Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 4, pp. 935-945, Warsaw 2015 DOI: 10.15632/jtam-pl.53.4.935 VIBRATION OF A DISCRETE-CONTINUOUS STRUCTURE UNDER MOVING LOAD WITH ONE OR TWO CONTACT POINTS Anna Kumaniecka Cracow University of Technology, Institute of Mathematics, Kraków, Poland e-mail: pukumani@cyf-kr.edu.pl Michał Prącik Cracow University of Technology, Institute of Applied Mechanics, Kraków, Poland e-mail: mp@sparc2.mech.pk.edu.pl What is of particular importance in view of the development and operation of fast railway transport is the overhead system vibration excited by pantograph motion. The problems discussed in the paper are related to continuous system vibration under moving loads. Mo- delling the catenary-pantographsystemis connectedwithmotionof two subsystems,namely: continuous (contact wire) and discrete (pantograph). In the paper, the results of research on dynamical phenomena caused by the interaction between the pantograph and catenary are presented. The stiffness of the catenary wire is taken into account. The dynamical phe- nomena occurring in the system are described by a set of partial and ordinary differential equations. The solution to these equations has been obtained using approximate numerical methods. Keywords: dynamics of pantograph, stiffness of catenary system 1. Introduction The theoretical problem discussed in the paper is a technical problem connected with the dyna- mics of systems undermoving loads (Bajer andDyniewicz, 2012; Bogacz andSzolc, 1993; Fryba, 1999; Szolc, 2003).As a vibratingdiscrete-continuous structure, thepantograph-catenary system has been chosen for analysis. In any high voltage electric traction system, the current needed for operating a train is collected from an overhead contact system by some form of sliding electrical contact. Such a systemusually consists of ahorizontalwirewithwhich thepantographmakes continuous contact, and a catenary cable slung between the supports from which the contact wire is suspended at intervals by vertical dropper wires. The complex behaviour of the catenary-pantograph systemhas been the focus of attention of many researchers for several years (Poetsch et al., 1997; Wu and Brennan, 1998). Over the last sixty years, many studies of the catenary-pantograph dynamic behaviour have been undertaken (Kumaniecka and Grzyb, 2000; Kumaniecka and Nizioł, 2000; Zhang et al., 2002). In the past years, many researchers attempted to improve current collection quality in order to reduce wear and maintenance costs of both the overhead line and pantograph. Numerous studies on rail vehicles proved that the processes describing their dynamic state have a complex andnon-periodiccharacter (Poetsch et al., 1997;Kumaniecka, 2007).To improve thepantograph- -catenary interface, it is essential to understandbetter the complex behaviour of this couple.The pantograph-catenary interaction at high speed is the critical factor for reliability and safety of high speed railways. The large amplitudes of transversal vibration of themessenger and contact wires can result in pantograph strip wear, loss of contact or disturbance of mutual interaction. 936 A. Kumaniecka,M. Prącik With an increase in the train speed, the catenary-pantograph systemwith its dynamic beha- viour proved to be a very important component for new train systems required to run at higher speed. This speed can be limited by the power supply through the overhead catenary system. The key point that describes the efficiency of the current collection is the contact force. The zero value of this force induces the brake in the current collection, but too large value can result in wear of the contact wire and pantograph strips. The aim of the paper is to obtain a better understanding of the pantograph-catenary system dynamics. The emphasis of studies is placed on the description of contact loss and proper description of contact wire stiffness. A relatively simple analytical model presented in the paper is appropriate to gain physical insight into the pantograph-catenary system. In the paper, an analytical method for calculating the response of a catenary to a uniformly moving pantograph is presented. To the authors’ knowledge, the loss of contact of such a system has not been investigated so far. Thepaper is organized in five Sections. Following Introduction 1, themodels of the catenary- -pantograph system including the contactwire,messengerwire, droppers, supporting towers and the pantograph itself are described in Section 2. InSection 3 an analyticalmethod for calculating the responseof a catenary to auniformlymovingpantograph is presented.The simulation results are given in Section 4. Final concluding remarks are formulated in Section 5. 2. Modelling of the catenary-pantograph system The presented models belong to the class of continuous systems excited by a uniformlymoving load. In the literature, many physical and a lot of different analytical models of the catenary- -pantograph systemhavebeenproposed.Both the contact and carryingwire are one-dimensional systems. The contact and carrying cables have beenmodelled by infinite or non-infinite homoge- nous strings or Bernoulli-Euler beams. The pantograph has beenmodelled by an oscillator with two or four degrees of freedom. Such systemswere studied in the past by a number of researchers employing different methods. A review paper describing the pantograph-catenary systems was presented byPoetsch et al. (1997) and byKumaniecka (2007). The dynamic interaction between a discrete oscillator with four degrees of freedom and a continuous beam was also studied by Kumaniecka and Prącik (2011). The simplifiedmodel of the catenarywithonecontact point introduced in thepaper, shown in Fig. 1, is composed of two parallel infinitely long homogenous beams (the contact and carrying cables) connected by lumped elements (suspension rods), which are positioned equidistantly along thebeams.Theupperbeam(carrying cable) is fixedatperiodically spacedelastic supports. The lower beam (contact wire) is suspended from the upper beam by visco-elastic elements. These elements are used as a model of suspension rods. They are periodically placed at points along thebeams. It is assumed that the distance between the supports of carrying cable is equal l and between the droppers 2lw. In the adopted model, the bending stiffness of the contact wire is taken into account (Wu and Brennan, 1999; Kumaniecka and Prącik, 2011). The system in question is subjected to a concentrated force (model of pantograph), which is applied to the lower beam. This load moves along the lower beam at a constant velocity v. Between the contact wire and pantograph there also appears the friction force. In the pre- sented study, the friction force is neglected. The physical model of the catenary with two contact points is presented in Fig. 2. Themathematical model for a physical model of the catenary system adopted in this paper and shown in Fig. 2, was discussed in detail and presented in the monograph by Kumaniecka (2007). Vibration of a discrete-continuous structure under moving load... 937 Fig. 1. Physical model of catenary with one contact point Fig. 2. Physical model of catenary with two contact points Motion of the catenary in the vertical plane is governed by equations E1J1 ∂4w1 ∂x4 −N1 ∂2w1 ∂x2 +ρ1 ∂2w1 ∂t2 −p+pF −pm1 =0 E2J2 ∂4w2 ∂x4 −N2 ∂2w2 ∂x2 +ρ2 ∂2w2 ∂t2 +p−pm2 =0 (2.1) where the following notation is used: E1, E2 – Young’s modulus of lower and upper beam, respectively, Ji – cross-sectional moment of inertia (i = 1,2), Ni – tensile force in the beams, ρi – mass density, wi(x,t) – transversal displacements, x – spatial coordinate measured along the non-deformed axis of beams, t – time. The functions w1(x,t) and w2(x,t) describe the lower and upper beam transversal displacements, respectively. The loads p(x,t) acting on the beams and caused by internal forces in the springs and damping elements are treated as continuous. They can be expressed in the form p(x,t)= ∑ (n) {cb[w2(x,t)−w1(x,t)]+ bb[ẇ2(x,t)− ẇ1(x,t)]}δ(x−xn) (2.2) where: cb is the coefficient of spring elasticity, bb – damping coefficient, xn – coordinates of droppers spacing (concentrated masses), xn = 2lw(2s−1), s ∈ N, 2lw – distance between the droppers, δ – Dirac’s function. The interaction force between the pantograph and contact wire pF can be described by the term pF(x,t)= F(t)δ(x−vt) (2.3) 938 A. Kumaniecka,M. Prącik The reaction force pmi (i =1,2) that comes fromconcentratedmasses m spaced on the lower and upper beams acting at points xn can be treated as distributed and written in the form: — for the lower beam pm1(x,t)= ∑ (n) mẅ1(x,t)δ(x−xn) (2.4) — for the upper beam pm2(x,t)= ∑ (n) mẅ2(x,t)δ(x−xn) (2.5) The boundary and initial conditions adopted for numerical simulation have been based on the assumed vibration model of a linear system (data from identification research). In the present paper, the pantograph has been modelled as an oscillator with four degrees of freedom. Themodel refers to a real system designed by engineers from SchunkWien GmbH. The basic pantograph is the standard WBL-85/3kV. The collector strips are represented by masses m1L and m1P, the equivalent masses of the frames are denoted by m2 and m3. The masses are connected by springs c11 and c22 to provide a nominally constant uplift force. The aerodynamic force is taken into account (Bacciolone et al., 2005). The physical model of the pantograph investigated in our studies is shown in Fig. 3. Fig. 3. Model of pantograph Themathematical model for a physical model of the pantograph adopted in this paper and shown in Fig. 3 was discussed in detail and presented in themonograph byKumaniecka (2007). Inmany real pantograph systems, the springsare guided in telescopic sliders,which gives reasons to apply dry friction elements in the physical model. The structure of the simulation model has been based on the formal notation of motion in form of ordinary differential equations. Motion of the pantograph in the vertical plane is governed by equations m1Lẍ1L = m1Lg−|F1|sgn(ẋ1L − ẋ2)− c11(x1L −x2)+PL(x,t) m1P ẍ1P = m1Pg−|F1|sgn(ẋ1P − ẋ2)− c11(x1P −x2)+PP(x,t) m2ẍ2 = m2g−|F2|sgn(ẋ2− ẋ3)+ |F1|[sgn(ẋ1P − ẋ2)+ sgn(ẋ1L − ẋ2)] −c22(x2−x3)+ c11(x1P −x2)+ c11(x1L −x2)−Faer m3ẍ3 = m3g− c33x3− b33ẋ3−|F3|sgn(ẋ3)−Fstat + c22(x2−x3)+ |F2|sgn(ẋ2− ẋ3) (2.6) where: x1,x2,x3, ẋ1, ẋ2, ẋ3, m1,m2,m3 are displacements, velocities and masses of the ele- ments, respectively, F1,F2,F3 – friction forces, Faer,Fstat – aerodynamic and static forces, Vibration of a discrete-continuous structure under moving load... 939 PL(x,t),PP(x,t) – excitation forces, interaction forces between the pantograph and contact wire. The displacement of the pan-head is themain factor for dynamic performance of the panto- graph, and it is related to the contact forces directly. 3. Analytical model of the system vibration The catenary motion can be described by means of partial differential equations (2.1), which govern small vertical vibrations of each beamin thevicinity of their equilibriumstate, inducedby the transversal forcemoving along the lower beam. Somedetails are presented in themonograph by Kumaniecka (2007) and in the paper by Kumaniecka and Prącik (2011). The mathematical model for a physical model of the pantograph was discussed by Prącik and Furmanik (2000). The interaction between the pantograph and contact wire is limited to a set of two parallel forces PL(x,t)δ(x−vt) PP(x,t)δ(x−vt+xLP) (3.1) In theabove equation,xdenotes the spatial horizontal co-ordinate, t time,xLP is thedistance between the shoes. In the case of one contact point, the function describing the contact force can be expressed as F(x,t)= PL(x,t)+PP(x,t) (3.2) According to the results obtained by the authors (2011) and others (Wu and Brennan, 1999), the forces PL(x,t), PP(x,t) can be connected with harmonic changes of catenary stiffness and given in the form PL(x,t)= k0    [ 1−αcos (2πv L t )] (x1L0−x1L) for x1L0 > x1L 0 for x1L0 ¬ x1L PP(x,t)= k0    [ 1−αcos (2π L (vt+xLP) )] (x1P0−x1P) for x1P0 > x1P 0 for x1P0 ¬ x1P (3.3) In equations (3.3), the following notation is used: x1L, x1P denote vertical displacements of the pantograph contact shoes, x1L0, x1P0 – vertical displacements of two points on thewire that are in contact withmasses m1L and m1P , respectively, k0, α – are stiffness coefficients (Wu and Brennan, 1999), given by formulae k0 = kmax +kmin 2 α = kmax −kmin kmax +kmin (3.4) where L is the length of one span and kmax, kmin are the largest and the smallest stiffness values in the span, respectively. When the model with two contact points is investigated, the contact forces are written as PL(x,t)= k0    [ 1−αcos ( 2πv l t )] [x1L0−w1(x,t)] for x1L0 > w1(x,t) 0 for x1L0 ¬ w1(x,t) PP(x,t)= k0    [ 1−αcos ( 2π l (vt+xLP) )] [x1P0−w1(x+xLP , t)] for x1P0 > w1(x+xLP , t) 0 for x1P0 ¬ w1(x+xLP , t) (3.5) 940 A. Kumaniecka,M. Prącik where w1(x,t) is the transversal displacement of the lower beam, x1L, x1P are coordinates of strips motion (see Fig. 3), x1L ≡ w1(x,t)δ(x−vt) and x1P ≡ w1(x+xLP , t)δ(x−vt+xLP). The equations of catenary motion model (2.1) are presented in the monograph by Kuma- niecka (2007). After substituting relations (3.3) or (3.5), in the case of one or two contact points, respectively, to the equations of motion, they include some parameters associated with stiffness of the contact wire k0, α. In the case of one contact point, the solutions to set of equations (2.1) have been taken in the form of waves w1(x,t)= ∑ p1 ∑ r1 Ar1 sin x−vt p1lw x−vt p1lw sin(ωr1t−ϕr1) w2(x,t)= ∑ p2 ∑ r2 Ar2 sin x−vt p2lw x−vt p2lw sin(ωr2t−ϕr2) (3.6) where pi are associated with themovingmodes and ri with the standingmodes for i =1,2, and the coefficients Ar1, Ar2 could be determined numerically using a collocation method. To solve equations ofmotion (2.1) and (2.6) for two contact points, it is necessary to employ another expression for functions which describe transversal displacements of the lower beam w1(x,t)= ∑ p1 ∑ r1 Ar1 sin x−vt p1lw x−vt p1lw sin(ωr1t−ϕr1)+ ∑ p1 ∑ r1 Br1 sin x+xLP−vt p1lw x+xLP−vt p1lw sin(ωr1t−ϕr1) (3.7) 4. Numerical analysis On the basis of the given mathematical model, a simulation program applying the package VisSimAnalyze ver. 3.0 has beenbuilt.Numerical simulations have been carried out for different data sets. The numerical calculations have been done for the following parameters of the system m1L = m1P =7.93kg m2 =8.73kg m3 =10.15kg F1 =2.0N F2 = F3 =2.5N Faer =30.0N Fstat =600N Faer =30N c3 =60Ns/m xLP =1.0m The parameters of the system correspond to the parameters of the real overhead power lines for high speed trains (data set for pantographWBL 85-3kV/PKP). The block schemes of simulation ofmass displacements and contact forces of the pantograph model at Fstat =600N and 500N, Faer =30N are presented in Figs. 4 and 5. To investigate the phenomena of contact loss in the overhead system, numerical simulations for the contact force less than the limit forcewere carried out byKumaniecka andPrącik (2011). In Fig. 6, the loss of contact between the pantograph and catenary is illustrated. The results of simulations of the variability of contact forces of the pantograph (with two contact points), when the vertical displacement amplitude between the contact points spaced by 1m, is equal 0.03m, are presented in Fig. 7. The simulations have been done for velocity v = 55.55m/s using the blockscheme of simulation similar to that presented in Fig. 4 but utilizing a different expression for w1(x,t) (respectively to equation (3.7)). Based on the results of simulations, we can conclude that the pantograph with two contact strips guarantees better interaction (strips are detached convertible). The analysis of the simulation results (see Fig. 7) shows that the response of the system in question is not harmonic, it consists of standing and moving modes. To conclude, it can be Vibration of a discrete-continuous structure under moving load... 941 Fig. 4. Block scheme of simulation of mass displacements and contact forces of the pantographmodel at Fstat =600N, Faer =30N; in the case of one contact point, when x1L ∼= x1P Fig. 5. Block scheme of simulation of contact forces and contact loss at Fstat =500N, Faer =30N; in the case of one contact point, when x1L ∼= x1P 942 A. Kumaniecka,M. Prącik Fig. 6. Contact force for uplift force Fstat =500N; loss of contact in the case of one contact point Fig. 7. Variability of contact forces of the pantographwith two contact points stated that themotion of the contactwire has awavy character. The calculations have confirmed that the travelling force is a source of waves propagating leftwards and rightwards at different frequencies (Snamina, 2003; Bogacz and Frischmuth, 2013). The domination of lower frequency modes is visible (see Fig. 8). The same effect is visible in the case of a pantograph with one contact point. For a two contact points pantograph, the modal damping is more effective. Fig. 8. Spectrum FFT of vibration displacements In Figs. 9 and 10, some results of the lower beam vibration in the case of one or two con- tact points are shown. As can be seen in Fig. 10, the critical amplitude value of vibration displacement, critical as referred to displacement values x1L0 = x1P0 = 0.03m adopted as an example (for the data set taken for numerical simulations at the velocity v = 55.55m/s), has been exceeded. The examples of the results presented above have been obtained with the catenary stiffness parameters and the velocity of pantographmotion v =55.55m/s. At higher velocities taken for Vibration of a discrete-continuous structure under moving load... 943 Fig. 9. Results of simulations of the beam displacement function w1(x,t) for p1 =1, r1 =1,2; in the case of one contact point Fig. 10. Simulations of lower beam vibration of the catenarymodel in the case of two contact points simulations and the analysis of subsequent excited mode vibration frequencies of the catenary in 3D graphs of displacements. there can be seen a more marked interaction in the case of two contact points pantograph. Also two maxima and minima of waves moving parallel, positioned at an angle to the time axis, are visible. 5. Final conclusions The state-of-the-art of the theoretical and experimental investigations indicates the need for continuation of the research to improve the modelling of the catenary-pantograph system. In the present paper, a simplified model of the pantograph and catenary has been proposed. The equations of motion are based on a beam model with one or two concentrated varying forces moving along the contact wire at a constant velocity. The structure of the simulation model is based on a formal notation of motion in form of partial and ordinary differential equations. For the simulation, software package VisSim has been applied. The paper has discussed the application of the stiffness formula to the analysis of pantograph-catenary interaction. On the basis of the results of simulations of the lower beam displacements (Figs. 9 and 10) the following conclusions can be drawn: • Thewave sequence is associated with the pantographmotion (the compound of threemo- des ofmovingwaves and three standing ones). The frontalmaximumand reverseminimum of the displacement are visible. The ratio of the absolute value of themaximum to that of the minimum is equal to 9. 944 A. Kumaniecka,M. Prącik • Damping of the displacement amplitude excited in the lower beam is time and space variable. For example, a reduction of the beam displacement maximum by about 20dB occursafter approximately 0.6s, in thecase of analysis on thespanof lengthof about100m. The results of analysis of the simulation performed on the adopted model of mutual inter- action between the pantograph and catenary have shown the domination of lower frequencies components in the spectrum of the lower beam vibration displacements, similarly to the case of one contact point system (Fig. 8). This fact has been also indicated by others scientists (Poetsch et al., 1997; Szolc, 2003). On the basis of the simulation results of the lower beam displacements, it can be concluded that there is awave sequence associatedwith the pantograph motion. Damping of the lower beam vibrations, caused by the moving pantograph, is variable in time and space. The described phenomena should be taken into account in real applications in high speed railways. References 1. Bajer C.I., Dyniewicz B., 2012,Numerical Analysis of Vibrations of Structures under Moving Inertial Load, Springer 2. 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