Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 4, pp. 869-878, Warsaw 2008

DYNAMICS OF THE CATENARY MODELLED BY

A PERIODICAL STRUCTURE

Anna Kumaniecka

Cracow University of Technology, Institute of Mathematics, Cracow, Poland

e-mail: pukumani@cyf-kr.edu.pl

Jacek Snamina

Cracow University of Technology, Institute of Applied Mechanics, Cracow, Poland

e-mail: js@mech.pk.edu.pl

In the paper, a dynamic analysis of a two-level model of the catenary
which takes into accountperiodic distribution of hangers and supports is
provided. An analytical method is proposed for calculating the response
of the catenary to a uniformly moving pantograph. The model of the
catenary is composed of two strings (the contact and carrying cables)
connected by lumped mass-spring-dashpot elements equidistantly posi-
tioned along the strings. These elements are assumed to be visco-elastic.
The pantograph is modelled by a concentrated force whichmoves along
the contact cable. The force exerted by the pantograph varies harmoni-
cally. This model is capable of describing coupled wave dynamics of the
catenary. The proposed method of calculations is based on the Fourier
transformation and, therefore, is applicable only to linear models of the
catenary. In the analysis, the periodicity condition is used. The spectral
analysis is carried out. General results are illustrated by a numerical
example in which the effect of wave propagation is visible.

Key words: speed railways, pantograph-catenary system, wavemotion

1. Introduction

The pantograph-catenary interaction at high speeds is the critical factor for
reliability and safety of high speed railways. In the past years, many rese-
archers studied how to improve current collection quality in order to reduce
wear andmaintenance costs of both overhead line and pantograph. To impro-
ve the pantograph-catenary interface it is essential to better understand the



870 A. Kumaniecka, J. Snamina

complex behaviour of this couple. Transportation efficiency is for the Polish
Railways a constant objective to improve the service to users and competi-
tiveness in Europe. Current collection is, among other railway subsystems,
a major functionality. Any failure can have important financial consequen-
ces. Therefore, researchers have initiated numerous projects to improve the
pantograph-catenary interface.

Recently, it has been proposed to replace conventional droppers by more
sophisticated rubber or friction dampinghangers. Introduction of the hangers,
which aremuch stiffer in comparison with the conventional droppers, leads to
much more intense interaction between the contact and carrying cables. To
account for this interaction, coupled vibration of the catenary cables should
be examined.Numerous studies on rail vehicles carried out over the years have
proved that the processes describing their dynamic state have a complex and
non-periodic character.

The aim of the paper is to present a dynamic analysis of a two-level mo-
del of the catenary taking into account periodic distribution of hangers and
supports. The theoretical problem, which is discussed in the paper, belongs
to technical problems connected with dynamics of the periodic systems under
moving loads. The emphasis of these studies is placed on the description of
wave propagation (Snamina, 2003). Both contact and carrying wire are one-
dimensional system, in which mechanical waves excited by moving and fixed
sources can propagate.

Thepresentedmodel belongs to the class of periodically non-homogeneous,
continuous systems excited by a uniformly moving load. Such systems were
studied in the past by a number of researchers employing different methods.
Bogacz et al. (1993) based their approach on the Flouquet theorem. Mead
(1971), Jezequel (1981) and Szolc (2003) applied the Fourier series technique.
Thedynamic interactionbetweenadiscrete oscillator of twodegrees of freedom
and a continuous string was studied by Kumaniecka and Snamina (2005).
They applied the theory of one-dimensional elastic wave to the description of
transient processes caused by an impulsive load. Using the Laplace transform,
they solved the equation of motion of the contact wire. Metrikine and Bosch
(2006) applied the periodicity condition and analyzed deflection of the contact
cable and the contact force between the cable and rubber or friction hangers
to estimate the fatigue life of hangers. More reference on that subject can be
found in the book by Frýba (1999).

In the paper, an analytical method for calculating the response of a ca-
tenary to a uniformly moving pantograph is presented and, because of easier
physical interpretation, the periodicity conditionmethod is chosen. To the au-



Dynamics of the catenary modelled... 871

thor’s knowledge, the response of such a system to amoving load has not been
investigated so far.

Theaimof thepaper is to obtainabetterunderstandingof thepantograph-
catenary system dynamics. A relatively simple analytical model, presented in
the paper, is appropriate to gain the physical insight into the pantograph-
catenary system. The paper is organised in five sections. Following the intro-
duction, the two-level model of the overhead contact system is presented in
Section 2. In Section 3, equations of system vibration are derived and their
solution using the method of Fourier’s transform is obtained. In Section 4,
some results of numerical calculations are presented. Concluding remarks are
formulated in Section 5.

2. Model of the system

In the literature, many models of the catenary-pantograph system have been
proposed. The contact and carrying cables have been modelled by infinite or
non-infinitehomogenous stringsorBernoulli-Eulerbeams.Thepantographhas
beenmodelledbya concentrated force or by anoscillator of two or fourdegrees
of freedom. A review paper describing the pantograph-catenary systems was
presented by Poetsch et al. (1997) and by Kumaniecka (2004).

Fig. 1. Physical model of the system

The simplified model of the catenary, introduced in the paper shown in
Fig.1, is composed of twoparallel, infinitely longhomogenous strings (the con-
tact and carrying cables) connected by lumpedmass-spring-dashpot elements
(the suspension rods), which are positioned equidistantly along the strings.
The upper string (the carrying cable) is fixed at periodically spaced support



872 A. Kumaniecka, J. Snamina

points (supports) x = nl, n = 0,±1,±2, . . .. The lower string (the contact
wire) is suspended from the upper string by means of visco-elastic elements.
These elements are used as amodel of suspension rods. They are periodically
placed at points x = klw, k = 0,±1,±2, . . . along the strings. It is assumed
that the distance l is a multiple of lw. It can be written then l= rlw, r∈N.

Thus the system in question has two spatial periods. The larger period is
introduced by the supports of the carrying cable, whereas the smaller period
is associated with the droppers.

The system in question is subjected to a concentrated force (model of the
pantograph), which is applied to the lower string. This load moves along the
lower string at a constant velocity v and oscillates in time harmonically. The
force appearing between the contact wire and the pantograph is the moving
source of waves, and forces in the suspension elements are fixed sources of the
waves.

Between the contact wire and the pantograph there also appears the fric-
tion force. In the present study, the friction force is neglected.

A description of motion of the whole system can be obtained bymeans of
equations which govern small vertical motion of each spring about its equili-
brium and continuity conditions at the suspension and support points.

3. Equations of the system vibration

The analysis of processes associated with the pantograph motion is provi-
ded in the fixed co-ordinate (x,y) system. The equations which govern small
transverse vibrations of the strings about their equilibrium, induced by the
transversal force moving along the lower string, have the following form

ρ1A1
∂2w1
∂t2
−T1
∂2w1
∂x2

=F(t)δ(x−vt)

(3.1)

ρ2A2
∂2w2
∂t2
−T2
∂2w2
∂x2

=0

In the above equations x denotes the spatial horizontal co-ordinate, t – ti-
me, subscripts 1 and 2 are related to the lower and upper wire, respectively,
wi denotes the vertical displacement of the strings, ρi –material density, Ai –
cross-sectional area, ρiAi – mass per unit length of the strings, Ti – tension
of the strings, F(t) is the moving force and δ(·) is the Dirac delta function.



Dynamics of the catenary modelled... 873

At the suspension points, where the visco-elastic elements are fixed, x=
klw 6=nl, the following boundary conditions have to be satisfied

w1|x=kl+w
−w1|x=kl−w

=0 w2|x=kl+w
−w2|x=kl−w

=0

T1
(∂w1
∂x

∣∣∣∣
x=kl

+
w

−
∂w1
∂x

∣∣∣∣
x=kl

−

w

)
=
[
c(w1−w2)+ b

∂

∂t
(w1−w2)+m

∂2w1
∂t2

]∣∣∣∣
x=klw
(3.2)

T2
(∂w2
∂x

∣∣∣∣
x=kl

+
w

−
∂w2
∂x

∣∣∣∣
x=kl

−

w

)
=
[
c(w2−w1)+ b

∂

∂t
(w2−w1)

]∣∣∣∣
x=klw

The first two equations are the continuity conditions at the suspension
points. The second two follow from the dynamic equilibrium conditions of
vertical forces at the suspension points. These conditions can be obtained as
a result of balancing the inner forces acting on the left and right-hand sides of
the suspension points.

The fixation of the upper wire at x=nl can be described as follows

w2|x=nl =0 (3.3)

Additionally, for x=nl (supportpoints of theupper string), theboundary
conditions may be expressed in the form

w1|x=nl+ −w1|x=nl− =0
(3.4)

T1
(∂w1
∂x

∣∣∣∣
x=nl+

−
∂w1
∂x

∣∣∣∣
x=nl−

)=
(
cw1+ b

∂w1
∂t
+m
∂2w1
∂t2

)∣∣∣∣
x=nl

In Eqs. (3.1)-(3.4) the following notation is used: c, b – stiffness and dam-
ping coefficients of the suspension rods, m – equivalentmass of the suspension
rods, l and lw – spatial periods of the support and suspension points, respec-
tively, k,n∈N.

The model shown in Fig.1 is periodic, i.e. its parameters vary periodi-
cally with the co-ordinate x. In steady-state motion, under a harmonically
oscillating load

F(t)=F0e
iΩt (3.5)

that moves along the strings at a constant speed, the following periodicity
condition must be satisfied (Metrikine and Bosch, 2006)

wi(x,t) =wi
(
x+nl,t+

nl

v

)
e
−iΩnl
v i=1,2 (3.6)



874 A. Kumaniecka, J. Snamina

Physically, the periodicity condition implies that the displacement pattern
of the strings repeats itself in timewith the period l/v. Displacements at a gi-
ven point x at themoment t are strictly connected with displacements at the
point x+nl at themoment t+nl/v. The time nl/v and distance nl are asso-
ciated with the load that moves along the lower string at a constant speed v.
The factor exp(−iΩnl/v) introduces a phase shift between the displacements.
The steady-state solution was obtained analytically by transforming the

problem into a frequency domain by means of Fourier transform. The frequ-
ency analysis of systemmotion can be then carried out.
Denote the Fourier transform of the displacement wi(x,t) by w̃i(x,ω),

i=1,2

w̃i(x,ω)=

∞∫

−∞

wi(x,t)e
−iωt dt (3.7)

Application of this transform to equations (3.1) gives

∂2w̃1
∂x2
+
ω2

c21
w̃1 =−

F0
T1v
e
i(Ω−ω)
v
x

∂2w̃2
∂x2
+
ω2

c22
w̃2 =0 (3.8)

where ci =
√
Ti/(ρiAi) are the wave speeds in the strings i=1,2.

These equations are linear with respect to w̃i(x,ω), i=1,2.
ApplyingFourier’s transformation, boundary conditions (3.2) can bewrit-

ten as

w̃1|x=kl+w
− w̃1|x=kl−w

=0 w̃2|x=kl+w
− w̃2|x=kl−w

=0

T1
(∂w̃1
∂x

∣∣∣∣
x=kl

+
w

−
∂w̃1
∂x

∣∣∣∣
x=kl

−

w

)
=

= [c(w̃1− w̃2)+ iωb(w̃1− w̃2)−mω
2w̃1]
∣∣
x=klw

(3.9)

T2
(∂w̃2
∂x

∣∣∣∣
x=kl+w

−
∂w̃2
∂x

∣∣∣∣
x=kl−w

)
= [c(w̃2− w̃1)+ iωb(w̃2− w̃1)]

∣∣
x=klw

Application of Fourier’s transformation to equations (3.3) and (3.4), gives

w̃2|x=nl =0 w̃1|x=nl+ − w̃1|x=nl− =0
(3.10)

T1
(∂w̃1
∂x

∣∣∣∣
x=nl+

−
∂w̃1
∂x

∣∣∣∣
x=nl−

)
=(cw̃1+iωbw̃1−mω

2w̃1)
∣∣
x=nl

The periodicity condition, after application of Fourier’s transformation,
may be expressed in the form

w̃i(x,ω)= w̃i(x+nl,ω)e
−i(Ω−ω)nl

v (3.11)



Dynamics of the catenary modelled... 875

Solving equations (3.8), the functions which describe Fourier’s transforms
of displacements can be written as

w̃1(x,ω)=C
(k)
1 e

iωx
c1 +C

(k)
2 e

−iωx
c1 +Cse

i(Ω−ω)x
v

klw ¬x¬ (k+1)lw k=0,±1,±2, . . . (3.12)

w̃2(x,ω)=D
(k)
1 e

iωx
c2 +D

(k)
2 e

−iωx
c2

with the constant Cs

Cs =
F0v

T1

1

(Ω−ω)2− ω
2v2

c
2
1

(3.13)

Equations (3.12) in eachneighbouring intervals, constrainedby the support
points of the upper string, can be coupled usingperiodicity condition (3.11). It
is sufficient to express w̃i(x,ω), i=1,2 in the interval x∈ [0, l]. The solution
can be extended to the interval x∈ [l,2l] by employing periodicity condition
(3.11).
As it follows fromequations (3.12), there are 4runknowncoefficientswhich

should be determined. Taking into account boundary conditions (3.9) and
(3.10) the number of which in the integral x ∈ [0, l] is 4r+2, and substi-
tuting (3.12) and (3.13) into the boundary conditions, a set of 4r+2 linear
algebraic equations is obtained with respect to the unknown coefficients. Sin-
ce the boundary conditions for x = 0 and x = l have constants associated
with the solutions for neighbouring intervals x∈ [−l,0] and x∈ [l,2l], there
occurs 4r+4 unknown coefficients in the set of algebraic equations. Apart
from the equations following from the boundary conditions, some equations
resulting from the periodicity condition should be taken into consideration.
The linear set of equations with respect to the unknown constants is non-
homogeneous and can be readily solved numerically for given parameters of
the system and arbitrarily chosen values of ω. Substituting the determined
constants to equation (3.12), the Fourier transform w̃i(x,ω), i = 1,2 can be
derived. The displacement spectra for any points of the system can be found.
These amplitude spectra of the displacements are the base for analysis of the
system in question.

4. Numerical analysis

Theequations presented in theprevious section canbeused todescribemotion
of the considered system. In this section, a steady-state dynamical response



876 A. Kumaniecka, J. Snamina

of the catenary is studied. The numerical calculations are carried out for the
following parameters of the system

l=64m r=4 lw =16m

v=50m/s T1 =19600N T2 =9800N

c=1500kg/m Ω=2Hz ρ1A1 =2.175kg/m

ρ2A2 =1.5kg/m b=50Ns/m c1 =94.93m/s

c2 =80.83m/s m=0.4kg

These data correspond to the parameters of real overhead power lines for
high-speed trains.

Making use of equations (3.12), amplitude spectra of displacements of two
points were derived. The x-coordinate of each point is equal to 8m. The po-
ints are placed on the contact wire and the carrying cable. The results of
calculations are presented in Fig.2.

Fig. 2. Amplitude spectrum of displacement at x=8m: (a) contact wire,
(b) carrying cable

The presented spectra have a similar pattern. Motion of the contact and
carrying cable is dominated by two harmonic components. Their frequencies
are equal to 1.2Hz and 5.2Hz. Spectra of other points of the cable system are
similar. The harmonically varying concentrated force moves at the constant
velocity v = 50m/s, which is about a half of the wave speed in the contact
cable. The cyclic frequency Ω of the load is equal to 2Hz.



Dynamics of the catenary modelled... 877

5. Conclusions

In paper the application of one-dimensional elastic wave propagation theory
to the analysis of pantograph-catenary interaction is discussed. A simplified
model of the pantograph and catenary has been proposed. The equations of
motion are based on the string model with a concentrated, harmonically va-
rying force moving along the contact wire at a constant velocity.

Calculations have been done for the casewhen the load velocities are lower
than the speed of transverse waves in the cable. Concluding, we can state that
any time motion of the cable has a wavy character. The calculations confir-
med that the travelling force is a source of two waves propagating leftwards
and rightwards with different frequencies (Kumaniecka and Snamina, 2005).
The wave propagating in front of the load has a frequency greater than Ω,
whereas the wave propagating behind the load has a frequency lower than Ω.
Because of wave reflection and motion of hangers, the amplitude spectra of
each point show two dominant harmonic components. The described pheno-
mena should be taken into account in real applications concerning high-speed
railway structures.

References

1. BogaczR.,KrzyżyńskiT., PoppK., 1993,Ondynamics of systemsmodel-
ling continuous and periodic guideways,Archives of Mechanics, 45, 5, 575-593

2. FrýbaL., 1999,Vibration of Solid and StructureUnderMoving Loads,Telford,
London

3. Jezequel L., 1981, Response of periodic systems to a moving load, Journal
of Applied Mechanics – Transactions of the American Society of Mechanical

Engineers, 48, 603-618

4. Kumaniecka A., 2004, Problems of modelling of the catenary-pantograph
system,Proc. X Workshop PTSK Simulation in Researches and Development,
Cracow, 167-174

5. Kumaniecka A., Snamina J., 2005, Transient response of the pantograph –
catenary systemto impulsive loads,Machine Dynamics Problems,29, 2, 91-105

6. Mead D.J., 1971, Vibration response andwave propagation in periodic struc-
ture, Engineering for Industry – Transactions of the American Society of Me-
chanical Engineers, 93, 783-792



878 A. Kumaniecka, J. Snamina

7. Metrikine A.V., Bosch A.L., 2006, Dynamic response of a two-level cate-
nary to moving load, Journal of Sound and Vibration, 292, 676-693

8. Poetsch G., Evan J., Meisinger R., Kortum W., Baldauf W., Ve-
itl A.,Wallaschek J., 1997,Pantograph/CatenaryDynamics andControl,
Vehicle System Dynamics, 28, 159-195

9. Snamina J., 2003, Mechanical Wave Phenomena in Overhead Transmission
Lines, CracowUniversity of Technology,Monograph 287

10. Szolc T., 2003,Dynamic Analysis of Complex Discrete-ContinuousMechani-
cal Systems, IFTRReports,Warsaw

Dynamika sieci trakcyjnej modelowanej strukturą periodyczną

Streszczenie

W pracy przedstawiono wyniki analizy dynamicznej modelu sieci trakcyjnej
z uwzględnieniem charakterystycznej powtarzalności fragmentów konstrukcji zawie-
szenia przewodu jezdnego i wieszaków. Podano rozwiązanie problemuwspółdziałania
sieci z poruszającym się ze stałą prędkością odbierakiem prądu. Model sieci składa
się z dwóch nieskończenie długich strun oddziaływujących ze sobą poprzez sprężysto-
-tłumiące elementy rozłożone periodycznie. Oddziaływanie odbieraka prądu na dolną
strunę modelowane jest harmonicznie zmienną siłą. Stosując transformację Fourie-
ra, przeprowadzono analizę częstotliwościową ruchu układu. Wykorzystano warunek
periodyczności. Rezultaty rozważań analitycznych zobrazowano przykładem nume-
rycznym.

Manuscript received March 28, 2008; accepted for print July 9, 2008