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PANTOGRAPH/CATENARY CONTACT FORMULATIONS
Shubhankar Kulkarni1
Carmine M. Pappalardo2
Ahmed A. Shabana1
1Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, IL 60607, USA
2Department of Industrial Engineering, University of Salerno, Fisciano (Salerno), 84084, Italy
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ABSTRACT
In this investigation, the pantograph/catenary contact is examined using two different formulations. The first is an elastic contact formulation that allows for the catenary/panhead separation and for the analysis of the effect of the aerodynamic forces, while the second approach is based on a constraint formulation that does not allow for such a separation by eliminating the freedom of relative translation in two directions at the catenary/panhead contact point. In this study, the catenary system, including the contact and messenger wires, is modeled using the nonlinear finite element (FE) absolute nodal coordinate formulation (ANCF) and flexible multibody system (MBS) algorithms. The generalized aerodynamic forces associated with the ANCF position and gradient coordinates and the pantograph reference coordinates are formulated. The new elastic contact formulation used in this investigation is derived from the constraint-based sliding joint formulation previously proposed by the authors. By using a unilateral penalty force approach, separation of the catenary and panhead is permitted, thereby allowing for better evaluating the response of the pantograph/catenary system to wind loading. In this elastic contact approach, the panhead is assumed to have six degrees of freedom with respect to the catenary. The coordinate system at the pantograph/catenary contact point is chosen such that the contact model developed in this study can be used with both the fully parameterized and gradient deficient ANCF elements. In order to develop a more realistic model, the MBS pantograph model is mounted on a detailed three-dimensional MBS rail vehicle model. The wheel/rail contact is modeled using a nonlinear three-dimensional elastic contact formulation that accounts for the creep forces and spin moment. In order to examine the effect of the external aerodynamic forces on the pantograph/catenary interaction, two scenarios are considered in this investigation. In the first scenario, the crosswind loading is applied on the pantograph components only, while in the second scenario, the aerodynamic forces are applied on the pantograph components and also on the flexible catenary. In this study, the time-varying nonlinear aerodynamic forces are modeled, thereby capturing the influence of the aerodynamic forces on the dynamic behavior of the pantograph/catenary system. For the configuration considered in this investigation, it was found that the crosswind assists the uplift force exerted on the pantograph mechanism, increasing the mean contact force value. Numerical results are presented in order to compare between the cases with and without the wind forces.
Keywords: Multibody systems dynamics; absolute nodal coordinate formulation; pantograph/catenary interaction; wheels/rail contact; aerodynamic forces.
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1. INTRODUCTION
This investigation is focused on developing a MBS computational approach for the nonlinear
three-dimensional dynamic analysis of the pantograph/catenary system subject to aerodynamic
forces. The pantograph/catenary model developed in this study is implemented in a
computational MBS algorithm that allows for developing detailed three-dimensional railroad
vehicle models without the need for using co-simulation approaches.
1.1 Background
Pantograph/catenary systems are the most feasible way to power high speed trains which travel
at speeds higher than 300 km/h (Bruni et al., 2012; Facchinetti and Bruni, 2012; Facchinetti et
al., 2013). Pantographs are mechanical systems mounted on the top of the rail vehicles for the
purpose of collecting current from an overhead contact line carrying power. An uplifting
mechanism keeps the pantograph current collectors or the panhead in contact with the contact
wire. The dynamics of the pantograph-catenary system plays a crucial role in the current
collection quality, and therefore, accurate computational modeling is required to correctly predict
the dynamical behavior of this complex system.
The dynamics of the pantograph/catenary system is of paramount importance in maintaining
a consistent current collection quality in high-speed trains. The car body vibrations or adverse
weather conditions can affect the interaction between the pantograph panhead and the overhead
contact line, which may lead to serious problems including arcing or damage to the system
components. Unfavorable operating conditions or inefficient design can result in higher wear
rates and hence a shorter fatigue life of the panheads and the catenary system, thereby increasing
the maintenance costs significantly. The wear rates are governed by several parameters such as
the sliding speed, current intensity, contact force, as well as the nature of the materials in contact
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(Bucca and Collina, 2009). Hence, analyzing the contact forces between the pantograph and the
catenary is essential in order to ensure smooth power delivery and reduce the component wear.
1.2 Aerodynamic Forces
The weather conditions, such as temperature and wind, have a major impact on the
pantograph/catenary interaction (Bacciolone et al., 2006; Carnevale et al., 2015; Pombo et al.,
2009). High temperature can alter the static position and the tension in the catenary. Cold
weather conditions may result in formation of ice on the wires, leading to deformation. On the
other hand, wind forces can lead to severe vibrations of the catenary system and can also
influence the dynamics of the pantograph components directly. The drag and lift forces can cause
the mean contact force value to change. When pantographs with multiple collectors are
considered, the aerodynamic forces can introduce an unbalance between the front and rear
collectors, causing uneven wear. Very high wind forces can impact the catenary system
significantly causing it to oscillate with large amplitudes. The worn-out overhead contact line
may generate asymmetric drag and lift forces, resulting in the galloping motion of the catenary
system (Stickland and Scanlon, 2001; Stickland et al., 2003). Heavy cross-wind loads can cause
severe railroad vehicle vibrations that influence the pantograph/catenary interaction (Bacciolone
et al., 2008; Cheli et al., 2010). Hence, assessing the effect of aerodynamic forces on the
pantograph and catenary becomes important while designing the current collection systems so as
to maintain continuous and consistent contact between the panhead and the contact wire with an
optimum uplift force, thus minimizing the wear of the system components. To avoid this contact
force unbalance, spoilers are often placed on the collectors, or the panhead geometry is
optimized such that the effect of the aerodynamic pitching moment is minimized.
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1.3 MBS and FE Modelling of the Pantograph/Catenary Systems
The field of MBS dynamics has matured over the past few decades to a level that allows for
efficient modeling the dynamics of complex systems that contain both rigid and flexible bodies.
The MBS computational framework has proved to be very useful in a wide range of applications
including the automotive and aerospace industries, the machine industry, railroad vehicle
applications, among others. An MBS application consists of rigid and flexible bodies connected
by mechanical joints or subjected to motion constraints. Significant research efforts have been
devoted to developing new formulations to accurately capture the deformations of flexible bodies
in MBS applications. In the case of rigid body dynamics, the principal methods employed to
analytically describe the motion of a mechanical system are the relative or recursive coordinate
formulation or the augmented formulation (Roberson and Schwertassek, 1988; Pappalardo, 2015;
Wittenburg, 1977; Shabana 2010). On the other hand, formulations such as the floating frame of
reference (FFR) formulation are used to capture small deformations of bodies having large
displacements and large rotations. In particular, ANCF elements have proved to be very effective
in cases where large rotations and large deformations are considered (Pappalardo et al., 2016;
Shabana, 1998).
MBS dynamics virtual prototyping is an effective approach for studying and understanding
the pantograph/catenary interaction and for accurately estimating the contact forces (Gerstmayr
and Shabana, 2006; Pappalardo et al., 2015; Seo et al., 2005; Seo et al., 2006). This
computational approach allows for systematically including the effect of the aerodynamic forces
as well, which is important in determining the required uplift force for the pantograph
mechanism and also in defining the control strategy for the contact force. This is needed as the
drag and lift components of the aerodynamic forces can cause system instability by means of
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fluid- structure interaction resulting in unexpected behavior of the system or it may even damage
the system severely. Therefore, it is necessary to account for the aerodynamic forces in order to
have a more realistic model which can correctly predict the behavior of the pantograph system
(Pombo et al., 2009; Stickland and Scanlon, 2001).
1.4 Pantograph/Catenary Modeling Approaches
In previous investigations on the pantograph/catenary interaction, different methods were used to
model the system and compute the contact forces. The first method uses partial differential
equations to mathematically represent the catenary as a continuous string or beam (Arnold and
Simeon, 2000). Although the resulting equations are simple and easy to solve, they have
limitations in capturing many nonlinear effects which are important in the pantograph/catenary
interaction. A higher number of discretization points and smaller time steps are required to solve
the catenary equations in order to obtain accurate numerical results (Poetsch et al., 1997). The
second method uses linear finite elements to model the catenary (Collina and Bruni, 2010;
Massat et al., 2006; Pombo et al., 2009; Pombo et al., 2012). In these models, the moving contact
force is applied as an external force on the catenary contact wire (Ambrosio et al., 2009;
Ambrosio et al., 2012; Poetsch et al., 1997). In this case, a co-simulation technique is used
between an FE analysis code which models the catenary and an MBS code which models the
pantograph. Dynamic sensitivity analysis of the pantograph-catenary systems has also been
performed so as to have optimized pantograph designs. The catenary was modeled using a FE
software and the pantograph was represented by a linear spring-damper system (Park et al.,
2003). The third method uses ANCF finite elements to model the catenary to capture the
geometric nonlinearities more accurately (Lee and Park, 2012; Pappalardo et al., 2015; Seo et al.,
2005; Seo et al., 2007). The need for co-simulation is eliminated in this method as the flexible
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ANCF catenary and the pantograph mechanism systems are modeled using a single MBS
computational framework. A benchmark study of the pantograph/catenary system simulation
presented by Bruni et al. (2015) involves multiple cases where the pantograph is modeled in an
MBS environment and the catenary is modeled using linear Euler-Bernoulli beam elements. A
catenary model created with ANCF beam elements is also benchmarked in the study.
1.5 ANCF Catenary Model
In this study, ANCF elements are used to develop the catenary system model. Through a wide
range of applications, ANCF has emerged as an effective method for capturing the large
rotations and large deformations in MBS problems. ANCF elements use absolute nodal positions
and nodal position vector gradients as generalized coordinates to define the FE kinematics. This
particular choice of the generalized coordinates leads a constant mass matrix and zero Coriolis
and centrifugal inertia forces. The ANCF beam element, in particular, is consistent with the
geometrically exact continuum mechanics formulations and can capture the deformation of the
cross-section. The ANCF cable element, on the other hand, leads to a more efficient
implementation because it is a gradient deficient element, and therefore, the computation of the
element generalized elastic forces is less intensive. Furthermore, as shown in the literature
previously (Shabana, 2011), the ANCF elements can be conveniently used in the case of
structures with an initial curvature. Therefore ANCF elements can be used to model the catenary
system in the case of straight- and curved-track scenarios. However, these simulation scenarios
are outside the scope of this invetigation and can be a topic for future studies.
1.6 Pantograph/Catenary Contact Forces
There are mainly two methods used to model the pantograph/catenary contact forces, the first is
the constraint contact approach and the second is the elastic contact approach (Pappalardo et al.,
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2015). The constraint contact approach, referred to in this investigation as the sliding joint
formulation, imposes constraints on the relative motion at the contact point on the pantograph
panhead and the catenary. These constraints are formulated in terms of nonlinear algebraic
equations that are enforced at the position, velocity, and acceleration levels. The constraint
contact approach has the advantage of obtaining a smoother constraint force solution. However,
the constraint approach does not allow for the catenary/panhead separation, and therefore, cannot
be used to model the loss of contact which can be the result of the vehicle vibrations and/or
extreme weather conditions (Seo et al., 2005). The elastic contact formulation, on the other hand,
uses a unilateral spring-damper force model that allows for the catenary/panhead separation
(Khulief and Shabana, 1987; Lankarani and Nikravesh, 1994).
1.7 Contributions of this Investigation
It is the objective of this paper to develop an MBS computational framework for the dynamic
analysis of the pantograph/catenary systems. Two contact formulations are used in this
investigation to calculate the nonlinear pantograph/catenary interaction forces; the elastic contact
and constraint contact formulations. The new elastic contact formulation proposed in this
investigation does not eliminate any freedom of relative motion and produces unilateral contact
forces, thereby allowing for the catenary/panhead separation. The constraint contact formulation,
on the other hand, does not allow for the catenary/panhead separation and eliminates the freedom
of the relative translation in directions perpendicular the direction of the sliding. The results
obtained using the two different three-dimensional approaches are compared. Another
contribution of this paper is the integration of an aerodynamic force model with ANCF elements
in order to allow for the analysis of the cross wind forces. A constant wind velocity is considered
in this study. Accounting for the turbulence involves complex fluid structure interaction
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phenomenon and can be a part of future investigations (Carnevale et al., 2015). The use of the
ANCF aerodynamic force model with the elastic contact formulation allows for the development
of new MBS high speed railroad vehicle models using one computational environment without
the need for using co-simulation techniques. The use of the formulations and procedures
introduced in this paper is demonstrated using the gradient deficient ANCF cable element which
allows for obtaining an efficient solution as compared to fully parameterized ANCF beam
elements. In this investigation, the train is assumed to negotiate a straight track with a constant
speed. As mentioned earlier, using the same MBS framework, the current model can be
extrapolated in future studies to account for the pantograph-catenary interaction in case of a
curved track. High values of inertia forces during acceleration and deceleration situations can
have a significant effect on the pantograph-catenary contact forces. The wave propagations and
nonlinear dynamics in this scenario can be fully captured by the ANCF catenary system and the
nonlinear pantograph MBS model. The acceleration and deceleration can also affect the wear
rates of the pantograph-catenary materials (Bucca and Collina, 2009). These effects can also be
studied in the high fidelity models in the future investigations by the authors and others.
2. PANTOGRAPH/CATENARY ELASTIC CONTACT FORMULATION
In this section, the new elastic contact formulation used in this investigation to determine the
contact forces between the catenary wire and the panhead is developed. In this formulation, one
geometric parameter is introduced with an additional algebraic equation, ensuring that no degree
of freedom is eliminated. Given the system coordinates, this algebraic equation can be solved for
the geometric parameter that defines the location of the contact point on the catenary wire.
2.1 Contact Point
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The geometric parameter used to define the location of the contact point between the catenary
and the panhead is the wire arc length parameter s . Let ( )p p tq q be the vector of generalized
coordinates of the panhead, t is time, c c te e be the vector of the nodal coordinates of the
ANCF catenary element under consideration, 1 2 3
Tc c c cx x x x
r r r r be the gradient vector
tangent to the catenary wire at the contact point, , ,c c cx y z tr S e is the position vector of an
arbitrary point on the catenary, cS is the shape function matrix of the ANCF element under
consideration, and ,x y , and z are the element spatial coordinates. For the catenary wire, and
without any loss of generality, one can assume x x s . Furthermore, if the generalized
coordinates of the panhead and the catenary are known from the numerical integration of the
equations of motion, one can formulate the following algebraic equation:
0Ts c p c
e xC s s r r r (1)
Knowing the system generalized coordinates, the preceding equation can be considered as a
nonlinear algebraic equation which can be solved iteratively for the arc length parameter s . To
this end, a Newton-Raphson iterative procedure is employed to iteratively solve the equation
s se eC s s C , where s is the Newton-difference, and
Ts s c c cpT ce e x xs
C C s s s r r r r , where cp c p r r r . Assuming that 0se s
C ,
Eq. 1 defines the arc length s which can be used to define the location of the contact point on the
catenary wire c sr . It is clear that Eq. 1 is used to define the arc length parameter s , and
therefore, this equation does not impose any constraints on the motion of the pantograph or the
catenary. No degree of freedom of relative motion is eliminated and the panhead has six degrees
of freedom with respect to the catenary.
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2.2 Contact Frame
In order to determine the contact forces normal to the direction of sliding, one can define a
contact frame which has its longitudinal axes defined by a unit vector along the gradient vector
cxr and the other two axes are unit vectors perpendicular to the vector c
xr . In this investigation,
the axes of the contact frame is defined by the transformation matrix 1 2 3c c c c A a a a , where
1 2,c ca a , and 3ca are the three orthogonal unit vectors defined as
1 21 2 3
1 2
, ,c c c
c c cxc c cx
r n n
a a ar n n
(2)
and the vectors 1c cx ,r n , and 2
cn are
1 21 3
22
1 22 1 3
13 2 3
0
c cc cx xx x
c c c c c cx x x x
cc c cxx x x
, ,
r rr r
r r n r r n
rr r r
(3)
This formulation of the vectors leads to singularity if the vector cxr becomes parallel to the vector
0 1 0T
. In this special case which is not applicable to the catenary system under
consideration, the following three orthogonal vectors can be used (Gere and Weaver, 1965;
Shabana, 2013):
2
1 22
0 0
0 0
0 10
cx
c c c cx x , ,
r
r r n n (4)
By defining two axes perpendicular to the longitudinal direction, one can formulate the normal
contact forces. The frame defined by Eq. 3 or alternatively Eq. 4 does not require having a
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complete set of gradient vectors, and therefore, it can be used with both fully parameterized and
gradient deficient ANCF elements.
2.3 Contact Forces
In the case of the pantograph/catenary system, the panhead can lose contact with the catenary
wire. Therefore, when an elastic contact formulation is used, a unilateral force model must be
used in order to allow for the pantograph/catenary separation. In order to develop this unilateral
force mode, the following penetration variables along two orthogonal directions can be defined:
2 2 3 3,T Tc p c c p cl l a r r a r r (5)
Along each of the two directions 2ca and 3
ca , one can define a compliant force model
, , 2,3pck k kF l l k . This force model is function of selected stiffness and damping coefficients,
pckk , and pc
kc , respectively. Using the assumed force model, one can write the virtual work of the
compliant forces as 3
2,pc pc
k k k kkW F l l l
, which can be written in terms of the virtual
changes in the generalized coordinates of the panhead and catenary as
3
2,pc pc p ck k
k k k p ck
l lW F l l
q q
q q (6)
This equation defines the generalized forces associated with the panhead and catenary
generalized coordinates as
3 3
2 2, , ,
T T
p pc c pck kk k k k k kp ck k
l lF l l F l l
Q Q
q q (7)
These generalized forces are unilateral forces and are not applied to the panhead and catenary in
the case of separation.
The elastic force formulation presented in this section also allows for introducing
tangential forces that may include friction forces as the result of the catenary/panhead relative
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sliding. In the case of friction, for example, the tangential forces are function of the normal
forces. In this case the magnitude of the normal forces can be determined as
2 2
2 3pc pc pc
nF F F . The tangential friction force can be defined as 1pc pc c
t nF F a , where
is a friction coefficient.
3. CONSTRAINT CONTACT FORMULATION
Another approach for modeling the pantograph/catenary interaction is to neglect the effect of the
separation and assume that that panhead remains in contact with the catenary wire. While this
constraint approach may not be appropriate to use in the case of severe wind weather conditions,
it has the advantage that it leads to a smoother solution for the contact forces and such an
approach does not require assuming stiffness and damping coefficients. In the
pantograph/catenary constraint contact formulation, the constraints of a sliding joint that
eliminate two degrees of freedom of relative motion are applied. These constraint equations at a
contact point P can be written in the vector form, ( , , )s s p c sC C q e as s p c 0C r r , where
pr is the global position vector of the contact point P on the panhead, and cr is the global
position vector of the contact point P on the catenary. The sliding joint constraints can be
written more explicitly as
1
2
3
cT p c
s cT p c
cT P c
a
a 0
a
r r
C r r
r r
(8)
In this equation, 1 2,c ca a , and 3ca are the unit vectors that define the axes of the contact frame, as
described in the preceding section. The constraints of Eq. 8 eliminate only two degrees of
freedom of relative motion in direction perpendicular to the direction of the forward sliding. This
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is mainly due to the fact that an additional non-generalized coordinate, the arc length parameter,
is introduced and is used in the formulation of these equations. It is important also to reiterate
that the sliding joint formulation proposed by Seo et al. (2005) can only be used for fully
parameterized elements. On the other hand, the sliding joint formulation proposed by Pappalardo
et al. (2015) and discussed in this section uses only one gradient vector to define the constraint
equations, and therefore, it can be used with both fully parameterized and gradient deficient
ANCF elements.
One can show that a virtual change in the coordinates leads to s s
ss
qC q C 0 ,
where TT pT q e q is the vector of generalized coordinates of the panhead and the catenary
contact wire, s s s s p qC C q C e C q is the Jacobian matrix associated with the
vector of generalized coordinates q , and s s
ss C C is the Jacobian matrix associated with
the arc length parameter s (Pappalardo et al., 2015). One can also show that the virtual change
and time derivative of the first equation in Eq. 8, 1sC leads to (Pappalardo et al., 2015)
1 1
1 1
s s
s s
s s
C Cs , s
C C
q qq q . (1)
which show that one equation can be used to write the arc length parameter s in terms of the
system generalized coordinates, and therefore, the three scalar equations of Eq. 8 eliminate two
degrees of freedom only, as previously mentioned. The details of the sliding joint formulation are
presented by Pappalardo et al. (2015). The results obtained using the new elastic contact
formulation described in the preceding section will be compared with the results obtained using
the constraint contact formulation briefly discussed in this section. In the numerical comparative
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study presented in a later section of this paper, the guidelines for the contact force recommended
in the literature are used (EN50317, 2012; EN50367, 2006). In this investigation, the EN50367
recommendation is referenced, which specifies specific parameters for the pantograph operation.
For example, the mean contact force limit is specified as 20.00097 70NmF v , where v
represents the velocity of the train. All other relevant parameters given in Table 1 are considered
in this investigation, except for the percentage of real arcing parameter.
4. AERODYNAMIC FORCE FORMULATION
In this section, it is shown how a simple aerodynamic force model can be integrated with the
flexible body ANCF catenary and rigid body pantograph models in order to account for the cross
wind effect. In the case of severe weather simulation scenarios, accounting for the effect of
aerodynamic forces is necessary, as these forces can have a significant effect on the mean contact
force values. The drag and lift force components of the cross-wind tend to add to the uplift force
exerted on the pantograph mechanism (Pombo et al., 2009). This effect is referred to as the
aerodynamic uplift. For pantographs having multiple panheads, an imbalance might be
introduced due to the aerodynamic forces causing one of the collector strips to wear faster than
the others (Bacciolone et al., 2006; Carnevale et al., 2015; Pombo et al., 2009). The turbulence
created in the boundary layer close to the car body roof due to vortex shedding can excite the
pantograph components, affecting the current collection quality. The vortex shedding at the
panhead wake may generate high frequencies and the sparking level can increase significantly
(Bacciolone et al., 2006; Ikeda et al., 2009). Therefore, it is important to account for the
aerodynamic forces when designing the pantograph/catenary systems.
4.1 Aerodynamic Force Models
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There are two main approaches used for simulating the effect of the aerodynamic forces on the
pantograph/catenary systems. In the first approach, computational fluid dynamics (CFD) is used
to compute the aerodynamic forces on individual components of the pantograph and ultimately
assess the contribution of the wind forces to the total uplift force on the pantograph (Bacciolone
et al., 2007; Carnevale et al., 2015). In the second approach, the drag and lift coefficients for the
individual pantograph components are obtained from experimental studies and the aerodynamic
forces are computed using the equation 2ˆ0.5 A relCF v v , where F is the drag or lift force,
is the density of the fluid, AC the drag or lift pressure coefficient, relv is the relative velocity
between the wind and the pantograph component, and v̂ is the unit vector in the direction of the
drag or lift force (Pombo et al., 2009).
4.2 Implementation and Force Distribution
A new approach, which is analogous to the second approach described above, is used in this
investigation in order to apply the time-varying nonlinear aerodynamic forces on the components
of the rigid-body pantograph and the flexible-body ANCF catenary. The approximate drag and
lift coefficients for the individual pantograph components are obtained from previous
experimental and numerical studies (Carnevale et al., 2015). The aerodynamic coefficients are
functions of the angle of attack and the geometry. For the constant wind velocity and constant
train speed scenario considered in this study, the relative angle of attack does not significantly
change justifying the use of constant aerodynamic coefficient values (Pombo et al., 2009). The
values of the aerodynamic pressure coefficients are shown in Table 2. In order to compute the
aerodynamic forces on each body in this investigation, a mesh data file is created by the user for
each rigid or flexible body in the model. This mesh data file defines distribution of points at
which the aerodynamic forces are applied. Using this mesh data file, the effect of the moments of
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the aerodynamic forces is taken into consideration. In the computer implementation of the
aerodynamic forces, the aerodynamic mesh created from this set of aerodynamic data points is
read by the general purpose computer program Sigma/Sams (Systematic Integration of
Geometric Modeling and Analysis for the Simulation of Articulated Mechanical Systems) from
the user-provided data file. The data file contains thirteen parameters for each data point
selected; three Cartesian position coordinates that define the location of the point with respect to
the body coordinate system, nine direction cosines defining the orientation of the coordinate
frame for the data point with respect to the body coordinate system, and the area associated with
the data point. Since aerodynamic pressure coefficients are used in this study, the area can be
simply considered as unity for each data point. The cross-section on which the aerodynamic
analysis is performed is the cross-section which is aligned with the direction of motion of the
train.
In the case of flexible catenaries, the nodal locations are used to define the aerodynamic
data points. Hence the aerodynamic force is applied at each node of the flexible contact wire and
the messenger wire. The user also defines the aerodynamic drag and lift coefficients, the fluid
density, and the wind velocity vector.
4.3 Generalized Aerodynamic Forces
A streamlined wind flow is assumed in this investigation. The relative velocity between the body
on which the aerodynamic forces are to be applied and the wind is computed at each integration
time-step. The lift and the drag forces on a body are then calculated. The direction of these forces
depends on the direction of the relative velocity as shown in Fig. 1. The aerodynamic drag and
lift forces can be written, respectively, as follows:
2 21 1ˆ ˆ,
2 2D D r r L L r LC A C A
F v v F v v (10)
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where DF and LF are the drag and lift forces, respectively, is the fluid density, DC and LC are
the drag and the lift pressure coefficients per unit area, respectively, A is the area on which the
aerodynamic force is applied, rv is the vector of the relative velocity between the body and the
wind, and ˆ rv and ˆ Lv are unit vectors in the direction of the drag and the lift forces, respectively.
The generalized forces associated with the generalized coordinates as the result of the
application of the aerodynamic drag force iDF and lift force i
LF at a point i can be calculated as
1
N iT iD Di Q J F and
1
N iT iL Li Q J F , respectively, where N is the number of points, and iJ is
the Jacobian matrix of the position field defined by differentiation with respect to the vector of
generalized coordinates. That is, in the case of a rigid body, i i i i i i J r q I A u G and
in the case of a flexible ANCF body i i i i J r e S . In these equations, iA is the
transformation matrix that defines the orientation of the point coordinate system on the body in
the global coordinate system, iu is the skew symmetric matrix associated with the vector iu that
defines the location of the point with respect to the coordinate system of the body on which the
point is defined, and iG is the matrix that relates the angular velocity vector iω defined in the
body coordinate system to the time derivatives of the orientation coordinates iθ ; that is,
i i iω G θ , and iS is the matrix of shape functions associated with the ANCF element on which
the aerodynamic force is applied.
5. MBS/FE FORMULATION
This section describes the MBS models and formulations used in this investigation. The
pantograph model used in the numerical study is based on the Faiveley Transport CX
pantograph. The wheel/rail contact formulation used in this study is a three-dimensional elastic
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contact formulation that accounts for the wheel and rail profile geometry. The MBS equations of
motion, which are formulated in terms of the rigid body reference coordinates and the ANCF
elastic coordinates, allow for systematically including the effect of the pantograph/catenary and
wheel/rail interaction forces.
5.1 Railroad Vehicle Models
A detailed rigid-body rail vehicle model, shown in Fig. 2, is used in this investigation. The model
has one car body mounted on two bogies. Each bogie consists of six bodies, two wheelsets, two
equalizers, a bolster and a frame. The equalizers are connected to the frame, and the frame to the
bolster using compliant bushing elements that take into account the vertical displacements of the
train components. The wheelsets are connected to the equalizers using bearings. The bolsters are
attached to the car body using revolute joints. The vehicle model consists of 14 rigid bodies, 48
bushing elements, 8 bearings and 2 revolute joints. The rigid-body model parameters are
provided in Table 3. A three-dimensional wheel/rail contact formulation, discussed later in this
section, is used to define the wheel/rail contact forces. An MBS model of a Faiveley Transport
CX pantograph is analyzed in this investigation (Pappalardo et al., 2015; Pombo et al., 2009).
The inertia properties as well as the global initial position and the Euler angles that define the
body orientations are provided in Table 4. The body coordinate system for each component is
located at the body center of mass and is oriented such that the directions of the axes are aligned
with the principal inertia directions. The pantograph consists of six rigid bodies connected by
revolute and spherical joints. The joint data are provided in Table 5. Figure 3 shows a schematic
drawing of the pantograph-catenary system.
5.2 Wheel/Rail Contact
20
A three-dimensional non-conformal contact formulation is used in this investigation to evaluate
the wheel/rail interaction forces. This contact formulation, which allows for the wheel/rail
penetration and separation, allows for general description of the wheel/rail profile geometries.
The locations of the contact points on the wheels and rails are determined online by solving the
following set of nonlinear algebraic equations:
1
2,
1
2
, , , ,
Tr w r
Tr w r
w r r w r w
T rw
T rw
t
t r r
t r r
t
t
C q q s s 0n
n
(11)
In this equation, superscripts w and r refer to the wheel and rail, respectively, wr and rr are
the global position vectors of the potential contact points on the wheel and rail, respectively, 1wt
and 2wt are the tangents to the wheel surface at the contact point, wn is the normal to the wheel
surface at the contact point, 1rt and 2
rt are the two tangents to the rail surface, rn is the normal to
the rail surface at the contact point, rq is the vector of generalized coordinates of the rail, wq is
the vector of generalized coordinates of the wheel, rs is the vector of non-generalized
coordinates or surface parameters of the rail, and ws is the vector of non-generalized coordinates
or surface parameters of the wheel. Given the vectors of the generalized coordinates of the wheel
and rail wq and rq , the four nonlinear equations in Eq. 11 can be solved for each contact for the
four surface parameters, ws and rs using an iterative Newton-Raphson method which requires
the iterative solution of the following algebraic equations:
w,r w w w,r r r w,r C s s C s s C (12)
21
In this equation, ws and rs are the Newton differences. The solution of the nonlinear
algebraic equations of Eq. 11 defines the four wheel and rail surface parameters which can be
used to define the location of the contact points on the wheel and rail. The solution of these
equations also ensures that the coordinates of the contact point on the wheel and the rail surface
are the same along the tangents 1rt and 2
rt , and the tangent planes at the contact point are the
same for the wheel and the rail surfaces. Knowing the locations of the contact points on the
wheel and rail, the normal force can be calculated using a compliant force model. Knowing the
normal force, the wheel and rail material properties and geometries, and the creepages, the
tangential creep forces and spin moment can be calculated and used to define the generalized
forces acting on the wheel and rail.
5.3 ANCF Catenary Model
In general, a catenary system consists of two wires; a contact wire and a messenger wire. The
contact wire comes into contact with the pantograph panhead, while the messenger wire prevents
the contact wire from sagging due to its own-weight. The contact wire is connected to the
messenger wire using droppers. Supporting poles are placed at certain distances so as to support
the messenger wire and the contact wire. The contact wire is suspended using registration arms,
which have low inertia. The general construction of the catenary system is shown in the Fig. 4.
The contact wire is assumed to have a cross-sectional area in the range 65-150 mm2 and a span in
the range 50-65 m. The contact wire is usually made of high electrical conductivity materials
such as steel or copper alloys. The catenary system carries around 1000 V to 25000 V of power
depending on the type of trains for which it is used. The droppers are tensile elements and are
modeled to virtually have zero stiffness in compression to account for slackness. At high speeds,
the wave propagation speed in the contact wire becomes one of the important factors that have a
22
significant effect on the ability of the system to maintain a continuous contact between the
pantograph and the wire. The catenary pre-tension plays an important role in the current
collection quality and in defining the speed limit of the train. The pre-tension values typically
range between 14- 40 kN, depending on the operating speed of the train. In this investigation, the
contact and the messenger wires are modeled using ANCF Euler-Bernoulli beam elements
capable of capturing the catenary nonlinear dynamic behavior (Pappalardo et al., 2015; Seo et al.,
2005; Seo et al., 2007). The ANCF cable element used in this study to model the catenary has the
global position and one gradient vector as nodal coordinates (Gerstmayr and Shabana, 2005).
The global position of an arbitrary point on the element is defined as ,t tr x S x e , where
Tx y zx is the vector of the element spatial coordinates, S S x is the space-dependent
shape function matrix, and ( )te e is the vector of the time-dependent element nodal
coordinates. For a cable element, the vector of nodal coordinates is defined for node k as
, 1, 2TT Tk k k
x k e r r . The element shape function matrix can be written as
1 2 3 4 s s s sS I I I I , where
32 3
1 2
32 3
3 4
1 3 = + , s = 1 ,
2 4 4 8
1 3 = , s = 1
2 4 4 8
Ls
Ls
(13)
In this equation, L is the length of the element, and 2 1x
L . This ANCF cable element leads
to a constant mass matrix defined as 0
L Te eA dx M S S , where, e is the density of the cable
element, and eA is the cross-sectional area of the beam.
23
The ANCF catenary model used in this investigation is shown in Fig. 5. A simple
construction is used which consists of a messenger wire, a contact wire, supporting poles and
droppers. The droppers are modeled as spring-damper elements with properties chosen to be
representative of an actual dropper. For simplicity, the staggering of the messenger and the
contact wires is neglected in this investigation. The contact and messenger wires are modeled
using three-dimensional ANCF cable elements. The contact cable material is assumed to be
copper. As shown in Fig. 5, the catenary system has total 20 spans, each 56.25 m in length. The
pre-tension in the contact and the messenger wires is assumed to be 20 kN and 14 kN,
respectively. The effect of the pre-tension is introduced to the model by using appropriate values
for the nodal displacements. A preliminary static equilibrium analysis is performed to obtain a
stable and curved configuration of the catenary system. Using this configuration, the dynamic
simulations were performed for the full rail vehicle, including the pantograph/catenary system.
During the simulation, the state of every dropper is checked before every integration time-step in
order to determine whether the dropper is in tension or compression, and the dropper stiffness in
compression is neglected. The properties of the catenary model used in this investigation are
shown in Table 6.
5.4 MBS Equations of Motion
The constrained MBS equations of motion are formulated in terms of both rigid and very flexible
body coordinates. The vector of generalized and non-generalized coordinates of the system is
TT T Tr p q e s where rq defines the reference coordinates of the bodies, e is the vector of
ANCF coordinates for the flexible bodies, and s is the vector of non-generalized coordinates or
surface parameters used in the wheel/rail contact formulation. The MBS equations of motion can
be written in the augmented Lagrangian form as
24
r
r
Trr q rr
Teee e
Ts
cq e s
M 0 0 C Qq
Qe0 M 0 C
0s0 0 0 C
QλC C C 0
(14)
where λ is the vector of Lagrange multipliers, rrM is the mass matrix associated with the
reference motion of the bodies, eeM is the mass matrix associated with the ANCF coordinates,
rqC and eC are the constraint Jacobian matrices associated with the coordinates rq and e ,
respectively, the generalized forces associated with these coordinates are given by rQ and eQ ,
respectively, and cQ is a quadratic velocity vector which results from the differentiation of the
constraint equations twice with respect to time (Shabana, 2013).
6. NUMERICAL RESULTS AND DISCUSSION
The numerical results obtained using the proposed new pantograph/catenary elastic contact
formulation and new ANCF aerodynamic force model are presented in this section. While, the
new elastic contact formulation and the ANCF aerodynamic force model can be used with both
the fully parameterized and gradient deficient ANCF elements, the gradient deficient cable
element is used in this investigation in order to obtain more efficient simulations and also avoid
locking problems that characterize some fully parameterized elements. The effect of the
aerodynamic forces on the contact force is also evaluated by considering the cross-wind loading
which is applied on both the pantograph and the catenary system.
6.1 Numerical Results without Wind Loading
In this subsection, the simulation results for the case of no wind loading on the
pantograph/catenary system are presented. An MBS model representing the Faiveley CX
25
pantograph is used in this investigation. The rail vehicle is assumed to travel with a forward
velocity of 200 km/h on a straight track with no aerodynamic forces applied to the pantograph or
the catenary system. A simulation time of 10 seconds is considered. The uplift force exerted on
the pantograph mechanism is modeled as a vertical upward force applied on the lower arm with a
value of 1465 N. The contact force as function of time is shown in Fig. 6. Figure 7 shows the
results in Fig. 6 in the time window of 7 to 10 seconds. The transient effects in the catenary
model are clear at the beginning of the simulation. As can be observed from the results presented
in Fig. 6, during the time period 1 - 2 seconds, the transverse waves are dominant in the catenary
system and after approximately 2 seconds their effect diminishes. The contact force solution
from this point becomes mainly periodic, indicating that a steady state has been reached.
Therefore, in the steady state analysis of the contact force, the first two seconds of the simulation
should not be considered. Figure 7 shows large peaks of the contact force when the pantograph
passes under support pole positions and the droppers, with the peaks corresponding to the
support poles are much higher than the peaks corresponding to the droppers.
Figure 8 compares the contact force predicted using the constraint (sliding joint) and the
new elastic contact formulations. The numerical results for the contact force obtained using the
constraint contact formulation with no control action are the same as presented by Pappalardo et
al. (2015). Figure 9 shows a good agreement between the contact force results obtained using the
constraint contact formulation and the proposed new elastic contact formulation. The mean
contact force predicted using the elastic contact formulation is -107.79 N, compared to -110.427
N in the case of the constraint contact formulation. The standard deviation of the contact force
computed with the elastic contact formulation is found to be 36.7257 N, while it is 37.3033 N in
the case of the constraint contact formulation. The mean contact force and the standard deviation
26
of the contact force are within the limits specified by the standard EN50367, as previously
discussed.
6.2 Effect of the Wind Loading
The effect of the aerodynamic forces on the contact force is discussed in this section. The rail-
vehicle forward velocity is assumed to be 200 km/h and the uplift force on the pantograph
mechanism is assumed 1465 N. These values are kept the same as in the case in which the wind
loading is not considered. Two cases are considered; in the first case the crosswind loading is
applied on the pantograph components only, and in the second case the crosswind loading is
applied on both the pantograph and the catenary systems. In both cases, the wind is assumed to
have a velocity of 20 m/s at a yaw angle of 15 degrees and the effect of turbulence is neglected.
Figure 10 shows the contact force when the crosswind is applied on the pantograph
components only. The aerodynamic forces, which are applied on the four components of the
pantograph (the upper arm, lower arm, lower link and panhead shown in Fig. 2), contribute
significantly to the total aerodynamic uplift force. The aerodynamic forces are applied at the
center of gravity of each component, and the aerodynamic pressure coefficients of the drag and
lift used in this study are shown in Table 2. These coefficients are not dependent on the Reynolds
number and the aerodynamic uplift force depends on the square of the train velocity, as
confirmed by the trial tests given in the standard EN50317. The effect of the aerodynamic
pitching moment is not considered in this investigation. The time history of the drag and lift
forces acting on the panhead is shown in the Figs. 11 and 12, respectively. The drag and lift
forces are nonlinear in nature and are dependent on the velocity of the train and the wind loading
conditions. A positive lift force indicates that the panhead is lifted upwards, causing an increase
in the mean contact force. This effect can be observed from the results presented in Figs. 13 and
27
14, where the two cases in which the effect of the crosswind is considered and neglected are
compared. However, the upward lift of the panhead depends on many factors including the
collector geometry profile, use of spoilers, turbulence in wind, and others. The mean contact
force in the case of applying the crosswind load on the pantograph is found to be -120.22 N
while it is -107.79 N when the crosswind effect is neglected, which shows 11.53% difference.
The standard deviations are found to be similar in both cases, with a slight decrese of 2.80%
when the wind loads are applied on the pantograph.
Figure 15 shows the contact force when the aerodynamic forces are applied on the
pantograph components and the catenary system including the contact and the messenger wires.
The mean contact force and the standard deviation of the contact force are found to be -120.21 N
and 35.66 N, respectively. Figures 16 and 17 compare the contact force results obtained when
considering and neglecting the effect of the wind forces on the pantograph components and the
catenary. As shown in Table 7, there is an increase of 11.52% in the mean contact force value
and a decrease of 2.91% in the standard deviation. It can be observed, for the example
considered in this numerical study, that these values are similar to those values obtained by
applying the aerodynamic forces only on the pantograph components. Table 7 shows the values
for the mean contact force and standard deviation for all cases considered in this section.
7. SUMMARY AND CONCLUSIONS
In this investigation, a new pantograph/catenary elastic contact formulation and a new ANCF
aerodynamic force model are proposed. Unlike the constraint contact (sliding joint) formulation
(Pappalardo et al., 2015), the new elastic contact formulation allows for modeling the
catenary/panhead loss-of-contact scenarios. The proposed contact formulation is based on the
28
definition of a contact frame that requires a single gradient vector only, and therefore, such a
formulation can be used with both gradient deficient and fully parameterized ANCF elements.
The ANCF cable element, used in this investigation to model the overhead contact line and the
messenger wire, was found to be appropriate for capturing the geometric nonlinearities in the
catenary system. The MBS pantograph and the ANCF catenary models are developed in a single
computational framework, thereby eliminating the need for using a co-simulation technique that
requires the use of different computer codes. The numerical results obtained in this investigation
are shown to be consistent with the standard EN50317 recommendations and also show a good
agreement with the previously published results obtained using the constraint contact
formulation.
A new ANCF approach for modeling the time varying aerodynamic forces was also
proposed. This approach allows for applying aerodynamic forces on very flexible structural
components modeled using ANCF elements. Using this new approach, the drag and lift forces
can be applied simultaneously on the individual pantograph components and the contact and
messenger wires. For the example considered in this investigation, it was found that the
aerodynamic uplift force contributes to increasing the uplift force exerted on the pantograph
mechanism, increasing the mean contact force by about 11.53%, whereas the increase in the
mean contact force when the aerodynamic forces are applied on the pantograph components and
the catenary is about 11.52%. On the other hand, for the example considered in this
investigation, the value of the contact force standard deviation decreases by about 2.80% when a
crosswind is applied only on the pantograph, while the standard deviation value of the contact
force decreases by about 2.91% when the aerodynamic forces are applied to both the pantograph
and catenary.
29
30
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35
Table 1. Aerodynamics coefficients for the pantograph components
Parameter for Pantograph Acceptance Acceptance Norm
Mean contact force 20.00097 70NmF v
Standard deviation 0.3max mF
Maximum contact force 350NmaxF
Maximum catenary wire uplift at steady arm 120 mmupd
Maximum pantograph vertical amplitude 80mmz
Percentage of real arcing 0.2%NQ
Table 2. Aerodynamics coefficients for the pantograph components
Pantograph Component Drag Coefficient DC A Lift Coefficient LC A
Lower Arm 0.007 -0.0075
Lower Link 0.008 -0.001
Upper Arm 0.07 0.02
Panhead 0.05 0.02
36
Table 3. Rail vehicle model data
Body Mass (kg)
Inertia (kg·m2)
Initial Position (m) (xi0, y
i0, z
i0)
xxiI yyiI zziI
Wheelsets 2091 1098 191 1098
0, 0, 0.4570488
2.5908, 0, 0.4570488
12.573, 0, 0.4570488
15.1638, 0, 0.4570488
Equalizers 469 6.46 255 252
1.2954, 1.0287, 0.3049427
1.2954, -1.0287, 0.3049427
13.8684, 1.0287, 0.3049427
13.8684, -1.0287, 0.3049427
Frame 3214 1030 1054 2003 1.2954, 0, 0.5081427
13.8684, 0, 0.5081427
Bolster 1107 498 20.4 458 1.2954, 0, 0.7088
13.8684, 0, 0.7088
Car Body 24170 30000 687231 687231 1.8289, 0, 1.8289
37
Table 4. Pantograph model data
Body Mass (kg) Initial Position (m)
(xi0, y
i0, z
i0)
Initial Orientation
( , , )
Inertia
(kg·m2)
(Ixxi, Iyyi, Izzi)
Lower arm 32.18 11.26924156, 0,
3.84511275
/2 , 0.5528807212,
- /2 0.31, 10.43, 10.65
Upper arm 15.60 11.45796454, 0,
4.52440451
- /2, 0.2896816713,
/2 0.15, 7.76, 7.86
Lower Link 3.10 10.96436876, 0,
3.81940451
/2, 0.6234559506,
- /2 0.05, 0.46, 0.46
Upper link 1.15 11.58587608, 0,
4.49940451
- /2, 0.3028168645,
/2 0.05, 0.48, 0.48
Plunger 1.51 12.5, 0, 4.835 0, 0, 0 0.07, 0.05, 0.07
Panhead 9.50 12.5, 0, 4.945 0, 0, 0 1.59, 0.21, 1.78
Table 5. Pantograph model joints data
First Body Second Body Joint Constraint
Car Body Lower Arm Revolute
Lower Arm Upper Arm Revolute
Upper Arm Plunger Revolute
Car Body Lower Link Spherical
Upper Arm Lower Link Spherical
Plunger Upper Link Spherical
Lower Arm Upper Link Spherical
38
Table 6. Catenary Properties
Contact/Messenger Wire Geometry Elements per Span 9
Element Length (m) 6.25 Element Cross Section Area (mm2) 144
Total Number of Spans 20 Catenary System Material Properties
Contact Wire Density (kg/m3) 8960 Contact Wire Modulus of Elasticity (Pa) 1.2E+11
Messenger Wire Modulus of Elasticity (Pa) 1.2E+11 Other General Catenary Properties
Tension in Contact Wire (N) 20000 Tension in Messenger Wire (N) 14000
Dropper Stiffness kd (N/m) 200000 Dropper Damping Cd (N/m) 10000
Table 7: Numerical results for the contact force mean value and the standard deviation in the contact force
Wind Loading Mean Force (N) Std. Dev (N) ∆ Mean Force ∆ Force Std. Dev
No wind -107.79 36.73 / /
Wind only on the pantograph -120.22 35.70 11.53% -2.80%
Wind on the pantograph and the catenary
-120.21 35.66 11.52% -2.91%
39
Figure 1. The direction of the drag and the lift forces
Figure 2. Rail vehicle system schematic
40
Figure 3. Pantograph-catenary model schematic
Figure 4. The Catenary System
41
Figure 5. Catenary computational model
Figure 6. Contact force without wind loading
42
Figure 7. Contact force without wind loading- zoomed window
Figure 8. Contact force comparison between the sliding joint formulation and the penalty formulation ( Penalty Force Formulation, Sliding Joint Formulation)
43
Figure 9. Contact force comparison between the sliding joint formulation and the penalty formulation- zoomed window ( Penalty Formulation, Sliding Joint Formulation)
Figure 10. Contact force with cross wind loading on the pantograph components of 20 m/s at a yaw angle of 15 degrees
44
Figure 11. Drag force on the panhead
Figure 12. Lift force on the panhead
45
Figure 13. Contact force comparison with crosswind loading on the pantograph components of 20 m/s at a yaw angle of 15 degrees and with no wind loading ( Without wind loading,
With crosswind loading)
Figure 14. Contact force comparison with crosswind loading on the pantograph components of
20 m/s at a yaw angle of 15 degrees and with no wind loading- zoomed window ( Without wind loading, With crosswind loading)
46
Figure 15. Contact force with cross wind loading on the pantograph components and the catenary of 20 m/s at a yaw angle of 15 degrees
Figure 16. Contact force comparison with crosswind loading on the pantograph components and the catenary of 20 m/s at a yaw angle of 15 degrees and with no wind loading ( Without
wind loading, With crosswind loading)
47
Figure 17. Contact force comparison with crosswind loading on the pantograph components and the catenary of 20 m/s at a yaw angle of 15 degrees and with no wind loading- zoomed window
( Without wind loading, With crosswind loading)