sliding mode control with pd sliding surface for high-speed

Research ArticleSliding Mode Control with PD Sliding Surface forHigh-Speed Railway Pantograph-Catenary Contact Force underStrong Stochastic Wind Field

Yang Song12 Zhigang Liu12 Huajiang Ouyang3 Hongrui Wang12 and Xiaobing Lu12

1State Key Laboratory of Traction Power Southwest Jiaotong University Chengdu Sichuan 610031 China2School of Electrical Engineering Southwest Jiaotong University Chengdu Sichuan 610031 China3School of Engineering University of Liverpool Brownlow Street Liverpool L69 3GH UK

Correspondence should be addressed to Zhigang Liu liuzgswjtucn

Received 24 August 2016 Revised 19 November 2016 Accepted 27 November 2016 Published 18 January 2017

Academic Editor Sergio De Rosa

Copyright copy 2017 Yang Song et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As is well known the external disturbance (especially the stochastic wind load) has nonnegligible effect on the operation ofpantograph-catenary system which may cause the strong fluctuation in contact force as well as the increased occurrence of contactloss In order to improve the current collection quality of a high-speed railway pantograph-catenary systemunder a strong stochasticwind field a sliding mode controller with a proportional-derivative (PD) sliding surface for a high-speed active pantograph isproposed The nonlinear finite element procedure is employed to establish the catenary model The fluctuating wind speeds alongcatenary are simulated using empirical spectrums The buffeting forces exerted on contact and messenger wires are derived toconstruct the stochastic wind field along the catenary A PD sliding surface is properly determined to guarantee that themechanicalimpedance of pantograph head at the dominant frequencies of contact force decreases when the sliding surface approaches zeroThrough several numerical simulations with different wind velocities and wind angles the control performance of two popularcontrol laws (proportional switching law and constant switching law) is evaluated

1 Introduction

The recent decades have witnessed a rapid expansion of high-speed electrified railway inmany countries around the worldThe increase of the driving speed of high-speed trains leadsto many new technical issues One of them is the strongvibration of the pantograph-catenary system resulting inthe deterioration of the current collection quality Figure 1describes the schematic of a pantograph-catenary systemThe electric power is transmitted from the catenary to thelocomotive via a pantograph installed on the roof Obviouslythe sliding contact between the pantograph collector andthe contact wire is the most vulnerable point in this systemespecially in high-speed driving conditions Recently thecontact quality between catenary and pantograph has been ofgreat interest to many scholars as it directly determines thecurrent collection quality restricting the driving speed limitof high-speed trains An excessive contact force can aggravate

thewear and fatigue of contact wire and pantograph collectorIn contrast an inadequate contact force may increase thepossibility of the separation of the pantograph collector fromthe contact wire which may lead to the occurrence of arcingand the interruption of electric power transmission A stablecontact force between the pantograph and catenary should bemaintained to avoid the separation between the pantographhead and the contact wire as well as to reduce the wear ofpantograph strip and contact wire

As is well known mathematic modeling is an effectivemeans to study the pantograph-catenary complex dynamicbehaviour In order to obtain the exact initial configurationof catenary Lopez-Garcia et al [1] proposed a nonlinearcalculation procedure based on nonlinear cable equationsTur et al [2] proposed a shape-finding method for catenarysystem based on ANCF (absolute nodal coordinate formula-tion) In order to make sense of the dynamic behaviour ofpantograph-catenary the cosimulations of a finite element

HindawiShock and VibrationVolume 2017 Article ID 4895321 16 pageshttpsdoiorg10115520174895321

2 Shock and Vibration

Catenary

Pantograph

Messenger wire

Contact wire DropperSliding contact

Figure 1 Schematic of the pantograph-catenary system

procedure for catenary and multibody codes for pantographwere performed by Ambrosio et al [3] The damping ofcatenary was identified by Navik et al [4] using the timeseries sampled on existing catenaries An advanced detectionmethod for contact wire irregularities was proposed byWang et al [5] In order to exactly describe the geometricalnonlinearity of contactmessenger wire a nonlinear finiteelement procedure was performed based on ANCF [6] Andthe analytical formulations of nonlinear cable were utilizedto describe the large deformation of contactmessenger wireby Song et al [7] On the other hand in order to considerthe complex operation conditions of pantograph-catenarysystem Carnicero et al [8] analysed the effect of locomotivevibration on pantograph-catenary interaction Pombo et al[9] analysed the influence of aerodynamics on pantograph-catenary interaction Song et al [10] constructed the windfield along a catenary considering the stochastic wind fromdifferent directions and analysed the effect of stochastic windload on pantograph-catenary interaction The results of pre-vious studies indicated that the effect of locomotive vibrationon the contact force between catenary and pantograph waslimited In contrast the effect of strong stochastic wind loadwas very significant

Based on the study of pantograph-catenary dynamicsparameter optimization has been developed as a very generalway to improve the dynamic performance of pantograph-catenary systems A design optimization procedure for a pan-tographwas conducted by Lee et al [11] based on a sensitivityanalysis of the pantograph-catenary system Ambrosio et al[12] studied the suspension characteristics of a pantographhead and proposed a corresponding optimization strategyto decrease the fluctuation in contact force Roslashnnquist andNavik [13] used the frequency-analysis method to evaluatethe upgrade of a catenary and Navik et al [14] suggesteda new tension forces for a higher driving speed Consid-ering multiple-pantograph operation conditions Liu et al[15] studied the effect of interval between front and rearpantographs on dynamic performance

However a new design produced by the parame-ter optimization means the reconstruction of an existingpantograph-catenary system which incurs a big cost andsometimes may not be easy to realize in reality Besidesparameter optimization of catenary-pantograph systemscontrol of pantographs is another effective way to maintaina stable contact force Facchinetti et al [16] performed ahardware-in-the-loop experiment for an active pantographIn that work an actuator was placed in parallel with thecollector suspensions which could obtain better performance

but is not convenient to be realized in an existing railway lineGenerally an actuator can be installed on the lower frameof the pantograph which is more convenient to implementcompared to [16] and does not lead to a large change to theoriginal pantograph structure but poses a greater challenge tothe control algorithm because it is far away from the contactinterface

Generally the study of pantograph-catenary dynamicinteraction belongs to moving-load dynamics problems [17]Among them vibration suppression of beams excited bymoving loads has been of great interest to many scholars as ithas wide potentials in various engineering applications suchas bridge-vehicle interaction and vehicle-track interactionPi and Ouyang [18] used Lyapunov-based boundary controlmethod for suppressing the vibration of a multispan beamsubjected to moving masses Stancioiu and Ouyang [19]proposed an effective time-varying optimal control strategyfor a beam excited by a moving mass

However compared with the traditional moving-loadcontrol problems whose objectives are to decrease thevibration response the control objective of an active panto-graph is to reduce the fluctuation in contact force and thecontact loss between the pantograph collector and contactwire simultaneously without reducing the contact forcersquosmean value Considering the particularity of the pantograph-catenary system only few states of the pantograph and thecontact force can be detected which can be used as feedbackin a controller Recently various control algorithms for anactive pantograph were proposed by many experts [20ndash28]Normally due to the strong nonlinear effect of the catenaryon the pantograph it is very difficult to design the controllerbased on the whole complex catenary-pantograph modeldirectly In the stage of the controller design the catenary isalways simplified as a one degree-of-freedom (DOF) modelwith time-varying stiffness (shown in Figure 2(a)) to facilitatedevelopment of the control strategy Sometimes a time-varying mass is also included (shown in Figure 2(b)) InFigure 2 119896c(119905) and 119898c(119905) are the time-varying stiffness andmass of catenary 1198981 1198961 and 1198881 are the mass stiffness anddamping of the pantograph head respectively1198982 1198962 and 1198882are themass stiffness and damping of the pantograph framerespectively119891119888 is the pantograph-catenary contact force1198910 isthe static uplift force 119891119906 is the control force

Based on the simplified model shown in Figure 2(a)Lin et al [20] developed a control strategy for an activepantograph by LQR (linear quadratic regulator) Recentlythe robustness of an optimal control approach for an activepantograph was investigated considering the effect of actua-tor time delays [21] Rachid [22] developed an LMI (LinearMatrix Inequality) control law to keep the contact forceclose to a desired value OrsquoConnor et al [23] used thesteady state matrix Riccati equation to design a controller foran active pantograph which obtained over 50 reductionin the contact force variation Pisano and Usai [24] pro-posed an output feedback control strategy based on higher-order sliding modes and high-gain observers which had agood robustness and stability Considering the locomotivevibration Wang [25] developed a geometric framework tosuppress the vibration of pantograph-catenary system Based

Shock and Vibration 3

Catenary

Pantograph

fc

k1

k2

m1

m2

c1

c2 fuf0

k＝(t)

(a)

Catenary

Pantograph

fc

k1

k2

m1

m2

c1

c2 fuf0

k＝(t)

m＝(t)

(b)

Figure 2 (a) Pantograph and simplified catenary model with time-varying stiffness (b) Pantograph and simplified catenary model withtime-varying stiffness and mass

on the simplified model shown in Figure 2(b) Pisano andUsai [26] implemented a variable structure control (VSC)technique with sliding mode on a wire-actuated symmetricpantograph Sanchez-Rebollo et al [27] designed a PIDcontroller for an active pantograph which was the firstattempt to implement a controller with a validated FEMcatenary model

It is obvious that the simplified models shown in Figure 2can only reflect the unsmooth stiffness and mass distributionof the catenary without considering some other nonlineardisturbances from catenary to pantograph in the stage ofcontroller design On the other hand except the work doneby Sanchez-Rebollo et al [27] and Pappalardo et al [28]none of the other works implemented their controllers with amore realistic nonlinear validated catenary model Howeverthe validation of the controller with a realistic nonlinearcatenary model is very important in the theoretical study ofthe pantograph control before it can be evaluated through afield experiment in a real railway line Furthermore to thebest of the authorsrsquo knowledge the complex environmentalperturbations (especially the stochastic wind along catenary)have not been considered when the control strategy wasimplemented Due to these perturbations the fluctuationin contact force increases significantly and the separationbetween the pantograph head and contact wire occurs morefrequently The efficiency of the control strategy should beevaluated under these realistic complex operation conditionswhich definitely exist in a real railway line The robustness ofthe controller also needs a further investigation with strongexternal perturbations

This paper addresses these difficulties of pantographcontrol and proposes a sliding mode control (SMC) strategywith a PD sliding surface for a high-speed pantographCompared with other control strategies SMC seems to beconvenient to realize the control objective of keeping thecontact force close to a desired value through defining aproper sliding surface SMC was shown to be very suitablefor nonlinear time-varying systems For example Pi andOuyang [29] also found that SMC had a good performance insuppressing the vibration of a beam excited by amovingmass

In [26] SMC was firstly introduced in the active pantographcontrol and a very simple control law was proposed But theoversimplified model used in that work cannot guarantee itsvalidity in a more realistic pantograph-catenary model

To address the shortcomings of the previous studiesinstead of the oversimplified model in Figure 2 a nonlinearFEM catenary model under the stochastic wind field alongthe catenary is established The contact force between thepantograph and catenary is chosen as the feedback signalThe transfer function between the pantograph velocity andcontact force (called mechanical impedance) is introducedas an auxiliary indicator to determine the sliding surfaceparameters Then several numerical simulations with twopopular control laws (proportional switching law and con-stant switching law) are implemented to evaluate the controlperformance under a strong stochastic wind field In orderto verify the advantage of the proposed control strategy thecontrol performance is compared with EN 50318 referencemodel reported in the literature This work is a theoreticalattempt to introduce the PD sliding mode control strategyinto pantograph-catenary dynamics under more complexand realistic operation conditions than previous studies toimprove the current collection quality of high-speed railwaypantograph-catenary system

2 Establishment of Pantograph-CatenarySystem with Stochastic Wind Load

Based on the authorsrsquo previous work [7 10] the pantograph-catenary dynamic model with stochastic wind load as shownin Figure 3 is established in this section

21 Modeling of Catenary-Pantograph Normally a catenarysystem is comprised of three main components the contactwire the messenger wire and droppers as shown in Figure 3In order to properly describe the nonlinear behaviour understrong wind field the contactmessenger wire is discretizedas a number of nonlinear cable elements based on the FEMand the dropper is modeled by the nonlinear string elementwhich behaves like one-side spring Then a nonlinear finite


Contact wire

Wind

Pantograph

Figure 3 Pantograph-catenary system with wind load

element dynamic procedure is performed to solve the initialconfiguration and the dynamic response of catenary Theglobal stiffness matrix K119862(119905) of catenary can be formulatedthrough the FEM at any time instant t The global lumped-mass matrixM119862 and the global damping matrix C119862 can alsobe obtained The equation of motion for catenary can bewritten as

M119862ΔX119862 (119905) + C119862ΔX119862 (119905) + K119862 (119905) ΔX119862 (119905) = ΔF119862 (119909 119905) (1)

in which ΔX119862(119905) ΔX119862(119905) and ΔX119862(119905) are the incrementalvectors of the global acceleration velocities and displace-ments of catenaryΔF119862(119909 119905) on the right side is the incremen-tal vector of the excitation

A two-DOFs pantograph model (as shown in Figure 2)is adopted The contact between the pantograph head andthe contact wire is realized by a penalty method in which anassumption of ldquocontact stiffnessrdquo is definedThe contact forcecan be calculated based on a penalisation of the interpenetra-tion between the pantograph strip and the contact wire Thecontact force can be calculated by

119891119888 = 119870119878 (1199101 minus 119910119888) 1199101 ge 1199101198880 1199101 lt 119910119888 (2)

in which119870119878 is the contact stiffness 1199101 is the vertical displace-ment of the pantograph head 119910119888 is the vertical displacementof the contact wire in the contact pointThe static uplift forceis determined by a formula in a SIEMNS simulation reportmentioned in [7]

22 Validation of Pantograph-Catenary Model Severalnumerical simulations have been conducted in [7] to verifythe accuracy of the static configuration of catenary andthe validation of the dynamic solution Here the newestbenchmark proposed by Bruni et al [30] is adopted toconduct a further validation of the presented model Table 1shows the dynamic simulation results compared to thereference values in the benchmark It is observed that twosets of the results are only slightly different

Table 1 Validation of the present model according to the bench-mark results

Indicators Benchmark Simulation Error ()Speed (kms) 320 320 0Mean contact force (N) 169 17125 13Standard deviation (N) 5391 5521 24Max real value (N) 31322 32179 27Min real value (N) 6040 5684 minus59Banded SD 0ndash2Hz (N) 3827 3911 22Banded SD 0ndash5Hz (N) 4104 4052 minus13Banded SD 5ndash20Hz (N) 3480 3607 36PPA of the verticaldisplacement (mm) 4758 4531 48

Max uplift at support (mm) 5912 6017 18

Wind

D

o

FＬ F F９ F＄Ｌ

yＱＬ

F８

F＄

yＱ

zＱＬzＱ zＡ

yＡ

Figure 4 Sketch of contact wire section against wind load

23 Stochastic Wind Field Figure 4 shows the contact wirecross section under wind load 119880 and 120572 are the wind speedand angle of attack 119871 is the length of contact wire 119863 is thediameter of the cross section of contact wire According tothe derivation in [10] the buffeting forces acting on contactwire can be written as

119865D = 12120588air1198802119871119863[119862D (1205720) sdot 2119906 (119905)119880

+ [ 119862D (1205720) minus 119862L (1205720)] sdot 119908 (119905)119880 ] + 1

2sdot 120588air1198802119863119871119862D (1205720)

(3a)

119865L = 12120588air1198802119871119863[119862L (1205720) sdot 2119906 (119905)119880

+ [ 119862L (1205720) + 119862D (1205720)] sdot 119908 (119905)119880 ] + 1

2sdot 120588air1198802119863119871119862L (1205720)

(3b)

in which 120588air is the density of air 1205720 is the initial angle ofattack 119862D(1205720) and 119862L(1205720) are the drag and lift coefficients atthe angle of attack 1205720 which can be calculated through CFD


Figure 5 Nephogram of the flow velocity around the contact wirecross section in CFD calculation

or measured in a wind tunnel experiment 119908(119905) and 119906(119905) arethe time-histories of the vertical and horizontal wind velocity

By transforming (3a) and (3b) to the finite elementcoordinate the formulation of the aerodynamic forces thatcan be used in the FEMmodel directly is obtained as follows

119865X = 119865Dcos (1205720) minus 119865L sin (1205720) (4a)

119865Y = 119865D sin (1205720) + 119865L cos (1205720) (4b)

In order to obtain the wind load acting on the catenarythe aerodynamic coefficients 119862D(120572) and 119862L(120572) and thefluctuatingwind velocity 119906(119905) and119908(119905) should be determined

The CFD softwaremdashANSYS Fluentmdashis utilized to deter-mine the aerodynamic coefficients 119862D(120572) and 119862L(120572) Inthis paper the CuAg01AC120-type contact wire is chosenas the object of analysis Simulation conditions (includingthe boundary conditions and mesh properties) are definedaccording to [10] Figure 5 shows the nephogram of theflow velocity around the contact wire cross section in theCFD calculation The calculation results of the aerodynamiccoefficients are shown in Figure 6 [10]

The empirical formulations of the wind velocity spectrumare utilized to simulate the fluctuating wind velocities in dif-ferent directionsThe nondimensional form of theDavenportspectrum for horizontal wind speed 119878119911 is [31]

119899119878119911 (119899)11988121 = 4120581 1199092(1 + 1199092)43 (5)

in which 119909 = 400011989911988121 120581 is the surface drag coefficient 119899is the frequency of fluctuating wind and 1198811 is the velocity atreference height (10m) The Panofsky spectrum for verticalwind speed 119878119908 is [32]

119899119878119908 (119899 119911)11988121199111 = 11 119891(1 + 4119891)2 log1198902 (11991111199110) (6)

where 119891 = 1198991199111198811 1198811199111 is the reference velocity at height 1199111and 1199110 is the roughness length

The variance of the spatial distribution of the stochasticwind velocity is considered by defining the spatial correlationfunction using Wiener-Khinchin equation Through fourth-order AR model the time-histories of the vertical and hori-zontal wind velocities can be generated in which the stochas-tic feature of the fluctuating wind velocity in time-domainis properly considered In this paper an existing high-speedpantograph-catenary system constructed in China is studied[7] Consider that the longitudinal interval in simulation forthe fluctuating wind is 10m The simulation results of thefluctuating wind velocities and their spectrums at the firstpoint of this longitudinal interval are shown in Figure 7 (inwhich the steady wind speed is chosen as 20ms) It is foundthat the autopower spectrum shows excellent agreement withthe target spectrum The effect of the stochastic wind is alow-frequency disturbance whose dominant frequency isnormally less than 1Hz

3 Definition of the Sliding Surface

According to the literature [16] the contact force between thepantograph collector and the contact wire can be measuredthrough a contact force sensor which directly reflects thecontact quality of the pantograph-catenary system So thecontract force can be utilized as the feedback signal in thecontroller In order to realize the control objective a PDsliding surface is defined as follows

119878 (119905) = 119902p119890 (119905) + 119902d 119890 (119905) (7)

in which 119890(119905) = 119891119903 minus 119891119888(119905) 119891119903 is the desired value of contactforce (which is normally themean value of contact force)Thecontrol force 119891119906(119905) is determined by the sliding surface 119878(119905)to keep the sliding surface 119878(119905) close to 0 So the values ofthe parameters 119902p and 119902d are of great importance Normallythe parameters of a sliding surface can be determined byexperience or the placement of poles to guarantee the stabilityof the system However there seem to be no previous studiesor experience on how to determine the sliding surface Inthis paper the mechanical impedance of pantograph headis introduced as an auxiliary indicator to define the slidingsurface parameters

When the control force is exerted on the lower frame ofpantograph the equation of motion for the pantograph canbe written as

1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(8)

Normally the static uplift force 1198910 is very close to the meanvalue of contact force 119891119903 So it can be assumed that 1198910 = 119891119903The contact force can be expressed as

119891119888 (119905) = 1198910 + Δ119891119888 (119905) (9)


Angle of attack (∘)

WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms

806040200minus20minus40minus60minus8009

095

1

105

11

115

12

125

13

135

14C

＄

(a) 119862D

WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms

Angle of attack (∘)806040200minus20minus40minus60minus80

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

C

(b) 119862L

Figure 6 Simulation results of aerodynamic coefficients of contact wire cross section

Autopower spectrumTarget spectrum

10 20 30 40 50 60 70 80 90 1000t (s)

minus15

minus10

minus5

0

5

10

15

(m

s)

10minus2

100

102

S(Ｇ

2s

)

10minus2 10minus1 100 10110minus3

Frequency (Hz)

(a) Horizontal wind velocity

10 20 30 40 50 60 70 80 90 1000t (s)



Frequency (Hz)

10minus2

10minus1

100

101

102

S(Ｇ

2s

)

minus6

minus4

minus2

0246

(m

s)

(b) Vertical wind velocity

Figure 7 Simulation results of fluctuating wind velocity and their power spectrums

in which Δ119891119888(119905) is the fluctuation in contact force around itsmean value 1198910 The mechanical impedance 119867(119904) a transferfunction between the fluctuation of contact force Δ119865119888(119904) and

the velocity of pantograph head1198811(119904) is established by takingLaplace transformation of (8) and ignoring the constantterms as

119867(119904) = Δ119865119888 (119904)1198811 (119904) = 119898111989821199044 + (11988821198981 + 11988811198981 + 11988811198982) 1199043 + (11989611198981 + 11989621198981 + 11989611198982 + 11988811198882) 1199042 + (11988811198962 + 11988821198961) 119904 + 11989611198962(11989821199042 + (1198881 + 1198882) 119904 + 1198961 + 1198962 minus (1198961 + 1198881119904) (119902119889119904 + 119902119901)) 119904 (10)

in which 119904 is the Laplace constant As the control force119891119906(119905) isdetermined by the sliding surface 119878(119905) let 119878(119905) replace119891119906(119905) in

(10) to analyse the influence of the sliding surface parameterson the mechanical impedance of pantograph head in the


3 34 363228260

350

300

250

200

150

100

50

Time (s)

Con

tact

forc

e (N

)

Without windU = 20 = 60

U = 30

U = 20

= minus40

= minus10

(a) Contact force

5 10 15 20 25 30 35 40 45 50Frequency (Hz)

Catenary span

Dropper interval

Dominant frequency


10minus4

10minus2

100

102

104

Mag

nitu

de (d

B)

U = 30

U = 20

= minus40

= minus10

(b) Spectral density of contact force

Figure 8 Contact force and its spectral density

stage of the controller design In [33] transfer function119867(119904)is defined as the mechanical impedance of the pantographwhich can be assumed as an important reflection of thedynamic performance of the pantograph under the excitationof fluctuating wind A lower mechanical impedance of thepantograph head at the dominant frequencies of contactforce can guarantee a good current collection quality of thepantograph-catenary system In contrast a large mechanicalimpedance of the pantograph headmay lead to an increase ofthe fluctuation in contact force So in this section accordingto (10) the contribution of each of the parameters 119902p and 119902dto the mechanical impedance is analysed for the definition ofthe sliding surface

Before the analysis of the mechanical impedance ofpantograph head the range of the dominant frequency ofcontact force should be determined through the frequencyanalysis of the contact force Using the pantograph-catenarymodel established in Section 2 the simulation results ofcontact force are shown in Figure 8(a) in which the drivingspeed is 300 kmh and three sets of wind field parametersare adopted (which are U = 20ms with 120572 = 60∘ U = 20mswith 120572 = minus10∘ and U = 30ms with 120572 = minus30∘) The resultsof corresponding spectral density are shown in Figure 8(b)From the spectral density of contact force the frequencycomponents related to the catenary span and dropper intervalcan be clearly observed The dominant frequency of contactforce resides mainly in the range from 167Hz (which isrelated to the catenary span (50m) traversed by a pantographwith the driving speed of 300 kmh) to 20HzThe effect of theenvironmental wind leads to an increase of the input energyinto the pantograph-catenary system at most frequenciesbut does not result in a significant change of the range ofdominant frequency

By substituting 119904 = 1198952120587119891 to (10) the effect of 119902p and 119902don the mechanical impedance at the dominant frequenciesof contact force is analysed The effect of parameter 119902p isinvestigated firstlyThe Bode diagram of the transfer function

119867(119904) at different 119902p is plotted in Figure 9(a) It is clearly foundthat positive 119902p has a significant contribution to decreasingthe mechanical impedance at the dominant frequencies ofcontact force (especially from 15Hz to 20Hz)

Similar to the analysis of the effect of 119902p the mechanicalimpedance with different 119902d is plotted in Figure 9(b) It isclearly found that a positive 119902d can lead to a total increaseof the mechanical impedance at all frequencies which mayresult in a deterioration of the current collection qualitySanchez-Rebollo et al [27] have also drawn a similar con-clusion in their work a positive value of the differentialparameter gain in their PID controller can cause the insta-bility of the system and the increase of fluctuation in contactforce In contrast it can also be observed that a negative119902d can lead to the decrease of the mechanical impedance atall frequencies In order to observe the effect of 119902p and 119902don the mechanical impedance more clearly the surface ofthe mechanical impedance at the most significant frequency167Hz (which is corresponding to the catenary span) versus119902p and 119902d in the revised manuscript is shown in Figure 10Similar conclusion to the above analysis can be drawn Sothrough the analysis of the effect of each of the parameterson the mechanical impedance of pantograph head it isconcluded that the sliding surface should be comprised of apositive 119902p and a negative 119902d4 Numerical Simulations

As mentioned in the Introduction there are two control lawsto determine the control force through the sliding surface119878(119905) One is the constant switching law and the other is theproportional switching law

For the constant switching law the control force switchesbetween a lower and an upper bound which can be describedas

119891119906 (119905) = 119906 119878 (119905) gt 0minus119906 119878 (119905) lt 0 (11)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

qＪ = 0

qＪ = 2

qＪ = 4

qＪ = 8

qＪ = 14

qＪ = 20

100

101

102

103M

agni

tude

(dB)

(a) Mechanical impedance with different 119902p

2 4 6 8 10 12 14 16 18 20Frequency (Hz)

Mag

nitu

de (d

B)

104

102

103

101

q＞ = 0

q＞ = 002

q＞ = 001

q＞ = minus001q＞ = minus002

(b) Mechanical impedance with different 119902d

Figure 9 Mechanical impedance of pantograph head with 119902p and 119902d

05

1015

200

50

100

150

200

250

300

minus002minus004

minus006minus008

minus01 qＪ

050

100150200250300350400

Mag

nitu

de (d

B)

q＞

Figure 10 Surface ofmechanical impedance at 167Hz versus 119902p and119902d

in which 119891119906(119905) is the control force u is the bound of controlforce

For the proportional switching law the control forceis determined by a smooth proportional function and thesliding surface as

119891119906 (119905) = 1003816100381610038161003816120572119890 (119905) + 120573 119890 (119905)1003816100381610038161003816 sgn (119878 (119905)) (12)

where 120572 and 120573 are the control gains in the proportionalswitching law Compared with the constant switching law thecontrol force changes more smoothly against time

In order to determine the values of 119902p and 119902d in the slidingsurface a lot of simulations are conducted to show howthe sliding surface affects the control performance Figure 11shows the calculation results of standard deviation of contactforce versus 119902p and 119902d In this discussion the control law isselected as a proportional switching law with 120572 = 04 and120573 = minus2 times 10minus4 From Figure 11 it can be seen that when

5q＞ qＪ

34

33

32

31

30

29

28STD

of c

onta

ct fo

rce (

N)

Bad effect

Adopted inthe following analysis

minus001

minus0005

0 0

1015

20

Figure 11 Standard deviation of contact force versus 119902p and 119902d

119902119901 increases more than 0 the standard deviation generallycannot be largely influenced by the change of 119902p and 119902dIn the following analysis through comparing the values of119890(119905) and its derivative 119890(119905) the parameters of sliding surfaceare defined as 119902p = 20 and 119902d = minus001 for the DSA-380pantograph which shows a relatively good performance inthis analysis

41 Comparison of Two Control Laws without Wind LoadIn this section several numerical simulations are conductedto evaluate the control performance using the two controllaws For the constant switching law the control force boundis chosen as 10N 20N 30N 40N 50N 60N and 70Nseparately For the proportional switching law three sets ofcontrol gains are defined as follows

Case 1 120572 = 02 and 120573 = minus1 times 10minus4Case 2 120572 = 04 and 120573 = minus2 times 10minus4Case 3 120572 = 08 and 120573 = minus4 times 10minus4


Table 2 The results of standard deviation and maximum control force

No control Constant switching law Proportional switching law119906 = 10N 119906 = 20N 119906 = 30N 119906 = 40N 119906 = 50N 119906 = 60N 119906 = 70N Case 1 Case 2 Case 3

STD (N) 3357 3168 3046 3027 2996 3169 3299 3306 3089 2901 2625Mean value (N) 11784 11816 11857 11826 11813 11822 11927 11996 11738 11687 11688Max Ctrl force (N) 0 10 20 30 40 50 60 70 1821 3443 6506

200

180

160

140

120

100

80

60

40

20

Con

tact

forc

e (N

)

Time (s)

Without controlu = 20 N

u = 40 Nu = 60 N

36343232826

Figure 12 Contact force with constant switching law

The results of contact force with constant switching laware shown in Figure 12 It should be noted that the resultsare filtered with the frequency range of 0ndash20Hz accordingto the requirement of EN 50318 The results of contact forcewith the proportional switching law are shown in Figure 13(a)The corresponding control force is shown in Figure 13(b)From the results of contact force it is not very clear to seehow the controller contributes to decrease of the fluctuationin contact force The results of standard deviation of contactforce mean value of contact force and the correspondingmaximum control force using different control strategies areshown in Table 2 It can be clearly found that due to theeffect of the controller the standard deviation of contact forceexperiences a decrease For the constant switching law thebest result appears at u = 40NWhen u increases from 10N to40N the control effect increases accordingly However whenu exceeds 40N the control performance begins to becomeworse For the proportional switching law it can be foundthat from case 1 to case 3 the standard deviation experiencesa significant decrease which means an improvement of thecontrol performance Through the comparison between thetwo control laws it is found that the proportional switchinglaw has a better potential in decreasing the contact forcefluctuation The proportional switching law of case 3 canmake the standard deviation decrease to 2625 using themaximum control force of 6506N Through the analysis ofthe mean value of contact force it is found that the controller

cannot lead to a large change of the mean value of contactforce

From Figure 13 it can also be found that a large controlforce can only result in a small change of the contact forcefor instance at t = 278 s the control forces are approximately17N 31N and 58N with each of the cases which howeverlead to very small difference in the contact force (le10N)Thisphenomenon can be explained by the frequency responsebetween the control force and the contact force

As is well known the equation of motion for the panto-graph can be written as

1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(13)

In this discussion the effect of catenary on pantograph issimplified to simple time-varying stiffness 119870119904(119905) exerted onthe pantograph head So the contact force can be expressedby

119891119888 (119905) = 119870119904 (119905) 1199101 (119905) (14)

The equivalent stiffness 119870119904(119905) along the catenary can becalculated through applying a force moving along the contactwire which has been done in our previous work [7]

Substituting (14) to (13) the simplified equation ofmotionfor pantograph-catenary can be written as

1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119870119904 (119905) 1199101 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(15)

Through Laplace transformation of the above two equationsthe transfer function119867119906(119904) = 119865119888(119904)119865119906(119904) between the controlforce119865119906(119904) and contact force119865119888(119904) can be formulated As119870119904(119905)is time-varying its mean value (4000Nm) is adopted in thisdiscussion


Without controlCase 1 Case 3

Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826

(a) Contact forceCase 2Case 1 Case 3

26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force

Figure 13 Contact force and control force with proportional switching law

0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)

Figure 14 Frequency response between the control force 119865119906(119904) andcontact force 119865119888(119904)

The frequency response between the control force 119865119906(119904)and contact force 119865119888(119904) is shown in Figure 14 It can be seenthat the magnitude shows a sharp decrease with the increaseof the frequencyOndB scale when themagnitude is less than0 dB the response will be smaller than the inputThrough theFast Fourier Transform of the control force in Figure 13(b)the amplitudes of control force at different frequencies areshown in Figure 15 It can be seen that apart from the firstpeak appearing at 168Hz (whose amplitudes are only 123N82N and 5N resp in those three cases) almost all thedominant frequencies of control force are over 5Hz whoseeffect on contact force is definitely reduced by the pantographsystemThis analysis is consistent to our previous knowledgeof the pantograph which has a low natural frequency andserves as a low-pass filter for the disturbance transmittedfrom the bottom of pantograph to its head

Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)

Figure 15 Amplitude of control force at different frequencies

42 Evaluation of Control Performance with Wind LoadWhen the external disturbance is included the fluctuationin contact force would increase significantly and contactloss may occur So the control performance of the proposedcontroller is evaluated under stochastic wind field The windspeeds are chosen as 20ms and 30ms respectively Theangles of attack are chosen from minus60∘ to 0∘ The results ofstandard deviation of contact force with the wind speed of20ms and 30ms are shown in Figures 16(a) and 16(b)respectively The more closely the wind flows to the verticaldirection the larger the standard deviation of contact forcebecomes and the more significant the control performancebecomes Case 3 shows the best control performance indecreasing the fluctuation in contact force under different


0

Without controlCase 1Case 2

Case 3Constant switching law


25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)

minus10minus20minus30minus40minus50minus60

(a) Wind speed 20ms



0Angle of attack (∘)


40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms

Figure 16 Results of standard deviation of contact force with different wind speeds and angles of attack

15

1

05

0

Con

tact

loss

()

minus50 minus40 minus30 minus20 minus10 0minus60




(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms

Figure 17 Results of percentage of contact loss with different wind speeds and angles of attack

wind speeds and angles of attack For the comparisonwith theproportional switching law the constant switching controllerwith u = 40N is adopted in the simulation which shows thebest control performance among different constant switchinglaws in the above analysis It can be found that the constantswitching controller with u = 40N shows a better controlperformance than case 1 but worse control performance thancase 2 with each angle of attack

The strong wind load may cause the separation betweenthe pantograph and catenary The results of contact loss

percentage with different angles of attack under the windspeeds of 20ms and 30ms are shown in Figures 17(a) and17(b) respectively From Figure 17(a) it can be seen that thecontact loss can be totally eliminated by the proportionalswitching controller of case 2 and case 3 as well as theconstant switching controller with u = 40N under each angleof attackWhen thewind speed reaches 30ms themaximumpercentage of contact loss can increase to 142 It is obviousthat due to the effect of the controller the percentage ofcontact loss experiences a significant decline and drops to 0


Table 3 10 sets of control gains

Case 1 2 3 4 5 6 7 8 9 10120572 01 02 03 04 05 06 07 08 09 1120573 5 times 10minus5 1 times 10minus4 15 times 10minus4 2 times 10minus4 25 times 10minus4 3 times 10minus4 35 times 10minus4 4 times 10minus4 45 times 10minus4 5 times 10minus4

Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force

Figure 18 Results of contact force and control force with case 0 case 2 case 4 and case 8

even when the angle of attack increases by more than minus20∘The proportional switching controller of case 3 has the bestcontrol effect to reduce the contact loss compared with othercontrollers

In the following analyses the wind angle of minus60∘ andwind velocity of 30ms are adopted to study the effect of thecontroller on the contact force in more detail with more setsof control gains 10 sets of control gains are adopted for theanalysis in Table 3

The results of the contact force in case 0 case 2 case4 and case 8 are presented in Figure 18(a) Different fromthe nonsignificant effect of the controller in the aboveinvestigation without wind load (in Section 41) the controlperformance can be very clearly observed under the strongwind load displaying a significant reduction of the contactloss

The results of the corresponding control force of theproportional switching controllers are plotted in Figure 18(b)It can be found that a better control performance needs alarger consumption of the energy The largest control forceappears with case 8 When the wind speed reaches 30msthe maximum control force reaches more than 250N whichis more than twice as large as the static uplift force of thepantograph and may be difficult to implement in reality Soa proper set of control gains should be chosen accordingto the practical environmental situation and the limitationof the actuator On the other hand it can be observed thatthe controller introduces some negative effect on the contact

force which was not present originally This is because thecatenary system exhibits strong nonlinearity which is verydifficult to control As shown in Figure 18(a) during the timeduration of interest the performance of the controller is notvery good at a few isolated time instants However on thewhole the total fluctuation and contact loss are decreasedconsiderably

In order to make sense of the effect of the controlleron contact force in detail several indexes are introduced toevaluate the control performance which include the standarddeviation the percentage of contact loss the number ofoccurrences of contact loss the mean value of contact lossthemaximum interval of separation and themaximum valueof control force The results of standard deviation of contactforce and the maximum interval of separation are shownin Figures 19(a) and 19(b) respectively It can be seen thatthe increase of the control gains can decrease the standarddeviation of contact force which means a lower fluctuationin contact force Simultaneously the maximum interval ofseparation is effectively decreased by the controller withlarger control gains which can reduce the risk of interruptionof the electric transmission

The results of mean value of contact force and themaximumcontrol force are shown in Figures 20(a) and 20(b)respectively It can be seen that the mean value of contactforce is not largely affected by the controller which cannotlead to a significant additional wear of the contact wire Themaximum control force experiences a sharp increase as the


0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)

(a) Standard deviation

0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)

(b) Maximum interval of separation

Figure 19 The results of standard deviation of contact force and maximum interval of separation

Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)

(a) Mean value of contact force

11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0

(b) Maximum control force

Figure 20 The results of mean value of contact force and maximum control force

increase of the control gains (from case 0 to case 10) Incombination to the analysis of Figure 19 it can be found that abetter control performance requires a larger consumption ofthe control energy It should be noted that when the controlgains increase more than case 5 the maximum control forcebegins to be more than the value of uplift force of thepantograph which is quite difficult to be realized in reality

The results of percentage of contact loss and the numberof contact loss occurrences are shown in Figures 21(a) and21(b) respectively It can be seen that generally the contactloss and the number of its occurrences are decreased by thecontroller However the trends are not strictly consistent tothe results in Figure 19 In particular the controllers with case4 and case 6 have better performance in reducing the contactloss of the pantograph-catenary system compared to others

43 Comparison with an Existing Controller Reference [27]used a PID control strategy on EN 50318 reference model ofpantograph-catenary system based on a minimum variance

principle without any external disturbances and achieved a10 reduction in the contact force varianceThe controller inthat work was comprised of a positive proportion gain anda positive integration gain The maximum control force inthat strategy was limited to no more than 100N In orderto evaluate the efficiency of the proposed control strategy inthis paper the same reference model in EN 50318 is adoptedand the comparison of the control performance with [27]is conducted In the simulation the proportional switchinglaw is utilized and the control gains are chosen as 120572 = 09and 120573 = minus6 times 10minus4 which can guarantee that the range ofcontrol force is consistent with those in [27] The results ofthe contact force are shown in Figure 22(a) and those of thecontrol force are shown in Figure 19(b) A statistical analysisfinds that the contact force variance decreases from 1314N2(without control) to 940N2 (with control) The reduction ofthe contact force variance reaches more than 25 which issignificantly better than the previous result


Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)

(a) Percentage of contact lossCase

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce

(b) Number of contact loss occurrences

Figure 21 The results of Percentage of contact loss and number of contact loss occurrences

Time (s)

Without controlWith control

20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force

Figure 22 Results of the contact force and control force with EN 50318 reference model

5 Conclusions and Future Works

In this paper a pantograph-catenary interactionmodel undera strong stochastic wind field is constructed based on anonlinear finite element procedure In order to decrease thefluctuation in contact force and reduce the loss of contactunder a strong wind field a sliding mode control strategywith a PD sliding surface is presented The transfer functionbetween the contact force and the pantograph head veloc-ity (called mechanical impedance) is established A propersliding surface is defined to guarantee that the mechanicalimpedance of pantograph head decreases at the dominantfrequencies of the contact force when the sliding surfaceapproaches zero Then the proportional switching controllerwith several sets of control gains and the constant switchingcontroller are presented in the numerical simulations under

the stochastic wind field with different wind speeds andangles of attackThe results indicate that the proposed controlstrategy has a good performance in decreasing the contactforce fluctuation and the contact loss under strong wind fieldThe best results are obtained using the proportional switchinglaw with a set of large control gains However a bettercontrol performance needs a larger consumption of actuationenergy Through the comparison with the current literaturethe proposed control strategy shows a better performancein decreasing the contact force variance (25) compared toprevious method (10)

Competing Interests

The authors declare that they have no competing interests


Acknowledgments

This work was supported in part by the National NatureScience Foundation of China (U1434203 51377136 and51407147) Scientific Research and Development Programfor Railway Ministry (2013J010-B) Sichuan Province YouthScience andTechnology InnovationTeam (2016TD0012) andExcellent Doctoral Thesis Cultivation Program of SouthwestJiaotong University

References

[1] O Lopez-Garcia A Carnicero and V Torres ldquoComputationof the initial equilibrium of railway overheads based on thecatenary equationrdquo Engineering Structures vol 28 no 10 pp1387ndash1394 2006

[2] M Tur E Garcıa L Baeza and F J Fuenmayor ldquoA 3D absolutenodal coordinate finite element model to compute the initialconfiguration of a railway catenaryrdquo Engineering Structures vol71 pp 234ndash243 2014

[3] J Ambrosio J Pombo F Rauter et al ldquoA memory basedcommunication in the co-simulation of multibody and finiteelement codes for pantograph-catenary interaction simulationrdquoinMultibody Dynamics pp 231ndash252 Springer AmsterdamTheNetherlands 2009

[4] P Navik A Roslashnnquist and S Stichel ldquoIdentification of systemdamping in railway catenary wire systems from full-scalemeasurementsrdquo Engineering Structures vol 113 pp 71ndash78 2016

[5] H Wang Z Liu Y Song et al ldquoDetection of contact wireirregularities using a quadratic time-frequency representationof the pantograph-catenary contact forcerdquo IEEETransactions onInstrumentation and Measurement vol 65 no 6 pp 1385ndash13972016

[6] J H Lee T W Park H K Oh and Y G Kim ldquoAnalysis ofdynamic interaction between catenary and pantograph withexperimental verification and performance evaluation in newhigh-speed linerdquo Vehicle System Dynamics vol 53 no 8 pp1117ndash1134 2015

[7] Y Song Z Liu HWang X Lu and J Zhang ldquoNonlinear mod-elling of high-speed catenary based on analytical expressions ofcable and truss elementsrdquo Vehicle System Dynamics vol 53 no10 pp 1455ndash1479 2015

[8] A Carnicero J R Jimenez-Octavio C Sanchez-Rebollo ARamos and M Such ldquoInfluence of track irregularities in thecatenary-pantograph dynamic interactionrdquo Journal of Compu-tational andNonlinear Dynamics vol 7 no 4 Article ID 0410152012

[9] J Pombo J Ambrosio M Pereira F Rauter A Collinaand A Facchinetti ldquoInfluence of the aerodynamic forces onthe pantograph-catenary system for high-speed trainsrdquo VehicleSystem Dynamics vol 47 no 11 pp 1327ndash1347 2009

[10] Y Song Z Liu H Wang X Lu and J Zhang ldquoNonlinear anal-ysis of wind-induced vibration of high-speed railway catenaryand its influence on pantographndashcatenary interactionrdquo VehicleSystem Dynamics vol 54 no 6 pp 723ndash747 2016

[11] J-H Lee Y-G Kim J-S Paik and T-W Park ldquoPerformanceevaluation and design optimization using differential evolu-tionary algorithm of the pantograph for the high-speed trainrdquoJournal of Mechanical Science and Technology vol 26 no 10 pp3253ndash3260 2012

[12] J Ambrosio J Pombo and M Pereira ldquoOptimization of high-speed railway pantographs for improving pantograph-catenary

contactrdquoTheoretical and Applied Mechanics Letters vol 3 no 1Article ID 013006 2013

[13] A Roslashnnquist and P Navik ldquoDynamic assessment of existingsoft catenary systems using modal analysis to explore highertrain velocities a case study of aNorwegian contact line systemrdquoVehicle System Dynamics vol 53 no 6 pp 756ndash774 2015

[14] P Navik A Roslashnnquist and S Stichel ldquoThe use of dynamicresponse to evaluate and improve the optimization of existingsoft railway catenary systems for higher speedsrdquo Proceedings ofthe Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 230 no 4 pp 1388ndash1396 2015

[15] Z Liu P A Jonsson S Stichel and A Ronnquist ldquoImplicationsof the operation of multiple pantographs on the soft catenarysystems in Swedenrdquo Proceedings of the Institution of MechanicalEngineers Part F Journal of Rail and Rapid Transit vol 230 no3 pp 971ndash983 2016

[16] A Facchinetti L Gasparetto and S Bruni ldquoReal-time cate-nary models for the hardware-in-the-loop simulation of thepantographndashcatenary interactionrdquo Vehicle System Dynamicsvol 51 no 4 pp 499ndash516 2013

[17] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[18] Y Pi and H Ouyang ldquoLyapunov-based boundary control ofa multi-span beam subjected to moving massesrdquo Journal ofVibration and Control 2015

[19] D Stancioiu and H Ouyang ldquoOptimal vibration control ofbeams subjected to a mass moving at constant speedrdquo Journalof Vibration and Control vol 22 no 14 pp 3202ndash3217 2016

[20] Y-C Lin C-L Lin and C-C Yang ldquoRobust active vibrationcontrol for rail vehicle pantographrdquo IEEE Transactions onVehicular Technology vol 56 no 4 pp 1994ndash2004 2007

[21] Y-C Lin N-C Shieh and V-T Liu ldquoOptimal control for railvehicle pantograph systems with actuator delaysrdquo IET ControlTheory amp Applications vol 9 no 13 pp 1917ndash1926 2015

[22] A Rachid ldquoPantograph catenary control and observation usingthe LMI approachrdquo in Proceedings of the 50th IEEE Conferenceon Decision and Control and European Control Conference(CDC-ECC rsquo11) pp 2287ndash2292 IEEE Orlando Fla USADecember 2011

[23] D N OrsquoConnor S D Eppinger W P Seering and D NWormley ldquoActive control of a high-speed pantographrdquo Journalof Dynamic Systems Measurement and Control vol 119 no 1pp 1ndash4 1997

[24] A Pisano and E Usai ldquoOutput-feedback regulation of thecontact-force in high-speed train pantographsrdquo Journal ofDynamic Systems Measurement and Control vol 126 no 1 pp82ndash87 2004

[25] J Wang ldquoActive Control of Contact Force for a Pantograph-Catenary Systemrdquo Shock and Vibration vol 2016 Article ID2735297 7 pages 2016

[26] A Pisano and E Usai ldquoContact force regulation in wire-actuated pantographs via variable structure control andfrequency-domain techniquesrdquo International Journal ofControl vol 81 no 11 pp 1747ndash1762 2008

[27] C Sanchez-Rebollo J R Jimenez-Octavio and A CarniceroldquoActive control strategy on a catenary-pantograph validatedmodelrdquo Vehicle System Dynamics vol 51 no 4 pp 554ndash5692013


[28] C M Pappalardo M D Patel B Tinsley et al ldquoContact forcecontrol in multibody pantographcatenary systemsrdquo Proceed-ings of the Institution of Mechanical Engineers Part K Journalof Multi-Body Dynamics vol 230 no 4 pp 307ndash328 2016

[29] Y Pi and H Ouyang ldquoVibration control of beams subjected toa moving mass using a successively combined control methodrdquoAppliedMathematicalModelling vol 40 no 5-6 pp 4002ndash40152016

[30] S Bruni J Ambrosio A Carnicero et al ldquoThe results of thepantograph-catenary interaction benchmarkrdquo Vehicle SystemDynamics vol 53 no 3 pp 412ndash435 2015

[31] A G Davenport ldquoThe application of statistical concepts to thewind loading of structuresrdquo Proceedings of the Institution of CivilEngineers vol 19 no 4 pp 449ndash472 1961

[32] H A Panofsky and R A McCormick ldquoThe spectrum ofvertical velocity near the surfacerdquoQuarterly Journal of the RoyalMeteorological Society vol 86 no 370 pp 495ndash503 1960

[33] M Aboshi and K Manabe ldquoAnalyses of contact force fluctu-ation between catenary and pantographrdquo Quarterly Report ofRTRI vol 41 no 4 pp 182ndash187 2000

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Catenary

Pantograph

Messenger wire

Contact wire DropperSliding contact

Figure 1 Schematic of the pantograph-catenary system

procedure for catenary and multibody codes for pantographwere performed by Ambrosio et al [3] The damping ofcatenary was identified by Navik et al [4] using the timeseries sampled on existing catenaries An advanced detectionmethod for contact wire irregularities was proposed byWang et al [5] In order to exactly describe the geometricalnonlinearity of contactmessenger wire a nonlinear finiteelement procedure was performed based on ANCF [6] Andthe analytical formulations of nonlinear cable were utilizedto describe the large deformation of contactmessenger wireby Song et al [7] On the other hand in order to considerthe complex operation conditions of pantograph-catenarysystem Carnicero et al [8] analysed the effect of locomotivevibration on pantograph-catenary interaction Pombo et al[9] analysed the influence of aerodynamics on pantograph-catenary interaction Song et al [10] constructed the windfield along a catenary considering the stochastic wind fromdifferent directions and analysed the effect of stochastic windload on pantograph-catenary interaction The results of pre-vious studies indicated that the effect of locomotive vibrationon the contact force between catenary and pantograph waslimited In contrast the effect of strong stochastic wind loadwas very significant

Based on the study of pantograph-catenary dynamicsparameter optimization has been developed as a very generalway to improve the dynamic performance of pantograph-catenary systems A design optimization procedure for a pan-tographwas conducted by Lee et al [11] based on a sensitivityanalysis of the pantograph-catenary system Ambrosio et al[12] studied the suspension characteristics of a pantographhead and proposed a corresponding optimization strategyto decrease the fluctuation in contact force Roslashnnquist andNavik [13] used the frequency-analysis method to evaluatethe upgrade of a catenary and Navik et al [14] suggesteda new tension forces for a higher driving speed Consid-ering multiple-pantograph operation conditions Liu et al[15] studied the effect of interval between front and rearpantographs on dynamic performance

However a new design produced by the parame-ter optimization means the reconstruction of an existingpantograph-catenary system which incurs a big cost andsometimes may not be easy to realize in reality Besidesparameter optimization of catenary-pantograph systemscontrol of pantographs is another effective way to maintaina stable contact force Facchinetti et al [16] performed ahardware-in-the-loop experiment for an active pantographIn that work an actuator was placed in parallel with thecollector suspensions which could obtain better performance

but is not convenient to be realized in an existing railway lineGenerally an actuator can be installed on the lower frameof the pantograph which is more convenient to implementcompared to [16] and does not lead to a large change to theoriginal pantograph structure but poses a greater challenge tothe control algorithm because it is far away from the contactinterface

Generally the study of pantograph-catenary dynamicinteraction belongs to moving-load dynamics problems [17]Among them vibration suppression of beams excited bymoving loads has been of great interest to many scholars as ithas wide potentials in various engineering applications suchas bridge-vehicle interaction and vehicle-track interactionPi and Ouyang [18] used Lyapunov-based boundary controlmethod for suppressing the vibration of a multispan beamsubjected to moving masses Stancioiu and Ouyang [19]proposed an effective time-varying optimal control strategyfor a beam excited by a moving mass

However compared with the traditional moving-loadcontrol problems whose objectives are to decrease thevibration response the control objective of an active panto-graph is to reduce the fluctuation in contact force and thecontact loss between the pantograph collector and contactwire simultaneously without reducing the contact forcersquosmean value Considering the particularity of the pantograph-catenary system only few states of the pantograph and thecontact force can be detected which can be used as feedbackin a controller Recently various control algorithms for anactive pantograph were proposed by many experts [20ndash28]Normally due to the strong nonlinear effect of the catenaryon the pantograph it is very difficult to design the controllerbased on the whole complex catenary-pantograph modeldirectly In the stage of the controller design the catenary isalways simplified as a one degree-of-freedom (DOF) modelwith time-varying stiffness (shown in Figure 2(a)) to facilitatedevelopment of the control strategy Sometimes a time-varying mass is also included (shown in Figure 2(b)) InFigure 2 119896c(119905) and 119898c(119905) are the time-varying stiffness andmass of catenary 1198981 1198961 and 1198881 are the mass stiffness anddamping of the pantograph head respectively1198982 1198962 and 1198882are themass stiffness and damping of the pantograph framerespectively119891119888 is the pantograph-catenary contact force1198910 isthe static uplift force 119891119906 is the control force

Based on the simplified model shown in Figure 2(a)Lin et al [20] developed a control strategy for an activepantograph by LQR (linear quadratic regulator) Recentlythe robustness of an optimal control approach for an activepantograph was investigated considering the effect of actua-tor time delays [21] Rachid [22] developed an LMI (LinearMatrix Inequality) control law to keep the contact forceclose to a desired value OrsquoConnor et al [23] used thesteady state matrix Riccati equation to design a controller foran active pantograph which obtained over 50 reductionin the contact force variation Pisano and Usai [24] pro-posed an output feedback control strategy based on higher-order sliding modes and high-gain observers which had agood robustness and stability Considering the locomotivevibration Wang [25] developed a geometric framework tosuppress the vibration of pantograph-catenary system Based


Catenary

Pantograph

fc

k1

k2

m1

m2

c1

c2 fuf0

k＝(t)

(a)

Catenary

Pantograph

fc

k1

k2

m1

m2

c1

c2 fuf0

k＝(t)

m＝(t)

(b)











Contact wire

Wind

Pantograph



M119862ΔX119862 (119905) + C119862ΔX119862 (119905) + K119862 (119905) ΔX119862 (119905) = ΔF119862 (119909 119905) (1)



119891119888 = 119870119878 (1199101 minus 119910119888) 1199101 ge 1199101198880 1199101 lt 119910119888 (2)






Wind

D

o

FＬ F F９ F＄Ｌ

yＱＬ

F８

F＄

yＱ

zＱＬzＱ zＡ

yＡ



119865D = 12120588air1198802119871119863[119862D (1205720) sdot 2119906 (119905)119880

+ [ 119862D (1205720) minus 119862L (1205720)] sdot 119908 (119905)119880 ] + 1

2sdot 120588air1198802119863119871119862D (1205720)

(3a)

119865L = 12120588air1198802119871119863[119862L (1205720) sdot 2119906 (119905)119880

+ [ 119862L (1205720) + 119862D (1205720)] sdot 119908 (119905)119880 ] + 1

2sdot 120588air1198802119863119871119862L (1205720)

(3b)







119865Y = 119865D sin (1205720) + 119865L cos (1205720) (4b)




119899119878119911 (119899)11988121 = 4120581 1199092(1 + 1199092)43 (5)


119899119878119908 (119899 119911)11988121199111 = 11 119891(1 + 4119891)2 log1198902 (11991111199110) (6)





119878 (119905) = 119902p119890 (119905) + 119902d 119890 (119905) (7)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(8)


119891119888 (119905) = 1198910 + Δ119891119888 (119905) (9)



WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


095

1

105

11

115

12

125

13

135

14C

＄

(a) 119862D

WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

C

(b) 119862L



10 20 30 40 50 60 70 80 90 1000t (s)

minus15

minus10

minus5

0

5

10

15

(m

s)

10minus2

100

102

S(Ｇ

2s

)


Frequency (Hz)


10 20 30 40 50 60 70 80 90 1000t (s)



Frequency (Hz)

10minus2

10minus1

100

101

102

S(Ｇ

2s

)

minus6

minus4

minus2

0246

(m

s)





119867(119904) = Δ119865119888 (119904)1198811 (119904) = 119898111989821199044 + (11988821198981 + 11988811198981 + 11988811198982) 1199043 + (11989611198981 + 11989621198981 + 11989611198982 + 11988811198882) 1199042 + (11988811198962 + 11988821198961) 119904 + 11989611198962(11989821199042 + (1198881 + 1198882) 119904 + 1198961 + 1198962 minus (1198961 + 1198881119904) (119902119889119904 + 119902119901)) 119904 (10)




3 34 363228260

350

300

250

200

150

100

50

Time (s)

Con

tact

forc

e (N

)


U = 30

U = 20

= minus40

= minus10

(a) Contact force

5 10 15 20 25 30 35 40 45 50Frequency (Hz)

Catenary span

Dropper interval

Dominant frequency


10minus4

10minus2

100

102

104

Mag

nitu

de (d

B)

U = 30

U = 20

= minus40

= minus10










119891119906 (119905) = 119906 119878 (119905) gt 0minus119906 119878 (119905) lt 0 (11)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

qＪ = 0

qＪ = 2

qＪ = 4

qＪ = 8

qＪ = 14

qＪ = 20

100

101

102

103M

agni

tude

(dB)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

Mag

nitu

de (d

B)

104

102

103

101

q＞ = 0

q＞ = 002

q＞ = 001




05

1015

200

50

100

150

200

250

300

minus002minus004

minus006minus008

minus01 qＪ

050

100150200250300350400

Mag

nitu

de (d

B)

q＞




119891119906 (119905) = 1003816100381610038161003816120572119890 (119905) + 120573 119890 (119905)1003816100381610038161003816 sgn (119878 (119905)) (12)



5q＞ qＪ

34

33

32

31

30

29

28STD

of c

onta

ct fo

rce (

N)

Bad effect


minus001

minus0005

0 0

1015

20









200

180

160

140

120

100

80

60

40

20

Con

tact

forc

e (N

)

Time (s)


u = 40 Nu = 60 N

36343232826






1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(13)


119891119888 (119905) = 119870119904 (119905) 1199101 (119905) (14)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119870119904 (119905) 1199101 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(15)




Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826


26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force


0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)



Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)




0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Catenary

Pantograph

fc

k1

k2

m1

m2

c1

c2 fuf0

k＝(t)

(a)

Catenary

Pantograph

fc

k1

k2

m1

m2

c1

c2 fuf0

k＝(t)

m＝(t)

(b)











Contact wire

Wind

Pantograph



M119862ΔX119862 (119905) + C119862ΔX119862 (119905) + K119862 (119905) ΔX119862 (119905) = ΔF119862 (119909 119905) (1)



119891119888 = 119870119878 (1199101 minus 119910119888) 1199101 ge 1199101198880 1199101 lt 119910119888 (2)






Wind

D

o

FＬ F F９ F＄Ｌ

yＱＬ

F８

F＄

yＱ

zＱＬzＱ zＡ

yＡ



119865D = 12120588air1198802119871119863[119862D (1205720) sdot 2119906 (119905)119880

+ [ 119862D (1205720) minus 119862L (1205720)] sdot 119908 (119905)119880 ] + 1

2sdot 120588air1198802119863119871119862D (1205720)

(3a)

119865L = 12120588air1198802119871119863[119862L (1205720) sdot 2119906 (119905)119880

+ [ 119862L (1205720) + 119862D (1205720)] sdot 119908 (119905)119880 ] + 1

2sdot 120588air1198802119863119871119862L (1205720)

(3b)







119865Y = 119865D sin (1205720) + 119865L cos (1205720) (4b)




119899119878119911 (119899)11988121 = 4120581 1199092(1 + 1199092)43 (5)


119899119878119908 (119899 119911)11988121199111 = 11 119891(1 + 4119891)2 log1198902 (11991111199110) (6)





119878 (119905) = 119902p119890 (119905) + 119902d 119890 (119905) (7)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(8)


119891119888 (119905) = 1198910 + Δ119891119888 (119905) (9)



WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


095

1

105

11

115

12

125

13

135

14C

＄

(a) 119862D

WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

C

(b) 119862L



10 20 30 40 50 60 70 80 90 1000t (s)

minus15

minus10

minus5

0

5

10

15

(m

s)

10minus2

100

102

S(Ｇ

2s

)


Frequency (Hz)


10 20 30 40 50 60 70 80 90 1000t (s)



Frequency (Hz)

10minus2

10minus1

100

101

102

S(Ｇ

2s

)

minus6

minus4

minus2

0246

(m

s)





119867(119904) = Δ119865119888 (119904)1198811 (119904) = 119898111989821199044 + (11988821198981 + 11988811198981 + 11988811198982) 1199043 + (11989611198981 + 11989621198981 + 11989611198982 + 11988811198882) 1199042 + (11988811198962 + 11988821198961) 119904 + 11989611198962(11989821199042 + (1198881 + 1198882) 119904 + 1198961 + 1198962 minus (1198961 + 1198881119904) (119902119889119904 + 119902119901)) 119904 (10)




3 34 363228260

350

300

250

200

150

100

50

Time (s)

Con

tact

forc

e (N

)


U = 30

U = 20

= minus40

= minus10

(a) Contact force

5 10 15 20 25 30 35 40 45 50Frequency (Hz)

Catenary span

Dropper interval

Dominant frequency


10minus4

10minus2

100

102

104

Mag

nitu

de (d

B)

U = 30

U = 20

= minus40

= minus10










119891119906 (119905) = 119906 119878 (119905) gt 0minus119906 119878 (119905) lt 0 (11)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

qＪ = 0

qＪ = 2

qＪ = 4

qＪ = 8

qＪ = 14

qＪ = 20

100

101

102

103M

agni

tude

(dB)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

Mag

nitu

de (d

B)

104

102

103

101

q＞ = 0

q＞ = 002

q＞ = 001




05

1015

200

50

100

150

200

250

300

minus002minus004

minus006minus008

minus01 qＪ

050

100150200250300350400

Mag

nitu

de (d

B)

q＞




119891119906 (119905) = 1003816100381610038161003816120572119890 (119905) + 120573 119890 (119905)1003816100381610038161003816 sgn (119878 (119905)) (12)



5q＞ qＪ

34

33

32

31

30

29

28STD

of c

onta

ct fo

rce (

N)

Bad effect


minus001

minus0005

0 0

1015

20









200

180

160

140

120

100

80

60

40

20

Con

tact

forc

e (N

)

Time (s)


u = 40 Nu = 60 N

36343232826






1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(13)


119891119888 (119905) = 119870119904 (119905) 1199101 (119905) (14)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119870119904 (119905) 1199101 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(15)




Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826


26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force


0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)



Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)




0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Contact wire

Wind

Pantograph



M119862ΔX119862 (119905) + C119862ΔX119862 (119905) + K119862 (119905) ΔX119862 (119905) = ΔF119862 (119909 119905) (1)



119891119888 = 119870119878 (1199101 minus 119910119888) 1199101 ge 1199101198880 1199101 lt 119910119888 (2)






Wind

D

o

FＬ F F９ F＄Ｌ

yＱＬ

F８

F＄

yＱ

zＱＬzＱ zＡ

yＡ



119865D = 12120588air1198802119871119863[119862D (1205720) sdot 2119906 (119905)119880

+ [ 119862D (1205720) minus 119862L (1205720)] sdot 119908 (119905)119880 ] + 1

2sdot 120588air1198802119863119871119862D (1205720)

(3a)

119865L = 12120588air1198802119871119863[119862L (1205720) sdot 2119906 (119905)119880

+ [ 119862L (1205720) + 119862D (1205720)] sdot 119908 (119905)119880 ] + 1

2sdot 120588air1198802119863119871119862L (1205720)

(3b)







119865Y = 119865D sin (1205720) + 119865L cos (1205720) (4b)




119899119878119911 (119899)11988121 = 4120581 1199092(1 + 1199092)43 (5)


119899119878119908 (119899 119911)11988121199111 = 11 119891(1 + 4119891)2 log1198902 (11991111199110) (6)





119878 (119905) = 119902p119890 (119905) + 119902d 119890 (119905) (7)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(8)


119891119888 (119905) = 1198910 + Δ119891119888 (119905) (9)



WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


095

1

105

11

115

12

125

13

135

14C

＄

(a) 119862D

WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

C

(b) 119862L



10 20 30 40 50 60 70 80 90 1000t (s)

minus15

minus10

minus5

0

5

10

15

(m

s)

10minus2

100

102

S(Ｇ

2s

)


Frequency (Hz)


10 20 30 40 50 60 70 80 90 1000t (s)



Frequency (Hz)

10minus2

10minus1

100

101

102

S(Ｇ

2s

)

minus6

minus4

minus2

0246

(m

s)





119867(119904) = Δ119865119888 (119904)1198811 (119904) = 119898111989821199044 + (11988821198981 + 11988811198981 + 11988811198982) 1199043 + (11989611198981 + 11989621198981 + 11989611198982 + 11988811198882) 1199042 + (11988811198962 + 11988821198961) 119904 + 11989611198962(11989821199042 + (1198881 + 1198882) 119904 + 1198961 + 1198962 minus (1198961 + 1198881119904) (119902119889119904 + 119902119901)) 119904 (10)




3 34 363228260

350

300

250

200

150

100

50

Time (s)

Con

tact

forc

e (N

)


U = 30

U = 20

= minus40

= minus10

(a) Contact force

5 10 15 20 25 30 35 40 45 50Frequency (Hz)

Catenary span

Dropper interval

Dominant frequency


10minus4

10minus2

100

102

104

Mag

nitu

de (d

B)

U = 30

U = 20

= minus40

= minus10










119891119906 (119905) = 119906 119878 (119905) gt 0minus119906 119878 (119905) lt 0 (11)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

qＪ = 0

qＪ = 2

qＪ = 4

qＪ = 8

qＪ = 14

qＪ = 20

100

101

102

103M

agni

tude

(dB)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

Mag

nitu

de (d

B)

104

102

103

101

q＞ = 0

q＞ = 002

q＞ = 001




05

1015

200

50

100

150

200

250

300

minus002minus004

minus006minus008

minus01 qＪ

050

100150200250300350400

Mag

nitu

de (d

B)

q＞




119891119906 (119905) = 1003816100381610038161003816120572119890 (119905) + 120573 119890 (119905)1003816100381610038161003816 sgn (119878 (119905)) (12)



5q＞ qＪ

34

33

32

31

30

29

28STD

of c

onta

ct fo

rce (

N)

Bad effect


minus001

minus0005

0 0

1015

20









200

180

160

140

120

100

80

60

40

20

Con

tact

forc

e (N

)

Time (s)


u = 40 Nu = 60 N

36343232826






1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(13)


119891119888 (119905) = 119870119904 (119905) 1199101 (119905) (14)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119870119904 (119905) 1199101 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(15)




Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826


26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force


0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)



Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)




0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119865Y = 119865D sin (1205720) + 119865L cos (1205720) (4b)




119899119878119911 (119899)11988121 = 4120581 1199092(1 + 1199092)43 (5)


119899119878119908 (119899 119911)11988121199111 = 11 119891(1 + 4119891)2 log1198902 (11991111199110) (6)





119878 (119905) = 119902p119890 (119905) + 119902d 119890 (119905) (7)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(8)


119891119888 (119905) = 1198910 + Δ119891119888 (119905) (9)



WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


095

1

105

11

115

12

125

13

135

14C

＄

(a) 119862D

WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

C

(b) 119862L



10 20 30 40 50 60 70 80 90 1000t (s)

minus15

minus10

minus5

0

5

10

15

(m

s)

10minus2

100

102

S(Ｇ

2s

)


Frequency (Hz)


10 20 30 40 50 60 70 80 90 1000t (s)



Frequency (Hz)

10minus2

10minus1

100

101

102

S(Ｇ

2s

)

minus6

minus4

minus2

0246

(m

s)





119867(119904) = Δ119865119888 (119904)1198811 (119904) = 119898111989821199044 + (11988821198981 + 11988811198981 + 11988811198982) 1199043 + (11989611198981 + 11989621198981 + 11989611198982 + 11988811198882) 1199042 + (11988811198962 + 11988821198961) 119904 + 11989611198962(11989821199042 + (1198881 + 1198882) 119904 + 1198961 + 1198962 minus (1198961 + 1198881119904) (119902119889119904 + 119902119901)) 119904 (10)




3 34 363228260

350

300

250

200

150

100

50

Time (s)

Con

tact

forc

e (N

)


U = 30

U = 20

= minus40

= minus10

(a) Contact force

5 10 15 20 25 30 35 40 45 50Frequency (Hz)

Catenary span

Dropper interval

Dominant frequency


10minus4

10minus2

100

102

104

Mag

nitu

de (d

B)

U = 30

U = 20

= minus40

= minus10










119891119906 (119905) = 119906 119878 (119905) gt 0minus119906 119878 (119905) lt 0 (11)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

qＪ = 0

qＪ = 2

qＪ = 4

qＪ = 8

qＪ = 14

qＪ = 20

100

101

102

103M

agni

tude

(dB)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

Mag

nitu

de (d

B)

104

102

103

101

q＞ = 0

q＞ = 002

q＞ = 001




05

1015

200

50

100

150

200

250

300

minus002minus004

minus006minus008

minus01 qＪ

050

100150200250300350400

Mag

nitu

de (d

B)

q＞




119891119906 (119905) = 1003816100381610038161003816120572119890 (119905) + 120573 119890 (119905)1003816100381610038161003816 sgn (119878 (119905)) (12)



5q＞ qＪ

34

33

32

31

30

29

28STD

of c

onta

ct fo

rce (

N)

Bad effect


minus001

minus0005

0 0

1015

20









200

180

160

140

120

100

80

60

40

20

Con

tact

forc

e (N

)

Time (s)


u = 40 Nu = 60 N

36343232826






1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(13)


119891119888 (119905) = 119870119904 (119905) 1199101 (119905) (14)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119870119904 (119905) 1199101 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(15)




Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826


26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force


0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)



Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)




0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


095

1

105

11

115

12

125

13

135

14C

＄

(a) 119862D

WindWind

ＭＪ＞ = 20 msＭＪ＞ = 30 ms


minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

C

(b) 119862L



10 20 30 40 50 60 70 80 90 1000t (s)

minus15

minus10

minus5

0

5

10

15

(m

s)

10minus2

100

102

S(Ｇ

2s

)


Frequency (Hz)


10 20 30 40 50 60 70 80 90 1000t (s)



Frequency (Hz)

10minus2

10minus1

100

101

102

S(Ｇ

2s

)

minus6

minus4

minus2

0246

(m

s)





119867(119904) = Δ119865119888 (119904)1198811 (119904) = 119898111989821199044 + (11988821198981 + 11988811198981 + 11988811198982) 1199043 + (11989611198981 + 11989621198981 + 11989611198982 + 11988811198882) 1199042 + (11988811198962 + 11988821198961) 119904 + 11989611198962(11989821199042 + (1198881 + 1198882) 119904 + 1198961 + 1198962 minus (1198961 + 1198881119904) (119902119889119904 + 119902119901)) 119904 (10)




3 34 363228260

350

300

250

200

150

100

50

Time (s)

Con

tact

forc

e (N

)


U = 30

U = 20

= minus40

= minus10

(a) Contact force

5 10 15 20 25 30 35 40 45 50Frequency (Hz)

Catenary span

Dropper interval

Dominant frequency


10minus4

10minus2

100

102

104

Mag

nitu

de (d

B)

U = 30

U = 20

= minus40

= minus10










119891119906 (119905) = 119906 119878 (119905) gt 0minus119906 119878 (119905) lt 0 (11)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

qＪ = 0

qＪ = 2

qＪ = 4

qＪ = 8

qＪ = 14

qＪ = 20

100

101

102

103M

agni

tude

(dB)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

Mag

nitu

de (d

B)

104

102

103

101

q＞ = 0

q＞ = 002

q＞ = 001




05

1015

200

50

100

150

200

250

300

minus002minus004

minus006minus008

minus01 qＪ

050

100150200250300350400

Mag

nitu

de (d

B)

q＞




119891119906 (119905) = 1003816100381610038161003816120572119890 (119905) + 120573 119890 (119905)1003816100381610038161003816 sgn (119878 (119905)) (12)



5q＞ qＪ

34

33

32

31

30

29

28STD

of c

onta

ct fo

rce (

N)

Bad effect


minus001

minus0005

0 0

1015

20









200

180

160

140

120

100

80

60

40

20

Con

tact

forc

e (N

)

Time (s)


u = 40 Nu = 60 N

36343232826






1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(13)


119891119888 (119905) = 119870119904 (119905) 1199101 (119905) (14)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119870119904 (119905) 1199101 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(15)




Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826


26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force


0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)



Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)




0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










3 34 363228260

350

300

250

200

150

100

50

Time (s)

Con

tact

forc

e (N

)


U = 30

U = 20

= minus40

= minus10

(a) Contact force

5 10 15 20 25 30 35 40 45 50Frequency (Hz)

Catenary span

Dropper interval

Dominant frequency


10minus4

10minus2

100

102

104

Mag

nitu

de (d

B)

U = 30

U = 20

= minus40

= minus10










119891119906 (119905) = 119906 119878 (119905) gt 0minus119906 119878 (119905) lt 0 (11)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

qＪ = 0

qＪ = 2

qＪ = 4

qＪ = 8

qＪ = 14

qＪ = 20

100

101

102

103M

agni

tude

(dB)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

Mag

nitu

de (d

B)

104

102

103

101

q＞ = 0

q＞ = 002

q＞ = 001




05

1015

200

50

100

150

200

250

300

minus002minus004

minus006minus008

minus01 qＪ

050

100150200250300350400

Mag

nitu

de (d

B)

q＞




119891119906 (119905) = 1003816100381610038161003816120572119890 (119905) + 120573 119890 (119905)1003816100381610038161003816 sgn (119878 (119905)) (12)



5q＞ qＪ

34

33

32

31

30

29

28STD

of c

onta

ct fo

rce (

N)

Bad effect


minus001

minus0005

0 0

1015

20









200

180

160

140

120

100

80

60

40

20

Con

tact

forc

e (N

)

Time (s)


u = 40 Nu = 60 N

36343232826






1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(13)


119891119888 (119905) = 119870119904 (119905) 1199101 (119905) (14)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119870119904 (119905) 1199101 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(15)




Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826


26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force


0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)



Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)




0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 4 6 8 10 12 14 16 18 20Frequency (Hz)

qＪ = 0

qＪ = 2

qＪ = 4

qＪ = 8

qＪ = 14

qＪ = 20

100

101

102

103M

agni

tude

(dB)


2 4 6 8 10 12 14 16 18 20Frequency (Hz)

Mag

nitu

de (d

B)

104

102

103

101

q＞ = 0

q＞ = 002

q＞ = 001




05

1015

200

50

100

150

200

250

300

minus002minus004

minus006minus008

minus01 qＪ

050

100150200250300350400

Mag

nitu

de (d

B)

q＞




119891119906 (119905) = 1003816100381610038161003816120572119890 (119905) + 120573 119890 (119905)1003816100381610038161003816 sgn (119878 (119905)) (12)



5q＞ qＪ

34

33

32

31

30

29

28STD

of c

onta

ct fo

rce (

N)

Bad effect


minus001

minus0005

0 0

1015

20









200

180

160

140

120

100

80

60

40

20

Con

tact

forc

e (N

)

Time (s)


u = 40 Nu = 60 N

36343232826






1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(13)


119891119888 (119905) = 119870119904 (119905) 1199101 (119905) (14)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119870119904 (119905) 1199101 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(15)




Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826


26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force


0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)



Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)




0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













200

180

160

140

120

100

80

60

40

20

Con

tact

forc

e (N

)

Time (s)


u = 40 Nu = 60 N

36343232826






1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119891119888 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(13)


119891119888 (119905) = 119870119904 (119905) 1199101 (119905) (14)



1198981 d21199101 (119905)d1199052 + 1198881 (d1199101 (119905)d119905 minus d1199102 (119905)

d119905 )+ 1198961 (1199101 (119905) minus 1199102 (119905)) = minus119870119904 (119905) 1199101 (119905)

1198982 d21199102 (119905)d1199052 + 1198882 (d1199102 (119905)d119905 minus d1199101 (119905)

d119905 ) + 1198881 d1199102 (119905)d119905+ 1198962 (1199102 (119905) minus 1199101 (119905)) + 11989611199102 (119905) = 1198910 + 119891119906 (119905)

(15)




Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826


26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force


0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)



Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)




0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Case 2

Time (s)

20

40

60

80

100

120

140

160

180

200

Con

tact

forc

e (N

)

36343232826


26 28 3 32 34 36

Time (s)

minus80

minus60

minus40

minus20

0

20

40

60

Con

trol f

orce

(N)

(b) Control force


0 5 10 15 20 25 30Frequency (Hz)

minus40

minus30

minus20

minus10

0

10

20

Mag

nitu

de (d

B)



Case 1Case 2

Case 3

5 10 15 20 25 30 35 400Frequency (Hz)

0

2

4

6

8

10

12

14

Am

plitu

de o

f con

trol f

orce

(N)




0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0




25

30

35

40

45

50

55

60

65St

anda

rd d

evia

tion

of co

ntac

t for

ce (N

)


(a) Wind speed 20ms





40

50

60

70

80

90

100

110

120

130

Stan

dard

dev

iatio

n of

cont

act f

orce

(N)

(b) Wind speed 30ms


15

1

05

0

Con

tact

loss

()





(a) Wind speed 20ms





0

2

4

6

8

10

12

14

16

Con

tact

loss

()

(b) Wind speed 30ms








Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Time (s)

Case 0Case 2Case 4


0

50

100

150

200

250

300

350

400

450

Con

tact

forc

e (N

)

36343232826

(a) Contact force

minus300

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Con

trol f

orce

(N)

Time (s)

Case 2Case 4Case 8

36343232826

(b) Control force










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5 6 7 8 9 10Case

140

120

100

80

60

40

20

0

Stan

dard

dev

iatio

n (N

)


0 1 2 3 4 5 6 7 8 9 10Case

012

01

008

006

004

002

0

Max

imum

inte

rval

of s

epar

atio

n (m

)



Case0 1 2 3 4 5 6 7 8 9 10

140

120

100

80

60

40

20

0

Mea

n va

lue o

f con

tact

forc

e (N

)


11Case

1 2 3 4 5 6 7 8 9 10

Max

imum

cont

rol f

orce

(N)

350

300

250

200

150

100

50

0








Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Case0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16Pe

rcen

tage

of c

onta

ct lo

ss (

)


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Num

ber o

f con

tact

loss

occ

urre

nce



Time (s)


20

40

60

80

100

120

140

160

180

200

220

Con

tact

forc

e (N

)

424383634323

(a) Contact force

Con

trol f

orce

(N)

Time (s)424383634323

100

80

60

40

20

0

minus20

minus40

minus60

minus80

(b) Control force





Competing Interests



Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









sliding mode control with pd sliding surface for high-speed

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