# Effect of Pantograph’s Main Structure on the Contact Quality in High-Speed Railway

Authors: Jiangwen Wang; Guiming Mei
Journal: Shock and Vibration
 (2021)

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## Abstract

In general, the electrical energy is provided to the high-speed train through the pantograph-catenary sliding contact. The variation of the dynamic contact force is expected to be small enough to keep the good current collection quality and to extend the service life of the pantograph-catenary system. In this paper, the two tension wires of the catenary are discretized by the sine-series expansions, a multibody dynamics theory based on relative coordinates is adopted to describe the dynamic behavior of the pantograph, and the standard deviation (STD) of dynamic contact force is used as the indicator to evaluate the contact quality. The objective is to investigate how the variations of the pantograph’s main structure influence the contact quality, which may support the structural design and parameter optimization of the pantograph in high-speed railway.


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## Body

## 1. Introduction

The pantograph-catenary system (PCS) is the key subsystem of high-speed railway, shown in Figure1, which transmits the electrical energy from the traction substations to the moving high-speed trains. The catenary is arranged above the track centerline, and the pantograph is mounted on the roof of the train, forming the special sliding-contact-current-collection system in high-speed railway. The contact quality must be strictly required, which means that the dynamic contact force should not be too large or too small to avoid the impact or arc damage [1].Figure 1
Pantograph-catenary systems.To evaluate the contact quality, lots of methods have been developed to describe the interaction of the PCS [2–18]. Besides, the aerodynamic effect [19–23], the vibration caused by track or contact wire irregularity [24–26], and the technology of active control pantograph [27–33] have been intensively studied to illustrate the PCS.In the research field of parameters optimization of the PCS, Park et al. [33] gave a pantograph with improved parameters based on the linear spring-mass model suiting for the high-speed rail vehicle. Kim et al. [34] investigated the sensitivity of the PCS, confirming that the span length and the plunger spring constant are the two important design variables of the PCS. Zhou and Zhang [35] pointed out that the stiffness and damping of the pan-head and frame, the static lifted force, and the tension force of the contact wire have a significant influence on the dynamic performance of the PCS. Ambrósio et al. [36] redesigned the suspension characteristics of the pantograph head to improve the contact quality. Pombo and Ambrósio [37] studied the influence of the head mass, the head suspension stiffness, and the damper coefficient, indicating that the increase in the pantograph head suspension stiffness and lower damper will improve the stability of the contact interaction and decrease the potential for contact loss. Lee et al. [38] applied the response surface analysis method and the differential evolutionary algorithm to improve the current collection performance. Massat et al.[39] compared the differences between current mechanical suspension and pneumatic device with associated nonlinear stiffness of pantograph, pointing out that the impact of pneumatic suspension will decrease compared with the case of the lumped mass model.However, most research works mentioned above adopt the lumped mass model to handle the parameters optimization problems of the pantograph, shown in Figure2, and the meanings of the physical quantity are shown in reference [34], which did not concern the geometric of the pantograph’s main frame; thus, it is not clear enough how the variation of the pantograph’s main frame influences the dynamic contact quality.Figure 2
Typical linear spring-mass model of the pantograph.The objective is to evaluate the variation of the pantograph’s main frame to contact quality, focusing on the geometric parameters of the main frame. The multibody model of the pantograph based on relative coordinates theory [40–44] is developed to describe the variation of main frame. The dynamic vertical displacements of the two tension wires in the catenary are described by the Fourier sine-series expansions, respectively, and all general load vectors are acquired and solved in each time step directly. The paper is organized as follows:(a)
In Section2, the multibody dynamic model of the pantograph is established and the dynamic equations of the catenary are built(b)
In Section3, the initialization strategy of the catenary is given in detail and validated compared with the reference(c)
In Section4, the variation of the pantograph’s main frame and the research conditions are described carefully(d)
In Section5, the simulation results are given and the main conclusions are summed about how the main structure affects the contact quality


## 2. Methodology

### 2.1. Modeling of the Pantograph

The single-arm type pantograph object is simplified, keeping the main load-carrying components stay, including the base frame, the link, the lower frame, the upper frame, the supporting rack, the front contact strip, and the rear contact strip, shown in Figure3.Figure 3
Main load-carrying components of the single-arm type pantograph.Based on the relative coordinates theory in multirigid body dynamics [40–44], the relationship between the main components is described in Figure 4, and the meanings of the symbols are shown in Table 1.Figure 4
Relationship of the main frame based on relative coordinates theory.Table 1
The physical meaning of the symbols in Figure4.
SymbolsPhysical meaningO1, x1, y1Global coordinate systemO1, O2, O3, O4, O5Hinge pointse11,e21,e31Local coordinate fixed on linke12,e22,e32Local coordinate fixed on upper framee13,e23,e33Local coordinate fixed on supporting racke14,e24,e34Local coordinate fixed on lower frameL1, L2, L3, L4LengthC1, C2, C3, C4Center of massm1, m2, m3, m4MassJ1, J2, J3, J4Moment of inertia around the center of massθ10, θ20, θ30, θ40,Initial angleθ1, θ2, θ3, θ4Relative motion angleL5The length betweenO2 and O5TRaise torqueQL, QRLeft and right forces action on supporting rackgGravity accelerationThe relationship of the pantograph’s main structure based on the graph theory is shown in Figure5, and O5 is the cut-off hinge through which the nontree shape mechanical system is transferred to a tree shape system. The relation matrix of the main frame is(1)cij=c11c12c22c23c33c44.Figure 5
Description of pantograph’s main structure based on the graph theory.The dynamic equation of the pantograph’s main structure according to the theory by Roberson and Wittenburg [41–44] is(2)Aq¨=B,in which the expressions of A and B are(3)A=αT⋅mα+βT⋅mβ,B=αT⋅Fg−mu+βT⋅Mg−J⋅σ−ε+p⋅Ma,where A is the generalized mass matrix of the frame structure, B is the generalized load vector, q is the relative coordinates of the frame structure, αandβ are the transfer matrix, m and J are the mass matrix and inertia matrix of the system, Fg is the external force vector, Mg is the external moment vector to the mass center of the system, Ma is the external moment vector of the internal rigid body of each hinge point to the external rigid body, u,σ, andε are the transfer vectors, and p is the coordinate matrix of the revolutions.To describe the constraint equation, the related parameters and structure are shown in Figure6.Figure 6
Simplified graph from Figure4 to describe the constraint equation.La and Lb are the distance between O1 and O4 in x1 and y1 direction separately. The constraint equation of the main frame is(4)Φq,t=0.Then, the dynamic model of the pantograph’s main structure is established combining the Lagrange multiplierλ item.(5)Aq¨=B+ΦTλ.For modeling of pantograph head, each collector strip is simplified as a single spring-mass model, shown in Figure7.Figure 7
Left and right collector strip of the pantograph.The differential equations of collector strips are(6)mLy¨LH=bL,mRy¨RH=bR,bL=kLyLS−yLH+cLy˙LS−y˙LD−FL,bR=kRyRS−yRH+cRy˙RS−y˙RD−FR.Finally, the dynamic equation of the pantograph is assembled as follows:(7)AmLmRq¨y¨LHy¨RH=B+ΦqTλbLbR.


### 2.2. Modeling of Catenary

#### 2.2.1. Basic Matrix

The simple catenary is mainly composed of the contact wire, the messenger wire, the droppers, and the support devices. The supports of the messenger wire and the steady arms are equivalent to the springs connected to the fixed points, shown in Figure8. To describe the nonlinear behaviors of the droppers while compressing, the droppers are replaced by the spring with only tension stiffness, whose mass is added on the connecting points of each wire.Figure 8
Model of catenary.The dynamic displacement of each wire can be described by Fourier sine-series expansions [45].(8)yMx,t=∑m=1m0AmtsinmπxL,yCx,t=∑n=1n0AntsinnπxL,where yM and yC represent the dynamic displacement of the messenger wire and the contact wire separately, m0 and n0 are the maximum expansion orders, L is the total length of the catenary, Am and An are the amplitudes of the mth sine term and the nth sine term, x is the distance along the catenary, and t is the time.To establish the equation of the catenary, the second Lagrange equation is adopted:(9)ddt∂LM∂A˙m−∂LM∂Am=QAm,(10)ddt∂LC∂A˙n−∂LC∂An=QAn,(11)LM=TM−VM,(12)LC=TC−VC.TM and TN are the kinetic energy of each wire, VM and VN are the potential energy, and QAm and QAn are the mth and nth general load of each wire, where(13)TM=12∫0LρMy˙2Mdx+12∑i=1nDmMiy˙2Mxi,t,(14)VM=12FMW∫0L∂yM∂x2dx+12EMIM∫0L∂2yM∂x22dx,(15)TC=12∫0LρCy˙2Cdx+12∑i=1nDmCiy˙2Cxi,(16)VC=12FCW∫0L∂yC∂x2dx+12ECIC∫0L∂2yC∂x22dx.ρM and ρC are the line density of each wire, mMi and mCi are the mass of ith dropper added on each wire, nD is the dropper number, FMW and FCW are the tension forces of each wire, EM and EC are the elastic modulus, and IM and IC are the moments of inertia of each wire.Combining equations (14)–(16), the catenary system in the coordinate spaces {Am} and {An} is written as follows:(17)MMMCA¨MA¨C+CMCCA˙MA˙C+KMKCAMAC=QAMQAC,MM, MC, KM, and KC are the mass matrix and stiffness matrix of each wire which are expressed by(18)MM=MM1+MM2,(19)MM1i,k=∑j=1nDmMDjsiniπxDjLsinkπxDjL+∑j=1nSmMSjsiniπxSjLsinkπxSjL,(20)MM2i,i=ρML2,(21)KMi,i=FMWπ2i22L+EMIMπ4i42L3,(22)MC=MC1+MC2,(23)MC1i,k=∑j=1nDmMCjsiniπxDjLsinkπxDjL+∑j=1nSmCSjsiniπxSjLsinkπxSjL,(24)MC2i,i=ρCL2,(25)KCi,i=FCWπ2i22L+ECICπ4i42L3.AM is composed of Am, and AC is composed of An. The Rayleigh damping is also applied in coordinate spaces {Am} and {An}, and the damping matrices CM and CC are(26)CM=αMM+βKM,CC=αMC+βKC.


#### 2.2.2. Load Vector

QAM is composed of QAm, and QAC is composed of QAn, which is the generalized load vector in coordinate spaces {Am} and {An}. The concentrated forces Fi (NF is the number of Fi) and the distributed forces f¯x can be transferred by(27)Qk1=∑i=1NFFi∂yi∂qk1;Qk2=∂∫0Lf¯xyx,tdx∂qk2.For the messenger wire, the distributed force is−ρMg, and the concentrated forces consist of the dropper forces −FDk and the supporting forces of messenger wire FSMk. The general load expression {QAm} can be transferred by(28)QAm=∂∫0L−ρMgyMx,tdx∂Am+∑k=1nD∂−FDkyMxDk,t∂Am+∑k=1nS∂FSMkyMxSk,t∂Am.For the contact wire, the distributed force is−ρCg, and the concentrated forces consist of the dropper forces FDk, the supporting forces of messenger wire FSCk, and the dynamic contact force FC. The general load expression {QAn} can be transferred by(29)QAn=∂∫0L−ρCgyCx,tdx∂An+∑k=1nD∂FDkyCxDk,t∂An+∑k=1nS∂FSCkyCxSk,t∂An+∂FCyCxC,t∂An.


## 2.1. Modeling of the Pantograph

The single-arm type pantograph object is simplified, keeping the main load-carrying components stay, including the base frame, the link, the lower frame, the upper frame, the supporting rack, the front contact strip, and the rear contact strip, shown in Figure3.Figure 3
Main load-carrying components of the single-arm type pantograph.Based on the relative coordinates theory in multirigid body dynamics [40–44], the relationship between the main components is described in Figure 4, and the meanings of the symbols are shown in Table 1.Figure 4
Relationship of the main frame based on relative coordinates theory.Table 1
The physical meaning of the symbols in Figure4.
SymbolsPhysical meaningO1, x1, y1Global coordinate systemO1, O2, O3, O4, O5Hinge pointse11,e21,e31Local coordinate fixed on linke12,e22,e32Local coordinate fixed on upper framee13,e23,e33Local coordinate fixed on supporting racke14,e24,e34Local coordinate fixed on lower frameL1, L2, L3, L4LengthC1, C2, C3, C4Center of massm1, m2, m3, m4MassJ1, J2, J3, J4Moment of inertia around the center of massθ10, θ20, θ30, θ40,Initial angleθ1, θ2, θ3, θ4Relative motion angleL5The length betweenO2 and O5TRaise torqueQL, QRLeft and right forces action on supporting rackgGravity accelerationThe relationship of the pantograph’s main structure based on the graph theory is shown in Figure5, and O5 is the cut-off hinge through which the nontree shape mechanical system is transferred to a tree shape system. The relation matrix of the main frame is(1)cij=c11c12c22c23c33c44.Figure 5
Description of pantograph’s main structure based on the graph theory.The dynamic equation of the pantograph’s main structure according to the theory by Roberson and Wittenburg [41–44] is(2)Aq¨=B,in which the expressions of A and B are(3)A=αT⋅mα+βT⋅mβ,B=αT⋅Fg−mu+βT⋅Mg−J⋅σ−ε+p⋅Ma,where A is the generalized mass matrix of the frame structure, B is the generalized load vector, q is the relative coordinates of the frame structure, αandβ are the transfer matrix, m and J are the mass matrix and inertia matrix of the system, Fg is the external force vector, Mg is the external moment vector to the mass center of the system, Ma is the external moment vector of the internal rigid body of each hinge point to the external rigid body, u,σ, andε are the transfer vectors, and p is the coordinate matrix of the revolutions.To describe the constraint equation, the related parameters and structure are shown in Figure6.Figure 6
Simplified graph from Figure4 to describe the constraint equation.La and Lb are the distance between O1 and O4 in x1 and y1 direction separately. The constraint equation of the main frame is(4)Φq,t=0.Then, the dynamic model of the pantograph’s main structure is established combining the Lagrange multiplierλ item.(5)Aq¨=B+ΦTλ.For modeling of pantograph head, each collector strip is simplified as a single spring-mass model, shown in Figure7.Figure 7
Left and right collector strip of the pantograph.The differential equations of collector strips are(6)mLy¨LH=bL,mRy¨RH=bR,bL=kLyLS−yLH+cLy˙LS−y˙LD−FL,bR=kRyRS−yRH+cRy˙RS−y˙RD−FR.Finally, the dynamic equation of the pantograph is assembled as follows:(7)AmLmRq¨y¨LHy¨RH=B+ΦqTλbLbR.


## 2.2. Modeling of Catenary

### 2.2.1. Basic Matrix

The simple catenary is mainly composed of the contact wire, the messenger wire, the droppers, and the support devices. The supports of the messenger wire and the steady arms are equivalent to the springs connected to the fixed points, shown in Figure8. To describe the nonlinear behaviors of the droppers while compressing, the droppers are replaced by the spring with only tension stiffness, whose mass is added on the connecting points of each wire.Figure 8
Model of catenary.The dynamic displacement of each wire can be described by Fourier sine-series expansions [45].(8)yMx,t=∑m=1m0AmtsinmπxL,yCx,t=∑n=1n0AntsinnπxL,where yM and yC represent the dynamic displacement of the messenger wire and the contact wire separately, m0 and n0 are the maximum expansion orders, L is the total length of the catenary, Am and An are the amplitudes of the mth sine term and the nth sine term, x is the distance along the catenary, and t is the time.To establish the equation of the catenary, the second Lagrange equation is adopted:(9)ddt∂LM∂A˙m−∂LM∂Am=QAm,(10)ddt∂LC∂A˙n−∂LC∂An=QAn,(11)LM=TM−VM,(12)LC=TC−VC.TM and TN are the kinetic energy of each wire, VM and VN are the potential energy, and QAm and QAn are the mth and nth general load of each wire, where(13)TM=12∫0LρMy˙2Mdx+12∑i=1nDmMiy˙2Mxi,t,(14)VM=12FMW∫0L∂yM∂x2dx+12EMIM∫0L∂2yM∂x22dx,(15)TC=12∫0LρCy˙2Cdx+12∑i=1nDmCiy˙2Cxi,(16)VC=12FCW∫0L∂yC∂x2dx+12ECIC∫0L∂2yC∂x22dx.ρM and ρC are the line density of each wire, mMi and mCi are the mass of ith dropper added on each wire, nD is the dropper number, FMW and FCW are the tension forces of each wire, EM and EC are the elastic modulus, and IM and IC are the moments of inertia of each wire.Combining equations (14)–(16), the catenary system in the coordinate spaces {Am} and {An} is written as follows:(17)MMMCA¨MA¨C+CMCCA˙MA˙C+KMKCAMAC=QAMQAC,MM, MC, KM, and KC are the mass matrix and stiffness matrix of each wire which are expressed by(18)MM=MM1+MM2,(19)MM1i,k=∑j=1nDmMDjsiniπxDjLsinkπxDjL+∑j=1nSmMSjsiniπxSjLsinkπxSjL,(20)MM2i,i=ρML2,(21)KMi,i=FMWπ2i22L+EMIMπ4i42L3,(22)MC=MC1+MC2,(23)MC1i,k=∑j=1nDmMCjsiniπxDjLsinkπxDjL+∑j=1nSmCSjsiniπxSjLsinkπxSjL,(24)MC2i,i=ρCL2,(25)KCi,i=FCWπ2i22L+ECICπ4i42L3.AM is composed of Am, and AC is composed of An. The Rayleigh damping is also applied in coordinate spaces {Am} and {An}, and the damping matrices CM and CC are(26)CM=αMM+βKM,CC=αMC+βKC.


### 2.2.2. Load Vector

QAM is composed of QAm, and QAC is composed of QAn, which is the generalized load vector in coordinate spaces {Am} and {An}. The concentrated forces Fi (NF is the number of Fi) and the distributed forces f¯x can be transferred by(27)Qk1=∑i=1NFFi∂yi∂qk1;Qk2=∂∫0Lf¯xyx,tdx∂qk2.For the messenger wire, the distributed force is−ρMg, and the concentrated forces consist of the dropper forces −FDk and the supporting forces of messenger wire FSMk. The general load expression {QAm} can be transferred by(28)QAm=∂∫0L−ρMgyMx,tdx∂Am+∑k=1nD∂−FDkyMxDk,t∂Am+∑k=1nS∂FSMkyMxSk,t∂Am.For the contact wire, the distributed force is−ρCg, and the concentrated forces consist of the dropper forces FDk, the supporting forces of messenger wire FSCk, and the dynamic contact force FC. The general load expression {QAn} can be transferred by(29)QAn=∂∫0L−ρCgyCx,tdx∂An+∑k=1nD∂FDkyCxDk,t∂An+∑k=1nS∂FSCkyCxSk,t∂An+∂FCyCxC,t∂An.


## 2.2.1. Basic Matrix

The simple catenary is mainly composed of the contact wire, the messenger wire, the droppers, and the support devices. The supports of the messenger wire and the steady arms are equivalent to the springs connected to the fixed points, shown in Figure8. To describe the nonlinear behaviors of the droppers while compressing, the droppers are replaced by the spring with only tension stiffness, whose mass is added on the connecting points of each wire.Figure 8
Model of catenary.The dynamic displacement of each wire can be described by Fourier sine-series expansions [45].(8)yMx,t=∑m=1m0AmtsinmπxL,yCx,t=∑n=1n0AntsinnπxL,where yM and yC represent the dynamic displacement of the messenger wire and the contact wire separately, m0 and n0 are the maximum expansion orders, L is the total length of the catenary, Am and An are the amplitudes of the mth sine term and the nth sine term, x is the distance along the catenary, and t is the time.To establish the equation of the catenary, the second Lagrange equation is adopted:(9)ddt∂LM∂A˙m−∂LM∂Am=QAm,(10)ddt∂LC∂A˙n−∂LC∂An=QAn,(11)LM=TM−VM,(12)LC=TC−VC.TM and TN are the kinetic energy of each wire, VM and VN are the potential energy, and QAm and QAn are the mth and nth general load of each wire, where(13)TM=12∫0LρMy˙2Mdx+12∑i=1nDmMiy˙2Mxi,t,(14)VM=12FMW∫0L∂yM∂x2dx+12EMIM∫0L∂2yM∂x22dx,(15)TC=12∫0LρCy˙2Cdx+12∑i=1nDmCiy˙2Cxi,(16)VC=12FCW∫0L∂yC∂x2dx+12ECIC∫0L∂2yC∂x22dx.ρM and ρC are the line density of each wire, mMi and mCi are the mass of ith dropper added on each wire, nD is the dropper number, FMW and FCW are the tension forces of each wire, EM and EC are the elastic modulus, and IM and IC are the moments of inertia of each wire.Combining equations (14)–(16), the catenary system in the coordinate spaces {Am} and {An} is written as follows:(17)MMMCA¨MA¨C+CMCCA˙MA˙C+KMKCAMAC=QAMQAC,MM, MC, KM, and KC are the mass matrix and stiffness matrix of each wire which are expressed by(18)MM=MM1+MM2,(19)MM1i,k=∑j=1nDmMDjsiniπxDjLsinkπxDjL+∑j=1nSmMSjsiniπxSjLsinkπxSjL,(20)MM2i,i=ρML2,(21)KMi,i=FMWπ2i22L+EMIMπ4i42L3,(22)MC=MC1+MC2,(23)MC1i,k=∑j=1nDmMCjsiniπxDjLsinkπxDjL+∑j=1nSmCSjsiniπxSjLsinkπxSjL,(24)MC2i,i=ρCL2,(25)KCi,i=FCWπ2i22L+ECICπ4i42L3.AM is composed of Am, and AC is composed of An. The Rayleigh damping is also applied in coordinate spaces {Am} and {An}, and the damping matrices CM and CC are(26)CM=αMM+βKM,CC=αMC+βKC.


## 2.2.2. Load Vector

QAM is composed of QAm, and QAC is composed of QAn, which is the generalized load vector in coordinate spaces {Am} and {An}. The concentrated forces Fi (NF is the number of Fi) and the distributed forces f¯x can be transferred by(27)Qk1=∑i=1NFFi∂yi∂qk1;Qk2=∂∫0Lf¯xyx,tdx∂qk2.For the messenger wire, the distributed force is−ρMg, and the concentrated forces consist of the dropper forces −FDk and the supporting forces of messenger wire FSMk. The general load expression {QAm} can be transferred by(28)QAm=∂∫0L−ρMgyMx,tdx∂Am+∑k=1nD∂−FDkyMxDk,t∂Am+∑k=1nS∂FSMkyMxSk,t∂Am.For the contact wire, the distributed force is−ρCg, and the concentrated forces consist of the dropper forces FDk, the supporting forces of messenger wire FSCk, and the dynamic contact force FC. The general load expression {QAn} can be transferred by(29)QAn=∂∫0L−ρCgyCx,tdx∂An+∑k=1nD∂FDkyCxDk,t∂An+∑k=1nS∂FSCkyCxSk,t∂An+∂FCyCxC,t∂An.


## 3. Initialization of the Catenary

In the initialization of the catenary, the large ratiosα and β are applied in the damping matrix. The strategy of the initialization process is shown in Figure 9.Figure 9
Initialization of the catenary.The two initialization processes consist of the contact wire and the messenger wire, which can also be described in Figure10.Figure 10
Initialization of the contact wire and the messenger wire.According to the parameters in reference [18], the static position of the contact wire and the static forces is shown in Figure 11. The dropper force results are very close compared with the OSCAR in reference [18] as shown in Table 2.Figure 11
Initialization of the contact wire and the messenger wire.
(a)(b)Table 2
Dropper forces results of the model against the OSCAR.
Force itemOSCAR value (N)Model value (N)Relative error (%)Dropper 1167.90163.03−2.90Dropper 250.6752.503.61Dropper 355.4755.13−0.61Dropper 447.5247.670.31Dropper 555.3955.08−0.56


## 4. Variation of the Main Frame

The size parameters of the pantograph’s main structure considered here are shown in Figure12 and listed in Table 3, and several important rules need to be noticed.Figure 12
Main size parameters considered in the simulation.Table 3
Initial size parameters.
ItemsDescriptionReference valueRangeL1Length of the link1.18 m1–15% to 1 + 15%L2Length of the upper frame2.21 mL4Length of the lower frame1.6 mL5Length betweenO2 and O50.29 mθ5Angle betweenL5 and L212.83°LaDistance betweenO1 and O4 along x1 direction0.72 mLbDistance betweenO1 and O4 along y1 direction0.15 m


### 4.1. Height Design

The most important height values related to the pantograph-catenary system are shown in Figure13, where H1 is the designed height value between the contact line and the steel rail surface. H1 is usually 5.3 m in China’s high-speed railway. H2 is the working height defined representing the vertical distance between the pantograph mounting plane on the top of insulators and the upper surface of contact strips. H3 is the height value decided by the size of the pantograph’s main frame, which is set as a constant value of 1.6 m in this paper to analyze the effect of the main frame’s size.Figure 13
Important height values relating to the pantograph-catenary system.


### 4.2. Variation Rules

The size control of the pantograph’s main frame should comply with equation (30) to satisfy the designed height H3.(30)L1cosθ10+L5cosθ5+θ20−L4cosθ40−La=0,L1sinθ10+L5sinθ5+θ20−L4sinθ40−Lb=0,L1sinθ10+L2sinθ20−H3=0.The initial position of the pantograph’s main frame to keepH3 as a constant value at different size parameters is shown in Figure 14. Several conditions failed to satisfy the rules when the changing ratio is larger. L1 failed when the change ratio is −15%, and L4 failed when the change ratios are 10% and 15%, shown in Figures 14(a) and 14(c).Figure 14
The initial position of the pantograph’s main frame (the red lines or hinges in this figure are the related variable parameters): (a) variation ofL1 (failed at −15%);. (b) variation of L2; (c) variation of L4 (failed at 10% and 15%); (d) variation of La; (e) variation of Lb; (f) variation of L5; (g) variation of θ5 (the angles between two red lines).
(a)(b)(c)(d)(e)(f)(g)The moments of inertia of the componentsIi changed following the size parameters, and each component is assumed to be a uniform bar.(31)Ii=112miLi2,where Ii is the ith components’ moment of inertia, Li is the length, and mi is the constant parameter; Ii will change as Li changes, as shown in equation (31).


### 4.3. Static Contact Force and Pantograph Raise Torque

When designing the pantograph-catenary system, the static contact force plays an important role and needs to be carefully considered. For all conditions simulated in this paper, the static force is set as a constant value 70 N. Thus, the torque raising value of the pantograph will be redesigned in each size parameter condition. The force analysis diagram is shown in Figure15.Figure 15
Static force analysis.For collector strips loaded on the upper frame,(32)FO3=FStatic+mLg+mRg.For the upper frame,(33)∑Fyup=0;∑Fxup=0;∑MO2up=0.For the link,(34)∑Fylin=0;∑Fxlin=0;∑MO1lin=0.For the lower frame,(35)∑MO4lo=0.The computed raising torque values are shown in Table4 with different parameters.Table 4
Torque valueT/Nm with variable parameters.
L1L2L4L5θ5LaLb1–15%╳990.81079.71296.21073.4770.21061.11–10%778.21020.21115.51214.81074.8889.41066.71–5%969.11049.21122.81142.61076.4989.671072.311078.11078.11078.11078.11078.11078.11078.11 + 5%1138.61106.7939.41020.21079.81158.81083.91 + 10%1165.01135.3╳968.11081.71234.71089.91 + 15%1164.81163.7╳921.01083.71307.51095.97“╳” means the failure to the height rules and will not be considered in the simulation procession.


### 4.4. Others Need to Be Noticed

(1)
The total mass of the pantograph’s main is constant, and the target force is a constant value of 70 N, and the changes of the torque values are due to the variation of gravity center of the pantograph caused by size changing.(2)
The aerodynamics effects are always treated as the constant value corresponding to speed, which are not considered in pantograph-catenary interaction.(3)
The catenary parameters simulated in the paper are based on EN50318:2002 [46], and the initial parameters of the pantograph are simplified based on pantograph SSS400+ [47].(4)
The effects of the vertical movement of the train body to contact quality are not considered here.


## 4.1. Height Design

The most important height values related to the pantograph-catenary system are shown in Figure13, where H1 is the designed height value between the contact line and the steel rail surface. H1 is usually 5.3 m in China’s high-speed railway. H2 is the working height defined representing the vertical distance between the pantograph mounting plane on the top of insulators and the upper surface of contact strips. H3 is the height value decided by the size of the pantograph’s main frame, which is set as a constant value of 1.6 m in this paper to analyze the effect of the main frame’s size.Figure 13
Important height values relating to the pantograph-catenary system.


## 4.2. Variation Rules

The size control of the pantograph’s main frame should comply with equation (30) to satisfy the designed height H3.(30)L1cosθ10+L5cosθ5+θ20−L4cosθ40−La=0,L1sinθ10+L5sinθ5+θ20−L4sinθ40−Lb=0,L1sinθ10+L2sinθ20−H3=0.The initial position of the pantograph’s main frame to keepH3 as a constant value at different size parameters is shown in Figure 14. Several conditions failed to satisfy the rules when the changing ratio is larger. L1 failed when the change ratio is −15%, and L4 failed when the change ratios are 10% and 15%, shown in Figures 14(a) and 14(c).Figure 14
The initial position of the pantograph’s main frame (the red lines or hinges in this figure are the related variable parameters): (a) variation ofL1 (failed at −15%);. (b) variation of L2; (c) variation of L4 (failed at 10% and 15%); (d) variation of La; (e) variation of Lb; (f) variation of L5; (g) variation of θ5 (the angles between two red lines).
(a)(b)(c)(d)(e)(f)(g)The moments of inertia of the componentsIi changed following the size parameters, and each component is assumed to be a uniform bar.(31)Ii=112miLi2,where Ii is the ith components’ moment of inertia, Li is the length, and mi is the constant parameter; Ii will change as Li changes, as shown in equation (31).


## 4.3. Static Contact Force and Pantograph Raise Torque

When designing the pantograph-catenary system, the static contact force plays an important role and needs to be carefully considered. For all conditions simulated in this paper, the static force is set as a constant value 70 N. Thus, the torque raising value of the pantograph will be redesigned in each size parameter condition. The force analysis diagram is shown in Figure15.Figure 15
Static force analysis.For collector strips loaded on the upper frame,(32)FO3=FStatic+mLg+mRg.For the upper frame,(33)∑Fyup=0;∑Fxup=0;∑MO2up=0.For the link,(34)∑Fylin=0;∑Fxlin=0;∑MO1lin=0.For the lower frame,(35)∑MO4lo=0.The computed raising torque values are shown in Table4 with different parameters.Table 4
Torque valueT/Nm with variable parameters.
L1L2L4L5θ5LaLb1–15%╳990.81079.71296.21073.4770.21061.11–10%778.21020.21115.51214.81074.8889.41066.71–5%969.11049.21122.81142.61076.4989.671072.311078.11078.11078.11078.11078.11078.11078.11 + 5%1138.61106.7939.41020.21079.81158.81083.91 + 10%1165.01135.3╳968.11081.71234.71089.91 + 15%1164.81163.7╳921.01083.71307.51095.97“╳” means the failure to the height rules and will not be considered in the simulation procession.


## 4.4. Others Need to Be Noticed

(1)
The total mass of the pantograph’s main is constant, and the target force is a constant value of 70 N, and the changes of the torque values are due to the variation of gravity center of the pantograph caused by size changing.(2)
The aerodynamics effects are always treated as the constant value corresponding to speed, which are not considered in pantograph-catenary interaction.(3)
The catenary parameters simulated in the paper are based on EN50318:2002 [46], and the initial parameters of the pantograph are simplified based on pantograph SSS400+ [47].(4)
The effects of the vertical movement of the train body to contact quality are not considered here.


## 5. Simulation Results and Discussion

The STDs of the dynamic contact force simulated related to varying size values at different running speeds from 200 km/h to 350 km/h are shown in Figure16.Figure 16
STD of the dynamic contact force: (a) variation ofL1 (failed at −15%); (b) variation of L2; (c) variation of L4 (failed at 10% and 15%); (d) variation of La; (e) variation of Lb; (f) variation of L5; (g) variation of θ5.
(a)(b)(c)(d)(e)(f)(g)Three basic laws can be concluded in Figure16.(1)
The STDs of dynamic contact force decrease as the size values ofL1, L2, and La increase(2)
The STDs of dynamic contact force increase as the size values ofL4 and L5 increase(3)
The size values ofLb and θ5 are not sensitive to the contact qualityTo evaluate the effect of variation on STD, the relative rate of changee is used:(36)e=σ−σ0σ0,where σ0 is the STD for the initial parameters.It is more obvious in Figure17 that the length of the link (L1), the upper frame (L2), and the size value along x-direction between O1 and O4 (La) are sensitive to the contact quality, which will improve current collection quality at the larger size values. The lower frame (L5) and the distance between O2 and O5 are also sensitive to contact quality, which will improve current collection quality at the smaller size values. The vertical distance between O1 and O4 (Lb) and the angle between L2 and L5 (θ5) are not sensitive enough to the contact quality compared with other size parameters.Figure 17
Relative change ratee at different speeds (basically, the red lines mean decrease, the blue lines mean increase, and the black lines mean small change): (a) 200 km/h; (b) 250 km/h; (c) 300 km/h; (d) 350 km/h.
(a)(b)(c)(d)The torque value applied to the lower frame changes with variable size values under the same target designed operation height and static contact force. The changing law of the torque value is in opposition as to the contact quality, shown in Figures17 and 18. When the size values of L1, L2, and La become lager, the raising torque values will increase and the STDs will decrease, indicating that the contact quality will improve when the torque value increases while the designed height and vertical static contact force are constant.Figure 18
Torque values to meet the static contact force 70 N and the designed height 1.6 m.


## 6. Conclusion

This paper established the dynamic mathematic model of pantograph-catenary systems and analyzed the variation of the pantograph’s main frame including the link, the upper frame, the lower frame, and the hinge positions. Two important conclusions can be given as follows:(1)
Most geometric values of the pantograph’s main frame are sensitive to contact quality and show different influence laws, which need to be carefully considered when designing a pantograph to suit a known catenary(2)
Larger pantograph-raising torque value is effective to improve the pantograph-catenary interaction when the static force value and the operation height are determined in advanceThe work here is important for the structure design and parameters optimization of the pantograph; however, more research works remain needed to be done consisting of how the aerodynamic force acts on the pantograph which is different from that acts on a mass-spring model, whether the flexibility of the upper frame needs to be concerned in parameters optimization of the pantograph, and how the mass distribution of the pantograph’s main frame influences the contact quality.



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*Source: 4037999-2021-08-05.xml*