Introduction
The catenary pantograph system is the way to power trains and vehicles that required an amount of electricity for moving. this system needs a stable contact in order to guarantee the supply of energy to the vehicles; furthermore a range of maximum contact force is dedfined by the norm EN50367 that specifies thresholds for pantograph acceptance, concernig the maximum contact force, mean force and maximum standard deviation. This work exhibits an analisys of the main parameters for the desing of the geometry and the dynamic behavior of a catenary pantograph system with the availment of the softwares: MSC Adams and Mathworks Matlab.
Objectives – mandatory
This work will show:
 A general method in order to identify the correct geometry of the models; the objective of this geometry is to allow the head of the pantograph to rise and lower without rotational displacement; this need comes from the possibility of a lower altitude of the catenary, for example, during a tunnel, and consequently the risk of a loss of contact.
 An analisys of the response of the system to an input defined by a sinusoidal oscillation of the catenary; the objective is to define the critical frequency value at which a loss of contact could appear; the letter is defined through a state space formulation of the problem.
 An analysis of the vibration response of the system have been carried out, defining the transfer functions of the model.
The modelling problem
More than one technical solution exists for the mechanism of the pantograph, the studied one is composed by two four bar linkages with a fixed joint between the first one head bar and the second one left bar; this solution guranties only one degree of freedom in order to adjust the pantograph high.
Simplified model: bodies and joint
The first fourbar linkage has a fixed geometry and it is linked to the ground with two revolute joints, one of these is controlled by a lifting device. The second one has variable geometry that will be defined with an optimization methods on the orientation and dimension of bodies; it is linked to the fisrt one with a fixed joint. The second four bar linkage has an extra arm support fixed on the right bar of the first four bar linkage in order to adjust its position. Here’s a table of the entire system of joints in which:
 R: revolute joint

F: fixed joint

C: cylindrical joint

S: spherical joint
Optimization
Design Variables
The first step concerns the conversion of the fixed joints into revolute with a rotational motor that fixes their orientations. This expedient allows to test several orientations during the optimization.
Afterwards, some extra bodies will be linked to the bodies of interest in order to generate a translational displacement that change the length of the body itself, these are also equipped with a translational motors that fix the location. The original members are linked with the first fourbar linkage and the extra members are linked with the head of the pantograph.
These expedient will allow to automate the changes of the geometry during the optimization. There is a wide range of parameters that can be optmized, the purpose required at least two parameters. The following three have been chosen:
 Angle of the left arm of fourbar linkage(figure 2).
 Position of the revulute joint of the rigth arm of the fourbar linkage defined through the angle of the additional base arm (figure 3).
 Length of the right arm of the fourbar linkage (figure 4).
All the costant functions of the motors are defined by design variables that will change their value during the optimization.
In order to go on with the optimization, a parameters to minimize is needed: the choice is the angular velocity of the head of the pantograph. The minimization is computed for working in a specific range of motion of the mechanism, defined by the lifting device.
Initial values
There could be errors during the optimization caused by the disassembly of the mechanism by too large displacements of the elements. In order to define the right initial values for the optimization, another process has been previously run; the mechanism used for the purpose is an open chain mechanism with a motor between the two links. With the same optimization objective, the new design varibles to optimized is the angular velocity of the new motor defined as the sum of the angular velocities in order to obtain zero rotation.
The simulation has run with the end time equal to half the end time of the close mechanism in order to allow the final values of this simulation replacing the design variables of the other one in a intermediate condition. After obtaining the right initial values, the optimization of the close chain model follows:
Complete model
The model analyzed was an uncomplete mechanism but enough for optimization problem. It is now represented a schematic complete mechanism:
The mechanism present the same joints and dimensions of the optimization results with a more realistic geometry for the structural support of the load located on the head of the pantograph. Most of the load is supported by the right bar of the first four bar linkage and the left bar of the second four bar linkage. The other bars of the linkages guide the orientation of the mechanism during the motion. In this analysis are shown two different models of the pantograph:
 Rigid body model: the two fourbar linkages are made with rigid elements and linked with a torsional springer in the right bars in order to be an equivalent structural flexiblity; the head of the pantograph is linked to the second fourbar linkage with a translational joint and a springer on the left side of the upper bar, in the intersection between the left bar and the head of the second four bar linkage; it is free to move vertically. At the base of the mechanism, the lifting device is linked to the right bar of the mechanism with a torsional springer that represents the flexibility of the actuator. This system has three d.o.f..
 FE part model: the two four bar linkages are modelled with FE part from Adams library and fixed togheter in order to control the movement with a single actuator. The head of the mechanism remains a rigid body with a springer as connection to the remainder of the mechanism. The base is controlled by the lifting device in the same way of the previous model.
In order to obtain a frequency response of the system, it has been used a sweep sine formula.  The catenary is modeled with a rigid body moving vertically through a linear actuator; the contact between the head and the catenary generates the excitation force.
Simulations and analysis of results
The head of the pantograph is linked to the support mechanism with a traslational joint in order to let it free to oscillate vertically; the parameters of the spring that linkes the head with the support is the only that reproduce existing pantograph values of mass stiffness and damping.
A mechanism of excitation is introduced to simulate the action of the catenary. The oscillation of the mechanism is suppposed to be a sine wave in the Y position, like the same ectitation of the device used in laboratory to test the pantograph. The contact force between the pantograph and the catenary will produce a force with variable amplitude that arises from the dynamics of the two system: formulation of conacts between the pantograph and catenary it’s needed. The contatcs between the two mechanism are defined by Adams as a continuous impact modelling. The simulation consists in a first part in which the position is adjusted and a second part of oscillation.
The excitation consists in a sine wave oscillation with constant amplitude; the conatct force increases with the amplitude of the oscillation and the increase of frequency, this behavior remains until a critical frequency causes the loss of contacts.
Critical Frequency
The main problem to solve in order to proceed with the pantograph system design is the loss of contact. During the activity, the catenary transmits an oscillation to the mechanism and the dynamic response of the letter could induce a separation that causes a stop of the supply of electricity and also the possibility of the manifestation of electric arcs that damage the head of the pantograph.
It would be usefull to know which is the critical frequency at a certain amplitude of oscillation or others input of displacement. The starting point is the computation of the state matrices of the system.
Initial condition analysis
This method assumes the catenary system as a rigid input of oscillations. In order to find the critical frequency, the response to analyze are the Y displacement and Y velocity of the head of the pantograph.
The contact force is a non linear input, caused by the non holonomic constraint of the contact; futhermore there’s no way to use an equivalent force function because of the unknown amplitude of the letter at certain frequency.
In order to find a method of analisys, the displacement and the velocity of the catenary have been used as the initial condition of a free response of the system; this is justified by the consideration that the pantograph head and the catenary displacement and velocity must be the same until the conatct loss.
The range of configuration used to analyzes the response of the system to all the pairs of displacementvelocity of a complete period of the sine wave. Only the i.c. of zero displacement and maximum velocity is reported because it is considered the most critical.
The assumption of this criterion is that the oscillation of the catenary introduces initial condition only at the head of the mechanism; the other d.o.f. remain at an equilibrium points and only react to the free respose.
If the oveshoot displacement of the free oscillation matches the excitation and the velocity of the system moving to an equilibrium point matches the velocity of the catenary, in this precise moment, the separation begins.
The analisys is computed in the software Matlab after exporting the state matrices from the Adams model and after a calibration of the initial condition through the measurements of the geometry in Adams.
This analisys works better when the oscillation is small and the frequency is high in order to consider the initial condition of the other d.o.f. constant, otherwise the statioinary behavior generates a stationary response of the d.o.f that fairly alterate the position and the velocity of the head; a good working range upper limit ends at 0,005 m. In figure (8) and (9) are shown the free responses of the displacement and the velocity of system in a not critical frequency, on the left and at the critical frequency, on the rigth. The medium load acting on the head is 150 N; the amplitude of oscillation is 0.001 m.
Springequivalent model
A second approach to this problem is based on the modeling of the catenary as a springdamper element linked on the head of the pantograph. In a compressed configuration, the system works in a realistic way but when the compression comes to zero, the model diverges from the reality. The moment of zero compression is considered the starting time for the contact loss. The analysis is computed in matlab with a state space model of the mechanism with catenary springdamper, considering the input as the displacement of the catenary, that transimt a force to the head of the pantoraph. In figure (10) is shown the influence of the oscillation on the compression force, both in a sub critical frequency and at the critical frequency. The medium load acting on the head is 150 N; the amplitude of oscillation is 0.001 m.
This model depends on the pantogrph parameters and springequivalent catenary parameters.
Comparison and coclusion
These two ways of analysis’ show the behavior of the mechanism in two different and quite opposite actions of the catenary; in order to understand and reproduce the real effects of the interaction, a finished element model is required for the catenary, furthermore it is neccessary to couple the dynamics of both the catenary and the pantograph with a contact model. With this analysis, it has been identified kind of borders of the range in wich could be the right interaction.
In order to find the critical frequency, it is possible to use the Matlab code with the state system model or simulate the model in Adams looking for the zero in the contact force or for the zero compression of the equivalent spring. Both the two model have been developed in Adams, and the results match.
Transfer functions
The transfer functions of the bodies have been extracted from the model; the analysis exposes the acceleration response with the frequency, in order to follow the results found on scientific papers. The sperimatal procedure to analyze the pantograph expects the availment of a three masses model linked with springdamper elements like the one in figure (12):
From the real experimental data of the pantograph, tested on a excitation machine, the stiffness and damping coeficients of the model are calibrated in order to replicate the frequency response of the real model; only the stiffness and damping of the third mass are the original values of the real pantograph; this in order to compute a study of the contact with the catenary system and improve a better contact average force. In figure (14) are shown the transfer functions of the model derived from a real pantograph, this extract comes from experimental data. In this work only the head of the two models replicates the exact values of a real pantograph as suggested in the esperimental procedure.
The model made by rigid bodies has been developed with values of mass, stiffness and damping that follow the magnitudes of the three mass model, but not the exact values, the head is the only one that has the same parameters. This procedure comes from the lack of data of real pantographes because of the experimental tests that work with three masses models. It shows these results:
The FE parts model shows these results:
The FE parts has been analyzed in matlab after a simulation with a sweep function; this comes from the impossibility of a linear analysis in MSC Adams with the FE parts.
Conclusion
 Through the optimization method, the correct geometry has been succesfully reached.
 In a small range of oscillation, the script that detects the critical frequency works well; it loses precision with higer amplitude and lower frequencies. Anyway this method allows to study the phenomena with few informations: the dynamic model and the features of the excitation, without the formulation of the force between the bodies. It could give an introductory estimation of the activity of the model, usefull for subsequent considerations.
It would be useful analizying the free response of the system ater a few cycles of work, in order to catch the behavior influenced by others initial conditions on the d.o.f. of the mechanism.  The spring equivalent model could represent a simplify model of the catenary,since it is studied as an elastic element; unfortunately the dynamics of the catenary system required a deeper and more complex analysis and this model loses accuracy.
 The transfer functions of both the rigid bodies model and FE parts model are not similar at all with the experimental data of tested model, especially the FE parts model; anyway the rigid bodies model seems to be a better way to approach the pantograph system instead of FE parts; the letters need more accurate informations of the structures that composed the mechanism and stiffness and damping combined values similar to the reality. The behavior of the head of the pantograph in the rigid body models follows a similar trend with the original one; this is an important achievement because it allows to continue with the analysis and improve an optimization criterion for minimization of the standard deviation of the contact force.
References
[1] Jorge Ambròsio, Joao Pombo, Manuel Pereira, “Optimization of highspeed railway pantographs for improving pantographcatenary contact”, IDMECIST Techincal Univ. of Lisbon Av. Rovisco Pais, 2013.
[2] Ricardo Miguel Graça Aquino Vieira, Jorge Ambròsio, Josè Firmino Aguilar Madeira, “High Speed Train Pantograph Models Identification”, Techincal Univ. of Lisbon Av. Rovisco Pais, 2016.