theoriginal bifurcation diagram for the left side ampli-tudes of the bogie frame with respect to the track cen-terline in different colors. Then the resultant bifurca-tion diagram is illustrated in Fig. 9 where the symmet-ric/asymmetric motion states at different speeds can beeasily judged. If there is a periodic motion, the num-ber of periodic motions should be determined from

The “resultant bifurcation diagram” method and its application to bifurcation behaviors

Fig. 9 Bifurcation diagram of the bogie frame (the resultantdiagram)

Fig. 7 or Fig. 8, or branches in one color in Fig. 9.In the following analysis, the bifurcation diagramsare the resultant bifurcation diagrams if there are nospecial instructions. At the same time, the projectedPoincaré map at some speeds are still constructed bythe Poincaré section

∏1.

There are various motions and different symme-try states in the bifurcation diagram Fig. 9. So itis divided into three ranges for clarity in the analy-sis, 124.5 m/s < V < 135.0 m/s, 147.0 m/s < V <

152.0 m/s and 152.0 m/s < V < 157.0 m/s. The de-tailed motions and the symmetry states in each speedrange are in the following.

4.2.1 In the speed range 124.5 m/s < V < 135.0 m/s

An enlarged portion of Fig. 9 in the speed interval124.5 m/s < V < 135.0 m/s is showed in Fig. 10.It can be seen from the diagram that there exists asymmetric periodic oscillation below the speed V =125.03 m/s since the solution branches in different col-ors are identical and superposed with each other.

Figure 11 shows phase trajectories and power spec-tra for periodic motions at V = 125.0 m/s for the bogieframe. Figure 11(a) shows the phase trajectories show-ing the lateral displacements of the leading wheel axleof the bogie versus the lateral displacements of thebogie frame. It is composed of a single closed curveand forms a periodic attractor, which indicates thatthe system oscillates periodically. At the same time,the collision between the wheel axles and the side railis also visible. The leading wheel axle of the bogiehits the side rail with two consecutive flange contacts

Fig. 10 Enlarged portion of Fig. 9 in the speed interval124.5 m/s < V < 135.0 m/s

with two different amplitudes within one period be-fore the wheel axle moves over to flange contacts withthe other side rail also with two different amplitudes.The corresponding amplitudes of the contact with theleft and right side rail are equal in value. Figure 11(b)shows the power spectra of the lateral displacementsof the bogie frame and the abscissa is frequency. Thesampling frequency is 104 Hz. The spectral lines aresome separate and discrete peaks including the basefrequency f0 = 3.896 Hz and its harmonics 3f0 and5f0 etc. Both characteristics in Fig. 11 indicate a sym-metric periodic attractor at this velocity.

When the speed is larger than V = 125.03 m/s,there are a lot of dense points except for some char-acteristic windows of periodic solutions in differentcolors on the bifurcation diagram in Fig. 10. It meansthat the symmetric periodic attractor suddenly disap-pears and the asymmetric chaotic attractor suddenlyappears from the state space as the forward speed in-creases through V = 125.03 m/s through a blue skycatastrophe [26] in the beginning. Figure 12 shows thebifurcation diagram of the bogie frame in the speedrange 124.2 m/s < V < 125.2 m/s with increasingspeed and decreasing speed, respectively. The asym-metric chaotic motion is delayed to the speed V =124.373 m/s where the chaotic attractor is suddenlydestroyed as the forward speed decreases also througha blue sky catastrophe. In other words, there is an ob-vious hysteresis in this small speed range.

Figure 13 shows phase trajectories and power spec-tra at V = 125.5 m/s for the bogie frame. The coordi-nates are the same as on Fig. 11. The phase trajecto-ries on the yt − yw phase plane is bounded and some-

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Fig. 11 Phase trajectories and power spectra for periodic motions at V = 125.0 m/s for the bogie frame

Fig. 12 Bifurcation diagram of the bogie frame with increasing speed and decreasing speed

what tangle in Fig. 13(a). It seems to have a structurelike a stretched, twisted and folded band in this pro-jection and forms a chaotic attractor. The power spec-tra in Fig. 13(b) have a broad band structure aroundthe peaks. Both figures indicate chaotic motions in thesystem at this speed. There is no doubt that the compu-tation of Lyapunov exponents is the strongest evidencefor chaotic motions. It indicates chaos if one of thecharacteristic exponents is larger than zero. Figure 14shows the convergence of the three largest Lyapunovexponents with the elapse of time at V = 125.5 m/s.The time step is 0.5 s in the computation. It can be seenfrom the diagram that the approximation to the threelargest Lyapunov exponents is 0.25, 0.0 and −0.27.The largest Lyapunov exponent stays positive which

is an obvious characteristic of chaotic oscillations inthe bogie system at this velocity.

As the speed further increases, the bandwidths ofthe chaotic motions increase in Fig. 10 until the speedreaches V = 131.47 m/s where the bandwidths in theleft side of track centerline decrease suddenly, whilethe bandwidths in the right one increase suddenly.The size of chaotic attractor suddenly increases ordecreases as the forward speed is increased throughV = 131.47 m/s. Maybe the chaotic attractor collideswith an unstable periodic solution that is in the interiorof the basin of attraction. That is an interior crisis [26].After that the new bandwidths increase slowly withthe speed increases. The asymmetric chaotic attrac-tors disappear and the asymmetric characteristic win-

The “resultant bifurcation diagram” method and its application to bifurcation behaviors

Fig. 13 Phase trajectories and power spectra for chaotic periodic motions at V = 125.5 m/s for the bogie frame

Fig. 14 Convergence of the three largest Lyapunov expo-nents with elapse of time at V = 125.5 m/s, LE1 = 0.2544,LE2 = −0.0011, LE3 = −0.2729

dows of periodic solutions appear in the speed range132.53 m/s < V < 133.44 m/s.

Figure 15 shows the projected Poincaré map at V =133.0 m/s. The ordinate and abscissa are the lateraldisplacements of the leading wheel axle of the bogieand the lateral displacements of the bogie frame, re-spectively. There are eight points in the diagram whichindicate 8 periodic motions.

When the speed increases above V = 133.44 m/s,the asymmetric periodic attractors disappear andasymmetric chaotic attractors appear. This can beconfirmed by the projected Poincaré map at V =133.6 m/s in Fig. 16. There are dense points in theprojected Poincaré map in Fig. 16 which form the dis-

Fig. 15 The projected Poincaré map for 8 periodic motions atV = 133.0 m/s

connected stripes somewhat infinite nesting, layeredstructure.

As the speed continues to increase above V =133.85 m/s, the asymmetric chaotic attractors dis-appear and then the asymmetric periodic attractorsappear in a large speed interval. The asymmetricperiodic attractors change into asymmetric 2 peri-odic solutions from asymmetric 4 periodic solutionsthrough an inverse period doublings at the bifurca-tion speed V = 134.09 m/s. The phase trajectoriesaround this bifurcation point are illustrated in Fig. 17which shows the trajectories at V = 134.0 m/s andV = 134.2 m/s. There are four closed curves on thisphase plane in the phase trajectories at V = 134.0 m/sin Fig. 17(a), while there are only two closed curvesat V = 134.2 m/s in Fig. 17(b). It indicates that the

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four periodic solutions have changed to two periodicsolutions.

4.2.2 In the speed range 147.0 m/s < V < 152.0 m/s

Figure 18 shows the enlarged part of Fig. 9 in the speedrange 147.0 m/s < V < 152.0 m/s, and the asymmet-ric oscillations are strongly off-centered to the trackcenterline. It can be seen from the diagram that a se-ries of period doublings starts as the speed increases.The first period doublings appears at V = 147.1 m/swhere the asymmetric 2 periodic solutions go into theasymmetric 4 periodic solutions, the second perioddoublings is at V = 150.68 m/s after which the asym-metric 8 periodic solutions appear and the third one is

Fig. 16 The projected Poincaré map for chaotic motions atV = 133.6 m/s

at V = 150.99 m/s. If the speed continues to increasewith small steps, the corresponding solutions of perioddoublings will appear, and the system will come intochaotic motions in the end.

A projection of the Poincaré sections at some keyvelocities showing the period doublings to chaos aredepicted in Fig. 19. It can be found from the diagramthat, with the speed increasing through the different bi-furcation points, there are two points in Fig. 19(a), fourpoints in Fig. 19(b), eight points in Fig. 19(c) at theprojected Poincaré map. The periods follow the ruleof 2n until n = 3 (n ≥ 1 is the times of period dou-blings) with the increasing speed. The system ends inthe chaotic attractors in Fig. 19(d) at last.

Fig. 18 Enlarged part of Fig. 9 in the speed interval147.0 m/s < V < 152.0 m/s

Fig. 17 Phase trajectories at V = 134.0 m/s and V = 134.2 m/s

The “resultant bifurcation diagram” method and its application to bifurcation behaviors

Fig. 19 A projection of the Poincaré sections at some key velocities showing the period doublings to chaos

4.2.3 In the speed range 152.0 m/s < V < 157.0 m/s

Figure 20 shows the enlarged portion of Fig. 9 in thespeed interval 152.0 m/s < V < 157.0 m/s. As can beseen from the diagram, the bandwidths of the asym-metric chaotic attractors increase with the increasingspeed. There is a small characteristic window of pe-riodic solutions where an asymmetric periodic motionevolves to an asymmetric aperiodic motion in a narrowspeed range near the speed V = 152.4 m/s. After thatthe system will go into the asymmetric chaotic motionuntil the speed reaches V = 155.52 m/s where the sys-tem evolves to the asymmetric 3 periodic motions. Theevidence can be presented from the projected Poincarémap at V = 156.0 m/s in Fig. 21 where three points onthe yt − yw phase plane are good evidences of 3 peri-odic motions.

Fig. 20 Enlarged portion of Fig. 9 in the speed interval152.0 m/s < V < 157.0 m/s

It can also be seen on Fig. 20 that the asymmet-ric 3 periodic motions continue after the speed of V =

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155.52 m/s until the speed reaches V = 156.87 m/swhere the periodic motions disappear suddenly. Afterthat speed, the system seems to come into symmet-ric chaotic motions and the bandwidths of the chaoticattractors increase with the increasing speed. Someevidence should be presented to support the conjec-ture. Figure 22 shows the projected Poincaré map ofthe symmetric chaotic attractors at V = 157.0 m/s.The Poincaré section of the left diagram is

∏1 and

the right is∏

2. The two diagrams will be identicalif the right diagram is rotated 180 degrees around theaxis of yt = 0 or yw1 = 0. This proves the existenceof the symmetric chaotic attractors at this speed fromthe other side. It seems that the ‘resultant bifurca-tion diagram’ method in the paper can not only revealthe forms of motions in the system, but also deter-mine the symmetry of motions around the track cen-terline.

Fig. 21 The projected Poincaré map at V = 156.0 m/s

5 Conclusion

In the paper, the ‘resultant bifurcation diagram’ methodis constructed for conveniently judging motions to besymmetric or asymmetric around the track centerlinein symmetric wheel-rail system. The time response ofthe system can be calculated by the numerical inte-gration method and the dynamic Poincaré section andits symmetric section for symmetric wheel-rail sys-tem are established. After that, two bifurcation dia-grams constructed, respectively, by the Poincaré sec-tion and its symmetric section are superimposed. Inthis way, the obtained resultant bifurcation diagramcan reveal the possible motion states and their evolu-tion processes, and judge the symmetry/asymmetry ofmotions around the track centerline.

A two-axle railway bogie is taken as the researchobject and the ‘resultant bifurcation diagram’ methodis used to investigate the symmetric/asymmetric bi-furcation behaviors and chaotic motions of the sym-metric bogie running on an ideal straight and perfecttrack. The system nonlinearities mainly stem from thecreep forces in the wheel-rail contact surface and theflange forces between the oscillating wheels and therails. The continuation algorithm is applied to calcu-late the Hopf bifurcation point and trace the periodicsolutions after the bifurcation point, and moreover thestable solution branches as well as unstable solutionbranches are determined. It is indicated that the stablestationary solution and the stable periodic solutionscoexist for the sub-critical Hopf bifurcation appearingin the system. To what motion the system eventually

Fig. 22 The projected Poincaré map of the symmetric chaotic attractors at V = 157.0 m/s

The “resultant bifurcation diagram” method and its application to bifurcation behaviors

tends is greatly related to the initial disturbances ofthe system.

It is also indicated in the results that there are alot of symmetric periodic motions for lower speeds,and then the bogie system evolves to the asymmet-ric chaotic or asymmetric periodic motions for a largespeed range, and return again to the symmetric mo-tions with narrow speed range in the end. The break ofsymmetry in the system starts with a blue sky catastro-phe in the beginning.

Acknowledgements This research was supported by Open-ing Fund of State Key Laboratory of Traction Power, South-west Jiaotong University (Grant No. TPL1106), and also sup-ported by National Natural Science Foundation of China (GrantNo. 11072204, 11102030, 10902092) and the Fundamental Re-search Funds for the Central Universities.

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