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Pantograph-Catenary Dynamics: An Analysis of Models and Simulation Techniques

Pantograph-Catenary Dynamics: An Analysis of Models and Simulation Techniques Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-225$16.00 2001, Vol. 7, No. 2, pp. 225±238 Swets & Zeitlinger Pantograph-Catenary Dynamics: An Analysis of Models and Simulation Techniques * * M. SCHAUB and B. SIMEON ABSTRACT In this paper we analyze models and simulation techniques for the interaction of pantograph and catenary. Detailed models for catenary and pantograph and the propagation of waves are ®rst investigated. Next, the semi discretization by the ®nite element method and the time integration are described. In this context numerical techniques like GGL-stabilization and superconvergent patch recovery are applied. The latter yields an error estimation for the ®nite element grid and shows the critical points of the system. Keywords: interaction pantograph/catenary, partial differential-algebraic equation (PDAE), error estimation, superconvergent patch recovery, GGL-stabilization. 1 INTRODUCTION Pantograph and catenary dynamics is the most critical part in the energy transmission of high-speed trains. Often, oscillations in the contact wire occur such that the contact force between pantograph and catenary varies strongly and the contact may even get lost. Therefore, constructive changes for both pantograph and catenary are under development. Design criteria include the permanent contact of pantograph head and contact wire at high speed and the reduction of both aeroacoustic noise and wear [9]. Dynamical simulation plays an important role in this development since prototypes and measure- ments are very expensive. In the present paper we investigate mathematical models and numerical simulation techniques for this coupled system. Our approach is based on a Center for Scienti®c Computing and Mathematical Modelling, University of Karlsruhe, Karlsruhe, Germany. 226 M. SCHAUB AND B. SIMEON descriptor formulation where the contact condition is treated as a unilateral constraint and where a continuous model for the catenary is combined with a mechanical multibody system for the pantograph. This results in partial differential equations (PDE's, catenary) and differential-algebraic equations (DAE's, pantograph). We show how numerical techniques from the ®nite element method and from differential-algebraic equations can be effectively applied and present some simulation results. The paper is organized as follows: First, we introduce shortly the coupled system for the equations of motion and analyze its structure, followed by a description of the solution behavior. Next, space and time discretization techniques are discussed. In particular, the quality of the FE grid is assessed in terms of the superconvergent patch recovery of Zienckiewicz / Taylor [13]. Finally, several simulation runs at different speeds demonstrate the dynamic behavior. It turns out that contact losses appear if the velocity exceeds 200 km / h. 2 MATHEMATICAL MODEL 2.1 Equations of Motion In this section we introduce the mathematical models for pantograph and catenary. As extension of the benchmark problem in [1, 12] more detailed models are used which cover the three-dimensional geometry correctly, while the motion of the catenary is still restricted to the vertical direction. In detail, the pantograph model contains seven degrees of freedom and ®ve masses, see Figure 1, and describes, in addition to the vertical motion, n ˆ 2 contact points and the rotation of the pan head (q ), see [8] for more details. The latter is necessary to take the zigzag-course of the contact wire into account. The model for the catenary consists of carrier, contact wire and n ˆ 14 droppers, see Figure 2. Furthermore, we model the registration arm at the midpoint as a spring damper system, in contrast to those at the boundary, which are ®xed. The carrier is modelled as a string, but we use an Euler-Bernoulli-beam for the contact wire to get a continuous ®rst derivative at the contact point. The equations of motion contain the density , cross section area A, damping SYSTEM OF PANTOGRAPH AND CATENARY 227 Fig. 1. Pantograph model with ®ve masses and seven degrees of freedom. constant , normal force T and the bending stiffness EI in case of the beam. Moreover, w and w denote the displacement of the carrier resp. contact wire. c w These displacements satisfy, see [1], A w  ‡ w _ ˆ T w ÿ  A gÿ F …1† c c c c c c c c dc; j jˆ1 0000 00 A w  ‡ w _ ˆÿ E I w ‡ T w ÿ  A g w w w w w w w w w w w w n n d s X X ÿ F ‡ F ‡ F : …2† dw; j pw; j r jˆ1 jˆ1 Fig. 2. System of catenary and pantograph. 228 M. SCHAUB AND B. SIMEON The constraint forces F , F , F and F are point forces and therefore dc; j dw; j pw; j r contain a delta function. The dropper forces F and F include also the dc; j dw; j inertia terms of droppers and Lagrange multipliers  , which combine these d; j two equations. For example the force F is given by dw; j F …x; t†ˆ …xÿ x † m ‡ 1=2 l  A gÿ  …t† ; dw; j d; j s d; j d d d; j where x and l denote the position resp. the length of the dropper j and m d; j d; j s the mass of its suspension clamps. The equation for the pantograph model M  q ˆÿD q _ ÿ C q ÿ B  ‡ F …3† p p p p p p p p contains also Lagrange multipliers  , which represent the contact forces p; j between contact wire and pantograph. These couplings lead to the following constraint conditions 0  w …x ; t†ÿ w …x ; t†‡ l ; j ˆ 1; .. . ; n …4† w d; j c d; j d; j d 0  w …x …t†; t†ÿ b q ; j ˆ 1; .. . ; n ; …5† w p; j p; j p s which are unilateral since the droppers have the possibility to slacken and the pantograph can loose the contact to the catenary. In the last inequality x …t† p; j denotes the positions of the pantograph strips. The equations (1) ± (5) include two partial differential equations, one ordinary differential equation and two algebraic equations. Hence they form a partial differential algebraic equation (PDAE), see [4]. 2.2 Solution Behavior Before we start with a description of the numerical techniques used for simulation we discuss the solution behavior of the system, in particular the propagation, re¯ection and transmission of waves. For the wave equation we know that the velocity of waves is constant for all frequencies. In the case of the above equations, wave equation with damping Aw  ‡ w _ ÿ T w ˆ 0 …6† and Euler-Bernoulli-beam with damping and normal force 0000 00 EIw ÿ T w ‡ Aw  ‡ w _ ˆ 0; …7† SYSTEM OF PANTOGRAPH AND CATENARY 229 ÿdt we can search for moving waves with the ansatz w…x; t†ˆ e f…x ct†.We set d ˆ =2A and obtain f…x c t†ˆ C cos k…x c t†‡ C sin k…x c t†…8† s 1 s 2 s for (6) and f…x ct†ˆ C cosh k …x c t†‡ C sinh k …x c t†‡ 1 1 1 2 1 1 ‡ C cos k …x c t†‡ C sin k …x c t†…9† 3 2 2 4 2 2 for (7) respectively with the velocities 2 2 T T EI 2 2 2 c ˆÿ ‡ and c ˆ ‡ ÿ  ; s 1=2 1=2 2 2 2 2 2 2 4 A k A 4 A  A A 1=2 2 2 2 2 where  ˆ k and  ˆÿk . 1 1 2 2 Compared to the simple wave equation, the damping introduces, in addition to the decaying solution behavior, a nonlinear term for the wave velocities. These are no longer the same for all frequencies but increase for higher frequencies. This dependence of the propagation velocity on frequencies is called dispersion. The same effect appears by inserting a bending stiffness in (7), which is therefore also known as dispersive wave equation. The previous result yields information about the behavior of waves in contact wire and carrier but we do not know what happens at the droppers, where the two wires in¯uence each other. For this reason we want to study a single moving wave. Before we can do this, we have to determine the compatibility conditions 000 ‡ 000 ÿ ÿm w  …x ; t†‡  …t†ˆ EI‰w …x ; t†ÿ w …x ; t†Šÿ s w d d w d w d 0 ‡ 0 ÿ ÿ T‰w …x ; t†ÿ w …x ; t†Š …10† w d w d 0 ‡ 0 ÿ m w  …x ; t†‡  …t†ˆ T‰w …x ; t†ÿ w …x ; t†Š; …11† s c d d c d c d which connect the constraint force  to the jumps in ®rst and third derivative of the displacement, for details see [11,12]. Normally, the second term at the right hand side of (10) vanishes because of the continuous ®rst derivative of the beam model. But with respect to a single wave this property is not ful®lled, so we have to deal with both parts. Combining (10) and (11) yields a coupled compatibility condition for carrier and contact wire without external forces, which we will use for the derivation of re¯exion and transmission coef®cients. 230 M. SCHAUB AND B. SIMEON For this purpose we consider a single wave in the contact wire and four i…!t kx† resulting waves, all of the form …x; t†ˆ e , see Figure 3. Inserting i i this in the coupled compatibility condition yields after some calculations an expression for the relations between the amplitudes of resulting waves and initial wave, whose absolute values de®ne the re¯exion and transmission coef®cients. Furthermore, with the continuity condition …x ; t†‡ …x ; t†ˆ …x ; t†ˆ …x ; t†ˆ …x ; t† 1 d 2 d 3 d 4 d 5 d the waves can be expressed in terms of each other. We obtain for an initial wave in the contact wire p 2 2 q q R ˆ and T ˆ wc wc 2 2 2 2 …† ‡ ‡ … ‡ † ‡ w c w c and for an initial wave in the carrier p 2 2 q q R ˆ and T ˆ cw cw 2 2 2 2 … ‡ † ‡ … ‡ † ‡ w c w c with …!†ˆ m ! …!†ˆ Tk …!† c c …!†ˆ k …!†…EIk …!†‡ T† w w Here, R and R denote the re¯exion coef®cients and T and T the wc cw wc cw transmission coef®cients, which are equal for the three resulting waves. Fig. 3. Sketch of wave directions. SYSTEM OF PANTOGRAPH AND CATENARY 231 Fig. 4. Re¯exion and transmission factors, left: initial wave in contact wire, right: initial wave in carrier. However, since the wave numbers are different in contact wire and carrier, the transmission waves have different wave lengths and propagation speeds. These results are also illustrated in Figure 4. Both pictures show that the behavior of low and high frequencies is quite different. For low frequencies the re¯exion and transmission coef®cients are similar and the in¯uence of the bending stiffness is observable. In constrast, for higher frequencies the re¯exion coef®cient is very high and the transmission coef®cient becomes negligible. From the numerical point of view this means that high frequencies appear only near the pantograph, not in the whole catenary, since the transmission waves are damped out soon. Therefore we have to use small elements in the section of the actual position of the pantograph, but we can choose larger elements for the other parts of the catenary. 3 SIMULATION 3.1 Semi-discretization To solve the above system, we use the method of lines and start with the semi- discretization by ®nite elements. For this purpose, we multiply the equations of motion (1)±(5) with a testfunction v resp. v and integrate over x. This w c leads to the weak form of the equations, and by projection onto a ®nite 232 M. SCHAUB AND B. SIMEON dimensional subspace, e.g., a ®nite element space, we obtain the semi discretization in terms of the differential algebraic equation (DAE) M  q ‡ D q _ ˆÿS q ‡ b ÿ H c c c c c c c d T T M  q ‡ D q _ ˆÿ…S ‡ K †q ‡ b ‡ H  ‡ F …t† w w w w w w w w d p w w M  q ˆÿD q _ ÿ C q ÿ B …q † ‡ F p p p p p p p p p 0  H q ÿ H q ‡ l w w c c d 0  F …t†q ÿ B…q †: …12† w w p Details can be found in [1, 12]. We introduce the mass matrix M, damping matrix D and stiffness matrix S and collect the applied forces in vector b, while the indices w, c and p denote contact wire, carrier and pantograph. These equations can again be summarized to the more convenient, linear time- variant form M q ‡ Dq _ ˆÿSq‡ b‡ G…t† 0  G…t†q‡ z …13† T T T with the vectors qˆ…q ; q ; q † , bˆ…b ; b ; F† , zˆ…l ; 0† and w c p w c d ˆ… ; † . d p 3.2 Error Estimation with Superconvergent Patch Recovery To examine the accuracy of the semidiscretization in space, we apply the superconvergent patch recovery (SPR), which yields an error estimation for the stresses. It is based on the superconvergence of gauss points, see [13] for the theoretical background. SPR calculates an approximation polynom through the gauss points of two neighbouring elements and computes the difference between this polynom and the ®nite element solution, see Figure 5. The polynom 0 m T u…x†ˆ P…x†dˆ‰1; x; .. . ; x Š‰d ; d ; .. . ; d Š 1 2 m‡1 should therefore minimize the potential 2m ˆ …w …x ; t †ÿ P…x †d† g;i k g;i iˆ1 with m ˆ 1 for linear elements and m ˆ 3 for kubic elements. Hence, the constant or quadratic ®rst derivative of the ®nite element solution is uniquely SYSTEM OF PANTOGRAPH AND CATENARY 233 Fig. 5. SPR-sketch for linear ®nite elements. determined by the Gauss-points. The degree of the approximation polynom m‡ 1 correspond to the degrees of freedom of the ®nite element solution. The latter is given by twice the degree of the elements minus the number of coupling conditions. Therefore, the linear polynom is exact but the kubic one is a least mean square approximation. The polynom is then obtained by differentiation of  with respect to d, which leads to the linear equation Ad ˆ b with 2m 2m X X T T 0 A ˆ P…x † P…x † and b ˆ P…x † w …x ; t †: g; i g; i g; i g; i k iˆ1 iˆ1 This equation can be solved numerically, see [11]. To avoid approximation polynoms through elements next to a discontinuity, we do not use SPR at critical points like droppers and registration arm. Instead we calculate the middle value of the neighboring approximation polynoms at these points. To demonstrate the behavior of SPR we study a Euler-Bernoulli-beam and a 2x sin and discretize the equations with four string under load q…x†ˆq ®nite elements. Afterwards, we calculate two approximation polynoms for the ®rst and the last two elements and compare the exact error in the derivatives to the estimated error, see Figure 6. One can see that the difference between exact and estimated error is very small in the middle of the approximation polynoms, but higher at the boundaries. However, since we normally calculate the estimated error only at the node in the middle, the result is satisfying. 3.3 Time Integration The resulting DAE (13) has the differential index three. This property leads to order reduction and other numerical dif®culties, see [7]. We apply the GGL- stabilization of Gear, Gupta, Leimkuhler, see [3, 7], which uses the velocity 234 M. SCHAUB AND B. SIMEON Fig. 6. Exact (solid line) and estimated error (dashed line) for string and beam with load q…x†: constraint condition and adds an Lagrange multiplier  to the system. Then we can append the displacement constraint condition as an invariant. For simplicity we demonstrate this for active constraints only, leading to Mq _ ˆ Mv‡ G …t† …14† Mv _ ˆÿDvÿ Sq‡ b‡ G …t† …15† 0 ˆ G…t†q‡ G…t†v …16† 0 ˆ G…t†q‡ z: …17† The system (24)±(27) can be integrated by implicit methods like BDF2 or half-explicit methods. The unilateral constraints are assumed to be active at the beginning. If a constraint force  reaches zero, the corresponding constraint becomes inactive and the time step is repeated. The calculation is continued until the constraint becomes active again. SYSTEM OF PANTOGRAPH AND CATENARY 235 4 SIMULATION RESULTS The numerical solution is shown in Figure 7 and Figure 8. Figure 7 displays the motion of the pantograph at a velocity v ˆ 48 m=s. On the left picture, one can see the displacement of the two contact strips q and the two masses p4;5 at the lower part of the pantograph q and q . The behavior of the contact p1 p2 strips is very similar, their displacement is relatively small and without strong oscillations. The motion of the other two masses is comparable to the motion of a beam under a constant moving force, see [5]. This shows that the in¯uence of the droppers is negligible for the overall motion of the pantograph. The right picture displays the rotation of the pan head q and the contact strips p3 q . Due to small motion of the pan head, the linearization of the equations of p6;7 motion for the pantograph is suf®cient. The rotation of the two contact strips is again very similar and follows the zigzag-course of the contact wire. Figure 8 shows the contact forces for the velocities v ˆ 48 m=s and v ˆ 63 m=s. They are not smooth but oscillate, which leads to fading of the contact strips. No slackening of droppers or contact losses occur for v ,but at the higher velocity v we obtain both slackening of droppers and contact losses of the pantograph. Remarkably, the last strip looses the contact more often than the ®rst because of the vibrations generated by the ®rst. Furthermore, the droppers directly before the registration arms in the middle and at the right side slacken both twice. In this situation the pantograph is at its highest position and is pressed down to the registration arm. This leads to a relatively high contact force as one can see in Figure 8. Fig. 7. Translatory and rotatory degrees of freedom of the pantograph for v ˆ 48 m=s: 0 236 M. SCHAUB AND B. SIMEON Fig. 8. Contact forces for v ˆ 48 m=s and v ˆ 63 m=s. 1 2 Figure 9 displays the SPR error estimation for the contact wire. The left picture indicates that the error at the droppers is much higher than for the areas in between. Therefore, we developed a non-equidistant grid where new nodes were introduced near the droppers, see Figure 10. The error becomes remark- Fig. 9. SPR error estimation at t ˆ 0:472 s for v ˆ 48 m=s with equidistant and nonequidistant grid. Fig. 10. Non-equidistant grid. SYSTEM OF PANTOGRAPH AND CATENARY 237 ably smaller at the droppers. This shows that one can improve the calculation, if a non equidistant grid is used. However, at the position of the pantograph at x …t† 25 m the error is still high. To avoid this, an adaptive grid re®nement is necessary which, at the moment, costs too much additional effort. 5 CONCLUSIONS Effects like contact losses and slackening of droppers can only be resolved by numerical methods that take the local character of the interaction between pantograph and catenary into account. The mathematical model introduced here allows the application of such methods and offers much ¯exibility, in contrast to approaches that intertwine modeling and simulation. Moreover, the SPR error indicator shows critical parts of the proposed FE discretization and could be a good basis for adaptive strategies. ACKNOWLEDGEMENTS We would like to thank DLR Oberpfaffenhofen very much, in particular M. Arnold, for the support and the fruitful cooperation. REFERENCES 1. Arnold, M. and Simeon, B.: Pantograph and Catenary Dynamics: A Benchmark Problem and its Numerical Solution. To Appear in Appl. Numer. Math, 34 (2000), 345±362. 2. Courant, R. and Hilbert, D.: Methoden der Mathematischen Physik I. Springer-Verlag, Heidelberg New York, 1968. 3. Gear, C.W., Gupta, G.K. and Leimkuhler, B.J.: Automatic Integration of the Euler- Lagrange Equations with Constraints. J. Comp. Appl. Math. 12 / 13 (1980), 77±90. 4. G unther, M. and Wagner, Y.: Index Concepts for Linear Mixed Systems of Differential- Algebraic and Hyperbolic-Type Equations. To Appear in SIAM J. Sci. Comp, 22/5 (2000), 1610±1629. 5. Hagedorn, P.: Technische Schwingungslehre. II. Lineare Schwingungen kontinuierlicher mechanischer Systeme. Springer-Verlag, Berlin, Heidelberg, 1989. 6. Hairer, E., Nrsett, S.P. and Wanner, G.: Solving Ordinary Differential Equations. I. Nonstiff Problems. Springer-Verlag, Berlin, Heidelberg, New York, 2. Au¯age, 1993. 7. Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, Heidelberg, New York, 1996. 8. Petri, K.: Vergleichende Untersuchung von Berechnungsmodellen zur Simulation der Dynamik von Fahrleitung-Stromabnehmer-Systemen. Dissertation, Heinz-Nixdorf-Institut, Universit at-GH Paderborn, Germany, 1996. 238 M. SCHAUB AND B. SIMEON 9. Poetsch, G., Evans, J., Meisinger, R., Kortu È m, W., Baldauf, W., Veitl, A. and Wallaschek, J.: Pantograph / Catenary Dynamics and Control. Vehicle System Dynamics 28 (1997), pp. 159±195. 10. Poetsch, G. and Wallaschek, J.: Simulating the Dynamic Behavior of Electrical Lines for High-Speed Trains on Parallel Computers. In: Proceedings of the International Symposium on Cable Dynamics, S. 565±572. 11. Schaub, M.: Simulation der Dynamik von Fahrleitung und Stromabnehmer. Diploma thesis, TU Darmstadt, 1999. 12. Simeon, B. and Arnold, M.: Coupling DAE's and PDE's for Simulating the Interaction of Pantograph and Catenary. Math. and Computer Model. of Systems 6(2000), pp. 129±144. 13. Zienckiewicz, O.C. and Taylor, R.L.: The Finite Element Method. McGraw Hill, London, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical and Computer Modelling of Dynamical Systems Taylor & Francis

Pantograph-Catenary Dynamics: An Analysis of Models and Simulation Techniques

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Abstract

Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-225$16.00 2001, Vol. 7, No. 2, pp. 225±238 Swets & Zeitlinger Pantograph-Catenary Dynamics: An Analysis of Models and Simulation Techniques * * M. SCHAUB and B. SIMEON ABSTRACT In this paper we analyze models and simulation techniques for the interaction of pantograph and catenary. Detailed models for catenary and pantograph and the propagation of waves are ®rst investigated. Next, the semi discretization by the ®nite element method and the time integration are described. In this context numerical techniques like GGL-stabilization and superconvergent patch recovery are applied. The latter yields an error estimation for the ®nite element grid and shows the critical points of the system. Keywords: interaction pantograph/catenary, partial differential-algebraic equation (PDAE), error estimation, superconvergent patch recovery, GGL-stabilization. 1 INTRODUCTION Pantograph and catenary dynamics is the most critical part in the energy transmission of high-speed trains. Often, oscillations in the contact wire occur such that the contact force between pantograph and catenary varies strongly and the contact may even get lost. Therefore, constructive changes for both pantograph and catenary are under development. Design criteria include the permanent contact of pantograph head and contact wire at high speed and the reduction of both aeroacoustic noise and wear [9]. Dynamical simulation plays an important role in this development since prototypes and measure- ments are very expensive. In the present paper we investigate mathematical models and numerical simulation techniques for this coupled system. Our approach is based on a Center for Scienti®c Computing and Mathematical Modelling, University of Karlsruhe, Karlsruhe, Germany. 226 M. SCHAUB AND B. SIMEON descriptor formulation where the contact condition is treated as a unilateral constraint and where a continuous model for the catenary is combined with a mechanical multibody system for the pantograph. This results in partial differential equations (PDE's, catenary) and differential-algebraic equations (DAE's, pantograph). We show how numerical techniques from the ®nite element method and from differential-algebraic equations can be effectively applied and present some simulation results. The paper is organized as follows: First, we introduce shortly the coupled system for the equations of motion and analyze its structure, followed by a description of the solution behavior. Next, space and time discretization techniques are discussed. In particular, the quality of the FE grid is assessed in terms of the superconvergent patch recovery of Zienckiewicz / Taylor [13]. Finally, several simulation runs at different speeds demonstrate the dynamic behavior. It turns out that contact losses appear if the velocity exceeds 200 km / h. 2 MATHEMATICAL MODEL 2.1 Equations of Motion In this section we introduce the mathematical models for pantograph and catenary. As extension of the benchmark problem in [1, 12] more detailed models are used which cover the three-dimensional geometry correctly, while the motion of the catenary is still restricted to the vertical direction. In detail, the pantograph model contains seven degrees of freedom and ®ve masses, see Figure 1, and describes, in addition to the vertical motion, n ˆ 2 contact points and the rotation of the pan head (q ), see [8] for more details. The latter is necessary to take the zigzag-course of the contact wire into account. The model for the catenary consists of carrier, contact wire and n ˆ 14 droppers, see Figure 2. Furthermore, we model the registration arm at the midpoint as a spring damper system, in contrast to those at the boundary, which are ®xed. The carrier is modelled as a string, but we use an Euler-Bernoulli-beam for the contact wire to get a continuous ®rst derivative at the contact point. The equations of motion contain the density , cross section area A, damping SYSTEM OF PANTOGRAPH AND CATENARY 227 Fig. 1. Pantograph model with ®ve masses and seven degrees of freedom. constant , normal force T and the bending stiffness EI in case of the beam. Moreover, w and w denote the displacement of the carrier resp. contact wire. c w These displacements satisfy, see [1], A w  ‡ w _ ˆ T w ÿ  A gÿ F …1† c c c c c c c c dc; j jˆ1 0000 00 A w  ‡ w _ ˆÿ E I w ‡ T w ÿ  A g w w w w w w w w w w w w n n d s X X ÿ F ‡ F ‡ F : …2† dw; j pw; j r jˆ1 jˆ1 Fig. 2. System of catenary and pantograph. 228 M. SCHAUB AND B. SIMEON The constraint forces F , F , F and F are point forces and therefore dc; j dw; j pw; j r contain a delta function. The dropper forces F and F include also the dc; j dw; j inertia terms of droppers and Lagrange multipliers  , which combine these d; j two equations. For example the force F is given by dw; j F …x; t†ˆ …xÿ x † m ‡ 1=2 l  A gÿ  …t† ; dw; j d; j s d; j d d d; j where x and l denote the position resp. the length of the dropper j and m d; j d; j s the mass of its suspension clamps. The equation for the pantograph model M  q ˆÿD q _ ÿ C q ÿ B  ‡ F …3† p p p p p p p p contains also Lagrange multipliers  , which represent the contact forces p; j between contact wire and pantograph. These couplings lead to the following constraint conditions 0  w …x ; t†ÿ w …x ; t†‡ l ; j ˆ 1; .. . ; n …4† w d; j c d; j d; j d 0  w …x …t†; t†ÿ b q ; j ˆ 1; .. . ; n ; …5† w p; j p; j p s which are unilateral since the droppers have the possibility to slacken and the pantograph can loose the contact to the catenary. In the last inequality x …t† p; j denotes the positions of the pantograph strips. The equations (1) ± (5) include two partial differential equations, one ordinary differential equation and two algebraic equations. Hence they form a partial differential algebraic equation (PDAE), see [4]. 2.2 Solution Behavior Before we start with a description of the numerical techniques used for simulation we discuss the solution behavior of the system, in particular the propagation, re¯ection and transmission of waves. For the wave equation we know that the velocity of waves is constant for all frequencies. In the case of the above equations, wave equation with damping Aw  ‡ w _ ÿ T w ˆ 0 …6† and Euler-Bernoulli-beam with damping and normal force 0000 00 EIw ÿ T w ‡ Aw  ‡ w _ ˆ 0; …7† SYSTEM OF PANTOGRAPH AND CATENARY 229 ÿdt we can search for moving waves with the ansatz w…x; t†ˆ e f…x ct†.We set d ˆ =2A and obtain f…x c t†ˆ C cos k…x c t†‡ C sin k…x c t†…8† s 1 s 2 s for (6) and f…x ct†ˆ C cosh k …x c t†‡ C sinh k …x c t†‡ 1 1 1 2 1 1 ‡ C cos k …x c t†‡ C sin k …x c t†…9† 3 2 2 4 2 2 for (7) respectively with the velocities 2 2 T T EI 2 2 2 c ˆÿ ‡ and c ˆ ‡ ÿ  ; s 1=2 1=2 2 2 2 2 2 2 4 A k A 4 A  A A 1=2 2 2 2 2 where  ˆ k and  ˆÿk . 1 1 2 2 Compared to the simple wave equation, the damping introduces, in addition to the decaying solution behavior, a nonlinear term for the wave velocities. These are no longer the same for all frequencies but increase for higher frequencies. This dependence of the propagation velocity on frequencies is called dispersion. The same effect appears by inserting a bending stiffness in (7), which is therefore also known as dispersive wave equation. The previous result yields information about the behavior of waves in contact wire and carrier but we do not know what happens at the droppers, where the two wires in¯uence each other. For this reason we want to study a single moving wave. Before we can do this, we have to determine the compatibility conditions 000 ‡ 000 ÿ ÿm w  …x ; t†‡  …t†ˆ EI‰w …x ; t†ÿ w …x ; t†Šÿ s w d d w d w d 0 ‡ 0 ÿ ÿ T‰w …x ; t†ÿ w …x ; t†Š …10† w d w d 0 ‡ 0 ÿ m w  …x ; t†‡  …t†ˆ T‰w …x ; t†ÿ w …x ; t†Š; …11† s c d d c d c d which connect the constraint force  to the jumps in ®rst and third derivative of the displacement, for details see [11,12]. Normally, the second term at the right hand side of (10) vanishes because of the continuous ®rst derivative of the beam model. But with respect to a single wave this property is not ful®lled, so we have to deal with both parts. Combining (10) and (11) yields a coupled compatibility condition for carrier and contact wire without external forces, which we will use for the derivation of re¯exion and transmission coef®cients. 230 M. SCHAUB AND B. SIMEON For this purpose we consider a single wave in the contact wire and four i…!t kx† resulting waves, all of the form …x; t†ˆ e , see Figure 3. Inserting i i this in the coupled compatibility condition yields after some calculations an expression for the relations between the amplitudes of resulting waves and initial wave, whose absolute values de®ne the re¯exion and transmission coef®cients. Furthermore, with the continuity condition …x ; t†‡ …x ; t†ˆ …x ; t†ˆ …x ; t†ˆ …x ; t† 1 d 2 d 3 d 4 d 5 d the waves can be expressed in terms of each other. We obtain for an initial wave in the contact wire p 2 2 q q R ˆ and T ˆ wc wc 2 2 2 2 …† ‡ ‡ … ‡ † ‡ w c w c and for an initial wave in the carrier p 2 2 q q R ˆ and T ˆ cw cw 2 2 2 2 … ‡ † ‡ … ‡ † ‡ w c w c with …!†ˆ m ! …!†ˆ Tk …!† c c …!†ˆ k …!†…EIk …!†‡ T† w w Here, R and R denote the re¯exion coef®cients and T and T the wc cw wc cw transmission coef®cients, which are equal for the three resulting waves. Fig. 3. Sketch of wave directions. SYSTEM OF PANTOGRAPH AND CATENARY 231 Fig. 4. Re¯exion and transmission factors, left: initial wave in contact wire, right: initial wave in carrier. However, since the wave numbers are different in contact wire and carrier, the transmission waves have different wave lengths and propagation speeds. These results are also illustrated in Figure 4. Both pictures show that the behavior of low and high frequencies is quite different. For low frequencies the re¯exion and transmission coef®cients are similar and the in¯uence of the bending stiffness is observable. In constrast, for higher frequencies the re¯exion coef®cient is very high and the transmission coef®cient becomes negligible. From the numerical point of view this means that high frequencies appear only near the pantograph, not in the whole catenary, since the transmission waves are damped out soon. Therefore we have to use small elements in the section of the actual position of the pantograph, but we can choose larger elements for the other parts of the catenary. 3 SIMULATION 3.1 Semi-discretization To solve the above system, we use the method of lines and start with the semi- discretization by ®nite elements. For this purpose, we multiply the equations of motion (1)±(5) with a testfunction v resp. v and integrate over x. This w c leads to the weak form of the equations, and by projection onto a ®nite 232 M. SCHAUB AND B. SIMEON dimensional subspace, e.g., a ®nite element space, we obtain the semi discretization in terms of the differential algebraic equation (DAE) M  q ‡ D q _ ˆÿS q ‡ b ÿ H c c c c c c c d T T M  q ‡ D q _ ˆÿ…S ‡ K †q ‡ b ‡ H  ‡ F …t† w w w w w w w w d p w w M  q ˆÿD q _ ÿ C q ÿ B …q † ‡ F p p p p p p p p p 0  H q ÿ H q ‡ l w w c c d 0  F …t†q ÿ B…q †: …12† w w p Details can be found in [1, 12]. We introduce the mass matrix M, damping matrix D and stiffness matrix S and collect the applied forces in vector b, while the indices w, c and p denote contact wire, carrier and pantograph. These equations can again be summarized to the more convenient, linear time- variant form M q ‡ Dq _ ˆÿSq‡ b‡ G…t† 0  G…t†q‡ z …13† T T T with the vectors qˆ…q ; q ; q † , bˆ…b ; b ; F† , zˆ…l ; 0† and w c p w c d ˆ… ; † . d p 3.2 Error Estimation with Superconvergent Patch Recovery To examine the accuracy of the semidiscretization in space, we apply the superconvergent patch recovery (SPR), which yields an error estimation for the stresses. It is based on the superconvergence of gauss points, see [13] for the theoretical background. SPR calculates an approximation polynom through the gauss points of two neighbouring elements and computes the difference between this polynom and the ®nite element solution, see Figure 5. The polynom 0 m T u…x†ˆ P…x†dˆ‰1; x; .. . ; x Š‰d ; d ; .. . ; d Š 1 2 m‡1 should therefore minimize the potential 2m ˆ …w …x ; t †ÿ P…x †d† g;i k g;i iˆ1 with m ˆ 1 for linear elements and m ˆ 3 for kubic elements. Hence, the constant or quadratic ®rst derivative of the ®nite element solution is uniquely SYSTEM OF PANTOGRAPH AND CATENARY 233 Fig. 5. SPR-sketch for linear ®nite elements. determined by the Gauss-points. The degree of the approximation polynom m‡ 1 correspond to the degrees of freedom of the ®nite element solution. The latter is given by twice the degree of the elements minus the number of coupling conditions. Therefore, the linear polynom is exact but the kubic one is a least mean square approximation. The polynom is then obtained by differentiation of  with respect to d, which leads to the linear equation Ad ˆ b with 2m 2m X X T T 0 A ˆ P…x † P…x † and b ˆ P…x † w …x ; t †: g; i g; i g; i g; i k iˆ1 iˆ1 This equation can be solved numerically, see [11]. To avoid approximation polynoms through elements next to a discontinuity, we do not use SPR at critical points like droppers and registration arm. Instead we calculate the middle value of the neighboring approximation polynoms at these points. To demonstrate the behavior of SPR we study a Euler-Bernoulli-beam and a 2x sin and discretize the equations with four string under load q…x†ˆq ®nite elements. Afterwards, we calculate two approximation polynoms for the ®rst and the last two elements and compare the exact error in the derivatives to the estimated error, see Figure 6. One can see that the difference between exact and estimated error is very small in the middle of the approximation polynoms, but higher at the boundaries. However, since we normally calculate the estimated error only at the node in the middle, the result is satisfying. 3.3 Time Integration The resulting DAE (13) has the differential index three. This property leads to order reduction and other numerical dif®culties, see [7]. We apply the GGL- stabilization of Gear, Gupta, Leimkuhler, see [3, 7], which uses the velocity 234 M. SCHAUB AND B. SIMEON Fig. 6. Exact (solid line) and estimated error (dashed line) for string and beam with load q…x†: constraint condition and adds an Lagrange multiplier  to the system. Then we can append the displacement constraint condition as an invariant. For simplicity we demonstrate this for active constraints only, leading to Mq _ ˆ Mv‡ G …t† …14† Mv _ ˆÿDvÿ Sq‡ b‡ G …t† …15† 0 ˆ G…t†q‡ G…t†v …16† 0 ˆ G…t†q‡ z: …17† The system (24)±(27) can be integrated by implicit methods like BDF2 or half-explicit methods. The unilateral constraints are assumed to be active at the beginning. If a constraint force  reaches zero, the corresponding constraint becomes inactive and the time step is repeated. The calculation is continued until the constraint becomes active again. SYSTEM OF PANTOGRAPH AND CATENARY 235 4 SIMULATION RESULTS The numerical solution is shown in Figure 7 and Figure 8. Figure 7 displays the motion of the pantograph at a velocity v ˆ 48 m=s. On the left picture, one can see the displacement of the two contact strips q and the two masses p4;5 at the lower part of the pantograph q and q . The behavior of the contact p1 p2 strips is very similar, their displacement is relatively small and without strong oscillations. The motion of the other two masses is comparable to the motion of a beam under a constant moving force, see [5]. This shows that the in¯uence of the droppers is negligible for the overall motion of the pantograph. The right picture displays the rotation of the pan head q and the contact strips p3 q . Due to small motion of the pan head, the linearization of the equations of p6;7 motion for the pantograph is suf®cient. The rotation of the two contact strips is again very similar and follows the zigzag-course of the contact wire. Figure 8 shows the contact forces for the velocities v ˆ 48 m=s and v ˆ 63 m=s. They are not smooth but oscillate, which leads to fading of the contact strips. No slackening of droppers or contact losses occur for v ,but at the higher velocity v we obtain both slackening of droppers and contact losses of the pantograph. Remarkably, the last strip looses the contact more often than the ®rst because of the vibrations generated by the ®rst. Furthermore, the droppers directly before the registration arms in the middle and at the right side slacken both twice. In this situation the pantograph is at its highest position and is pressed down to the registration arm. This leads to a relatively high contact force as one can see in Figure 8. Fig. 7. Translatory and rotatory degrees of freedom of the pantograph for v ˆ 48 m=s: 0 236 M. SCHAUB AND B. SIMEON Fig. 8. Contact forces for v ˆ 48 m=s and v ˆ 63 m=s. 1 2 Figure 9 displays the SPR error estimation for the contact wire. The left picture indicates that the error at the droppers is much higher than for the areas in between. Therefore, we developed a non-equidistant grid where new nodes were introduced near the droppers, see Figure 10. The error becomes remark- Fig. 9. SPR error estimation at t ˆ 0:472 s for v ˆ 48 m=s with equidistant and nonequidistant grid. Fig. 10. Non-equidistant grid. SYSTEM OF PANTOGRAPH AND CATENARY 237 ably smaller at the droppers. This shows that one can improve the calculation, if a non equidistant grid is used. However, at the position of the pantograph at x …t† 25 m the error is still high. To avoid this, an adaptive grid re®nement is necessary which, at the moment, costs too much additional effort. 5 CONCLUSIONS Effects like contact losses and slackening of droppers can only be resolved by numerical methods that take the local character of the interaction between pantograph and catenary into account. The mathematical model introduced here allows the application of such methods and offers much ¯exibility, in contrast to approaches that intertwine modeling and simulation. Moreover, the SPR error indicator shows critical parts of the proposed FE discretization and could be a good basis for adaptive strategies. ACKNOWLEDGEMENTS We would like to thank DLR Oberpfaffenhofen very much, in particular M. Arnold, for the support and the fruitful cooperation. REFERENCES 1. Arnold, M. and Simeon, B.: Pantograph and Catenary Dynamics: A Benchmark Problem and its Numerical Solution. To Appear in Appl. Numer. Math, 34 (2000), 345±362. 2. Courant, R. and Hilbert, D.: Methoden der Mathematischen Physik I. Springer-Verlag, Heidelberg New York, 1968. 3. Gear, C.W., Gupta, G.K. and Leimkuhler, B.J.: Automatic Integration of the Euler- Lagrange Equations with Constraints. J. Comp. Appl. Math. 12 / 13 (1980), 77±90. 4. G unther, M. and Wagner, Y.: Index Concepts for Linear Mixed Systems of Differential- Algebraic and Hyperbolic-Type Equations. To Appear in SIAM J. Sci. Comp, 22/5 (2000), 1610±1629. 5. Hagedorn, P.: Technische Schwingungslehre. II. Lineare Schwingungen kontinuierlicher mechanischer Systeme. Springer-Verlag, Berlin, Heidelberg, 1989. 6. Hairer, E., Nrsett, S.P. and Wanner, G.: Solving Ordinary Differential Equations. I. Nonstiff Problems. Springer-Verlag, Berlin, Heidelberg, New York, 2. Au¯age, 1993. 7. Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, Heidelberg, New York, 1996. 8. Petri, K.: Vergleichende Untersuchung von Berechnungsmodellen zur Simulation der Dynamik von Fahrleitung-Stromabnehmer-Systemen. Dissertation, Heinz-Nixdorf-Institut, Universit at-GH Paderborn, Germany, 1996. 238 M. SCHAUB AND B. SIMEON 9. Poetsch, G., Evans, J., Meisinger, R., Kortu È m, W., Baldauf, W., Veitl, A. and Wallaschek, J.: Pantograph / Catenary Dynamics and Control. Vehicle System Dynamics 28 (1997), pp. 159±195. 10. Poetsch, G. and Wallaschek, J.: Simulating the Dynamic Behavior of Electrical Lines for High-Speed Trains on Parallel Computers. In: Proceedings of the International Symposium on Cable Dynamics, S. 565±572. 11. Schaub, M.: Simulation der Dynamik von Fahrleitung und Stromabnehmer. Diploma thesis, TU Darmstadt, 1999. 12. Simeon, B. and Arnold, M.: Coupling DAE's and PDE's for Simulating the Interaction of Pantograph and Catenary. Math. and Computer Model. of Systems 6(2000), pp. 129±144. 13. Zienckiewicz, O.C. and Taylor, R.L.: The Finite Element Method. McGraw Hill, London,

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Mathematical and Computer Modelling of Dynamical SystemsTaylor & Francis

Published: Jun 1, 2001

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