Original Article
Static form-finding of normal and
defective catenaries based on the
analytical exact solution of the
tensile Euler–Bernoulli beam
Farzad Vesali
1
, Mohammad Ali Rezvani
1
,
Habibolah Molatefi
1
and Markus Hecht
2
Abstract
The aim of this research is to propose and develop an analytical exact solution for finding the static equilibrium con-
figuration of a catenary before and after incurring defects such as tension loss or a broken dropper. The procedure
includes considering the steady-state solution of the dynamic motion equation of the contact wire and the messenger
cable. The wire and the cable are considered as tensile Euler–Bernoulli beams. The stiffness matrix of the beam is
configured and is used to calculate the dropper’s dead load. Progressively, a novel method is proposed to find the
equilibrium configuration of the same catenary after the defect. The results prove that the tension loss in the messenger
cable is more precarious than the tension loss in the contact wire. The broken dropper causes a significant sag in the sub-
span and increases the static forces of the adjacent droppers. A comparison with field measurements justifies the
accuracy of the results of the proposed model.
Keywords
Catenary, static form-finding, broken dropper, tension loss, exact solution
Date received: 8 April 2018; accepted: 12 September 2018
Introduction
Background
Most of the high-speed trains collect electrical power
from the overhead catenary system. Therefore, the
dynamic interaction between the pantograph and the
catenary should be studied in order to analyze stable
current collection and to ensure a good contact qual-
ity. Since the static initial condition of the catenary
including the shape of the contact wire and the mes-
senger cable has significant role on the interaction
between the pantograph and the catenary system,
any practical model needs to estimate the initial
shape of the contact wire and the messenger cable.
Formulation of the problem of interest for this
investigation
Several models are available for simulating the
dynamic interaction of the pantograph and the caten-
ary.
1
Facchinetti and Bruni
2
stated that any method
to determine the static position of the catenary needs
to include some key modeling issues in the panto-
graph and catenary interaction. Additionally, the
literature proves that among the many methods avail-
able for the calculation of the static force of droppers,
the outcomes are not the same. The differences in the
results are considerable.
1
A comparison between the
static forces (N) of droppers that are calculated using
various finite element model (FEM) or finite differ-
ences method (FDM) software programs proves a
high standard deviation of force of each dropper
between the results. Therefore, the aim of this
research is to develop an accurate, fast method for
calculating the static forces of droppers for the caten-
ary systems.
1
School of Railway Engineering, Iran University of Science and
Technology, Tehran, Iran
2
Department of Rail Vehicle, Technical University of Berlin, Berlin,
Germany
Corresponding author:
Mohammad Ali Rezvani, Iran University of Science and Technology,
Narmak, Tehran 16846-1311, Iran.
Email: [email protected]
Proc IMechE Part F:
J Rail and Rapid Transit
2019, Vol. 233(7) 691–700
!IMechE 2018
Article reuse guidelines:
sagepub.com/journals-permissions
DOI: 10.1177/0954409718808990
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Literature survey
PantoCat
3
is a model that uses an optimization tech-
nique to minimize the function that quantifies the
distance between the static deformed geometry of
the contact wire and its specified position. SPOPS
4
uses an iterative method to find the length of each
dropper and the initial configuration of the catenary.
PCaDA
4
uses an iterative procedure in which a non-
linear optimization is used in each step to drive the
catenary geometry and tension in the contact wire
and the messenger cable. The above-mentioned soft-
ware and other software perform simulations that
are based on FEM or FDM. Therefore, it makes
sense when the iterative or the optimization methods
are used in these software in order to calculate the
static form of the catenary. There are several defin-
itions for form-finding or estimation of the initial
equilibrium state of a form-active structure.
In other words, it is a method of finding an optimal
shape of a structure that is in a state of static equi-
librium.
5
The existing form-finding methods for dis-
crete networks, such as the force density method,
dynamic relaxation, updated reference strategy and
a variety of other methods, are discussed and com-
pared.
6
In the overhead catenary system, the length
of the dropper should be adjusted in such a way that
the contact wire sets in a planned profile. In fact,
when dealing with laying the catenaries, the shape
of one element in the assembly is defined and the
shape of the other elements in addition to the inter-
action forces needs to be calculated. In this field, it is
then rather natural to calculate the initial equilib-
rium configuration of the railway catenary. For
example, an accurate, robust and fast method
based on analytical catenary equations for a two-
dimensional form-finding problem was previously
proposed by Lopez-Garcia et al.
7
Such et al.
8
extended this method to three-dimensional tension
cable structures. In addition, the geometry variation
method,
9
the separated model method,
10
the negative
sag method,
11
the optimization methods on con-
straint problems,
12
the minimal error of the dropper
tension,
13
minimizing the distance between the static
deformed geometry of the contact wire and its spe-
cified position
3
are able to evaluate the static config-
uration of the catenary. Recently, Yang et al.
14
proposed a dynamic equilibrium method to compute
the initial equilibrium configuration of the railway
catenary. The initial configuration of the catenary
is used not only to simulate the dynamic reaction
of the pantograph and catenary
15
but it can also
be used to make geometric optimization on the
catenary parameters.
16
In addition, the exact solu-
tion of the static configuration of a defective caten-
ary is useful to monitor the catenary by pantograph,
i.e. when it is possible to simulate the effect of any
defect on pantograph acceleration, it would be pos-
sible to measure the pantograph acceleration and
estimate the type of defect in the catenary.
Contribution of this study
Prior studies have attempted to calculate the static
force of droppers in the catenary systems with notice-
able low accuracies. However, this study proposes not
only the exact solution for calculating the static force
of droppers but also the method for evaluating the
static form of catenary after three types of defects,
including tension loss in messenger cable, tension
loss in contact wire and the breaking of one dropper.
Organization of this article
Primarily, the analytical model of the catenary is
developed, and the stiffness matrix of the contact
wire is calculated. This is required in order to find
the forces in the droppers. Configuration of the con-
tact wire and the messenger cable is then generated
by relying on the steady-state solution of the
dynamic equation of the tensile beam. Therefore,
according to the classification by Veenendaal and
Block,
6
the method that is presented in this research
can be considered as a combination of the stiffness
matrix method and the dynamic equilibrium method.
Then, the calculated force of each dropper is
used for evaluating the static form of the catenary
after defect.
Description of the catenary model
This research proposes a two-dimensional analytical
exact solution for form-finding of catenaries. To serve
the purpose, the following questions need to be
worked out:
Q1: What is the size of the force in each dropper that
can set the contact wire on its proper configuration
(Figure 1(a))? To elaborate further, when the long
tensile beam is considered as the model of the con-
tact wire and it is possible to apply force in loca-
tion of dropper connection points, what is the best
force distribution to put the contact wire in a
demanded profile? Should it be distributed uni-
formly, or its distribution depends on the distance
between the droppers?
Q2: What will be the configuration of the contact wire
and the messenger cable after applying forces of
the droppers? Clearly, after applying the calculated
forces of the droppers, the configuration of the
contact wire should be the same as in the
demanded profile.
Q3: What will be the new equilibrium shape of the
contact wire and the messenger cable after experi-
encing the defects such as the loss of tension of the
contact wire or the messenger cable? i.e. what
would happen if one dropper breaks (Figure 1(b)).
In Figure 1, Tmc and Tcw are the tension of the
messenger cable and the contact wire, respectively.
ksand kdr are the stiffnesses of the support and the
692 Proc IMechE Part F: J Rail and Rapid Transit 233(7)
registration arm. kdri is the stiffness of the ith dropper.
fdri is the static force or the dead load of the ith drop-
per that will be calculated in this research.
For the modeling purposes, the messenger cable
and the contact wire are considered within the tensile
Euler–Bernoulli beam theory. The droppers are con-
sidered as linear springs. The dropper clamps and the
registration arms are considered as concentrated
masses. The vertical stiffness of the registration arm
is a function of the tension of the contact wire (TcwÞ,
length of the stager (dstagger), length of the registration
arm (lRegistration) and span length (lspanÞ, as given in
equation (1)
kr¼2Tcw dstagger
lRegistration lspan
þ1
lspan
ð1Þ
The parameters that are used in equation (1) are
presented in Figure 2 and the definitions are given in
the Appendix.
The properties for the contact wire and the
messenger cable that are schematically shown in
Figure 1 are presented in Table 1. These properties
are extracted from the standard document EN
50318, 2002.
Formulation and the mathematic model
Both the contact wire and the messenger cable are
considered as tensile beams and the droppers are
modeled as nonlinear springs with significant stiffness
in tension and zero stiffness in compression. Since all
droppers in the static form of the catenary are under
tension, the nonlinearity of the droppers can be
ignored in the static form-finding. The differential
equation governing the tensile beam can be described
according to equation (2), and the related boundary
condition compatible with the modeling that is used in
Figure 1 is presented in equation (3). The boundary
condition contains zero deflection and momentum in
supports
Aw,tt þCw,tTw,xx þEIw,xxxx ¼FgþFdxðÞ,04x4l,t50
ð2Þ
w0, tðÞ¼0, wl,tðÞ¼0, w,xx 0, tðÞ¼0, w,xx l,tðÞ¼0
ð3Þ
In the above equations, wðx,tÞis the beam trans-
verse displacement in time tand space x,Ais the
mass per unit length of the beam, Cis the beam
Figure 1. (a) The basis for calculating dropper forces to set the contact wire on its objective profile. (b) The basis for recalculating
the equilibrium configuration of the catenary after the dropper breakage and tension variation.
Figure 2. The vertical stiffness of the registration arm and the zigzag in the contact wire.
Vesali et al. 693
damping coefficient, Tis the axial load applied to the
beam that is positive in tension and EI is the beam
flexure rigidity. The gravity load (Fg) and the concen-
trated force exerted on the droppers or the registra-
tion arm (Fd) are also involved on the right-hand side
of the equation. According to the method of the sep-
aration of variables, the deflection function of the
beam can be described as in equation (4)
wðx,tÞ¼X’ixðÞqiðtÞð4Þ
In equation (4), qitðÞis the Rayleigh quotient or the
temporal term and ’ixðÞis the spatial term or the
mode shape function of the ith mode shape. By con-
sidering the proper boundary conditions, the modal
analysis of the beam can be accomplished, and the
natural frequencies and mode shapes of the beam
can be extracted. In equation (2), it is noticed that
the droppers’ and the support forces appear in the
left-hand side of the equation; therefore, they are
not considered in the mode shapes and natural fre-
quencies of the contact wire and the messenger
cable. Figure 3 presents the first, second, third and
the last mode shapes and the related natural frequen-
cies of both the messenger cable and the contact wire.
The final form of the catenary is configured based
on coupling the two individual beams (messenger
cable and contact wire) via droppers. In fact, the
mode shapes and the natural frequencies of the final
form of the catenary are like the ones presented in
Figure 3 because the dropper forces and the support
forces are considered as excitation forces at the left-
hand side of the differential equation.
Expansion of the eigenfunctions can be used in
order to change the partial equations of a tensile
beam to a system of ordinary deferential equation
(equations (5) to (7))
€
qitðÞþi_
qitðÞþ!i2qitðÞ¼nZl
0
’ixðÞðFgþFdÞdx,t40
ð5Þ
n¼1
ARl
0’i2xðÞdxð6Þ
i¼C
Að7Þ
If the excitation forces on the right-hand side of
equation (5) are caused by a spring, damper or a
mass, they will shift to the left-hand side of the equa-
tion and add to the coefficients of qitðÞ,_
qitðÞand €
qitðÞ.
In static form-finding, _
qitðÞand €
qitðÞare zero and qitðÞ
can be calculated according to equations (8) and (9)
qitðÞ¼nZl
0
’ixðÞðFgþFdÞdx,t40ð8Þ
n¼1
!i2ARl
0’i2xðÞdxð9Þ
Figure 3. The first three and the last mode shapes and natural frequencies of the messenger cable (MC) and the contact wire (CW)
that are considered in the model.
Table 1. Properties of the contact wire (CW) and the messenger cable (MC).
17
Parameter Value Parameter Value Parameter Value
Span length (m) 60 Mass/unit length of CW (kg/m) 1.35 Mass/unit length of MC (kg/m) 1.07
Encumbrance (m) 1.2 Tension of CW (kN) 20 Tension of MC (kN) 16
Pre-sag at mid-span (mm) 0 Bending stiffness of CW (N m
2
) 195.0 Bending stiffness of MC (N m
2
) 131.7
Stagger (mm) 200 Mass of dropper clamps on CW (g) 0 Mass of dropper clamps on MC (g) 0
No. of spans 10 Mass of stager (g) 400 No. of droppers per span 9
Stiffness of dropper (kN/m) 10 Stiffness of registration arm (N/m) 340 Mass of droppers (kg) 0
Dropper position in each span (m) 5 10.5 17 23.5 30 36.5 43 49.5 55
694 Proc IMechE Part F: J Rail and Rapid Transit 233(7)
When qitðÞis calculated, the deflection of the tensile
beam (the messenger cable or the contact wire) can be
calculated using equation (4).
Calculating the droppers’ dead load
Calculations for the static equilibrium configuration
of the contact wire and the messenger cable after
incurring some common damage to the catenary,
such as the dropper break or tension loss in the con-
tact wire or the messenger cable, are exercised in this
section.
The static shape of the contact wire has a major
role on the dynamic interaction of the pantograph
and catenary. In catenary design, it is mostly
common to keep the contact wire in a straight line
while in some designs, a small pre-sag is considered.
9
In analytical simulation, the static shape of the con-
tact wire is determined by calculating the droppers’
dead load. For the purposes of this research, the drop-
pers’ dead load is calculated using the vector form of
the linear spring formula (equations (10) and (11))
Fdr
!
nd1¼Kcw
½
ndnd wcw
!
nd1ð10Þ
wcw
!
nd1¼ðwgcw
!wocw
!Þnd1ð11Þ
The properties related to the contact wire and the
messenger cable are denoted by subscripts cw and
mc, respectively. In equation (10), nd is the total
number of the droppers in the section, Fdr
!is the
droppers’ dead load, Kcw is the contact stiffness
matrix. wgcw
!is the vertical deflection of the contact
wire due to the gravity at dropper connection points
and wocw
!is the objective profile of the contact wire
at the dropper connection points. In order to calcu-
late the stiffness matrix of the contact wire, a unique
load is applied to the droppers’ connection points
and the deflection of the contact wire is calculated
based on equations (4) and (8). The vertical stiffness
of the registration arm (krin equation (1) and
Figure 4) is considered in the calculation of the stiff-
ness matrix.
w2,1 is the vertical deflection of the connection
point of the second dropper due to the application
of a unique load at the connection point of the first
dropper. The stiffness matrix of the contact wire is
calculated using equation (12)
Kcw
½¼
w1,1 w1,2 ...w1,3
w2,1 w2,2 ....
.
.
.
.
..
.
...
..
.
.
wnd,1 wnd,2 ...wnd,nd
2
6
6
6
6
6
4
3
7
7
7
7
7
5
1
ð12Þ
The force vector of the droppers can be calculated by
inserting Kcw from equation (12) into equation (10).
Then, the shape of the contact wire and the messenger
cable can be achieved by applying this force in upward
and downward directions, respectively. Figure 5 pre-
sents the shape of the contact wire and the messenger
cable after applying the calculated dead load of the
droppers. The particulates of the catenary for this exam-
ple are according to the EN 50318 standard documents.
In Figure 5(b), it is shown that applying the calcu-
lated dropper forces can set the contact wire on its
objective profile. The force distribution is almost uni-
form and the droppers close to the support should
carry higher dead load in order to set the contact
wire in a straight horizontal line.
Static form-finding of a defective catenary
A further aim of this study is form-finding of defective
catenaries. Tension loss in the messenger cable and in the
contact wire and the breaking of droppers are considered
as three common defects that can happen in catenaries.
A simple configuration of a catenary by considering
spring and mass components is presented in Figure 6.
In Figure 6, Kmc
½and Kcw
½represent the stiffness
matrix of the messenger cable and the contact
wire, respectively. The procedures for calculating the
elements of these matrices are described in the prior
sections of this article. Kdr
½is a matrix that contains
the stiffness of each dropper on its main diagonal elem-
ents (equation (13)). Mm
!and Mc
!are the vectors that
Figure 4. The schematic for applying a unique load on the contact wire for calculating the stiffness matrix.
Vesali et al. 695
describe the equivalent mass of the messenger cable
and the contact wire (in addition to all attached
masses such as the clamp masses, etc.) at connection
points of the droppers for the messenger cable and the
contact wire, respectively. In order to achieve the
equivalent mass of the messenger cable and the contact
wire, the stiffness matrix and the vertical deflection due
to the gravity are needed (equations (14) and (15)). In
Figure 6, y1
!is a vector that contains the vertical deflec-
tion of the dropper connection points on the messenger
cable measured from the reference axes of the messen-
ger cable, y2
!is a vector that contains the vertical
deflection of dropper connection points on the contact
wire measured from the reference axes of the messenger
cable. yo
!is a vector that contains the objective position
of the dropper connection points on the contact wire
measured from the reference axes of the contact wire
(in a straight contact wire without pre-sag, yo
!¼0). ~
H
in Figure 6 is a vector that does not change due to the
defect and describes the offset between the contact wire
and the messenger cable. ~
Hacts as the main hint to find
the equilibrium configuration of the catenary after the
defect. y1
!,y2
!and yo
!are presented in Figure 1(a)
Kdr
½¼
kdr 000
0kdr 00
00kdr 0
000kdr
2
6
6
43
7
7
5ð13Þ
Mm
!¼Kmc
½wgmc
!ð14Þ
Mc
!¼Kcw
½wgcw
!ð15Þ
The contents of vector ~
Hshould be calculated in
the catenary before considering the defect. For this
purpose, the system of equation (16) should be
solved with y1
!and y2
!as the unknown variables
Kmc
½y1
!¼Mm
!þMc
!þKcw
½yde
!ð16Þ
Kdr
½y1
!þKdr
½y2
!¼Mc
!þKcw
½yde
!ð17Þ
Figure 5. (a) Static shape of the catenary after applying the calculated dead load of the droppers, (b) equilibrium configuration of the
contact wire and (c) static force distribution of the droppers.
Figure 6. A simple model of a catenary by considering the
stiffness matrix and the mass matrix.
696 Proc IMechE Part F: J Rail and Rapid Transit 233(7)
~
Hcan be calculated from equation (18)
~
H¼y2
!þyde
!ð18Þ
The hint is that ~
Hwill be constant before and after
any defect. In a defective catenary, depending on the
type of the defect, Kdr
½,Kcw
½and Kmc
½can change.
For example, in the case of a broken dropper, Kdr
½will
change and one array of it turns to zero. When there is
a tension loss in the messenger cable or the contact
wire, the corresponding stiffness matrix will change. It
should be noted that by changing the stiffness matrix,
the equivalent mass should be recalculated according to
equations (14) and (15). In equilibrium equation of a
defective catenary, the variables y1
!,y2
!and yde
!are
unknown. However, since ~
His calculated previously,
the system of equations (19) and (20) can be derived
Kmc þK0
dr
y0
1
!K0
dr
y0
2
!¼Mm
!ð19Þ
K0
dr
y0
1
!þK0
dr þKcw
y0
2
!¼Mc
!þKcw
½
~
Hð20Þ
In these equations, it is assumed that one dropper
is broken and Kdr
½has changed to K0
dr
.y0
1
!and y0
2
!
are the position of the droppers’ connection points on
the messenger cable and the contact wire after the
defect, respectively
F0
dr
!¼K0
dr
ðy0
2
!y0
1
!Þð21Þ
By calculating y0
1
!and y0
2
!, the force vector of the
droppers (F0
dr) can be calculated according to equation
(21). The shape of the contact wire and the messenger
cable can be achieved by applying these forces in the
upward and downward directions, respectively.
Results
As an example, a mechanical section that includes 10
spans is considered. The properties of the catenary are
the same as in Table 1. Primarily, the effect of having
a broken dropper at the middle of the second span is
studied.
Figure 7 presents the equilibrium configuration of
a catenary before and after sustaining a defect of miss-
ing one dropper at the middle of the second span.
The zoomed area in this figure shows that when a
dropper breaks, the messenger cable rises up 22 mm
and the contact wire sags 13 mm at the location of the
broken dropper. The sag between the two droppers in
this catenary before the defect was 2.8 mm, and the
sag under registration arm was 7.8 mm. In addition,
the forces from the adjacent dropper to the broken
dropper increased from 86 N to 130 N (around 50%).
The model that is used to draw the results in
Figure 7 is for a catenary system that includes 10
spans while each span contains nine droppers. It com-
prises 360 m of a contact wire and 300 m of a messen-
ger cable. The dropper forces that are calculated for
this model exhibit symmetry for the whole model
along its mid-point. This means that as an example
the outcome for the second span in this model is as
symmetric as the eighth span. However, if the model
includes nine spans, the fifth span will behave sym-
metrically along its mid-point.
In the next step, the effect of 5% tension loss in the
messenger cable is studied.
The data shown in Figure 8 demonstrate that by
5% reduction in the tension of the messenger cable,
the mid-point of the messenger cable and the contact
wire in the second span sags for 20 mm. The forces of
the droppers close to the end of the span increase, but
the forces of the droppers located at the middle of the
span decrease.
As the last step, the effect of 10% reduction in the
tension in the contact wire is studied.
The effect of 10% reduction in the tension in the
contact wire is illustrated in Figure 9. This set of
data presents that changing the position of the messen-
ger cable due to this defect is negligible. The sag in the
contact wire at the position of the registration arm
increases from 7.7 mm to 8.5 mm. Therefore, the
Figure 7. The effect of breaking one dropper at the middle of the second span on the shape of the contact wire and the messenger
cable and the force on adjacent droppers (the black (dark) line: before the defect; the red (grey) line: after the defect).
Vesali et al. 697
effect of 10% reduction in the tension of the contact
wire is, generally, insignificant. When the objective pro-
file of the contact wire is zero, changing the tension in
the contact wire cannot have a significant effect on the
equilibrium configuration of the contact wire com-
pared with the tension loss in the messenger cable.
Using this method, a combination of different
defects, such as the tension loss in both the messenger
cable and the contact wire or the breaking of the two
droppers at the same time, can be studied as well. In
addition, the effect of the added mass on the contact
wire or the messenger cable can be considered.
Table 2. A comparison between the static force (N) of droppers in various FEM or FDM software programs, and the predictions by
the exact analytical method of this research for the French LN2 or the Italian C270 systems.
SW name
Drop. 1,
x¼4.50
Drop. 2,
x¼10.25
Drop. 3,
x¼16.00
Drop. 4,
x¼21.75
Drop. 5,
x¼27.5
Total force of
all droppers
% Error (weight of
contact wire)
PrOSA – – – – – – –
PantoCat 169.45 49.14 55.44 47.38 55.42 698.24 7.213
SPOPS 198.10 53.06 54.24 54.17 54.15 773.29 2.760
CaPaSIM 161.98 52.14 51.90 51.98 51.69 687.69 8.615
PCaDA 195.57 52.15 52.10 52.08 52.06 755.86 0.444
Gasen-do 171.32 50.09 55.89 47.80 55.84 706.04 6.177
OSCAR 167.90 50.67 55.47 47.52 55.39 698.51 7.177
PCRUN 162.61 52.43 56.20 48.55 56.20 695.78 7.540
CANDY 155.56 52.24 56.58 48.01 54.80 679.58 9.693
PACDIN 164.14 50.30 55.30 47.34 55.29 689.45 8.381
Standard deviation 14.067 1.255 1.614 2.442 1.527 30.621 –
This research (exact
analytical method)
197.73 47.40 55.74 47.25 55.81 752.1 0.057
SW: software.
Figure 8. The effect of 5% reduction in the tension of the messenger cable on the equilibrium configuration of the catenary (the
black (dark) line: before the defect; the red (grey) line: after the defect).
Figure 9. The effect of 10% reduction in the tension in the contact wire on the equilibrium configuration of the catenary (the black
(dark) line: before the defect; the red (grey) line: after the defect).
698 Proc IMechE Part F: J Rail and Rapid Transit 233(7)
Validation
For validation purposes, the calculated static force of
the droppers, for the example case is compared with
the results from some other software . The data in
Table 2 are extracted from Facchinetti and Bruni
2
that presented different software calculations for
the static force of each dropper for the French LN2
or the Italian C270 systems.
1
The last row in Table 2
is the result of this research for the dropper force calcu-
lations, for the same French LN2 or the Italian C270
systems, that is based on the exact analytical method.
The results prove that the error in this method is
insignificant in comparison with the other methods.
In fact, based on the actual measurements, the whole
weight of the contact wire and its attached clamps in 10
spans is 7525.21 N and the summation of the dropper
force is 7520.89 N. The small error exists due to the
weight of the contact wire carried by the support of
the contact wire at both sides of the section.
In order to evaluate the accuracy of the results that
are achieved from the analytical model, a field test is
organized. In this test, the tension of the messenger
cable is reduced by removing the weights of the ten-
sion wheel. The distance between the contact wire at
the dropper clamp position and the track centerline is
measured using a laser measuring device with a pre-
cision of 0.1 mm. The measured distances are com-
pared with the initial distance, that is, the distance
before reducing the tension. The deviation from the
initial position is reported in Table 3.
Since any variation in the shape of the dropper due
to the breakage of one dropper or reduction of ten-
sion in the contact wire is insignificant, the field meas-
urement for such case is ignored. The calculated and
the measured data that are presented in Table 3 prove
that the proposed analytical method of this research
can properly estimate the shape of the catenary.
The degree of precision in the calculations is a credit
to the modeling procedure.
Conclusions
In this article, an exact analytical solution for form-
finding of normal and defective catenaries is pro-
posed. For normal catenaries, the dead load or the
static force in the droppers is calculated such that
the contact wire can be in its objective profile. Since
the vertical offset between the contact wire and the
messenger cable does not change during the defect,
the equilibrium configuration of the catenary can be
achieved by recalculating the system equations. The
proposed method does not include any iterative pro-
cedure. It is fast and accurate. This method can be
used within any other simulation model (FEM or
FDM) since the output of this method is the droppers’
dead loads. The results of catenary form-finding after
incurring any of the three common defects show that,
for example, breaking of a dropper can create sag,
twice as big as the biggest possible sag in a normal
catenary. In addition, the shape of the catenary is
more sensitive to the varying tension in the messenger
cable compared to the contact wire.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of
this article.
Funding
The author(s) disclosed receipt of the following financial
support for the research, authorship, and/or publication
of this article: This research was supported by the office
for ‘‘National Master Plan for High Speed Trains’’ at Iran
University of Science and Technology. The authors are
grateful for the support awarded.
ORCID iD
Mohammad Ali Rezvani http://orcid.org/0000-0003-
1819-7772
Habibolah Molatefi http://orcid.org/0000-0002-7584-
9637
References
1. Finner L, Poetsch G, Sarnes B, et al. Program for caten-
ary–pantograph analysis, PrOSA statement of methods
and validation according EN 50318. Veh Syst Dyn 2015;
53: 305–313.
2. Facchinetti A and Bruni S. Special issue on the panto-
graph–catenary interaction benchmark. Veh Syst Dyn
2015; 53: 303–304.
3. Ambro
´sio J, Pombo J, Antunes P, et al. PantoCat state-
ment of method. Veh Syst Dyn 2015; 53: 314–328.
4. Cho YH. SPOPS statement of methods. Veh Syst Dyn
2015; 53: 329–340.
5. Lewis WJ. Tension structures: form and behaviour.
London: Thomas Telford, 2003.
6. Veenendaal D and Block P. An overview and compari-
son of structural form finding methods for general net-
works. Int J Solids Struct 2012; 49: 3741–3753.
7. Lopez-Garcia O, Carnicero A and Torres V.
Computation of the initial equilibrium of railway over-
heads based on the catenary equation. Eng Struct 2006;
28: 1387–1394.
Table 3. Deviation of the contact wire from its initial position at the location of the droppers after 5% reduction
in the messenger wire tension.
Dropper 1 Dropper 2 Dropper 3 Dropper 4 Dropper 5
Field measurement (mm) 9.5 13.7 17.5 19.7 20.1
This research (mm) 10.1 14.2 17.8 20.1 20.5
Vesali et al. 699
8. Such M, Jimenez-Octavio JR, Carnicero A, et al. An
approach based on the catenary equation to deal with
static analysis of three dimensional cable structures.
Eng Struct 2009; 31: 2162–2170.
9. Cho YH, Lee K, Park Y, et al. Influence of contact wire
pre-sag on the dynamics of pantograph–railway caten-
ary. Int J Mech Sci 2010; 52: 1471–1490.
10. Jo
¨nsson P-A, Stichel S and Nilsson C. CaPaSIM state-
ment of methods. Veh Syst Dyn 2015; 53: 341–346.
11. Zhou N, Lv Q, Yang Y, et al. TPL-PCRUN Statement
of methods. Veh Syst Dyn 2015; 53: 380–391.
12. Tur M, Garcı´a E, Baeza L, et al. A 3D absolute nodal
coordinate finite element model to compute the initial
configuration of a railway catenary. Eng Struct 2014;
71: 234–243.
13. Massat J-P, Balmes E, Bianchi J-P, et al. OSCAR state-
ment of methods. Veh Syst Dyn 2015; 53: 370–379.
14. Yang C, Zhang W, Zhang J, et al. Static form-finding
analysis of a railway catenary using a dynamic equilib-
rium method based on flexible multibody system for-
mulation with absolute nodal coordinates and controls.
Multibody Syst Dyn 2017; 39: 221–247.
15. Song D, Jiang Y and Zhang W. Dynamic performance
of a pantograph–catenary system with consideration of
the contact surface. Proc IMechE, Part F: J Rail and
Rapid Transit 2018; 232: 262–274.
16. Gregori S, Tur M, Nadal E, et al. An approach to geo-
metric optimisation of railway catenaries. Veh Syst Dyn
2018; 56: 1162–1186.
17. EN 50318:2002. Railway applications - Current collec-
tion systems - Validation of simulation of the dynamic
interaction between pantograph and overhead contact
line; 14–16.
Appendix
Notation
EI bending stiffness
Fdri ith dropper force
Fgweight load
~
HEncumbrance
kdri stiffness of ith dropper
kri stiffness of ith registration arm
ksi stiffness of ith support
Kcw
½ stiffness matrix of contact wire
Kmc
½ stiffness matrix of messenger cable
Mm
!mass of messenger cable at dropper
connection points
Mc
!mass of contact wire at dropper con-
nection points
Tmc tension of messenger cable
Tcw tension of contact wire
wgcw
!vertical deflection of contact wire due
to gravity at dropper connection points
wocw
!objective profile of contact wire at
dropper connection points
y1
!vertical deflection of the dropper con-
nection points on the messenger cable
measured from the reference axes of the
messenger cable
y2
!vertical deflection of the dropper con-
nection points on the contact wire
measured from the reference axes of the
messenger cable
y0
!objective position of the dropper con-
nection points on the contact wire
measured from the reference axes of the
contact wire
Amass per unit length of wire
700 Proc IMechE Part F: J Rail and Rapid Transit 233(7)
