Journal of System Design and Dynamics

Journal of System Design and Dynamics

Vol. 1, No. 3, 2007

Contact Model for The Pantograph-Catenary Interaction* Frederico Grases RAUTER**,***, João POMBO**, Jorge AMBRÓSIO**, Jérôme CHALANSONNET***, Adrien BOBILLOT*** and Manuel Seabra PEREIRA** ** IDMEC–Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal *** Dir. Innovation & Recherche - SNCF 45 Rue de Londres, 75008 Paris, France E-mail : [email protected]

Abstract In the great majority of railway networks the electrical power is provided to the locomotives by the pantograph-catenary system. The single most important feature of this system consists in the quality of the contact between the contact wire(s) of the catenary and the contact strips of the pantograph. The work presented here proposes a new methodology to study the dynamic behavior of the pantograph and of the interaction phenomena in the pantograph-catenary system. The catenary is described by a detailed finite element model while the pantograph is described by a detailed multibody model. The dynamics analysis of each one of these models uses different time integration algorithms: the finite element model of the catenary uses a constant time step Newmark type of integration algorithm while the multibody model uses a variable order and variable time step Gear integration algorithm. The gluing element between the two models is the contact model, it is through the representation of the contact and of the integration schemes applied to the referred models that the needed co-simulation is carried on. The work presented here proposes an integrated methodology to represent the contact between the finite element and multibody models based on a continuous contact force model that takes into account the co-simulation requirements of the integration algorithms used for each subsystem model. The discussion of the benefits and drawbacks of the proposed methodologies and of their accuracy and suitability is supported by the application to the real operation scenario considered and the comparison of the obtained results with experimental test data. Key words: Multibody Dynamics, Contact Modeling, Co-Simulation, Pantograph Catenary Interaction

1. Introduction

*Received 1 June, 2007 (No. 07-0239) [DOI: 10.1299/jsdd.1.447]

The pantograph-catenary system is still today the most reliable form of collecting electric energy for running trains, when high speed operational conditions are considered. This system should ideally run with relatively low contact forces, in order to minimize wear and damage of the contacting elements of the system, and no contact loss should be observed, so that the power supply would be constant and no electric arching would be observed. Unfortunately such ideal state is impossible to achieve with the pantograph technology presently in operation. An actively controlled pantograph could be an answer to minimize contact loss and to maintain the contact forces within an acceptable operational envelope.

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Due to the increasing importance of the pantograph-catenary interface with the developments in high-speed trains a large number of works have been developed to study this system. Several 2D finite element models for the catenary-pantograph system dynamics have been presented, as for example introducing nonlinear effects in the droppers and the possible contact losses between the pantograph and the catenary (1), using lumped mass models of the pantograph, and studying the wave propagation problem on the catenary for the catenary-pantograph dynamics (2). In the works mentioned above not only the representation of the contact forces is not discussed but also no reference is made on how the integration algorithms are able to handle the contact loss and impact between registration strip and contact wire. A co-simulation strategy between the code PROSA, where a catenary is described by the finite difference method and the SIMPACK commercial multibody code used to simulate the pantograph (3) has also been proposed recently. All models involved in this work are 3D but the catenaries are hard coded, and therefore, the models and programs can hardly be used for different catenary systems. A very thorough description of the pantograph catenary system that includes a 2D model for the catenary based on the finite element method, and a pantograph model based on a multibody approach (4) has also been presented. Works have also been developed to study the contact phenomena between the pantograph and the catenary using 3D models using unilateral constraints and penalty methods. For example, catenary contact wire can be modeled by a finite element model based on the absolute nodal coordinate formulation and the pantograph is a full 3D multibody model (5). The contact is represented by a kinematic constraint between contact wire and registration strip and no loss of contact is represented. None of the models used has been validated in this study. Although major efforts have been done to provide reliable models of pantographs and catenaries and a very good insight on the dynamics of the pantograph-catenary interaction has been offered by the studies referred, currently there are no accepted general numerical tools designed to simulate the pantograph-catenary system in nominal, operational, and deteriorated conditions. Here it is understood that operating conditions must take into account the wear effects and the deteriorated conditions include extreme climatic conditions, material defects or mechanical problems. However, several important efforts have been reported to understand the mechanisms of wear in catenaries and the effect of defect conditions on the dynamics of the complete system (6)(7)(8)(9). In addition the capability to analyze real three-dimensional models of the catenary and pantograph is very limited and no validated models exist for either of the subsystems. The European project EUROPAC project (European Optimized Pantograph Catenary Interface) aims at developing general software that is able to overcome the shortcomings just listed and in the process to develop accurate models for different types of catenaries and pantographs. The software and models are to be used not only for maintenance and design but also to appraise the interoperability between different systems by verifying compliance with standards on the mean contact force value, standard deviation or maximum uplift at supports. The work now presented addresses the choice of the methods used to develop and simulate the pantograph models and to interface them with the code used for catenary analysis in a co-simulation environment.

2. Multibody Dynamics In this work the methods presented to describe multi-body systems are based on the use of Cartesian coordinates, which lead to a set of Differential-Algebraic equations that need to be solved. In particular, the numerical issues that result from the use of this Cartesian coordinates, such as the existence of redundant constraints, the possibility of achieving singular positions or the drift of the constraint position and velocity equations that lead to violation of such equations, are also solved.

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Revolute joint

Spring

body i

body n

Ball joint

body 2

External forces

body 3

body 1

Damper

Figure 1 Generic multibody system

2.1 Multibody Equations of Motion A typical multibody model is defined as a collection of rigid or flexible bodies that have their relative motion constrained by kinematic joints and is acted upon by external forces. The forces applied over the system components may be the result of springs, dampers, actuators or external applied forces describing gravitational, contact/impact or other forces. A wide variety of mechanical systems can be modeled as the schematic system represented in Fig. 1. Let the configuration of the multibody system be described by n Cartesian coordinates q, and a set of m algebraic kinematic independent holonomic constraints Φ be written in a compact form as

Φ ( q,t ) = 0

(1)

Differentiating Eq. (1) with respect to time yields the velocity constraint equation. After a second differentiation with respect to time the acceleration constraint equation is obtained

Φq q = υ

(2)

= γ Φq q

(3)

where Φq is the Jacobian matrix of the constraint equations, υ is the right side of velocity equations, and γ is the right side of acceleration equations, which contains the terms that are exclusively function of velocity, position and time. The equations of motion for a constrained multibody system (MBS) of rigid bodies are written (10)

= g + g (c ) Mq

(4)

is the vector that contains the state where M is the system mass matrix, q accelerations, g is the generalized force vector, which contains all external forces and moments, and g(c) is the vector of constraint reaction equations. The joint reaction forces can be expressed in terms of the Jacobian matrix of the constraint equations and the vector of Lagrange multipliers (10)

g ( c ) = −ΦTq λ

(5)

where λ is the vector that contains m unknown Lagrange multipliers associated with m holonomic constraints. Substitution of Eq. (5) in Eq. (4) yields

+ ΦTq λ = g Mq

(6)

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In dynamic analysis, a unique solution is obtained when the constraint equations are considered simultaneously with the differential equations of motion with proper set of initial conditions (10). Therefore, Eq. (3) is appended to Eq. (6), yielding a system of differential and λ. This system is given by algebraic equations that are solved for q

M  Φ q

r   g  ΦTq  q   =   0   λ   γ 

(7)

, together with velocities vector, In each integration time step, the accelerations vector, q q , are integrated in order to obtain the system velocities and positions at the next time step. This procedure is repeated up to final time will be reached. The set of differential algebraic equations of motion, Eqs. (7) does not use explicitly the position and velocity equations associated to the kinematic constraints, Eqs. (1) and (2), respectively. Consequently, for moderate or long time simulations, the original constraint equations are rapidly violated due to the integration process. Thus, in order to stabilize or keep under control the constraints violation, Eq. (7) is solved by using the Baumgarte stabilization method (11) or the augmented lagrangean formulation (12) and the integration process is performed using a predictor – corrector algorithm with variable step and order(13). Furthermore, due to the long time simulations typically required for pantograph-catenary interaction analysis, it is also necessary to implement constraint violations correction methods, or even the use of the coordinate partition method (10)(14) for such purpose. 2.2 Multibody Modeling of the Pantograph In general, a pantograph consists of a collection of bodies and mechanical elements, as depicted in Fig. 2, attached to a railway carbody that is moving along the track. Two modeling strategies can be used to define the reference motion of the pantograph: define the kinematics of the pantograph base; define the motion of the railway vehicle and simply fix the pantograph in the top of the train carbody. Once the kinematics of the pantograph is compatible with the geometries of the catenary and railway both strategies are acceptable.

G s5Q

η6

ξ6

ζ6 Q

z y x

G s7P

G s7Q

ξ9

ξ5

G s5P

P P

ζ9

η5

ζ5

Q

η9

ζ7 ξ7

ζ1 CM

ζ3 η3 ξ 3 η4

ξ4 ζ4

ζ8 η7

η1 ξ1

ζ3 η3 ξ 3

η8 ξ8

η10

ξ10

ζ10

Figure 2 Graphical representation of a pantograph.

Although the moving components of the pantograph have some level of flexibility in this model they are represented as rigid bodies. Deformable elements, such as springs and dampers, are used to model relevant internal forces that represent interactions between rigid bodies of the system. In existent pantographs a good number of non-linear elements differ from standard kinematic and compliance elements used in traditional mechanical systems, as those exemplified in Fig. 3. Bodyj

Pj

Bodyi Pi Qi

Qj

α1

Figure 3 Revolute and translational joints geometric operational limits used for the pantograph model.

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C(v) K(x)

F

Figure 4 Model for the low-frequency damper system

There are different types of force elements that do not have a linear type behavior in the pantograph model. Among these, some relevant force elements, as that represented in Fig. 4, include friction effects besides the elastic forces and the damping. The analysis of the pantograph-catenary contact requires that the motion of the carbody must be defined. A set of control points that are representative of the trajectory of the CM of the carbody and the correspondent roll angle at each point are defined. With these nodal points, the computational tool uses a piecewise cubic interpolation scheme to parameterize the carbody trajectory and orientation, as a function of the traveled distance. Then, a prescribed motion constraint is used in order to enforce the carbody to follow the prescribed trajectory with the cant defined by the roll angle, as depicted in Fig. 5. z

y

Point 2

x

Point 1 zG

ζ ϕ ζϕ

η

ϕ

ηϕ

Point N

ξ ≡ ξϕ

P

Trajectory

yG xG

Figure 5 Prescribed motion constraint of the carbody

3. Finite Element Modeling The second part of the electric collecting system is the catenary. This system does not exhibit large displacements or large rotations, being modeled by linear finite elements. The FE code used here for the dynamic analysis of the catenary has been developed by SNCF to study models of different catenaries. The main catenary elements, the contact and messenger wires are modeled by using pre-tensioned Euler-Bernoulli beams. The code uses a Newmark algorithm to integrate the FE equations of motion. In what follows it is important to review the features of these algorithms. At any given time step the algorithm proceeds by first predicting the displacements and velocities for the new time step by using the information of the last completed time step as

∆t d t +∆t = dt + ∆t v t + (1 − 2β ) at 2 v t + ∆t = v t + ∆t (1 − γ ) at 2

(8)

Based on the position and velocity predictions for the FE mesh and on the pantograph predicted position and velocity the contact forces are evaluated for t+∆t and the FE mesh accelerations are calculated from the equilibrium equation

( M + γ ∆t C + β ∆t K ) a 2

t + ∆t

= ft +∆t − Cv t +∆t − Kd t + ∆t

(9)

Then, with the acceleration at t+∆t the positions and velocities of the finite elements at time t+∆t are corrected by

d t + ∆t = d t +∆t + β ∆t 2 at +∆t v t +∆t = v t +∆t + γ ∆t at +∆t

(10)

These procedure is repeated until a stability value is reached for a given time step.

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3.1 Finite Element Modeling Issues for the Catenary It is not the objective of this work to discuss in detail models for the catenaries but simply to highlight some of their features. Catenaries are complex periodical structures, as that represented in Fig. 6. Examples of typical structural elements involved in the catenary model are the contact, stitch and messenger wires, droppers and registration arms. Depending on the catenary system there are other elements that may have to be considered. In any case, the contact wire is the responsible for the contact between catenary and pantograph and therefore the element that provides electrical power. The messenger wire prevents excessive sag caused by the contact wire weight. Both of these wires are connected by vertical, tensile force droppers. Carrier cable Stay Support

Stitch wire

Console

Contact wire Steady arm Registration arm

Figure 6 Catenary three-dimensional mesh in the Finite Element software OSCAR (by SNCF)

A typical catenary is characterized by having nonlinear type of behavior. For example the stiffness of the messenger and contact wires is dependent of its current deformation state and the droppers have high tensile stiffness, but no resistance to compression. The overall vertical stiffness of the catenary is function of the behavior of each of its elements. Furthermore, the contact loss with the pantograph introduces also nonlinear effects on the behavior of the catenary. At this point it must be referred that there is a large variety of catenary systems used worldwide. Even in a single European country there are different types of catenaries in use with different particularities in their construction. This raises the complexity of having a general software tool to model catenaries.

4. Contact The pantograph and catenary systems are related due to the interaction between the registration strips of the pantograph and the contact wires of the catenary. The contact force due to pantograph-catenary interaction, regarding present operating conditions and pantograph and catenary technology, is characterized by a low-frequency oscillating force with high relative amplitude. Railway industry measurement data shows that reasonable values for the contact force are, for a train running at approximately 80 m/s: a mean value of 200N oscillating between 300N and 100N. Loss of contact in particular points of the catenary may also occur. The modeling of contact occurring between the pantograph and the catenary is a challenging problem as the location of the contact point is dependent on time and on space. An approach is the use of an arc-length parameter to determine the exact location of the contact point on line (5). In this work reference coordinates are used to describe all rigid body motion, the catenary system’s large deformation is described by absolute nodal coordinates and non-generalized arc-length parameters enter in the formulation of the sliding joint between the pantograph and the catenary. The occurrence of contact loss implies nonlinear impact effects that must be accounted for in the model, and therefore a force model must be used instead of the sliding joint approach.

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4.1 Contact Force Model The contact model used here is based on a contact force model with hysteresis damping for impact in multibody systems (15)(16). In this work, the Hertzian type contact force including internal damping can be written as

 3(1 − e 2 ) δ  FN = K δ n 1 + (−)  4 δ  

(11)

where K is the generalized stiffness contact, e is the restitution coefficient, δ is the relative penetration velocity and δ ( − ) is the relative impact velocity. The proportionality factor K is obtained from the Hertz contact theory as the external contact between two cylinders with perpendicular axis. 4.2 Co-Simulation Between the Pantograph and the Catenary Multibody Codes The analysis of the pantograph-catenary interaction is done with two independent codes, the pantograph code, developed using a multibody methodology reported in this work, and the catenary code, developed using finite elements. Both of these programs work as stand-alone, i.e., each code integrates separately the equations of motion of each sub-system. The structure of the communication scheme is shown in Fig. 7. The MB code provides the FEM code with the position and velocity of the pantograph’s registration strip. The FEM code calculates the contact force, using Eq. (11), and its application points in the pantograph and catenary, using a geometric interference model. These forces are applied to the catenary, in the finite element code, and to the pantograph model, in the MB code.

f contact

FEM - Catenary

rstrip , rstrip

Contact catenary - pantograph

Position,Velocity

 δ  = K 1 + 34 (1 − e2 ) ( − )  δ n n δ  

P f contact , s′strip

Force, Point

MB - Pantograph ζi ηi

Z

X

G ri G ri P

G siP

ξi

G f contact

P

Y

Figure 7 Structure of the communication scheme between the MB code and the FEM code

The numerical integration algorithms used by FEM codes are typical Newmark family algorithms (17). The FEM code needs a prediction of the positions and velocities not only of the catenary but also of the pantograph in a forthcoming time before advancing to a new time step. A predicted contact force is calculated and, using the finite element method equations of equilibrium, the catenary accelerations are computed for the new time-step. The calculated acceleration values are used to correct the initially predicted positions and velocities of the catenary. The MB code uses a Gear multi-step multi-order integration algorithm (18)(19). To proceed with the dynamic analysis, the MB code needs information about the positions and velocities of the pantograph components and also the contact force and its application point coordinates at different time instants during the integration time period, and not only at its start and end. The compatibility between the two integration algorithms imposes that the state variables of the two subsystems are readily available during the integration time and that a prediction of the contact forces is available at any given time step. Several strategies can be envisaged to tackle this co-simulation problem such as the gluing algorithms (20) or other co-simulation procedures (21). However, the problem that it is tackled in this work is characterized by eventual contact loss between the two systems, almost a problem of intermittent contact, and therefore, a different approach is envisaged. The key of the synchronization procedure between the MB and FE codes is the time integration, which must be such that it is ensured the correct dynamic analysis of the pantograph-catenary system, including the loss and regain of contact.

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5. Application Example The computational simulation described in this work aims at modeling the behavior of a real pantograph and catenary system used in French high speed lines. The velocity imposed to the pantograph system is of 80 m.s-1. 5.1 Pantograph Lumped Mass Model The pantograph model used in this work is described in Fig. 8. A local reference frame (ξηζ) is rigidly attached to the CM of each body in such a way that the axes are aligned with the principal inertia directions of the rigid bodies. The initial conditions for the bodies of the system are given by the location of the CM and by the orientation of the local reference frames (ξηζ) with respect to the pantograph coordinate system (xyz). Upper mass c23

fstatic k23

Middle mass c12

k12 Lower mass

c01

k01

Carbody

Figure 8 Graphical representation of the pantograph model used

The relevant data required for the bodies is presented in Tables 1 to 4. The system is composed by two subsystems: the carbody (Subsystem 1) and the pantograph (Subsystem 2). Tables 1 and 2 present the characteristics of the carbody and Tables 3 and 4 the properties of the remaining bodies of the pantograph. In both cases the data described for the models should not be deemed as exact, but simply approximate. Table 1 Mass and inertia properties of each rigid body of the carbody (subsystem1). ID 1

Rigid

Inertia Properties (Kg.m2)

Mass

Body

(Kg)

Iξξ / Iηη / Iζζ

Carbody

11160

14953.00 / 225365.00 / 224994.00

Table 2 Initial conditions of each rigid body of the carbocy (subsystem1). ID 1

Rigid

Initial Position (m)

Initial Orientation

Body

x0 / y0 / z0

e1 / e2 / e3

Carbody

-5.50 / 0.00 / -1.20

0.00 / 0.00 / 0.00

Table 3 Mass of each rigid body of the pantograph (subsystem2) ID

Rigid Body

Mass (Kg)

2

Lower mass

4.8

3

Middle mass

4.63

4

Top mass

8.5

The constant force applied to the pantograph model masses to achieve static equilibrium when in contact with the catenary is defined in Table 5. Spring-damper-actuator elements are used to model the forces transmitted among the rigid bodies that compose the pantograph system, with the characteristics presented in Tables 6 and 7.

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Table 4 Initial conditions of each rigid body of the pantograph (subsystem2) Initial Position (m)

Initial Orientation

x0 / y0 / z0

e1 / e2 / e3

Lower mass

0.35 / 0.00 / 1.69

0.00 / 0.00 / 0.00

3

Middle mass

0.45 / 0.00 / 1.69

0.00 / 0.00 / 0.00

4

Top mass

0.40 / 0.00 / 1.59

0.00 / 0.00 / 0.00

ID

Rigid Body

2

Table 5 Static force conditions of each rigid body of the pantograph (subsystem2) ID

Rigid Body

Static Force (N)

2

Lower mass

0.0

3

Middle mass

0.0

4

Top mass

200.0

Table 6 Characteristics of the spring elements used in the multibody model of the pantograph

ID

Attach Pts Local Coord (m)

Spring

Undef.

Bodies

Stiffness

Length

Connected

Body i

Body j

(N/m)

(m)

i

ξi/ηi/ζi

ξj/ηj/ζj

j

1

1

0.5

1

2

0/0/0

0/0/0

2

5400

0.3

2

3

0/0/0

0/0/0

3

6045

0.2

3

4

0/0/0

0/0/0

Table 7 Characteristics of the damper elements used in the multibody model of the pantograph

ID

Attach Pts Local Coord (m)

Damping

Bodies

Coeffic.

Connected

Body i

Body j

(N.s/m)

i

j

ξi/ηi/ζi

ξj/ηj/ζj

1

32

1

2

0/0/0

0/0/0

2

5

2

3

0/0/0

0/0/0

3

10

3

4

0/0/0

0/0/0

5.2 Catenary Finite Element Model The model of the catenary used in this work is described in Fig. 6 and corresponds to the Paris-Lyon French high speed railway track. All the structural elements that compose the real catenary are modeled, including the contact, messenger and stitch wires, droppers, registration and steady arms. The catenary static state is computed using a non-linear strategy since the stiffness of the wires depends on their actual displacements. 5.3 Results In Fig. 9 the numerical results of the model described are shown together with experimentally measured values of the pantograph-catenary system used in the high speed line, for a velocity of the pantograph on the catenary of 80 m.s-1. The correlation of the numerical and experimental results, according to the EN50318 standard (filtered at 20 Hz), is very good. The results include the effects of the structural elements of the catenary, due to the travel of the pantograph, but do not include, for example, the effects of singular defects. The experimental data was obtained over a commercial high-speed catenary in operating conditions, and therefore subjected to climatic defects, wear and other general defects.

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350

Filtered contact force [N] (20Hz)

300

250

200

150

100

50

0

621

675

729 Distance [m]

783

837

Figure 9 Numerical results (solid lines) and experimental measurements (interrupted line) for the French high speed line, for a train speed of 80 m.s-1

6. Conclusions In this work a force model is employed for modeling the contact force between the pantograph and the catenary allowing the description of continuous contact and of its loss. The analysis is performed using two independent codes in a co-simulation environment. A linear finite element code is used to model and analyze the behavior of the catenary while a multibody code is used to describe the dynamics of the pantograph. The contact model is responsible not only for the interaction between pantograph and catenary but also for the co-simulation procedure. The results of an application with the model of a real high speed setup, under operational conditions, demonstrate not only the suitability of methodologies and models but also, the good correlation between numerical and experimental results. Future developments on the models developed and on the methodologies applied include the use of flexible bodies for the pantograph model, introduction of defect conditions on the catenary, inclusion of transversal winds and further research on specific contact models.

Acknowledgements The work presented here has been developed in the framework of the European funded project EUROPAC (European Optimized Pantograph Catenary Interface, contract nº STP4CT-2005-012440) with the partners Société nationale des chemins de fer français (LEADER) , Alstom Transport, ARTTIC, Banverket, Ceské dráhy akciová společnost, Deutsche Bahn, Faiveley Transport, Mer Mec SpA, Politecnico di Milano, Réseau ferré de France, Rete ferroviara italiana, Trenitalia SpA, Union internationale des chemins de fer, Kungliga Tekniska Högskolan. The work has also been supported by Fundação para a Ciência e a Tecnologia, scholarship SFRH/BD/18848/2004.

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