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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)