Transient wave propagation analysis of a pantograph- catenary system

Transcription

1 Journal of Physis: Conferene Series PAPER OPEN ACCESS Transient wave propagation analysis of a pantograph- atenary system To ite this artile: Kyohei Nagao and Arata Masuda 216 J. Phys.: Conf. Ser View the artile online for updates and enhanements. Related ontent - Extension of Sampling Theorem and Its Appliations Tjundewo Lawu and Mitsuhiro Ueda - Appliation of Haar wavelets to timedomain BEM for the transient salar wave equation Kazuhiro Koro and Kazuhisa Abe - A diagnosti method of ion minority onentrations based on IBW propagation C Riardi, P Cantu and M Fontanesi This ontent was downloaded from IP address on 2/1/218 at 1:3

2 Transient wave propagation analysis of a pantographatenary system Kyohei Nagao and Arata Masuda Department of Mehanial and System Engineering, Kyoto Institute of Tehnology, Matsugasaki, Sakyo-ku, Kyoto, Japan masuda@kit.a.jp Abstrat. This paper proposes a systemati method to analyze the dynami response of an overhead atenary with pantographs moving at onstant speed. The overhead atenary is modeled as a onedimensional infinite-length string, whih is periodially supported by hangers. On the other hand, the pantograph is a sub-struture moving at a onstant speed, whih is modeled as a lumped mass system ontating the atenary. In this study, the whole system is divided into elements in the manner of the transfer matrix method. Then, the relationship among traveling waves in every element is systematially obtained in the Laplae domain following the method of reverberation-ray matrix. Sine the governing equation of the system hanges periodially with time, the analysis of the temporal evolution of the system an be realized by repeating a single period analysis starting from the instant when the pantograph omes into a unit ell by means of the reverberation-ray matrix analysis followed by the inverse Laplae transform. When the pantograph reahes the opposite hanger, the whole elements are shifted bakward, and the atenary response of the forehead element is used as the initial ondition of the next period. 1. Introdution An overhead atenary system is a omplex, elongated, almost periodi and infinite struture whih supplies urrent to a train vehile through pantographs. The pantograph whih is a strutural system moving along the atenary experienes omplex dynami interation with the atenary system espeially in high speed range. Sine the stability of the ontat fore between the atenary and the pantograph is a ritial fator for reliability of the urrent olletion and redution of the maintenane ost, fully understanding of the oupled behavior of the atenary-pantograph system is essential for high-speed railway. The atenary-pantograph system has the following harateristis whih may lead to diffiulties in developing an analysis method: it an be modeled as a struture with infinite length; the pantograph itself is a dynamial system interating with the atenary as a moving sub-struture; not only its steady-state response but also the transient response is of interest. A number of studies were onduted in the past on analyses of similar types of oupled systems. Gilbert and Davies [1] studied the steady-state responses of the atenary system modeled as a string supported by ontinuous elasti medium whose stiffness periodially varied along the line. The pantograph was modeled both as zero impedane, i.e., a simple fore, and as a lumped mass system. Sott and Rothman [2] presented a numerial study using a lumped parameter model of atenary and pantographs. Smith Content from this work may be used under the terms of the Creative Commons Attribution 3. liene. Any further distribution of this work must maintain attribution to the author(s and the title of the work, journal itation and DOI. Published under liene by Ltd 1

3 Figure 1. Analysis model of pantograph-atenary system. and Wormley [3] presented an analysis of periodially supported ontinuous beam with moving load. Metrikine el al. [4] analytially derived steady-state solutions of periodially supported struture under a moving load using periodiity ondition. The same approah was applied to a two-level atenary under a moving load [5, 6]. These analyses did not take the dynamial behavior of the pantograph into aount. On the other hand, Bitzenbauer and Dinkel [7] dealt with an infinite beam supported by homogeneous elasti medium oupled with a moving multi-degree-of-freedom struture based on double Fourier transform. This work did not take aount of the spatial periodiity. Shimogo et al. [8] formulated a steady-state response of a periodially suspended atenary with a moving pantograph assuming periodiity ondition. To the best of authors knowledge, however, there is no existing methods presented in the literature whih an deal with all of the three harateristis. In this paper, a Laplae transform-based semi-analytial approah to this hallenge is presented. The atenary is modeled as an infinite-length string periodially supported by hangers whih are modeled by lumped masses, springs and dampers. The pantograph is modeled as a lumped mass system moving at a onstant speed ontating the atenary. The whole system is divided into elements on the basis of transfer matrix method [9, 1]. Then, the relationship among traveling waves in every element is systematially obtained in the Laplae domain in terms of a method of reverberation-ray matrix [11, 12, 13]. The method of reverberation-ray matrix is a numerial method originally developed to analyze transient elasti waves in a planar truss [11] in whih a lot of members are inter-onneted to form a network-like struture. It is based on traking of wave propagation in the struture using the exat solutions of traveling waves, therefore it is a numerial sheme free from disretization errors and intrinsially stabile. Sine the governing equation of the system hanges periodially with time, the analysis of the temporal evolution of the system an be realized by repeating a single period analysis starting from the instant when the pantograph omes into a unit ell by means of the reverberation-ray matrix analysis followed by the inverse Laplae transform. Sine all the formulations are onstruted based on wave omponent representation, wave propagation phenomena suh as refletion, transmission at disontinuities and Doppler effet at the pantograph an be effetively taken into aount. 2. Modeling 2.1. Governing equations Typial atenary system onsists of a ontat wire from whih the pantograph reeives eletriity, a arrying wire whih suspends the ontat wire, and periodially plaed hangers whih vertially onnet two wires. In this study, the arrying wire is omitted from the model for the simpliity, and the flexibility of the arrying wire is represented by the flexibility of the hangers. Furthermore, only single pantograph is onsidered. It should be noted here that it would be relatively easy to implement a full pantograph-atenary model involving the arrying wire and multiple pantographs beause the methodology developed in this study is systemati and independent of any speifi model strutures. 2

4 The analysis model in this study is illustrated in figure 1. It onsists of a ontat wire supported by hangers and a moving pantograph. The ontat wire is modeled as an infinite-length string whih is governed by a wave equation given by ρ 2 y(x, t y(x, t t 2 T x 2 = (1 where y(x, t is the vertial displaement of the wire at horizontal loation x and time t; ρ and T are the linear density and tension of the ontat wire, respetively. The internal fore of the wire is defined as y(x, t V(x, t = T x The hanger are modeled by uniform single-degree-of-freedom lumped mass systems whose equation of motion is given by d 2 y h (t dy h (t m h dt 2 + h + k h y h (t = V hl (t V hr (t (3 dt where y h (t is the vertial displaement of the onneting point of the hanger with the ontat wire; m h, h, and k h denote the effetive mass, the effetive damping oeffiient, and the effetive stiffness of the hangers; V hl (t and V hr (t denote the fores the hanger reeives from the left and right neighbor wires at the onneting point, respetively. The pantograph is also modeled as a single-degree-of-freedom lumped mass system whose equation of motion is given by d 2 y p (t dy p (t m p dt 2 + p = F + V pl (t V pr (t (4 dt where y p (t is the vertial displaement of the ontat point of the pantograph with the ontat wire; m p, p, and F are the effetive mass, the effetive damping oeffiient, and the vertial thrust fore of the pantograph; V pl (t and V pr (t denote the fores the pantograph reeives from the left and right neighbor wires at the ontat point, respetively Transfer matrix and its wave omponent representation Let us onsider a part of the wire of length l. Taking the Laplae transform of equation (1 gives 2 ŷ(x, s x 2 (2 s2 ŷ(x, s = g(x, s (5 2 g(x, s = 1 {sy(x, + ẏ(x, } (6 2 where ( ˆ denotes the variable in the Laplae domain, ( denotes the temporal derivative, and = T/ρ. Solving the seond-order ordinary differential equation (5, the general solution at arbitrary loation is derived as ŷ(x, s = 1 (se s x + 2 (se s x + Y(x, s (7 where Y(x, s is the partiular solution of equation (5 given by Y(x, s = h 1 (x, se s x + h 2 (x, se s x (8 where h 1 (x, s = 2s h 2 (x, s = 2s x x e s x g(x, sdx (9 e s x g(x, sdx (1 3

5 From equations (7 and (2, the state variables at x = l are derived as a linear funtion of the state variables at x = as [ ] ŷ(l, s osh ( s = ˆV(l, s l T s sinh ( s l [ ] [ ] T s sinh ( s l osh ( ŷ(, s Y(l, s s l + ˆV(, s T Y(l,s (11 x where ŷ p (x, s is the partiular solution of equation (5, and osh ( s T w = l T s sinh ( s l T s sinh ( s osh ( s l l (12 is the transfer matrix. Diagonalization of the transfer matrix leads to the wave omponent representation as follows: [ ] [ ] [ ] ŵ f (l, s ŵ = Λ f (, s + Φ ŵ b (l, s ŵ b (, s 1 Y(l, s T Y(l,s (13 x where ŵ f (x, s and ŵ b (x, s are the forward and bakward propagating wave omponents, respetively. The matries appearing in the right-hand-side are defined as [ e s Λ = l ] e s l (14 and [ 1 1 Φ = T s T s ] (15 3. Method 3.1. Loal oordinate systems and wave omponents The ontat wire is divided into elements span-by-span. In order to develop an analysis method based on the method of reverberation-ray matrix [11, 12], the following naming rules of joints and elements, and the definition of loal oordinate systems are adopted. The onneting points of the hangers and the ontat point of the pantograph are onsidered as joints denoted by letters, 1,..., I, J, K,..., N 1, N as depited in figure 1. The element bounded by joints I and J is named element IJ. Beause the joint J is the ontat point moving in the positive diretion, the element I J and the element JK are time-varying elements with dereasing and inreasing length, respetively. Then, two sets of dual loal oordinate systems are introdued for eah element IJ, with the one loated at joint I (labeled by supersript IJ and the other at joint J (labeled by supersript JI as illustrated in figure 2. Figure 2. Loal oordinate systems. The wave omponents traveling on the wire are lassified into two groups: departing waves and arriving wave. For eah joint J, the wave omponents departing from J to K and arriving at J from K are designated as d JK and a JK, respetively, as illustrated in figure 3, where all the omponents related to the elements I J and JK are shematially denoted. 4

6 Figure 3. Wave omponents Phase matrix For eah element IJ, the wave omponents departing from the joint I will arrive at the joint J, and vie versa. These relationships among the wave omponents traveling through a ertain element are desribed by a phase matrix Time-invariant element Let us onsider an element HI with time-invariant length. Rearranging equation (13 and onsidering the loal oordinates, the arriving waves are related to the departing waves and initial onditions as a HI (s = P HI (s d IH (s + q HI (s (16 where and a IH (s = P IH (s d HI (s + q IH (s (17 P HI (s = P IH (s = e s l (18 q HI (s = 2s q IH (s = e s l 2s l l e s x g HI (x, sdx (19 e s x g HI (x, sdx ( Time-varying element Let us onsider the pantograph s ontat point J moving with a onstant veloity of v and adjaent joints I and K as illustrated in figure 4. Beause the elements IJ and JK have time-varying lengths, speial treatment is required in handling these elements. For the strething element IJ, the wave field is first desribed in the stationary oordinate IJ as ŷ IJ (x IJ, s = d IJ (se s xij + a IJ (se s xij (21 Figure 4. Time-varying elements. 5

7 whih is obtained from equation (7. Applying a oordinate transformation x IJ expression of the same field on the moving oordinate JI in the time domain as = vt x JI yields an y JI (x JI, t = 1 2πi = 1 2πi σ+i σ i σ+i σ i ŷ IJ (vt x JI, s e s t ds (22 } {d IJ (s e s (vt xji + a IJ (s e s (vt xji e s t ds (23 Taking the Laplae transform again results in ŷ JI (x JI, s = ( + v aij + v s e +v s xji ( v dij v s e v s xji (24 From equation (24, the arriving waves are related to the departing waves as ( + v a IJ (s = P IJ (s d JI s ( a JI (s = P JI (s d IJ v s where P IJ (s = + v P JI (s = v Similarly, for the shrinking element JK, the arriving waves are related to the departing waves and initial onditions as ( a JK (s = P JK (s d KJ + v s + q JK (s (29 ( v a KJ (s = P KJ (s d JK s + q KJ (s (3 where and q JK (s = e s +v l 2s q KJ (s = v 2s (25 (26 (27 (28 P JK (s = + v e +v s l (31 P KJ (s = v e s l (32 l l e +v s x g (x KJ, + v s dx (33 e s x g KJ (x, sdx (34 Equations (25, (26, (29 and (3 imply that arriving waves at a ertain frequeny are related to the departing waves at another frequeny. This orresponds to the Doppler effet that arises at the ontat point of the moving pantograph. Due to this effet, the relationships among wave omponents in the time-varying elements are not frequeny-independent in ontrast to the ordinary transfer matrix method and the method of reverberation-ray matrix. Hene, one have to onsider augmented vetors a IJ and d JI of arriving and departing wave omponents at disretized Laplae points sampled to alulate the 6

8 Bromwih integral for numerial inverse Laplae transform [14, 15]. Thus, equations (25 and (26 an be expressed as a IJ = P JI d IJ (35 and equations (29 and (3 as a JK = P JK d KJ + q JK (36 Then, all equations an be assembled into the global phase matrix equation as a = P d + q ( Sattering matrix The relationships among the arriving and departing wave at a ertain joint are desribed by the sattering matrix. For the joint I orresponding to the onneting point of the hanger, the sattering matrix is derived from the transfer matrix representation aross the onneting point, whih is given by where [ T h = z IH = T h z IJ + f h (38 1 ] m h s 2 + h s + k h 1 [ ] f h = fˆ h (s (39 (4 ˆ f h (s = (m h s 2 + h sy h ( + m h ẏ h ( (41 whih is derived via the Laplae transform of the equation of motion (3. Taking the oordinate transformation using the eigenvetor matrix Φ given by equation (15 leads to an equivalent representation written by the wave omponents as w IH = Φ 1 T h Φw IJ + Φ 1 f h (42 Then, rearranging equation (42 and gathering all the Laplae points in the same way as the previous subsetion, one will have the relationship between arriving and departing wave omponents in the form of d I = S I a I + f I (43 where S I is alled loal sattering matrix. For the joint J orresponding to the ontat point of the pantograph, the sattering equation is derived from the transfer matrix representation aross the ontat point, whih is given by z JI = T p z JK + f p (44 where [ 1 T p = m p s 2 + p s 1 [ ] f p = fˆ p (s ] (45 (46 ˆ f p (s = (m p s 2 + p sy p ( + m p ẏ p ( F s (47 7

9 whih is derived via the Laplae transform of the equation of motion (4. Taking the same way as desribed above, one will have d J = S J a J + f J (48 whih is the loal sattering equation at the ontat point of the pantograph. Finally, assembling all the loal sattering equations, the global sattering matrix equation is derived as d = Sa + f (49 Note that the vetors d in the above equation and d in equation (37 have the same elements but sequened in different orders. The relationship between them an be formulated as where U is the permutation matrix Reverberation-ray matrix Combining equations (37, (49 and (5 results in d = Ud (5 d = (I R 1 (Sq + F (51 where R = SPU is alled reverberation-ray matrix [11]. It would be hard to alulate the right-hand-side of equation (51 beause the size of the reverberationray matrix R an be huge and the inversion an be ill-posed. Instead, the Neumann series expansion of (I R 1 an be applied as suggested by Howard and Pao [11, 12] as follows (I R 1 = I + R + R R N +... (52 After getting all of the departing waves by alulating equation (51, the arriving waves are derived as a = PUd + q (53 It should be noted that multiplying R by the wave omponents means to satter the waves one more time. Thus, trunation of the expansion at N + 1-th term implies to take aount of wave omponents whih are sattered up to N times Infinite boundary onditions, spatial shift and ontinuation In this study, the pantograph is assumed to run along the ontat wire with infinite length. Of ourse only finite number of wave omponents an be dealt with, the model has to be trunated with finite number of spans. Assuming that the pantograph P is running between the hangers I and J, let us think of a region of interest bounded by hangers I M and J + M. The total span number in the region is 2M + 1. By taking suffiiently large M, it would be reasonable to assume that the waves inoming from outside of the region of interest are negligibly small. Therefore, at both boundaries of the region, this assumption an be used as the boundary onditions. As long as the pantograph remains between the hangers I and J, the formulation desribed in the previous subsetions is valid and the responses of the atenary and the pantograph an be alulated. When the pantograph reahes the hanger J and omes into the next span, however, the formulation an no longer be applied beause the strutural adjaent relation of the pantograph with the hangers hanges. Thus, it is neessary to alter the adjaent order from I, P, J, K to I, J, P, K. In order to avoid the strutural alteration, the analysis is interrupted at t = l/v when the pantograph reahes the hanger J. The displaement and veloity of the wire in the region of interest at this moment are derived by inverse Laplae transform. These are shifted bakward by one span length, then used as the initial onditions for the next time interval. The initial onditions for the span at the positive end of the region is set to zero. This proess is repeated as illustrated in figure 5, going bak and fore between temporal and Laplae domain. 8

10 Figure 5. Repetitive spatial shift and ontinuation. 4. Examples 4.1. Steady-state response of zero-impedane pantograph In this ase, the mass and the damping oeffiient of the pantograph are set suffiiently small so that the alulation results are ompared with the previous study by Metrikine et al. [4]. The parameter values are listed in Table 1. The analysis was started with zero initial onditions and ontinued until the response onverges to steady-state. Table 1. Parameter values. Parameter Value Parameter Value m p 1 6 kg m h 1 kg p 1 6 N s/m h 2 N s/m F 1 N k h 2 N/m 1 m/s T 1 N l 5 m M 2 spans The omparison is shown in figure 6 for various speed ratio v/. In the figure, the displaement of the ontat wire in the steady-state is plotted. The results from the present study are plotted in blue, while the results using the formulation proposed by Metrikine et al. [4] are in red. Metrikine el al. [4] pointed out two findings: When the load veloity was relatively small (up to v/ =.3, the displaement pattern was almost symmetri with respet to the loading point (pantograph. With inreasing veloity the pattern beame more and more asymmetri and for v/ =.9 the displaement before the load beame negligibly small. Maximum displaement of the wire grew, as the load veloity beame higher. 9

11 (a v/ =.1. (b v/ =.3. ( v/ =.6. (d v/ =.9. Figure 6. Displaement of the atenary over 1 span distane from the pantograph. Present study in blue line; results using the formulation proposed by Metrikine et al. [4] in red line. From figure 6, the present study agrees with these findings apart from the ase v/ =.9, where the displaement before the pantograph is disturbed possibly beause of numerial instability of the inverse Laplae transform Transient response of pantograph with impedane In this ase, the parameters of the pantograph are set more realistially as listed in Table 2. Sine the pantograph with impedane an interat with the waves propagating on the wire, the multiple sattering and the Doppler effet are fully onsidered in this analysis. The analysis was started with zero initial onditions and ontinued until the response onverges to steady-state. Table 2. Parameter values. Parameter Value Parameter Value m p 6 kg m h 1 kg p 35 N s/m h 2 N s/m F 73.5 N k h 2 N/m 1 m/s T 1 N l 5 m M 2 spans The transient displaement responses for v/ =.1 and v/ =.3 are shown on spatial-time plane in figure 7. Beause the pantograph applies onstant upthrust to the atenary, the wire undergoes step input 1

12 at t =. Then, repeating pass-by of the hanger at every 5 m makes the pantograph respond periodially, with wave motion spreading along the wire. Again, the maximum displaement in the steady-state beomes larger as the veloity inreases, but the variation of the displaement beomes smaller. These findings have to be further investigated through more omprehensive parameter study in future. (a v/ =.1. (b v/ =.3. Figure 7. Spatial-time transient response of ontat wire. 5. Conlusions In this paper, a systemati analysis method for the dynami response of an overhead atenary and a moving pantograph ontating the atenary. The atenary is modeled as a one-dimensional infinitelength string periodially suspended by hangers. The pantograph is modeled as a single-degree-offreedom struture moving at a onstant speed. The whole system is divided into elements, and the relationships among the traveling waves are systematially obtained in the Laplae domain following the method of reverberation-ray matrix. Sine the governing equation of the system hanges periodially with time, the analysis of the temporal evolution of the system an be realized by repeating a single period analysis inheriting the initial onditions with spatial shifting. Although the qualitative harateristis of the steady-state solutions agreed with those in the previous study, some numerial instability probably 11

13 ause by the inverse Laplae transform was observed. The reason for this instability must be fully understood in the future work in order to improve the reliability of the proposed method. Referenes [1] Gilbert G and Davies H E H 1966 Pro. IEE [2] Sott P R and Rothman M 1974 IEEE Trans. Ind. Appl [3] Smith C C and Wormley D N 1975 J. Dyn. Syst. Meas. Control [4] Metrikine A V, Wolfert A F M and Vrouwenvelder A C W M 1999 HERON [5] Metrikine A V and Bosh A L 26 J Sound Vib [6] Kumanieka A and Smanina J 28 J. Theor. Appl. Meh [7] Bitzenbauer J and Dinkel J 22 Arh. Appl. Meh., [8] Shimogo T, Yoshida K and Abe N 1984 Trans. JSME C [9] Holzer H 1921 Die Berehnung der Drehsenwingungen (Berlin: Springer [1] Faulkner M G and Hong D P 1985 J. Sound Vib [11] Howard S M and Pao Y H 1998 J. Eng. Meh [12] Pao Y H, Chen W Q and Su X Y 27 Wave Motion [13] Jiang J Q, Chen W Q and Pao Y H 211 J. Vib. Control [14] Levin D 1975 J. Comp. Appl. Math [15] Hwang C, Lu M J and Shieh L S 1991 Comput. Math. Appl