dynamic analysis of a high-speed train

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 1, JANUARY 2008 107

Dynamic Analysis of a High-Speed TrainSonia Leva, Member, IEEE, Adriano Paolo Morando, and Paolo Colombaioni

Abstract—Studies in modern high-speed traction need to de-velop detailed equations to formulate a complete electromechan-ical model that represents the dynamic coupling of electricaland mechanical phenomena. Starting from Lagrange and Parktheories, and after recalling the general criteria for the setup ofthe two distinct mechanical and electrical subsystems, the newformalization of a complete dynamic model with specific referenceto a train dedicated to high-speed services is hereby presented.Simulations are carried out with relation to two conditions ofparticular interest in the application side, i.e., the steady-statecondition and the three-phase short-circuit fault at a speed of100 km/h. Representative of the dynamics of each single compo-nent, the investigation in particular focuses on the transmissionorgans (rotor, Cardan joint, and gear case) and the relevanttorques associated with the short-circuit dynamics.

Index Terms—Dynamics, electromechanical analysis, high-speed train, torque control, traction motor drives.

I. INTRODUCTION

M ECHANICAL models of high-speed railway trains havebeen studied and consolidated in the past. Supported

by modern methods of numerical calculations, they are nowfrequently used for industrial applications. Similar results havelong been found in the electrical field too. Using the theory ofspace phasors, the analysis of time-variant electrical systemshas led to Park equations. Their use allows a numerical exami-nation of the magneto-electrical dynamics of machines.

The mechanical and electrical subsystems have been sepa-rately investigated until today: the electrical with an imposedresistant torque [1], [2] and the mechanical by means of themodal analysis without electric torque or in the more recentapplications only with an imposed traction torque [3]–[5].This approach had many objective motivations. One of thosemotivations is cultural. An inadequate osmosis between the twocultures (mechanical and electrical) has been directed toward alack of exchange of information between these two sectors. Theother motivation is applicative. The size order of the mechanicaland electrical time constants of the linearized system is sodifferent that more emphasis was placed on the two distincttransients in two subsequent and separate time intervals. Themechanical transient starts when the electrical one is effectivelyconcluded. Although they are substantially correct in their

Manuscript received March 15, 2006; revised November 13, 2006 andApril 12, 2007. The review of this paper was coordinated by Prof. D. Lovell.

S. Leva and A. P. Morando are with the Electrical Engineering Depart-ment, Politecnico di Milano, 20133 Milan, Italy (e-mail: [email protected];[email protected]).

P. Colombaioni is with BorgWarner Morse TEC Europe, 20043 Milan, Italy(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.901858

methodology, these readings, especially regarding the afore-mentioned problems of fatigue, can actually lead to numericaldeductions that need some corrective touches. This is eventruer today when we take into account both the performancesthat the trains are reaching and the complexity of the controlarchitecture.

In this paper, a mathematical electromechanical model fora high-speed train is elaborated upon, where the electricaland mechanical variables are no longer separate and become“undistinguishable.” In fact, taking into account the levelreached by the Lagrange (mechanical model) and Park (elec-trical model) approaches, the newly developed models can onlyresult in joining the two distinct systems of equations in a singleglobal model that becomes representative, in a united electricaland mechanical pair form, of the dynamics of the completedrive system.

The complete electromechanical model will be able to reacha detailed calculation, in every possible condition of the train’smotion (e.g., steady-state condition, starting and short-circuittransient), of the mechanical torque that really acts on the trans-mission and, with this, the periodic stresses that are present.These stresses, which may seem irrelevant to the value con-tained in their amplitude, actually turn out to be crucial tothe frequency that characterizes them. In fact, they can excitethe natural modes of vibration of the mechanical system and,at the same time, can determine intolerable fatigue and/orstress on the transmission. On the electrical side, such detailedknowledge allows us to closely study the effect of mechanicaloscillations on the electrical quantities (i.e., fluxes, currents, andvoltages).

In the following, after recalling the mechanical equations(Section II) and the electrical model (Section III), a newcomplete dynamic model of a high-speed train is presented(Section IV). Furthermore, the structure of the computationprogram (Section V)—which was developed in Simulink—and some considerations about the method of integration(Section VI) are presented. In Sections VII and VIII, sim-ulations relating to the steady-state condition and the three-phase short-circuit fault are carried out at a speed of100 km/h. The obtained results based on the Park–Lagrangemodel had been compared with the results obtained by adoptingseparate mechanical and electrical models [1], [4].

The train chosen for the analysis is constituted of a vari-ant of the ETR-470 (Fig. 1). It is characterized by the sameaerodynamic profile as the classic tilting train FS ETR-480 andCisalpino AG ETR-470. This train, with its high technologicalinnovation, which is remarkable for the uncommon complexityof its mechanical parts, is in fact a test bench for the setup ofthe complete model and for a comparative dynamic analysis ofthe train.

0018-9545/$25.00 © 2008 IEEE

108 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 1, JANUARY 2008

Fig. 1. Tilting train ETR-470.

II. MECHANICAL MODEL

A. Formulation

The following analysis is, however, limited to establishinga simplified 2-D scheme. The mathematical model of the me-chanical system is designed to describe the torsional chain fromthe engine to the wheels (including the interactions with theelectrical system and the wheel–rail contact) and, lengthwise,the relations between the axles, bogie, and box. Within thisformulation, no consideration is made about the axial motionof the central part of the Cardan transmission and the rollingmotions of the bogie and the box.1

Such simplifications do not have a negative influence onthe results obtained with the bidimensional model presentedhere. In fact, the most relevant harmonic stresses acting on thetransmission organs are those that are related to the rotationirregularities due to the nonhomokinetic behavior of the Cardanand to the rotation, which is a result of the harmonic pollutionproduced by the electrical part of the drive system, of the driveshaft.

The formulation of the dynamic equations referring to themechanical part of the model under exam analysis is based ona form of the following type of Lagrange’s equations [6]:

∂

∂t

(∂T

∂xi

)− ∂T

∂xi+

∂D

∂xi+

∂U

∂ xi=

∂L∗

∂x. (1)

Defined in this regard, starting from the configuration of themechanical system [7] (Fig. 2) to the degrees of freedom of thesystem (Table I), we must correlate to them, in (1), the kineticenergies T , potential U , and dissipative function D, deductthe Lagrange component of the active stress L, and, finally,calculate the derivative both for the chosen degrees of freedomxi and for the time.

Thus, we can obtain a system of 19 differential equationsof the second order in 19 unknown quantities. Once we have

1In the simplified model presented here, the further hypothesis that thetotal weight is equally shared between the two bogies is introduced. It is nowpossible to reduce the model of the vehicle to one single bogie, as depicted inFig. 2. The reasons for these simplifications arise from the following needs: areasonable numeric calculation complexity, objective difficulties for identifyingthe parameters, and reduction of the order of the model.

calculated, using the data supplied by the manufacturer, theelements of the matrix of inertia [M ], rigidity [C], and damping[K],2 these equations, in which the vector F of the activestresses is correlated to that of the degrees of freedom w, theycan be compacted in the following matrix form:

[M ] · w + [K] · w + [C] · w = F . (2)

In the recent years, different models to simulate the slippingforces between the wheels and the rails have been developed[8], [9]. A detailed description of the friction forces will requireto use a large number of parameters, but for the purpose ofthis paper, we can consider the utilization of a simplified lawto be sufficient. In particular, it is based on the followingassumptions.

• The status of the contact area is considered constantand dry.

• The influence of the normal force has been neglected.• The shape of the contact area has been considered

negligible.• It has been considered that the friction torque and the spin

do not affect the longitudinal friction.• The pseudoslipping has been considered to be defined only

by the single longitudinal term ε = εx = (Vx − Ωr0)/|V |,where Vx represents the component of the speed V ofthe wheel center in the longitudinal direction x (vehicledirection).

The longitudinal component Fx of the contact force can becalculated based on the Kalker model [9], [10], with

Fx = − X0 · εx√1 +

(X0·εx

µMAX

)2· Q (3)

where Q is the equivalent mass on each wheel, X0 repre-sents the initial slope of the slipping curve, and µMAX is themaximum value of the friction coefficient µ. The longitudinalcomponent of the contact force is applied at the kinematic pointthat represents the center of the contact area. This point isconsidered joined at the wheel.3

From (3), which describes the longitudinal component ofthe contact force, and taking into consideration the followingequation for pseudoslipping:

εx = (αvij − αij)/αvij (4)

the Lagrange component of the contact torque acting on thegeneric degree of freedom αvij can be expressed with thefollowing formulation:

T(αvij) =−Q · X0r0αvij − αij

αvij

/√1+

(Xo

µmax· αvij−αij

αvij

)2

(5)

2The data are given in Appendix A.3The virtual displacement δx of this point can be expressed as δx = x − r0 ·

sin α ∼= x − r0 · α. In particular, for a generic wheel identified by the index ij,it is possible that the virtual displacement δxij can be written as δxij = r0 ·αvij − r0 · αij , where the longitudinal translation of the wheel ij is expressedin terms of the angular displacement xij = r0 · αvij .

LEVA et al.: DYNAMIC ANALYSIS OF A HIGH-SPEED TRAIN 109

Fig. 2. Configuration of the mechanical system.

TABLE IMECHANICAL DEGREES OF FREEDOM

while on the generic degree of freedom, αij will be

T(αij) =Q · X0r0αvij−αij

αvij

/√1+

(Xo

µmax· αvij−αij

αvij

)2

.

(6)

The system of (2) is very complex. For space constraintreasons, we list only the motion equations of the degree offreedom linked to the organs of transmission.

1) Motor-induced rotation αr

(Jrot + Jc1)αr + Kcardαr − τKcardαu + τKcardαp

+ Ccardαu − τCcardαu + τCcardαp

= Te − Jc1α2r

ϑ20

2sin(2αr)

+ Ccard

(ϑ0

2

)2

[sin(2αr) − sin(2αr + ϕ)]

+ Kcardαrϑ2

0

2[cos(2αr) − cos(2αr + ϕ)] .

Te is the electromagnetic torque developed by the motorand represents an input for the mechanical system.


2) Output transmission rotation αu

(Jrid+Jc2τ2)αu−Jc2τ

2αp+(Kd+Ks+Kcardτ2)αu

− Kdα21 − Ksα22 − Kcardταr − Kcardτ2αp

+ (Cd + Cs + Ccardτ2)αu − Cdα21 − Csα22

+ −Ccardταr − Ccardτ2αp

= −Jc2α2rτϑ2

0 sin(2αr + ϕ)

− Ccard

(ϑ0

2

)2

τ [sin(2αr) − sin(2αr + ϕ)]

− Kcardαrϑ2

0

2τ [cos(2αr) − cos(2αr + ϕ)] .

3) Reducing the gear case rotation αp[Jpon+Jc2τ

2+Mm(ap tan ϑ0)2]Jc2αpτ

2c2−Jc2τ

2αu

− 12Mmr0ap tan ϑ0αv21 −

12Mmr0ap tan ϑ0αv22

+ Mmap(tan ϑ0)2zm + Mmamap(tan ϑ0)2βm

+(Kcardτ2 + Krh

2r

)αp − Kcardτ2αu − Kcardταr

+ Krhrx+ − Krhrh

+r β+

− 12r0

(Krhr

bpc

ba+(Kxm1+Kxm2)ap tan ϑ0

)αv21

+ (Kxm1 + Kxm2)ap(tan ϑ0)2zm

− 12r0

(Krhr

bpl

ba+(Kxm1+Kxm2)ap tan ϑ0

)αv22

+ (Kxm1 + Kxm2)ap tan ϑ0x∗

+ (Kxm1 + Kxm2)ap tan ϑ0(am tan ϑ0 − hm)βm

− Ccardτ2αu + Ccardταr

+[Ccardτ2+Crh

2r+(Cxm1+Cxm2)(ap tan ϑ0)2

]αp

+ (Cxm1 + Cxm2)ap tan ϑ0x∗

− 12r0

(Crhr

bpc

ba+ (Cxm1 + Cxm2)ap tan ϑ0

)αv21

+ (Cxm1 + Cxm2)ap(tan ϑ0)2zm

− 12r0

(Crhr

bpl

ba+(Cxm1+Cxm2)ap tan ϑ0

)αv22

+ Crhrx+ − Crhrh

+r β+

+ (Cxm1 + Cxm2)ap tan ϑ0(am tan ϑ0 − hm)βm

= +Jc2α2rτϑ2

0 sin(2αr + ϕ)

+ Ccard

(ϑ0

2

)2

τ [sin(2αr) − sin(2αr + ϕ)]

+ Kcardα2r

ϑ20

2τ [cos(2αr) − cos(2αr + ϕ)] .

4) Wheel 12 rotation α12

Jpα12 + C1α12 − C1α11 + K1α12 − K1α11

= QX0r0

αv12−α12αv12√

1 +(

Xo

µmax· αv12−α12

αv12

)2.

B. Modal Analysis

Using the developed model, it is possible to perform a modalanalysis of the convoy. With this analysis, we can evaluate thenatural frequencies and the mode of shapes of the mechanicalsystem. The modal analysis is the method habitually usedto investigate the dynamic behavior of mechanical systems[4], [5].

As a matter of fact, the modal analysis of the convoy al-lows us to explain the mode shapes in terms of the relativedisplacement of each single degree of freedom. In detail, foreach natural frequency of the system, it is possible to identifythe degree of freedom that shows the higher displacement.

Such an analysis will additionally permit to understand po-tential resonance phenomena that can be encountered along theanalysis of the whole system.

Analyzing the dissipative system and taking into account allthe damping bodies in the convoy, we obtain a linear second-order homogeneous differential equation. From this equation,which is expressed in normal form, we obtain

w = −[M ]−1[K] · w − [M ]−1[C] · w. (7)

Equation (7) can also be considered as the matrix system oftwo linear first-order equations[

ww

]=

[0 [I]

−[M ]−1[K] −[M ]−1[C]

] [ww

]. (8)

In this case, the evaluation of eigenvalues in general leadsto complex conjugate numbers. The imaginary parts of thesenumbers, in accordance with the subsequent equation

f0i =√

m(λi)/(2π) (9)

are related to the natural frequencies of the system, while thereal parts e(λi), for reasons of stability, have to be negatives.The eigenvectors represent the damped natural modes; they arecomplex numbers, and their magnitudes represent the ampli-tudes of the displacements as far as the phases identify thetemporal gaps between the degrees of freedom.

Finally, the analysis is performed considering that the vehicleis running at a constant speed V . In this condition, the contactforce expressed by (3) becomes Fx = −X0 · Q · εx. In theinvestigation of the transient analysis around the steady-statesolution, it will be εxij = r0(αvij − αij)/V , from which theequations of the external torques acting on the rotational degreeof freedom (αij) and the displacement αvij of the wheel ijwill become Tij(αij)

= X0Qr20(αvij − αij)/V and Tij(αvij)

=−X0Qr2

0(αvij − αij)/V , respectively.The addition of the terms due to the contact force at the linear

dynamic equations in the normal force will generate

[M ]w + ([K] + [KF ]) w + [C]w = 0 (10)

in which the elements different from zero in the matrix [KF ]are equal to ±X0Qr2

0/V .The natural modes associated with the damped system have

been calculated at speeds of 5, 100, and 200 km/h.


TABLE IINATURAL MODES ASSOCIATED WITH THE DAMPED SYSTEM WITH FORCES LINEARIZED AROUND A SPEED OF 100 km/h

Next to the analysis of results for the different simulations, itis possible to draw the following conclusions.

First at all, the frequency associated to the torsional vibrationof the motor was 23.70 Hz when the contact forces were notconsidered, and it decreases to 19.57 Hz at 5 km/h, 19.60 Hzat 100 km/h, and 19.77 Hz at 200 km/h once we consider thecontact force effect.

Furthermore, without taking into account the friction forces,the yawing motions are decoupled from the torsional vibrationof the axles as follows:

• yawing motored axle: 46.87 Hz; torsional vibration mo-tored axle: 72.06 Hz;

• yawing carrying axle: 47.83 Hz; torsional vibration carry-ing axle: 32.48 Hz.

On the contrary, taking also into consideration the frictionforces at 5 km/h, the two frequency modes are coupled asfollows:

• yawing and torsional vibration of the motored axle:54.07 Hz;

• yawing and torsional vibration of the carrying axle:76.24 Hz.

This is due to the fact that, at low speed, the nondiagonalterms of the matrix [KF ] have high values.

Finally, a particular interest resides in the frequencies corre-lated with the bodies constituting the power train chain at thespeed of 100 km/h (see Table II).

• The frequency associated to the torsional vibration of themotor is 19.60 Hz. αr shows the higher displacement.

• The frequency associated to gear case pitching is 93.99 Hz.αp and αr show the higher displacement.

• The frequency associated with the stator rotation is14.49 Hz. αstat shows the higher displacement.

• The frequency associated with yawing and torsional vibra-tion of the motored axle is 52.92 Hz. αv11, αv12, α11, andα12 show the higher displacement.

III. ELECTRICAL MODEL

The model of the magneto-electric part of the electric drive,which takes on stator ψs = ψsd + jψsq and rotor ψr = ψrd +jψrq fluxes as state variables, can be written using Park equa-tions in the following way (the electric system equations arewritten in terms of space phasor in the stationary referenceframe by using the stator and rotor flux) [1], [11]:

¯ψs = vs − α ψs + βψr¯ψr = δψs − (γ − j ωm) ψr

Te =[pLm m

ψs ψ∗

r

/(Lk Lr)

] (11)

where Ls and Lr are the stator and rotor inductances, vs isthe stator voltage, Lm is the magnetizing inductance, ωm isthe instantaneous mechanical angular speed, p is the numberof pole pairs, Te is the developed electromagnetic torque of theelectrical machines, and

Lk = Ls − L2

m/Lr

α = Rs/Lk, β = RsLm/LkLr

γ = RrLs/LkLr, δ = RrLm/LkLr.(12)


Fig. 3. Electrical model.

TABLE IIIELECTRIC DEGREES OF FREEDOM

Fig. 3 shows the electrical model according to the degrees offreedom, where[

ψs

ψr

]=

[Ls Lm

Lm Lr

]·[

isir

].

IV. COMPLETE ELECTROMECHANICAL MODEL

According to the degrees of freedom of the mechanicalsystem, the angular speed present in the electrical equations is

ωm/p = αr − αstat ⇒ ωm = p(αr − αstat). (13)

On the basis of this substitution, Park equations, in terms ofreal (subscript d) and imaginary (subscript q) components, takeon the following form:

ψsd + αψsd − βψrd = vsd

ψsq + αψsq − βψrq = vsq

ψrd − δψsd + γψrd = −p(αr − αstat)ψrq

ψrq − δψsq + γψrq = p(αr − αstat)ψrd

(14)

where the stator and rotor direct and quadrature flux compo-nents represent the electric degrees of freedom (see Table III).The matrix formulation is

ψ =

ψsd

ψsq

ψrd

ψrq

[I] =

1 0 0 00 1 0 00 0 1 00 0 0 1

Fe =

vsd

vsq

−p(αr − αstat)ψrq

p(αr − αstat)ψrd

[E] =

α 0 −β 00 α 0 −β−δ 0 γ 00 −δ 0 γ

.

(15)

Finally, we have

[I]4,4 ψ4,1

+ [E]4,4 ψ4,1

= Fe4,1. (16)

Replacing the expression for the torque Te =pLmψsqψrd − ψsdψrq/(LkLr) in (2),4 we obtain thefollowing mechanical system in which Fm = F (w, w, ψ):

[M ]19,19w19,1 + [K]19,19w19,1 + [C]19,19w19,1 = Fm19,1.(17)

Whereas the electrical part of the model is formalized interms of equations of the first order, the mechanical one isexpressed by differential equations of the second order. Wemust then transform the mechanical system of 19 differentialequations of the second order in an equivalent system of 38differential equations of the first order. Placing w = y in theequations of the system, we get

[MM ]38,38z38,1 + [CC]38,38z38,1 = FFm38,1 (18)

where

z38,1 =[

w19,1

y19,1

]

[MM ]38,38 =[

[I]19,19 [0]19,19

[0]19,19 [M ]19,19

]

FFm38,1 =[

019,1

Fm19,1

]

[CC]38,38 =[

[0]19,19 −[I]19,19

[C]19,19 [K]19,19

]. (19)

The two systems (electrical and mechanical) can now beunited to form a single system composed of 42 differentialequations of the first order, i.e.,

[Mem]42,42x42,1 + [Cem]42,42x42,1 = Fem42,1 (20)

where

[Mem]42,42 =[

[MM ]38,38 [0]38,4

[0]4,38 [I]4,4

]

x42,1 =[

z38,1

ψ4,1

]

[Cem]42,42 =[

[CC]38,38 [0]38,4

[0]4,38 [E]4,4

]

Fem42,1 =[

FFm38,1

Fe4,1

]. (21)

The analysis of the matrix relations obtained with this pro-cedure depicts how the two mechanical and electrical systems,which are uncoupled as far as the linear components are con-cerned, exclusively interact through the terms that belong to theLagrange component of the active stress.

V. STRUCTURE OF THE CALCULATION PROGRAM

The design of the calculation program implies a reformula-tion of (20) in the following form:

x = [Mem]−1 (Fem − [Cem]x) . (22)

4In particular, the electromagnetic torque is present in the equations thatcorrespond to αr and αstat.


Fig. 4. Flow chart of the calculation model.

The flow chart of the calculation model (Fig. 4) shows howthe vector of the forces, which represents the terms enclosedbetween the round brackets in (22), is initially multiplied bythe inverse of the Mem matrix, thus supplying the accelerationsof the mechanical degrees of freedom and the derivatives ofthe magnetic fluxes. These quantities are then integrated, thusleading to a vector containing the speed, movements, andfluxes. These results are used to trace to the vector of the forces.With this model, we can derive the accelerations, speed, move-ments, and forces of every mechanical and electrical degree offreedom.

VI. METHOD AND INTEGRATION CONDITIONS

The numeric simulation uses the method of integration withvariable steps “ode23s” (stiff-Mod.Rosenbrock) [12]. The max-imum size of the step is assumed to be equal to 0.0001 s.This method of resolution of the one-step type is based onthe modified Rosembrock formula of the second order. This

choice derives from the fact that the problem to study is ofthe stiff type, which means that its solution, while it canchange in a much shorter time interval compared to that forintegration, actually takes on an effective interest over a longertime interval. Methods that are inappropriate for stiff problemswould not be effective over intervals where the solution soughtslowly changes because they use steps small enough to resolveeven the most rapid changes [12], [13].

For the study of the startup transient, which is preliminary tothe analysis of the steady-state condition, we choose the “refineoutput” option, because if the output is very irregular, it allowsus to save a whole number (given the “refine factor” parameter)of additional points between one integration step and the nextone [12], [14].

Concerning the three-phase short-circuit transient, the sim-ulation will be starting from the steady-state condition at thespeed of 100 km/h, as previously elaborated upon. This willallow the identification of the initial state that will be followedby the short circuit.


Fig. 5. Torque acting on αr at a constant speed of 100 km/h.

TABLE IVMAIN HARMONIC COMPONENTS OF TORQUE ACTING ON αr

VII. STEADY-STATE ANALYSIS

The electromechanical model that was developed and usedin this paper does not implement the control system that corre-sponds to the inverter. For this reason, we assume in advancean operation condition for the actuator, which is related to theparticular situation that we wanted to simulate. Since one ofthe main aims of this paper is to characterize the mechanicalstresses on the transmission, we supposed the inverter workwith a square wave, which is surely a severe mechanical con-dition. We also assumed that the train is running at a constantspeed of 100 km/h.

With the aforementioned conditions, the electrical torquein the motor is a periodic function with an average value of358.73 N · m. Performing a Fourier analysis on this torque,it is possible to observe that the fundamental component hasa frequency of 450 Hz with amplitude of 132.15 N · m. Thefrequency of 450 Hz corresponds to six times the operating fre-quency of the inverter (75 Hz). This torque, which is generatedby the motor, is balanced on one side by the transmission andon the other side by the elements that join the stator with thebody of the train.

Analyzing the torsion tensions present on the rotor and theCardan shaft (Fig. 5), it is possible to notice that the resultingtorque oscillates around the zero value with a trend, as shownin Table IV; it shows the spectrum of frequencies of the torqueacting on αr.

As a result, also in a steady-state condition, the rotor and theCardan shaft are exposed to a high level of torque fluctuation(in particular at frequencies of 450 and 50 Hz). It is possible toidentify the inverter as a cause of torque fluctuation at 450 Hzand the Cardan shaft as a source of torque oscillation at 50 Hz.Although these frequencies are not close to the natural fre-

Fig. 6. Torque acting on αu at a constant speed of 100 km/h.

Fig. 7. Torque acting on αp at a constant speed of 100 km/h.

quencies of these machine elements (19.60 Hz), they haveamplitudes and frequencies that may cause damages for fatigue.

In addition, the torque on the degrees of freedom αu and αp

shows oscillations around the zero (Figs. 6 and 7). Analyzingthe spectrum of frequencies of these two degrees of freedom,we observe that the contributions at high frequencies, whichare generated by the inverter, have been filtered by the systemrotor–Cardan (as shown in Table V). However, these harmoniccomponents have small amplitudes and frequencies that are notclose to the natural ones.

Furthermore, the analysis of stress fluctuations on the trans-mission chain is completed by studying the trend of the torquegenerated by the friction between wheel and rail. In detail,it has been considered that the torque operates on driverwheel 21. The spectrum of frequencies (Fig. 8) shows a wavewith the fluctuation less accentuated compared to those presentin the other components. Furthermore, it is also possible to ob-serve that we practically have no contribution at high frequen-cies (Table V). The torque on the rotational degree of freedomof driver wheel 21 (α21) shows oscillation almost exclusively


TABLE VMAIN HARMONIC COMPONENTS OF TORQUE ACTING ON αu AND αp,

AND BY THE FRICTION BETWEEN WHEEL AND RAIL

Fig. 8. Spectrum of frequencies generated by the friction between wheel andrail at a constant speed of 100 km/h.

at a frequency of 49.82 Hz; hence, they are generated by theCardan. This frequency is close to the frequency previouslycalculated by the modal analysis.

Finally, the analysis of the electrical quantities shows thatthe harmonic stresses generated by the mechanical system, inparticular those due to the nonhomokinetic behavior of theCardan shaft, exist both in the rotor flux and in its derivativewith respect to time. Such components are missing in the fluxof the stator because of the weak feedback between stator androtor.

It is remarkable to underline that only with the analysisof the whole electromechanical system is it then possible toquantitatively characterize the mutual influence of the electricalsystem with the mechanical system. Such interaction would notappear if we separately conduct the analysis of the electricaland mechanical systems [4], [10].

VIII. THREE-PHASE SHORT-CIRCUIT FAULT ANALYSIS

A sudden short circuit normally causes severe electric andmechanical oscillations, and may also produce a significanttorque oscillation [3]. Thanks to the electromechanical com-plete model, it is now possible to study the effects due to theelectrical short circuit on the mechanical part. Starting fromthe steady-state condition analyzed in Section VII, which isidentified by an initial speed of 100 km/h, we simulate the short-circuit fault with the imposition of the simultaneous cancelingof the three-phase stator voltages.

In the instants immediately following the fault, the electrictorque is subjected to quite high peaks. As a consequence of

Fig. 9. Electromagnetic torque during the short-circuit fault.

Fig. 10. Torque acting on the rotor during short-circuit fault.

no power supply, the torque becomes equal to zero when thetransient is finished (Fig. 9). The dynamic torque that followsthe short circuit reaches values of approximately 10–20 timeshigher than those before the fault. Its transient overelongationhas consequences over all of the system’s components; inparticular, acting on the transmission organs, it may cause anoverstress that is very dangerous for the mechanical structures.

With particular reference to the transmission organs, the mostimportant actions are those affecting the degrees of freedom,i.e., motor-induced rotation (αr), transmission output rotation(αu), and gear case rotation (αp). After the short circuit, thetorques acting on the transmission organs reach peaks whosemodules are approximately 25 times higher to the preexistingone before the fault for the rotor, 100 times higher for thetransmission output, and 60 times higher for the gear case(Figs. 10–12).

It is also remarkable that, once the short-circuit transient isfinished, one force characterized by an oscillating motion is stillleft (Fig. 13). This stress can be attributed to the interactionof the only mechanical components of the system, given thatthe air gap torque is zero. Its effect reflects on the kinematicsmagnitude of speed (Figs. 14 and 15) and acceleration (Figs. 16and 17), which, even when the transient is over, will continueto have this oscillating motion. Once the fault is finished, themotor is no longer supplied: the action of the electrical torquestops, and what remains is only that of the resistant torque.


Fig. 11. Torque acting on the transmission output during short-circuit fault.

Fig. 12. Torque acting on the bridge during short-circuit fault.

Fig. 13. Torque acting on the rotor after short-circuit fault.

With regard to the degrees of freedom αr and αu, it can benoticed that the force has a negative average value (Fig. 13).It follows that the acceleration also has a negative averagevalue (Fig. 16), and as these degrees of freedom belong tothe kinematics transmission chain of motion, the speed willdecrease over time (Fig. 15).

On the other hand, the degree of freedom αp does not havea large motion, as it does not belong to the kinematics trans-

Fig. 14. Angular speed of transmission output during short-circuit fault.

Fig. 15. Angular speed of the gear case (a) during and (b) after short-circuitfault.

mission chain of motion, but only has a slight motion aroundits own equilibrium position (Fig. 18). In this case, the forcehas a positive average value as the gear case is an element ofreaction, whereas acceleration and angular speed always havean oscillating gait around the zero value (Figs. 15 and 17).

IX. CONCLUSION

The results of the analysis proposed in this paper have con-firmed the strong presence of a mutual influence between themechanical and electrical variables: they are closely correlated


Fig. 16. Acceleration of transmission output during short-circuit fault.

Fig. 17. Acceleration of gear case (a) during and (b) after short-circuit fault.

to each other. The obtained results have demonstrated that theelectrical and mechanical parts cannot be considered as twoseparate worlds, and for this reason, they cannot be describedwith two distinct independent models.

In fact, the steady-state analysis has underlined stress dueto the torque harmonic components generated by the electricdrive under a nonsinusoidal condition operation and by thenonhomokinetic behavior of the Cardan shaft acting on theorgans of transmission, in particular on the Cardan shaft itself.At the speed of 100 km/h, the rotor torque shows harmoniccomponents with amplitude of 10% of its average value at

Fig. 18. Rotation of the gear case during short-circuit fault.

the frequency of 50 Hz and of 37% at the frequency of450 Hz. These stresses are also found in the Cardan joint,which, during normal operation, has often been subjected tobreakage for fatigue.

These results, which are based on Lagrange–Park theoriesand the use of Simulink, if compared to those previouslyobtained using only a mechanical model [4] and an electricalmodel [10], allow emphasis on the role of the electrical part onthe mechanical one; the electric drive generates a torque with ahigh number of harmonic components that, during the normalworking operation, may create problems of fatigue on themechanical parts, joining the stator with the body of the train.On the electrical side, the harmonic components generated bythe mechanical system also act on the electrical parts and, inparticular, on the rotor flux.

Furthermore, the analysis of the short-circuit fault has fo-cused on the problem of resistance due to the higher excur-sions of transient torques. In particularly critical conditions,for example, starting up or electrical faults (short circuit),transient torques arise with high amplitude: Following the shortcircuit, the torques acting on the transmission organs reachpeaks whose module for the rotor is approximately 25 timeshigher than the preexisting one in the moment before thefault, 100 times higher for transmission output, and 60 timeshigher for the gear case. These aspects must be consideredto come to a correct design of the mechanical and electricalorgans.

Finally, it is important to underline that such a model mayhave a double utility. From one side, it allows an accurate studyof problems strictly correlated to the dynamics of the bogie,considering the mechanical quantities provided by the electricalmotor (torque and angular speed) as inherent to the model itselfand no longer as an external input a priori. On the other side,it also allows the study of the influence of the mechanical partonto the electrical part, emphasizing all the problems relatedto the undesired harmonics in the power line, as well as tocarry out a diagnostic process that, starting from the analysis ofthe electrical quantities (which is much easier to monitor thanthe mechanical quantities), can identify the wear status of themechanical transmission chain.


APPENDIX

ELEMENTS OF THE MATRIX OF INERTIA [M ],RIGIDITY [C], AND DAMPING [K]

ACKNOWLEDGMENT

The authors would like to thank the engineers and designersfrom Fiat Ferroviaria (now Alstom), in particular A. Magnaniand M. Ponassi, for their precious and detailed cooperation.

REFERENCES

[1] P. Vas, Electrical Machines and Drives: A Space-Vector Theory Approach.Oxford, U.K.: Clarendon, 1992.

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[4] A. Elia, E. Ponassi, and P. Gatti, WCML Drive Line Torsional Calcu-lations. Savigliano, Italy: Fiat Ferroviaria S.p.A., Tech. Dept., 1999.n 23/99/GB.

[5] W. Thomson, Theory of Vibration With Applications, 4th ed. London,U.K.: Chapman & Hall, 1993.

[6] F. Schmelz, H. C. Seherr-Thoss, and E. Aucktor, Universal Joints andDriveshafts. Berlin, Germany: Springer-Verlag, 1992.

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[8] W.-S. Kim, Y.-S. Kim, J.-K. Kang, and S.-K. Sul, “Electro-mechanical re-adhesion control simulator for inverter-driven railway electric vehicle,” inProc. IEEE 34th Ind. Appl. Soc. Annu. Meeting, Oct. 3–7, 1999, vol. 2,pp. 1026–1032.

[9] J. J. Kalker, “On the rolling contact of two elastic bodies in the pres-ence of dry friction,” Ph.D. dissertation, Techn. Hogeschool, Delft, TheNetherlands, 1967.

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[12] Matlab The Language of Technical Computing. Ver. 5.3, OnlineManuals.

[13] K. Atkinson, Elementary Numerical Analysis. New York: Wiley, 1995.[14] L. F. Shampine and M. W. Reichelt, “The MATLAB ODE suite,” SIAM

J. Sci. Comput., vol. 18, no. 1, pp. 1–22, 1997.

Sonia Leva (M’01) received the M.S. and the Ph.D.degrees in electrical engineering from Politecnico diMilano, Milan, Italy, in 1997 and 2001, respectively.

Since 1999, she has been an Assistant Profes-sor with the Electrical Engineering Department,Politecnico di Milano. Her current research concernsthe electromagnetic compatibility, power quality, andfoundation of electromagnetic theory of the electricnetwork.

Adriano Paolo Morando received the M.S. degreein electrical engineering from Politecnico di Milano,Milan, Italy.

From 1984 to 1989, he was with ASEA BrownBoveri, Milan, where his activity was concerned withac drives for electrical traction. He is currently anAssociated Professor with the Electrical Engineer-ing Department, Politecnico di Milano. His currentresearch concerns the electromagnetic compatibility,power quality, and foundation of electromagnetictheory of the electric network.

Paolo Colombaioni received the M.S. degree inmechanical engineering from Politecnico di Milano,Milan, Italy, in 2001.

He was a Product Engineer with the TransmissionComponents Plant, Ithaca, NY. Since 2001, he hasbeen with BorgWarner Morse TEC Europe, Milan,where he is currently responsible for the Chain Engi-neering Group.

dynamic analysis of a high-speed train

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