Study on Dynamic Interaction of Railway Pantograph–Catenary Including Reattachment Momentum Impact

vibration

Article Study on Dynamic Interaction of Railway Pantograph–Catenary Including Reattachment Momentum Impact

Wenping Chu 1 and Yang Song 2,* 1 National Rail Transit Electriﬁcation and Automation Engineering Technique Research Center, Southwest Jiaotong University, Chengdu 611756, China; [email protected] 2 Department of Structural Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway * Correspondence: [email protected]

Received: 17 August 2019; Accepted: 6 February 2020; Published: 11 February 2020

Abstract: The pantograph–catenary system is responsible for the electric transmission to the locomotive via the sliding contact between the pantograph head and the contact wire. The separation of the pantograph head from the contact wire is the main source of arcing, which challenges the normal operation of an electrified railway. To properly describe the contact loss procedure using simulation tools, a mathematical model of the reattachment momentum impact between the pantograph head and the contact wire is proposed in this paper. The Euler–Bernoulli beam is adopted to model the contact and messenger wires, which are connected by lumped mass-spring droppers. The Lagrange multiplier method is utilised to describe the contact between the pantograph head and the contact wire. The momentum impact generated during the reattachment process is derived based on the principle of momentum conservation. Through several numerical simulations, the contact wire uplift and the contact force are evaluated with the reattachment impact. The analysis result indicated that the velocities of the contact wire and the pantograph head experience a sudden jump at the time instant of reattachment, which leads to a sudden increase of the contact force. When the reattachment impact is included, the maximum value and the standard deviation of contact forces show a significant increase. The effect of reattachment impact is more significant with the increase of the pantograph mass and stiffness.

Keywords: railway; catenary; pantograph; reattachment impact; Lagrange multiplier method; contact loss

1. Introduction In electrified railway industries, the catenary and pantograph are widely used for transmitting the electrical energy to the locomotives. With the increase of the train speed, the vibration of the catenary–pantograph becomes stronger, which increases the fluctuation of the contact force. The frequent contact loss between the pantograph head and the contact wire has been one of the technical issues challenging the current collection quality, as it is the main source of electric arcing and the interruption of the power transmission. Normally, the contact loss is caused by inadequate contact force. Thus, the contact force should be stable, and the contact loss should be avoided to ensure continuous contact of the pantograph and the catenary. To improve the current collection quality of the pantograph–catenary, many scholars have made significant contributions to improving the understanding of the pantograph–catenary dynamics. The relevant study has been reviewed in [1], in which an overview of the mathematical methods of modelling the catenary–pantograph system has been given. Various internal and external disturbances

Vibration 2020, 3, 18–33; doi:10.3390/vibration3010003 www.mdpi.com/journal/vibration Vibration 2020, 3 19

(such as the wind load [2,3], the vehicle vibration [4], the wave propagation [5] and the aerodynamic instability [6]) to the pantograph–catenary system have also been investigated to improve the robustness of the system. The methods of modelling the pantograph–catenary contact have been briefly reviewed in [7], in which the influence of different contact models on the catenary–pantograph dynamic response has been revealed. In [8], the results of nine pantograph–catenary models are compared to establish a world benchmark for the validation of simulation tools. At present, the most prevalent method to model the sliding contact of the pantograph and the catenary is the “penalty function method”, in which, the contact between the pantograph head and the contact wire is defined through an assumption of “contact stiffness”. The contact force is calculated based on the interpenetration between the pantograph head and the contact wire. This method is first introduced to model the pantograph–catenary contact by Collina and Bruni [9]. The penalty function method is developed and gradually becomes the dominant approach to model the contact of pantograph–catenary, as it is convenient to be included in the stiffness matrix [10]. Ambrósio et al. [11] present a modified penalty function method using the relative velocities of the two contact bodies. Based on this model, a co-simulation of finite element and rigid multi-body dynamic codes is performed, and the influence of the environmental wind on the pantograph–catenary current collection is investigated in [12,13]. In [14], the ACNF (absolute nodal coordinate formulation) method is adopted to model the catenary. The interface between the contact wire and the pantograph is described as a spring-damper model, in which the contact loss is assumed by the compression force. In [15], the non-linearity of the catenary is considered by a non-linear finite element approach, and its interaction with the pantograph is developed by the penalty function method. A moving mesh method is developed to improve computational efficiency [16]. However, the penalty function method may not describe the realistic contact condition between the pantograph head and the contact wire, because an assumption is defined that the two contact bodies can penetrate each other. Therefore Seo et al. [17] and Lee [18] use the Lagrange multiplier method to model the interaction between the pantograph and the catenary, which excludes the penetration assumption and obtained more reasonable results. The Lagrange multiplier method is widely used in studies of moving load problems [19]. Lee [20,21] firstly proposes a method to describe the separation between the moving mass and the beam. Subsequently, Ouyang et al. [22,23], Dahlberg [24], Cheng et al. [25] and Lee et al. [26,27] expand this idea to different application backgrounds. According to [25], after the separation between a mass and a beam, the reattachment may produce a momentum impact on the beam, which may result in a sudden jump of the velocity of the beam. Stăncioiu et al. [22] develop a simple model for the reattachment impact and investigate its influence on the dynamic behaviour of the moving mass-beam system. The results show that in some numerical examples, the impact on reattachment has a significant effect on the dynamic response. As is well known, the contact loss of the pantograph and the catenary is the main source of the deterioration of the current collection quality. To properly describe the separation and reattachment of the pantograph head and the contact wire is of great importance for understanding the pantograph–catenary dynamics. To the authors’ best knowledge, no studies have ever been undertaken to describe the separation and the reattachment for the pantograph–catenary. This shortcoming is tackled in this paper. A contact model of the pantograph–catenary is presented based on the Lagrange multiplier method. The contact is described in terms of constraint equations so that the contact loss can be fully described. The impact of the pantograph head on the contact wire at the reattachment point is considered by introducing additional velocity into the contact wire. Then, some numerical simulations at higher speed are performed to analyse the pantograph–catenary interaction considering the reattachment impact.

2. Modelling of Pantograph–Catenary with Reattachment Impact Normally, a railway catenary is comprised of two tensioned cables called messenger wire and contact wire. The contact wire directly contacts with the pantograph head to transmit electricity to the Vibration 2020, 3, 3 20 theVibration locomotive.2020, 3 The messenger wire is used to hang the contact wire to keep it level. The two wires 20 are connected via several droppers. In this section, the equation of motion for the pantograph– catenary is derived. To properly describe the high-frequencies behaviour, the beam element with bendinglocomotive. stiffness The is messengeradopted to wiremodel is usedthe contact to hang and the messenger contact wire wires to keep[28]. itThen level. a contact The two model wires is are developedconnected including via several the droppers.momentum In impact this section, during the the equation reattachment of motion process. for the pantograph–catenary is derived. To properly describe the high-frequencies behaviour, the beam element with bending 2.1.sti Equationﬀness is of adopted Motion to for model Catenary the contact and messenger wires [28]. Then a contact model is developed including the momentum impact during the reattachment process. The description of the catenary with a moving contact force is shown in Figure 1. v denotes the () speed2.1. Equationof the contact of Motion force for CatenaryFc vt, t traversing along the contact wire. Ta and Tb represent the tensionsThe of the description messenger of thewire catenary (the upper with beam) a moving and the contact contact force wire is (the shown lower in Figurebeam),1 .respectively.v denotes the The equation of motion for the messenger wire can be described by: speed of the contact force Fc(vt, t) traversing along the contact wire. Ta and Tb represent the tensions of the messenger wire∂∂∂∂422wwww (the upper beam) and the contact wire (the lower beam),nd respectively. The equation EIaaaa() xt,,,,+−+=−ρδ() xt T() xt C() xt Fii() x x of motion foraaaa the messenger422 wire can be described by:  ad ∂∂∂∂xtxti=1 nt jj4 2 2 nd +−Fxxδ∂()wa ∂ wa ∂ wa ∂wa P i i EItta (x, t) + ρa (x, t) Ta (x, t) + Ca (x, t) = F δ x x = ∂x4 ∂t2 ∂x2 ∂t a d j 1 − i=1 − nt P j j ∂2 ii+ F δ x x iiiiwa F=−() wxt()t ,, wxtt () k −() xtm , (1) abdaddj=1 − ∂t 2 dd (1) ∂2w i i 2 i i a i i F = wb x , t ∂wa x , t k 2 x , t m jjjjja d w d d ∂t d d =− −− a2 − Fwxtktattj (),,j j ∂ w()a xtm ttj j F = wa x , t k ∂ 2 x , t m t − t t −t ∂t2 t t w (0, t∂∂)ww= ∂wa (0, t) = 0; w (L, t) = ∂wa (L, t) = 0 wt()0,a ==aa() 0,∂x t 0; wLt () ,a ==() Lt∂ ,x 0 aa∂∂xx

Messenger wire Dropper Steady arm

y z x o Tb

F ()vt, t Registration arm c v Contact wire

Figure 1. Description of catenary with a moving contact force. Figure 1. Description of catenary with a moving contact force. Similarly, the equation of motion for the contact wire can be expressed by: Similarly, the equation of motion for the contact wire can be expressed by: nd 422∂4w ∂2w ∂2w ∂w P EI∂∂∂∂wwwwb (x, t) + ρ b (x, t) T b (x, t) + C b (x, t) =nd Fi δ x xi EIb ∂bbbbx()4 xt,,,,+−+=−ρδb ∂t2() xt Tb ∂x2 () xt Cb ∂t () xt Fiib () x xd bbbb422− i=1 bd− ∂∂∂∂nr xtxtP j j i=1 + F δ x x + Fc(vt, t)δ(x vt) nr r − r − +−+−Fj=jjδδ1 () x x F()() vt, t x vt (2)  rrc ∂2w j=1Fi = w xi , t w xi , t ki b xi , t mi b a d b d d ∂t2 d d 2 − −2 j ∂ wb j j ∂ iiF = x , t m iiiiwb =−r()()∂t2 r ()r −() (2) Fbadbdd wxt,, wxt k2 xtm dd , ( ) = ∂wb ( ) = ∂(t ) = ∂wb ( ) = wb 0, t ∂x 0, t 0; wb L, t ∂x L, t 0 ∂2w Fxtmjjj= b (), In Equationsrrr (1)∂t 2 and (2), ρa and ρb denote the linear densities of the messenger and contact wires, respectively. EIa and EI∂∂wwb denote the ﬂexural rigidities of both wires. Ca and Cb are the corresponding ()==bb() () ==() damping forw bothbb0, t wires. L denotes 0, t 0; the w total Lt , length of Lt one, section. 0 δ is the Dirac delta function. w (x, t) ∂∂xx a ( ) i i and wb x, t are the deﬂections of the messenger wire and the contact wire, respectively. Fa and Fb are = i j the forces of the ith dropper acting on both wires at the point x xd. Ft is the force of the jth steady = j j arm acting on the messenger wire at the point x xt. Fr is the force of the jth registration arm acting Vibration 2020, 3 21

= j j j on the contact wire at the point x xr. md and kd are the mass and stiffness of the ith dropper, which j j can only work in the traction condition. mt and kt are the mass and stiffness of the jth steady arm, j respectively. mt is the mass of the jth steady arm. nd, nt and nr are the total numbers of droppers, messenger wire supports and steady arms, respectively. The last equations in Equations (1) and (2) define the constraint conditions for each wire. In this work, the modal superposition method is adopted to solve Equations (1) and (2). The beam deflection can be represented in a modal expansion as [5]:

P∞ wa(x, t) = ψn(x)An(t) n=1 (3) P∞ wb(x, t) = ψn(x)Bn(t) n=1 where, An(t) and Bn(t) are the modal coordinates for the nth modal function of the messenger/contact wire ψn(x), which is determined by the constraint conditions. The general solution for the Euler–Bernoulli beam is:

W(x) = ϑ1chαx + ϑ2shαx + ϑ3 cos βx + ϑ4 sin βx (4) in which, α, β, ϑ1, ϑ2, ϑ3 and ϑ4 are determined by the boundary conditions. As the messenger and contact wires can be assumed to be simply-supported beams [29], it is obtained that,

ϑ1 = 0, ϑ2 = 0, ϑ3 = 0 and ϑ4 , 0 (5)

Thus β = nπ (n = 1, 2, 3, ...... ). Substituting Equation (3) into Equations (1) and (2) yields the L ∞ equation of motion for the catenary concerning the generalised coordinates An(t) and Bn(t).

2.2. Contact Model of Pantograph–Catenary As shown in Figure2, the pantograph is considered as a three-stage oscillator with three lumped masses (M1, M2, M3) connected by three springs with specific stiffness (K1, K2, K3) and damps (C1, C2, C3), respectively. The corresponding uplifts of the three masses are y1(t), y2(t) and y3(t) respectively. F0 is the uplift forces including the aerodynamic effect. The equation of motion for the pantograph is described by: d2 y dy dy M 1 (t) + C 1 (t) 2 (t) + K (y (t) y (t)) = F (vt t) 1 dt2 1 dt dt 1 1 2 c , − − − d2 y dy dy dy dy M 2 (t) + C 2 (t) 1 (t) + C 2 (t) 3 (t) + K (y y ) + K (y y ) = 2 dt2 1 dt dt 2 dt dt 1 2 1 2 2 3 0 (6) − − − − d2 y dy dy dy M 3 (t) + C 3 (t) 2 (t) + C 3 (t) + K (y y ) + K y = F 3 dt2 2 dt − dt 3 dt 2 3 − 2 3 3 0 When the pantograph head contacts with the contact wire, the catenary and pantograph interact with each other by the contact force. However, when the contact loss occurs, the motion of the catenary and the pantograph are independent of each other. The criterion for contact and separation can be governed by the gap g of the contact surface as expressed by:

g = w (vt, t) y (t) (7) b − 1 Vibration 2020, 3 22

When g > 0, the pantograph is separated from the contact wire. If g = 0, the pantograph contacts with the contact wire. When the contact occurs, the motion of the contact wire is governed by the following equation:

nd ∂4w ∂2w ∂2w ∂w P EI b (x, t) + ρ b (x, t) T b (x, t) + C b (x, t) = Fi δ x xi b ∂x4 b ∂t2 b ∂x2 b ∂t b d − i=1 − nr (8) P j j d2 y dy dy + F x x + M 1 (t) + C 1 (t) 2 (t) + K (y (t) y (t) (x vt) rδ r 1 dt2 1 dt dt 1 1 2 δ j=1 − − − − Vibration 2020, 3, 3 22

K1 C1 y2

M2 v

C2 K2 y3

K3 C3 F0

Figure 2.2. Pantograph–catenary contact model.

ItWhen can bethe observed pantograph that whenhead thecontacts contact with occurs, the the contact deﬂection wire, of the the catenary contact wire and at pantograph the contact pointinteract is equalwith each to the other displacement by the contact of the force. pantograph However, head. when The the contact contact force loss isoccurs, acting the on motion a moving of coordinate.the catenary The and relationship the pantograph of theacceleration, are independent velocity of andeach displacement other. The atcriterion the contact for pointcontact can and be describedseparation by: can be governed by the gap g of the contact surface as expressed by:

y (t) = w (x, t) 1 b x==−vt () () dy g wvttb , yt1 (7) 1 ( ) = ∂wb ( ) + ∂wb ( ) dt t ∂t vt, t v ∂x vt, t (9) d2 y 2 2 2 When g > 0 , the pantograph1 (t) = ∂isw bseparate(vt, t) +d2 vfrom∂ wb ( vtthe, t) contact+ v2 ∂ w bwire.(vt, t )If g =0 , the pantograph dt2 ∂t2 ∂t∂x ∂x2 contactsWhen with the the pantograph contact wire. separates When from the contact the contact occurs, wire the (g motion> 0), the of vibrations the contact of wire the catenary is governed and theby the pantograph following are equation: independent of each other. The equation of motion for the contact wire and the pantograph can∂∂∂∂422 bewwww expressed by: nd EIbbbb() xt,,,,+−+=−ρδ() xt T() xt C() xt Fii() x x bbbb422  bd ∂∂∂∂xtxt= nd i 1 nr ∂4w ∂2w ∂2w ∂w P P j j (8) EI b nr(x, t) + ρ b (x, t) T2 b (x, t) + C b (x, t) = Fi δ x xi + F δ x x b ∂x4 jjb ∂t2 db y∂112x2  dyb ∂t dy b d r r +−++F δδ()xx M− () t C() t −+−() ti=1 Kyt( − () ytj ()=1 ( xvt −− )  rr 11 2  112 d2 y j=1 dy dy dt dt dt M 1 (t) + C 1 (t) 2 (t) + K (y (t) y (t)) = 1 dt2 1 dt dt 1 1 2 0 − − (10) It cand2 y be observeddy that whendy the contactdy occurs,dy the deflection of the contact wire at the contact M 2 (t) + C 2 (t) 1 (t) + C 2 (t) 3 (t) + K (y y ) + K (y y ) = 2 dt2 1 dt dt 2 dt dt 1 2 1 2 2 3 0 point is equal to the displacement− of the pantograph− head. The contact− force is− acting on a moving d2 y dy dy dy coordinate.M 3 (Thet) + relationshipC 3 (t) of2 th(t)e acceleration,+ C 3 (t) + velocityK (y yand) + displacementK y = F at the contact point can 3 dt2 2 dt − dt 3 dt 2 3 − 2 3 3 0 be described by: If g changes from a positive value to zero, the reattachment occurs, and the pantograph starts to yt()= wxt (, ) contact with the contact wire.1 However,b xvt= as the description in [22], the momentum impact between dy ∂∂ww a mass and a beam during1 reattachment()tvttvvtt=+bb(),, has a noticeable() inﬂuence on the dynamic response of the beam. This reattachment impactdt is modelled∂∂ t in Section x 2.3 by introducing an additional velocity in(9) the contact wire during the reattachment.dy2 ∂∂∂222www 1 ()t=+bbb() vtt,2 v() vtt , + v2 () vtt , dt22∂∂∂∂ t t x x 2 2.3. Modelling of Reattachment Impact When the pantograph separates from the contact wire ( g >0), the vibrations of the catenary Consider that the pantograph separates from the contact wire at time instant t1 and reattaches to and the pantograph are independent of each other. The equation of motion for the contact wire and the contact wire at time instant t2. During the period t1 t2, the contact force between the pantograph the pantograph can be expressed by: ∼

∂∂∂∂422wwwwnd nr EIbbbb() xt,,,,+−+=−+−ρδδ() xt T() xt C() xt Fii() x x F j () x x j bbbb422  bdrr ∂∂∂∂xtxtij==11 dy2 dy dy 112()+−+−=() () () () MtCt112  tKytyt 112()0 dt dt dt (10) d2 y dy dy dy dy 2212()+−+−+−+−() () () 3 () ()= MtCttCttKyyKyy212  2 121223() 0 dt dt dt dt dt dy2 dydy dy 33()+−++−+=() 2 () 3() MtCt322  tCtKyyKyF 3232330() dt dt dt dt

If g changes from a positive value to zero, the reattachment occurs, and the pantograph starts to contact with the contact wire. However, as the description in [22], the momentum impact between

Vibration 2020, 3 23 and catenary is zero. When the reattachment occurs, the equation of motion for the contact wire at time instant t2 due to the reattachment impact is described by:

nd ∂4w ∂2w ∂2w ∂w P EI b (x, t) + ρ b (x, t) T b (x, t) + C b (x, t) = Fi δ x xi b ∂x4 b ∂t2 b ∂x2 b ∂t b d − i=1 − nr (11) P j j + Frδ x xr + pδ(x vt)δ(t t2) j=1 − − − where, p is the impulse caused by the impact of the pantograph head. The equation of motion for the pantograph head (the ﬁrst mass) can be expressed by:

2 ! d y1 dy1 dy2 M (t) + C (t) (t) + K (y (t) y2(t)) = pδ(t t2) (12) 1 dt2 1 dt − dt 1 1 − − −

The inﬂuence of the impact p is the jump of the velocity of the contact wire at the time instant t2. i j By substituting Equation (3) into Equation (11), the contribution of Fb and Fr to the nth mode of the contact wire is expressed by:

2 R L i i R L ∂ wb i i i i i i F δ x x ψn(x)dx = x , t m w x , t wa x , t k δ x x ψn(x)dx 0 b − d 0 ∂t2 d d − b d − d d − d nd 2 nd nd P P d Bn(t) P P P P = ∞ mi xi xi ∞ B ki xi xi + ∞ A ki xi xi dt2 dψm d ψn d m dψm d ψn d m dψm d ψn d m=1 i=1 − m=1 i=1 m=1 i=1 2 (13) R L j j R L ∂ wb j j j F δ x x ψn(x)dx = x , t m δ x x ψn(x)dx 0 r − r 0 ∂t2 r r − r nd 2 P P d Bn(t) j j j = ∞ m x x dt2 rψm r ψn r m=1 j=1

Substituting Equation (13) into Equation (11), the equation of motion for the contact wire can be obtained as:

2 2 nd nr d Bn(t) P d Bn(t) P P j + 2 ∞ ( mi ψ xi ψ xi + m ψ xi ψ xi ) dt2 ρaL dt2 d m d n d r m d n d m=1 i=1 j=1 nd C dBn(t) EI 4 T 2 P P + a + b nπ + b nπ B + 2 ∞ B ki ψ xi ψ xi (14) ρa dt ρ L ρ L n ρaL m d m d n d b b m=1 i=1 P Pnd 2 ∞ A ki ψ xi ψ xi 2 pψ (vt)δ(t t ) = 0 ρaL m d m d n d ρ L n 2 − m=1 i=1 − b −

According to the distribution theory [30], the incremental velocities of the pantograph head and the contact wire are determined by the total mass. The velocity jump of the contact wire and the + pantograph head from the t2− to t2 can be derived by:

∂w (vt ,t+) ∂w (vt ,t ) b 2 2 b 2 2− = p P∞ 2 ( ) ∂t ∂t ρ L ψn vt2 − b n=1 (15) dy (t+) dy (t ) 1 2 1 2− = p dt − dt − m1 At the reattachment point, the relationship of the velocities of the pantograph head and the contact wire can be expressed by:

dy ∂w ∂w ∂w 1 + = b ( ) = b + + b + t2 vt, t vt2, t2 v vt2, t2 (16) dt ∂t = + ∂t ∂x t t2 Vibration 2020, 3 24

∂wb + in which the high-order term v ∂x vt2, t2 that is neglected in [22] is taken into account here. During + the period from t− to t , the assumption is made that the deﬂection of the contact wire is not changed 2 2 = + (namely wb x, t2− wb x, t2 ). So the following equation can be obtained:

∂wb + X∞ dψn(vt2) vt , t Bn t− (17) ∂x 2 2 ≈ 2 dx n=1

The velocity jump of the contact wire in each mode is:

+ dBn t2 dBn t2− p ∆vn = = ψn(vt) (18) dt − dt −ρbL

Solving Equations (14)–(16), p can be derived explicitly as: ! dy (t ) dB (t ) ( ) 1 2− P∞ n 2− ( ) + P∞ dψn vt2 dt dt ψn vt2 v Bn t2− dx − n=1 n=1 p = (19) − P∞ 2 ψn(vt2) n=1 + 1 ρbL m1

Substituting Equation (18) into Equation (17) yields: ! dy (t ) dB (t ) ( ) 1 2− P∞ n 2− ( ) + P∞ dψn vt2 ( ) + dt dt ψn vt2 v Bn t2− dx ψn vt2 dBn t2 dBn t2− − n=1 n=1 ∆vn = = (20) dt − dt P∞ 2 ρbL ψn(vt2) + m n=1 1

The equation of motion for the messenger wire is similar to the contact wire, which can be written by: 2 2 nd nt d An(t) P d An(t) P P j + 2 ∞ ( mi ψ xi ψ xi + m ψ xi ψ xi ) dt2 ρaL dt2 d m d n d t m t n t m=1 i=1 j=1 nd C dAn(t) EI 4 T 2 P P + a + a nπ + a nπ A + 2 ∞ A ( ki ψ xi ψ xi (21) ρa dt ρa L ρa L n ρaL m d m d n d m=1 i=1 nt nd + P j i i ) 2 P∞ P i i i = ktψm xt ψn xt ρ L Bm kdψm xd ψn xd 0 j=1 − b m=1 i=1 Implementing the Lagrange multiplier method, the equation of motion for the pantograph–catenary interaction can be written by:

.. . T MU(t) + CU(t) + KU(t) + G (t) fc(t) = F(t) (22) . .. in which, U(t) is the displacement vector of the pantograph–catenary system. U(t) and U(t) are the corresponding velocity and acceleration vectors. G(t) is the restrained displacement vector on the contact surface. F(t) is the external force vector excluding the contact force. The constraint condition for Equation (21) is: G(t)U(t) = 0 (23) which can be extended to:

X∞ w (vt, t) y = Bn(t)ψn(vt) y = 0 (24) b − 1 − 1 n=1 Vibration 2020, 3 25

Equations (21) and (22) can be solved by the direct integration method. Considering a second-order integration through time, the following incremental equation of motion can be obtained as [31]: " #" #  h i  b2M + b1C + b0KG(t + ∆t) ∆U(t + ∆t)  F(t + ∆t) Mq2 + Cq1 + Kq0  =  −  (25) b0G(t + ∆t) 0 fc(t + ∆t)  G(t + ∆t)q  − 0 where, fc(t) is the contact force on both of the contact wire and pantograph head. q0, q1 and q2 are diﬀerential operators of second order:

. .. q = U(t) + ∆tU(t) + 1 (∆t)2U(t) 0 . .. 2 q = U(t) + ∆tU(t) (26) 1 .. q2 = U(t)

and 1 2 b0 = 2 (∆t) β0 b1 = ∆tβ1 (27) b2 = 1 in which, β0 and β1 are the coeﬃcients in the integration algorithm. In this paper, a linear acceleration 1 1 method with β0 = 3 and β1 = 2 is adopted. At the reattachment point, Equation (18) is adopted to govern the reattachment impact.

3. Analysis with Reattachment Impact The pantograph–catenary model is established using the parameters in Tables1 and2. The maximum speed is 350 km/h. Higher speeds are adopted in the simulation to reproduce the separation and reattachment between the contact wire and the pantograph head. The influence of the reattachment impact on the pantograph–catenary interaction is investigated. The electromagnetic force caused by the arcing during the separation process is neglected in this work, as it is only a small magnitude of the contact force [32]. Then at different speeds, the separation and reattachment between the pantograph head and the contact wire are analysed. Finally, the influence of the pantograph head parameters on the interaction performance is analysed.

Table 1. Simulation parameters of high-speed railway catenary.

Item Value Item Value Span 48 m Interval of droppers 5/9.5/9.5/9.5/9.5/5 m Contact wire tension 27 kN Simulated length 10 spans Messenger wire tension 21 kN Messenger wire type CuMg0.5 AC 120 Number of droppers 5 Contact wire type BZ II 120 Dropper stiﬀness 1 105 N/m Dropper mass 0.4 kg × Messenger wire support 1 107 N/m Steady arm mass 1.125 kg stiﬀness ×

Table 2. Simulation parameters of pantograph.

Item 1 2 3 M (kg) 6 7.12 5.8 C (Ns/m) 0 0 70 K (N/m) 9430 14,100 0.1 Maximum operating speed: 350 (km/h); F = 0.00097 v2 + 70 (N) 0 × Vibration 2020, 3, 3 26

Table 1. Simulation parameters of high-speed railway catenary.

Item Value Item Value Span 48 m Interval of droppers 5/9.5/9.5/9.5/9.5/5 m Contact wire tension 27 kN Simulated length 10 spans Messenger wire tension 21 kN Messenger wire type CuMg0.5 AC 120 Number of droppers 5 Contact wire type BZ II 120 Dropper stiffness 1 × 105 N/m Dropper mass 0.4 kg Messenger wire support 1 × 107 N/m Steady arm mass 1.125 kg stiffness

Table 2. Simulation parameters of pantograph.

Item 1 2 3 M (kg) 6 7.12 5.8 C (Ns/m) 0 0 70 K (N/m) 9430 14,100 0.1 Vibration 2020, 3 26 Maximum operating speed: 350 (km/h); F0 = 0.00097 × v2 + 70 (N)

3.1. Analysis with 380 km/h 3.1. Analysis with 380 km/h To reproduce the contact loss between the catenary and the pantograph, the higher speed v = To reproduce the contact loss between the catenary and the pantograph, the higher speed v = 380 380 km/h is adopted. The separation between the pantograph and the catenary firstly appears at 42.5 km/h is adopted. The separation between the pantograph and the catenary firstly appears at 42.5 m, m, (close to the fifth dropper in the first span). The uplift and the velocity of the pantograph head are (close to the fifth dropper in the first span). The uplift and the velocity of the pantograph head are presented in Figure 3a,b. It is seen that the reattachment impact causes a significant increase of the presented in Figure3a,b. It is seen that the reattachment impact causes a significant increase of the uplift and the velocity of the pantograph head when the pantograph head re-contacts the contact uplift and the velocity of the pantograph head when the pantograph head re-contacts the contact wire. wire. The velocity of the contact wire at the contact point are shown in Figure 4. As the Lagrange The velocity of the contact wire at the contact point are shown in Figure4. As the Lagrange multiplier multiplier method is implemented, the uplift of the pantograph head matches the displacement of method is implemented, the uplift of the pantograph head matches the displacement of the contact the contact point in the contact wire. It is seen that the reattachment causes a jump of the velocity of point in the contact wire. It is seen that the reattachment causes a jump of the velocity of the contact the contact wire. The contact forces are shown in Figure 5. It is also seen that due to the reattachment wire. The contact forces are shown in Figure5. It is also seen that due to the reattachment impact, the impact, the contact force experiences a sudden increase from 0 to 400 N within a short time. The contact force experiences a sudden increase from 0 to 400 N within a short time. The impact produces impact produces almost excessive 100 N at the peak of the contact force during the reattachment. almost excessive 100 N at the peak of the contact force during the reattachment. The overlarge contact The overlarge contact force may exceed the safety criterion and damage the contact wire. force may exceed the safety criterion and damage the contact wire.

0.15 With reattachment impact 0.145 No reattachment impact

0.14

0.135

0.13

0.125 Pantograph headuplift(m) Pantograph 0.12

0.115 38 40 42 44 46 48 50 Moving coordinate(m) Vibration 2020, 3, 3 27 (a) 0.7 With reattachment impact 0.6 No reattachment impact

0.5

0.4

0.3

0.2

0.1 Pantograph headPantograph velocity(m/s) 0

-0.1 38 40 42 44 46 48 50 Moving coordinate(m) (b)

FigureFigure 3. UpliftUplift and velocity of thethe pantographpantograph headhead at at contact contact points points (from (from 38 38 m m to to 50 50 m). m). (a )(a Uplift) Uplift of ofthe the pantograph pantograph head; head; (b )(b velocity) velocity of of the the pantograph pantograph head. head.

0.7 With reattachment impact No reattachment impact 0.6

0.5

0.4

0.3

0.2 Contact wire velocity(m/s) Contactwire 0.1

0 38 40 42 44 46 48 50 Moving coordinate(m) Figure 4. Velocity of the contact wire at contact points (from 38 m to 50 m).

400 With reattachment impact 350 No reattachment impact 300

250

200

150 Contactforce(N) 100

0 38 40 42 44 46 48 50 Moving coordinate(m) Figure 5. Contact force of the pantograph–catenary (from 38 m to 50 m).

Vibration 2020, 3, 3 27 Vibration 2020, 3, 3 27 0.7 0.7 With reattachment impact 0.6 WithNo reattachment reattachment impact impact 0.6 No reattachment impact 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1

Pantograph headPantograph velocity(m/s) 0.1

Pantograph headPantograph velocity(m/s) 0 0 -0.1 38 40 42 44 46 48 50 -0.1 38 40 42 Moving coordinate(m)44 46 48 50 Moving coordinate(m) (b) (b) Figure 3. Uplift and velocity of the pantograph head at contact points (from 38 m to 50 m). (a) Uplift Figure 3. Uplift and velocity of the pantograph head at contact points (from 38 m to 50 m). (a) Uplift Vibrationof the2020 pantograph, 3 head; (b) velocity of the pantograph head. 27 of the pantograph head; (b) velocity of the pantograph head.

0.7 With reattachment impact 0.7 NoWith reattachment reattachment impact impact 0.6 No reattachment impact 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 Contact wire velocity(m/s) Contactwire 0.2 Contact wire velocity(m/s) Contactwire 0.1 0.1 0 038 40 42 44 46 48 50 38 40 42 Moving coordinate(m)44 46 48 50 Moving coordinate(m) Figure 4. Velocity of the contact wire at contact points (from 38 m to 50 m). FigureFigure 4. 4. VelocityVelocity of the contact wire at contact points (from 38 m to 50 m).

400 400 With reattachment impact 350 WithNo reattachment reattachment impact impact 350 No reattachment impact 300 300 250 250 200 200 150 Contactforce(N) 150 Contactforce(N) 100 100 50 50 0 38 40 42 44 46 48 50 0 38 40 42 Moving coordinate(m)44 46 48 50 Moving coordinate(m) Figure 5. ContactContact force force of of the the pantograph–catenary pantograph–catenary (from 38 m to 50 m). Figure 5. Contact force of the pantograph–catenary (from 38 m to 50 m). 3.2. Analysis at Diﬀerent Operating Speeds

According to the current standard [33], the maximum contact force and the standard deviation of the contact force are the most important indicators to evaluate the current collection quality of the pantograph–catenary system. The maximum contact force should be limited to avoid the potential damage to the catenary. The standard deviation directly describes the fluctuation dispersion of the contact force around its mean value. Figure6a–c show the maximum contact force, contact force standard deviation and the contact loss percentage at different speeds, respectively. It is found that with the increase of the operating speed, the maximum contact force, the contact force standard deviation and the contact loss rate increase sharply. When the reattachment impact is included, the maximum contact force and the contact force standard deviation are generally larger than the results without the reattachment impact. More contact loss can be observed when the reattachment is included. It is worthwhile noting that the contact force is not filtered with 0~20 Hz to fully describe the contact loss. That is why the standard deviation in Figure6b is significantly larger than previous works [ 34]. The presence of the reattachment impact introduces more disturbance to the pantograph–catenary interaction, which increases the maximum contact force, and accordingly increases the fluctuation of the contact force. These also result in more contact losses of the pantograph–catenary interaction. Vibration 2020, 3, 3 28

3.2. Analysis at Different Operating Speeds According to the current standard [33], the maximum contact force and the standard deviation of the contact force are the most important indicators to evaluate the current collection quality of the pantograph–catenary system. The maximum contact force should be limited to avoid the potential damage to the catenary. The standard deviation directly describes the fluctuation dispersion of the contact force around its mean value. Figure 6a–c show the maximum contact force, contact force standard deviation and the contact loss percentage at different speeds, respectively. It is found that with the increase of the operating speed, the maximum contact force, the contact force standard deviation and the contact loss rate increase sharply. When the reattachment impact is included, the maximum contact force and the contact force standard deviation are generally larger than the results without the reattachment impact. More contact loss can be observed when the reattachment is included. It is worthwhile noting that the contact force is not filtered with 0~20 Hz to fully describe the contact loss. That is why the standard deviation in Figure 6b is significantly larger than previous works [34]. The presence of the reattachment impact introduces more disturbance to the pantograph–catenary interaction, which increases the maximum contact force, and accordingly Vibrationincreases2020, 3 the fluctuation of the contact force. These also result in more contact losses of28 the pantograph–catenary interaction.

700 With reattachment impact No reattachment impact 650

600

550

500

Max contact force(N) Max contact 450

400

350 365 370 375 380 385 390 395 400 405 410 415 Operating speed(km/h) (a) 130 14 With reattachment impact With reattachment impact No reattachment impact 120 No reattachment impact 12

10 110 8 100 6 Contact loss(%) Standard deviation Standard 90 4

80 2

70 0 365 370 375 380 385 390 395 400 405 410 415 365 370 375 380 385 390 395 400 405 410 415 Operating speed(km/h) Operating speed(km/h) (b) (c)

FigureFigure 6. Maximum 6. Maximum contact contact force, contactforce, contact force standard force standard deviation deviation and contact and loss contact at diﬀ erentloss at speeds. different (a) Maximumspeeds. (a contact) Maximum force; contact (b) standard force; ( deviation;b) standard (c )deviation; contact loss. (c) contact loss.

3.3.3.3. Analysis Analysis with with Diff erentDifferent Pantograph Pantograph Head Head Parameters Parameters In orderIn order to understand to understand the e fftheect effect of pantograph of pantograph head parametershead parameters on the on reattachment the reattachment impact, impact, a set a of simulationsset of simulations is performed is performed with diff wierentth different mass and mass stiffness and of stiffness the pantograph of the pantograph head. The head. contact The forces contact withforces different withM different1 are presented M1 are in presented Figures7 andin Figures8 at 370 7 km and/ h8 andat 370 380 km/h km/h, and respectively. 380 km/h, Inrespectively. Figure7a, In the contactFigure 7a, forces the remaincontact unchangedforces remain when unchanged the reattachment when the impact reattachment is included, impact as nois contactincluded, loss as no occurs.contact However, loss occurs. with the However, increase with of the the pantograph increase headof the mass, pantograph more contact head lossmass, appears, more contact and the loss differenceappears, of theand contact the difference forces evaluated of the by contact two methods forces becomes evaluated more by significant. two methods The similarbecomes trend more is alsosignificant. observed The in Figure similar8. trend When isM al1soreaches observed 11 kg, in theFigure reattachment 8. When M impact1 reaches causes 11 kg, big the fluctuations reattachment of contact forces. The maximum contact force, the contact force standard deviation and the contact loss with different M1 at 370 km/h and 380 km/h are shown in Figure9a–c, respectively. It is seen that all these indexes increase when the reattachment impact is included. Equation (18) shows that the increase of the pantograph head mass directly leads to an increase of the reattachment impact. Also, the previous study has revealed that a big pantograph head mass aggravates the current collection quality. The reattachment impact makes this negative effect even bigger. The contact forces with different pantograph head stiffness at 380 km/h are shown in Figure 10. The corresponding statistics are presented in Figure 11. Similar to the influence of M1, when the reattachment impact is presented, the fluctuation in contact force becomes larger and more contact loss can be observed. VibrationVibration 2020 2020, ,3 3, ,3 3 2929 impactimpact causescauses bigbig fluctuationsfluctuations ofof contactcontact forcforces.es. TheThe maximummaximum contactcontact force,force, thethe contactcontact forceforce standard deviation and the contact loss with different M1 at 370 km/h and 380 km/h are shown in standard deviation and the contact loss with different M1 at 370 km/h and 380 km/h are shown in FigureFigure 9a9a––c,c, respectively.respectively. ItIt isis seenseen ththatat allall thesethese indexesindexes increaseincrease whenwhen thethe reattachmentreattachment impactimpact isis included.included. EquationEquation (18)(18) showsshows thatthat thethe increaseincrease ofof thethe pantographpantograph headhead massmass directlydirectly leadsleads toto anan increaseincrease ofof thethe reattachmentreattachment impact.impact. Also,Also, thethe prpreviousevious studystudy hashas revealedrevealed thatthat aa bigbig pantographpantograph head mass aggravates the current collection quality. The reattachment impact makes this negative Vibrationhead mass2020 ,aggravates3 the current collection quality. The reattachment impact makes this negative29 effecteffect eveneven bigger.bigger.

No reattachment impact 300 NoWith reattachment reattachment impact impact 300 With reattachment impact 250 250 200 200 150 150 100 Contact force(N) Contact 100 Contact force(N) Contact 50 50300 310 320 330 340 350 360 370 380 390 400 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m) Moving(a) coordinate(m) M =3kg 400 (a) M 1=3kg 400 1 300 300 200 200 100

Contact force(N) 100 Contact force(N) 0 0300 310 320 330 340 350 360 370 380 390 400 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m) Moving coordinate(m) (b) M1=7kg 500 (b) M1=7kg 500 400 400 300 300 200 200 100 Contact Contact force(N) 100 Contact Contact force(N) 0 0 300 310 320 330 340 350 360 370 380 390 400 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m) Moving coordinate(m) (c) M1=11kg (c) M1=11kg Figure 7. Contact force with different M1 at 370 km/h. Figure 7. Contact force with different diﬀerent M1 at 370 370 km/h. km/h.

400 No reattachment impact 400 NoWith reattachment reattachment impact impact With reattachment impact 300 300 200 200 100 100 Contact force(N) Contact force(N) 0 0300 310 320 330 340 350 360 370 380 390 400 300 310 320 330 340Moving350 coordinate(m)360 370 380 390 400 Moving coordinate(m) (a) M =3kg 500 (a) M 1=3kg 500 1 400 400 300 300 200 200 100

Contact force(N) Contact 100 Contact force(N) Contact 0 0 300 310 320 330 340 350 360 370 380 390 400 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m) Moving coordinate(m) (b) M1=7kg 600 (b) M1=7kg 600 500 500 400 400 300 300 200 Contact force(N) 200 Contact force(N) 100 100 0 0300 310 320 330 340 350 360 370 380 390 400 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m) Moving coordinate(m) (c) M1=11kg (c) M1=11kg Figure 8. Contact force with diﬀerent M1 at 380 km/h.

Vibration 2020, 3, 3 30

Vibration 2020, 3 30 Figure 8. Contact force with different M1 at 380 km/h.

600 v = 380km/h No reattachment impact With reattachment impact 550

500

450 v = 380km/h v = 370km/h v = 370km/h 400 Maximum value(N) 350

300

250 3 4 5 6 7 8 9 10 11 12 M1(kg) (a)

140 v = 380km/h 14 With reattachment impact 130 No reattachment impact v = 380km/h No reattachment impact With reattachment impact 12 120

110 10 100 v = 380km/h 8 90 v = 370km/h v = 370km/h 80 6 v = 380km/h Contact loss(%) Standard deviation Standard 70 v = 370km/h 60 4 v = 370km/h

50 2 40

30 0 3 4 5 6 7 8 9 10 11 12 3 4 5 6 7 8 9 10 11 12 M1(kg) M1(kg) (b) (c)

Figure 9. Maximum contact force, contact force standard deviation and contact loss with different M1. Figure 9. Maximum contact force, contact force standard deviation and contact loss with diﬀerent M1. (a) Max(a contact) Max contact force; force; (b) standard (b) standard deviation; deviation; (c) contact(c) contact loss. loss. Vibration 2020, 3, 3 31 The contact forces with different pantograph head stiffness at 380 km/h are shown in Figure 10. The corresponding statistics are presented in Figure 11. Similar to the influenceWith reattachment of M1 impact, when the 400 No reattachment impact reattachment impact is presented, the fluctuation in contact force becomes larger and more contact 300 loss can be observed. 200

Contact force(N) Contact 100

0 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m)

(a) K1=2000(N/m) 500 400

300 200

Contact force(N) Contact 100

0 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m)

(b) K1=4000(N/m) 500 400

300 200

Contactforce(N) 100

0 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m) (c) K1=6000(N/m)

FigureFigure 10. 10.Contact Contact force force withwith didifferentﬀerent KK1 atat 380 380 km/h. km/h. 1

480

460

440

420

400

Maximum value(N) Maximum 380 No reattachment impact 360 With reattachment impact

340 2000 2500 3000 3500 4000 4500 5000 5500 6000 K1(N/m) (a)

92 4

90 3.5

88 3 86 2.5 No reattachment impact 84 With reattachment impact Contact loss(%) 2 No reattachment impact Standard deviation Standard 82 With reattachment impact

80 1.5

78 1 2000 2500 3000 3500 4000 4500 5000 5500 6000 2000 2500 3000 3500 4000 4500 5000 5500 6000 K1(N/m) K1(N/m) (b) (c)

Figure 11. Maximum contact force, contact force standard deviation and contact loss with different

K1. (a) Max contact force; (b) standard deviation; (c) contact loss.

Vibration 2020, 3, 3 31

With reattachment impact 400 No reattachment impact

300

200

Contact force(N) Contact 100

0 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m)

(a) K1=2000(N/m) 500 400

300 200

Contact force(N) Contact 100

0 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m)

(b) K1=4000(N/m) 500 400

300 200

Contactforce(N) 100

0 300 310 320 330 340 350 360 370 380 390 400 Moving coordinate(m) (c) K1=6000(N/m) Vibration 2020, 3 31 Figure 10. Contact force with different K1 at 380 km/h.

480

460

440

420

400

Maximum value(N) Maximum 380 No reattachment impact 360 With reattachment impact

340 2000 2500 3000 3500 4000 4500 5000 5500 6000 K1(N/m) (a)

92 4

90 3.5

88 3 86 2.5 No reattachment impact 84 With reattachment impact Contact loss(%) 2 No reattachment impact Standard deviation Standard 82 With reattachment impact

80 1.5

78 1 2000 2500 3000 3500 4000 4500 5000 5500 6000 2000 2500 3000 3500 4000 4500 5000 5500 6000 K1(N/m) K1(N/m) (b) (c)

FigureFigure 11. 11.Maximum Maximum contact contact force, force, contact contact force force standard standard deviation deviation and and contact contact loss loss with with diﬀerent differentK1. (aK)1 Max. (a) Max contact contact force; force; (b) standard (b) standard deviation; deviation; (c) contact (c) contact loss. loss.

4. Conclusions In this work, the analysis of the interaction of the catenary–pantograph is performed, including the effect of reattachment impact. The additional velocity of the contact wire in each mode caused by the reattachment impact is derived, and its effect is taken into account in the simulation of pantograph–catenary interaction. The effect of the reattachment impact on the interaction performance is investigated with different speeds and pantograph head parameters. The analysis results show that the reattachment impact of the pantograph head can lead to a sudden increase of the velocity for both the contact wire and the pantograph head, which increases the contact force at the reattachment point. It is necessary to consider the reattachment impact in simulating the contact loss procedure. Otherwise, the results of the maximum contact force, the contact force standard deviation and the contact loss rate may be too conservative. A low pantograph head mass is also suggested to reduce the reattachment impact. It should be pointed out that the analysis in this paper only focuses on the mechanical interaction performance. The arcing occurring in the separation process may cause electromagnetic forces, the effect of which on the interaction performance will be discussed in the future.

Author Contributions: Conceptualization, Y.S.; methodology, Y.S.; formal analysis, W.C.; investigation, W.C.; writing—Original draft preparation, W.C.; writing—Review and editing, Y.S.; visualization, W.C.; supervision, Y.S.; project administration, Y.S.; funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (U1734202 U51977182) and the Funding of Chengdu Guojia Electrical Engineering Co. Ltd. (No. NEEC-2018-A02). Vibration 2020, 3 32

Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

References

1. Poetsch, G.; Evans, J.; Meisinger, R.; Kortüm, W.; Baldauf, W.; Veitl, A.; Wallaschek, J. Pantograph/catenary dynamics and control. Veh. Syst. Dyn. 1997, 28, 159–195. [CrossRef] 2. Song, Y.; Liu, Z.; Wang, H.; Lu, X.; Zhang, J. Nonlinear analysis of wind-induced vibration of high-speed railway catenary and its influence on pantograph–catenary interaction. Veh. Syst. Dyn. 2016, 54, 723–747. [CrossRef] 3. Song, Y.; Liu, Z.; Duan, F.; Lu, X.; Wang, H. Study on wind-induced vibration behavior of railway catenary in spatial stochastic wind field based on nonlinear finite element procedure. J. Vib. Acoust. Trans. ASME 2018, 140, 011010. [CrossRef] 4. Wang, Z.; Song, Y.; Yin, Z.; Wang, R.; Zhang, W. Random response analysis of axle-box bearing of a high-speed train excited by crosswinds and track irregularities. IEEE Trans. Veh. Technol. 2019, 68, 10607–10617. [CrossRef] 5. Song, Y.; Liu, Z.; Duan, F.; Xu, Z.; Lu, X. Wave propagation analysis in high-speed railway catenary system subjected to a moving pantograph. Appl. Math. Model. 2018, 59, 20–38. [CrossRef] 6. Song, Y.; Liu, Z.; Wang, H.; Zhang, J.; Lu, X.; Duan, F. Analysis of the galloping behaviour of an electrified railway overhead contact line using the non-linear finite element method. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2018, 232, 2339–2352. [CrossRef] 7. Lopez-Garcia, O.; Carnicero, A.; Maroño, J.L. Influence of stiffness and contact modelling on catenary-pantograph system dynamics. J. Sound Vib. 2007, 299, 806–821. [CrossRef] 8. Bruni, S.; Ambrosio, J.; Carnicero, A.; Cho, Y.H.; Finner, L.; Ikeda, M.; Kwon, S.Y.; Massat, J.P.; Stichel, S.; Tur, M.; et al. The results of the pantograph-catenary interaction benchmark. Veh. Syst. Dyn. 2015, 53, 412–435. [CrossRef] 9. Collina, A.; Bruni, S. Numerical simulation of pantograph-overhead equipment interaction. Veh. Syst. Dyn. 2002, 38, 261–291. [CrossRef] 10. Song, Y.; Liu, Z.; Wang, H.; Lu, X.; Zhang, J. Nonlinear modelling of high-speed catenary based on analytical expressions of cable and truss elements. Veh. Syst. Dyn. 2015, 53, 1455–1479. [CrossRef] 11. Ambrósio, J.; Pombo, J.; Pereira, M.; Antunes, P.; Mósca, A. A computational procedure for the dynamic analysis of the catenary-pantograph interaction in high-speed trains. J. Theor. Appl. Mech. 2012, 50, 681–699. 12. Pombo, J.; Ambrosio, J. Environmental and track perturbations on multiple pantograph interaction with catenaries in high-speed trains. Comput. Struct. 2013, 124, 88–101. [CrossRef] 13. Pombo, J.; Ambrsio, J. Multiple pantograph interaction with catenaries in high-speed trains. J. Comput. Nonlinear Dyn. 2012, 7, 041008. [CrossRef] 14. Seo, J.H.; Kim, S.W.; Jung, I.H.; Park, T.W.; Mok, J.Y.; Kim, Y.G.; Chai, J.B. Dynamic analysis of a pantograph-catenary system using absolute nodal coordinates. Veh. Syst. Dyn. 2006, 44, 615–630. [CrossRef] 15. Song, Y.; Ouyang, H.; Liu, Z.; Mei, G.; Wang, H.; Lu, X. Active control of contact force for high-speed railway pantograph-catenary based on multi-body pantograph model. Mech. Mach. Theory 2017, 115, 35–59. [CrossRef] 16. Song, Y.; Liu, Z.; Xu, Z.; Zhang, J. Developed moving mesh method for high-speed railway pantograph-catenary interaction based on nonlinear finite element procedure. Int. J. Rail Transp. 2019, 7, 173–190. [CrossRef] 17. Seo, J.H.; Sugiyama, H.; Shabana, A.A. Three-dimensional large deformation analysis of the multibody pantograph/catenary systems. Nonlinear Dyn. 2005, 42, 199–215. [CrossRef] 18. Lee, K. Analysis of dynamic contact between overhead wire and pantograph of a high-speed electric train. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2007, 221, 157–166. [CrossRef] 19. Ouyang, H. Moving-load dynamic problems: A tutorial (with a brief overview). Mech. Syst. Signal Process. 2011, 25, 2039–2060. [CrossRef] Vibration 2020, 3 33

20. Lee, U. Separation between the flexible structure and the moving mass sliding on it. J. Sound Vib. 1998, 209, 867–877. [CrossRef] 21. Lee, U. Revisiting the moving mass problem: Onset of separation between the mass and beam. J. Vib. Acoust. Trans. ASME 1996, 118, 516–521. [CrossRef] 22. Stancioiu,ˇ D.; Ouyang, H.; Mottershead, J.E. Vibration of a beam excited by a moving oscillator considering separation and reattachment. J. Sound Vib. 2008, 310, 1128–1140. [CrossRef] 23. Ouyang, H.; Mottershead, J.E. A numerical-analytical combined method for vibration of a beam excited by a moving flexible body. Int. J. Numer. Methods Eng. 2007, 72, 1181–1191. [CrossRef] 24. Dahlberg, T. Moving force on an axially loaded beam—With applications to a railway overhead contact wire. Veh. Syst. Dyn. 2006, 44, 631–644. [CrossRef] 25. Cheng, Y.S.; Au, F.T.K.; Cheung, Y.K.; Zheng, D.Y. On the separation between moving vehicles and bridge. J. Sound Vib. 1999, 222, 781–801. [CrossRef] 26. Lee, K.; Chung, J. Dynamic analysis of a hanger-supported beam with a movingoscillator. J. Sound Vib. 2013, 332, 3177–3189. [CrossRef] 27. Lee, K.; Cho, Y.; Chung, J. Dynamic contact analysis of a tensioned beam with a moving massspring system. J. Sound Vib. 2012, 331, 2520–2531. [CrossRef] 28. Zou, D.; Zhou, N.; Rui Ping, L.; Mei, G.M.; Zhang, W.H. Experimental and simulation study of wave motion upon railway overhead wire systems. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2017, 231, 934–944. [CrossRef] 29. Zhang, W.H.; Mei, G.M.; Wu, X.J.; Chen, L.Q. A study on dynamic behaviour of pantographs by using hybrid simulation method. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2005, 219, 189–199. [CrossRef] 30. Brogliato, B. Nonsmooth Mechanics: Models, Dynamics and Control, 3rd ed.; Springer: London, UK, 2016. 31. Cháidez, S.T.; de la, S.T. Contribution to the Assessment of the Efficiency of Friction Dissipaters for Seismic Protection of Buildings; Universitat Politècnica de Catalunya: Barcelona, Spain, 2003. 32. Zhao, Y.Y.; Wu, G.N.; Gao, G.Q.; Wang, W.G.; Zhou, L.J. Study on electromagnetic force of pantograph-catenary system. Tiedao Xuebao J. China Railw. Soc. 2014, 36, 28–32. 33. EN 50367. Railway applications—Current Collection Systems—Technical Criteria for the Interaction between Pantograph and Overhead Line. 2012. Available online: https://standards.globalspec.com/std/10202019/EN% 2050367 (accessed on 31 May 2012). 34. Song, Y.; Liu, Z.; Ouyang, H.; Wang, H.; Lu, X. Sliding mode control with PD sliding surface for high-speed railway pantograph-catenary contact force under strong stochastic wind field. Shock Vib. 2017, 2017, 4895321. [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).