UC Berkeley UC Berkeley Previously Published Works Title The effect of high-frequency parametric excitation on a stochastically driven pantograph- catenary system Permalink https://escholarship.org/uc/item/4nh8m103 Journal Shock and Vibration, 2014 ISSN 1070-9622 Authors Huan, RH Zhu, WQ Ma, F et al. Publication Date 2014 DOI 10.1155/2014/792673 Peer reviewed eScholarship.org Powered by the California Digital Library University of California Hindawi Publishing Corporation Shock and Vibration Volume 2014, Article ID 792673, 8 pages http://dx.doi.org/10.1155/2014/792673 Research Article The Effect of High-Frequency Parametric Excitation on a Stochastically Driven Pantograph-Catenary System R. H. Huan,1 W. Q. Zhu,1 F. Ma,2 and Z. H. Liu3 1 Department of Mechanics, Zhejiang University, Hangzhou 310027, China 2 Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA 3 Department of Civil Engineering, Xiamen University, Xiamen 361005, China Correspondence should be addressed to Z. H. Liu; [email protected] Received 5 June 2013; Accepted 7 August 2013; Published 11 February 2014 Academic Editor: Reza Jazar Copyright © 2014 R. H. Huan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In high-speed electric trains, a pantograph is mounted on the roof of the train to collect power through contact with an overhead catenary wire. The effect of fast harmonic and parametric excitation on a stochastically driven pantograph-catenary system is studied in this paper. A single-degree-of-freedom model of the pantograph-catenary system is adopted, wherein the stiffness of the nonlinear spring has a time-varying component characterized by both low and high frequencies. Using perturbation and harmonic averaging, a Fokker-Planck-Kolmogorov equation governing the stationary response of the pantograph-catenary system is set up. Based on the transition probability density of the stationary response, it is found that even small high-frequency parametric excitation has an appreciable effect on the system response. ngAmo other things, it shifts the resonant frequency and often changes the response characteristics markedly. 1. Introduction been accepted that parametrically induced vibration of a pantograph-catenary system occurs mainly in the low- Anumberofimportantstructurescanbemodeledasa frequency region. Thus earlier studies usually ignored the stochastically driven nonlinear system subjected to both slow high-frequency parametric effect generated by the catenary and fast harmonic and parametric excitations. An example [7, 8]. Recent theoretical studies on dynamical systems, is the pantograph-catenary system in railway engineering. however, suggest that high-frequency parametric excitation High-speed electric trains often employ a pantograph to could shift the resonant frequency or equilibrium states [9– collect their currents from an overhead catenary system. 12], thus altering the stability [13–15]andotherresponse During operation, the pantograph is excited by forces due to train-body vibration, ambient air flow, contact wire characteristics [16]. irregularities, and other disturbances. These disturbances In this work, the effect of fast harmonic and parametric can be realistically taken as a stochastic excitation to the excitation on a stochastically driven pantograph-catenary pantograph. Owing to the stiffness variation between the system is investigated. A nonlinear single-degree-of-freedom support poles and the short-distance droppers, the catenary model of the pantograph-catenary system possessing low- may be regarded as a nonlinear spring with a time-varying and high-frequency time-varying stiffness is adopted. Using stiffness component. As a result, the combined pantograph- perturbation, an approximate equation governing only the catenary system is an example of a stochastically driven non- low-frequency motion is derived. Then, an averaging method linear system with both low- and high-frequency parametric based on harmonic functions [17–21] is applied to the excitations. low-frequency equation. Subsequently, a Fokker-Planck- There is fairly extensive literature on the dynamics Kolmogorov equation governing the stationary response of of pantograph-catenary systems [1–6]. However, previous the pantograph-catenary system is set up. Based on the studies have focused on deterministic excitations. It has transition probability density of the stationary response, the 2 Shock and Vibration random excitation () is introduced into the equation of the pantograph-catenary system. Incorporating stiffness nonlinearity, the equation of motion of the pantograph- K(Ω ,Ω ,t) s f catenary system can be written as ̃ 3 2 +̈ +̇ + +( − ) m (4) ×(1+cos (Ω) + cos (Ω)) = () , c where is the vertical displacement of the pantograph system and the fourth item in the left of4 ( ) represents the interaction of pantograph and catenary system. 2 2 Let = / and = where =+ − and is a nondimensional time. Equation (4)canbeconvertedinto Figure 1: Dynamic model of pantograph-catenary system. the nondimensional form +2 ++3 effectoffastparametricexcitationontheresonantfrequency 2 and the primary resonant response are studied. Finally, direct =[−11 +12 ] cos () (5) numerical simulations of the nonlinear model are performed 2 to validate the analysis presented. +[−21 +22] cos () 0 +ℎ () , 2 2 2. Model of Pantograph-Catenary System where =/,=/ ,and A commonly used model in railway engineering of the ̃ 2 = ,= ,= , pantograph-catenary system [3, 7, 8] is the single-degree- 2 11 2 of-freedom model shown in Figure 1.Sincethestiffness variationisrepeatedineachspanofthecatenary,thecatenary 2 = ( ) ,= , stiffness possesses periodic components. If we consider the 12 21 2 2 stiffness variation between the vertical droppers, the catenary (6) is usually taken as a spring with time-varying stiffness 2 = ( ) ,ℎ= , components given by [1, 7, 8] 22 0 2 2 (Ω,Ω,)=0 () (1 + cos (Ω) + cos (Ω)) , Ω Ω 2 2 (1) = = , = = . () where represents the speed of train and 0 is the speed- = (1) dependent average stiffness of the catenary. In the above Here is the nondimensional low-frequency excitation, and ≫1is the nondimensional high-frequency equation, , are unspecified coefficients and cos(Ω), excitation. If a train is travelling at a constant speed ,then cos(Ω) account for the stiffness fluctuations between the 2 2 2,11 , 12 , 21, 22 ,andℎ0 are all small constants. support poles and the vertical droppers, respectively. Let and denote, respectively, the span distance and dropper Suppose these constants are of the same order in a small distance. Then the frequencies of stiffness fluctuation can be parameter .System(5)isarandomlyexcitedDuffing expressed as oscillator with both slow and fast time-varying stiffness. The goal of the present work is to investigate the influence of the 2 2 fast parametric excitation on the characteristics of system (5). Ω = ;Ω = . (2) In general, ≫ and therefore Ω ≫Ω.Inmany 3. Approximate Equation Governing cases, the average stiffness 0() can be approximated by a the Slow Motion quadratic function of the train speed [8]suchthat Introduce two different time-scales: () = − 2, 0 (3) −1 0 == ;1 =, (7) where is the static average stiffness of the catenary and is a coefficient accounting for dynamic interactions of the where the slow time 1 and the fast time 0 are considered as coupled pantograph-catenary system. The value of can be new independent variables in (5). Separate () into a slow calculated by the finite element method. part (1) and a fast part (0,1) [10]sothat Due to random disturbances (such as train body move- ment, contact wire irregularity, and ambient air flow), a weak () =(1)+(0,1). (8) Shock and Vibration 3 It has been recognized that the behavior of system (5)is 3.0 mainly described by the slow part since is small 2 2 ⟨⋅⟩ =(∫ ⋅)/2 2.5 1 compared to .Let 0 0 0 be the time-ave- raging operator over one period of the fast time scale 2.0 0 with the slow time 1 fixed. Assume that and its ⟨()⟩ = derivatives vanish upon 0-averaging so that 0 ) a 1.5 (1).Substitute(8)into(5)toobtain ( p 2 2 −1 2 1+(1 +201 + 0)+2(1+0+1) 1.0 3 +(+)+(+) 0.5 =[− + 2](+) ( ) 11 12 cos 1 0.0 01 −1 ×[−21 + 22](+)cos (0)+ℎ0(1), a (9) Figure 2: Probability densities of amplitude of the original system (5)andoftheslowsystem(12). Line 1: no fast excitation; line 2: 22 ⋅ where =/ ( = 0,1; = 1,2).Averageequation = 0.8.Solidlinesfortheoriginalsystem(5) and dotted lines for (9)withrespectto0 and subtract the averaged equation the slow system (12). from (9); an approximate expression for is obtained by −1 considering only the dominant terms of order as 2 0=22 cos (0). (10) The periodic solution of system (13)hastheform[17] The stationary solution to first order for is (1)=cos (1), 1(1)=−] (,) sin (1), =− ( ). 22 cos 0 (11) (1)=(1)+, (14) Substitute (11)into(9) and apply 0-averaging. Retain domi- 0 nant terms of order to obtain where is the amplitude, is the phase angle, and 2 (22) 2 [ ] 3 2 +21+ 1+ + 3 1 2 ] (,) = = √(2 + )(1+ (2)), [ ] (12) cos 1 4 =[− + 2] ( )+ℎ ( ). (15) 11 12 cos 1 0 1 (2/4) = . Equation (12)governsonlytheslowmotion of system (2 +32/4) (5). Note that the fast excitation affects the slow behavior of 2 system (12) by adding (22) /2 to the linear stiffness. By The instantaneous frequency ](, ) can be approximated by numerical simulations, the probability densities () of the the finite sum: amplitude of the original system (5) and of the slow system (12)areplottedinFigure2. It is observed that the larger the ] (,) =0 () +2 () cos 2 +4 () cos 4 +6 () cos 6, fast excitation parameter 22 ⋅, the bigger the difference (16) between the two amplitudes. where 4. Effect of Fast Parametric Excitation 32 2 () = √(2 + )(1− ) Inthelastsection,anapproximateequationgoverningonly 0 4 16 the slow motion of system (5) is obtained by perturbation. In thefollowing,wewilldiscusstheeffectofthefastharmonic 32 33 () = √(2 + )( + ) excitation on this slow system in greater detail.