Research Article the Effect of High-Frequency Parametric Excitation on a Stochastically Driven Pantograph-Catenary System

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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 𝑐𝑤𝜉(𝑡) 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 𝑚𝑦+𝑐̈ 𝑦+𝑘̇ 𝑦+𝑘𝑦 +(𝐾𝑠 −𝐾𝑑𝑉 ) m (4) ×(1+𝛼cos (Ω𝑠𝑡) + 𝛽 cos (Ω𝑓𝑡)) 𝑦𝑤 =𝑐 𝜉 (𝑡) ,

𝑦 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 𝜔𝑛 = 𝑘/𝑚 and 𝜏=𝜔𝑛𝑡 where 𝑘=𝑘+𝐾𝑠 −𝐾𝑑𝑉 and 𝜏 is a nondimensional time. Equation (4)canbeconvertedinto Figure 1: Dynamic model of pantograph-catenary system. the nondimensional form 𝑦󸀠󸀠 +2𝜁𝑦󸀠 +𝑦+𝑏𝑦3 effectoffastparametricexcitationontheresonantfrequency 2 and the primary resonant response are studied. Finally, direct =[−𝐸11 +𝐸12𝑟𝑠 ] cos (𝑟𝑠𝜏) 𝑦 (5) numerical simulations of the nonlinear model are performed 2 to validate the analysis presented. +[−𝐸21 +𝐸22𝑟𝑓] cos (𝑟𝑓𝜏) 𝑦0 +ℎ 𝜉 (𝜏) ,

󸀠 󸀠󸀠 2 2 2. Model of Pantograph-Catenary System where 𝑦 =𝑑𝑦/𝑑𝜏,𝑦=𝑑𝑦/𝑑𝜏 ,and

A commonly used model in railway engineering of the 𝑐 ̃𝑘 𝛼𝐾 2𝜁 = ,𝑏= ,𝐸= 𝑠 , pantograph-catenary system [3, 7, 8] is the single-degree- 𝑚𝜔 𝑚𝜔2 11 𝑚𝜔2 of-freedom model shown in Figure 1.Sincethestiffness 𝑛 𝑛 𝑛 variationisrepeatedineachspanofthecatenary,thecatenary 𝛼𝐾 𝐿 2 𝛽𝐾 𝐸 = 𝑑 ( 𝑠 ) ,𝐸= 𝑠 , stiffness possesses periodic components. If we consider the 12 21 2 𝑚 2𝜋 𝑚𝜔𝑛 stiffness variation between the vertical droppers, the catenary (6) is usually taken as a spring with time-varying stiffness 𝛽𝐾 𝐿 2 𝑐 𝐸 = 𝑑 ( 𝑑 ) ,ℎ= 𝑤 , components given by [1, 7, 8] 22 0 2 𝑚 2𝜋 𝑚𝜔𝑛

𝐾(Ω𝑠,Ω𝑓,𝑡)=𝐾0 (𝑉) (1 + 𝛼 cos (Ω𝑠𝑡) + 𝛽 cos (Ω𝑓𝑡)) , Ω Ω𝑠 2𝜋𝑉 𝑓 2𝜋𝑉 (1) 𝑟𝑠 = = ,𝑟𝑓 = = . 𝜔𝑛 𝜔𝑛𝐿𝑠 𝜔𝑛 𝜔𝑛𝐿𝑑 𝑉 𝐾 (𝑉) where represents the speed of train and 0 is the speed- 𝑟 = 𝑂(1) dependent average stiffness of the catenary. In the above Here 𝑠 is the nondimensional low-frequency excitation, and 𝑟𝑓 ≫1is the nondimensional high-frequency equation, 𝛼, 𝛽 are unspecified coefficients and 𝛼 cos(Ω𝑠𝑡), excitation. If a train is travelling at a constant speed 𝑉,then 𝛽 cos(Ω𝑓𝑡) account for the stiffness fluctuations between the 2 2 2𝜁,11 𝐸 , 𝐸12𝑟 , 𝐸21, 𝐸22𝑟 ,andℎ0 are all small constants. support poles and the vertical droppers, respectively. Let 𝐿𝑠 𝑠 𝑓 and 𝐿𝑑 denote, respectively, the span distance and dropper Suppose these constants are of the same order in a small 𝜀 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 2𝜋𝑉 2𝜋𝑉 fast parametric excitation on the characteristics of system (5). Ω𝑠 = ;Ω𝑓 = . (2) 𝐿𝑠 𝐿𝑑

In general, 𝐿𝑠 ≫𝐿𝑑 and therefore Ω𝑓 ≫Ω𝑠.Inmany 3. Approximate Equation Governing cases, the average stiffness 𝐾0(𝑉) can be approximated by a the Slow Motion quadratic function of the train speed [8]suchthat Introduce two different time-scales: 𝐾 (𝑉) =𝐾 −𝐾 𝑉2, 0 𝑠 𝑑 (3) −1 𝑇0 =𝑟𝑓𝜏=𝜀 𝜏;1 𝑇 =𝜏, (7) where 𝐾𝑠 is the static average stiffness of the catenary and 𝐾𝑑 is a coefficient accounting for dynamic interactions of the where the slow time 𝑇1 and the fast time 𝑇0 are considered as coupled pantograph-catenary system. The value of 𝐾𝑠 can be new independent variables in (5). Separate 𝑦(𝜏) into a slow calculated by the finite element method. part 𝑤(𝑇1) and a fast part 𝜀𝜙(𝑇0,𝑇1) [10]sothat Due to random disturbances (such as train body move- ment, contact wire irregularity, and ambient air flow), a weak 𝑦 (𝜏) =𝑤(𝑇1)+𝜀𝜑(𝑇0,𝑇1). (8) Shock and Vibration 3

It has been recognized that the behavior of system (5)is 3.0 mainly described by the slow part 𝑤 since 𝜀𝜙 is small 2 2𝜋 𝑤 ⟨⋅⟩ =(∫ ⋅𝑑𝑇)/2𝜋 2.5 1 compared to .Let 𝑇0 0 0 be the time-averaging operator over one period of the fast time scale 2.0 𝑇0 with the slow time 𝑇1 fixed. Assume that 𝜀𝜙 and its 𝑇 ⟨𝑦(𝜏)⟩ = derivatives vanish upon 0-averaging so that 𝑇0 )

a 1.5 𝑤(𝑇1).Substitute(8)into(5)toobtain ( p 2 2 −1 2 𝐷1𝑤+(𝜀𝐷1 +2𝐷0𝐷1 +𝜀 𝐷0)𝜑+2𝜁(𝐷1𝑤+𝐷0𝜑+𝜀𝐷1𝜑) 1.0

3 +(𝑤+𝜀𝜑)+𝑏(𝑤+𝜀𝜑) 0.5 =[−𝐸 +𝐸 𝑟2](𝑤+𝜀𝜑) (𝑟 𝑇 ) 11 12 𝑠 cos 𝑠 1 0.0 01 −1 ×[−𝐸21 +𝜀 𝐸22𝑟𝑓](𝑤+𝜀𝜑)cos (𝑇0)+ℎ0𝜉(𝑇1), a (9) Figure 2: Probability densities of amplitude of the original system 𝑗 𝑗 𝑗 (5)andoftheslowsystem(12). Line 1: no fast excitation; line 2: 𝐸22 ⋅ where 𝐷𝑖 =𝜕/𝜕𝑇𝑖 (𝑖 = 0,1;𝑗 = 1,2).Averageequation 𝑟𝑓 = 0.8.Solidlinesfortheoriginalsystem(5) and dotted lines for (9)withrespectto𝑇0 and subtract the averaged equation the slow system (12). from (9); an approximate expression for 𝜙 is obtained by −1 considering only the dominant terms of order 𝜀 as

2 𝐷0𝜑=𝐸22𝑟𝑓𝑤 cos (𝑇0). (10) The periodic solution of system (13)hastheform[17] 𝜙 The stationary solution to first order for is 𝑤(𝑇1)=𝑎cos 𝜃(𝑇1), 𝐷1𝑤(𝑇1)=−𝑎] (𝑎,) 𝜃 sin 𝜃(𝑇1), 𝜑=−𝐸 𝑟 𝑤 (𝑇 ). 22 𝑓 cos 0 (11) 𝜃(𝑇1)=𝜙(𝑇1)+𝛾, (14) Substitute (11)into(9) and apply 𝑇0-averaging. Retain domi- 0 nant terms of order 𝜀 to obtain where 𝑎 is the amplitude, 𝛾 is the phase angle, and 2 (𝐸22𝑟𝑓) 2 [ ] 3 2 𝐷 𝑤+2𝜁𝐷1𝑤+ 1+ 𝑤+𝑏𝑤 𝑑𝜙 3𝑏𝑎 1 2 ] (𝑎,) 𝜃 = = √(𝜔2 + )(1+𝜂 (2𝜃)), [ ] (12) cos 𝑑𝑇1 4 =[−𝐸 +𝐸 𝑟2]𝑤 (𝑟 𝑇 )+ℎ 𝜉(𝑇 ). (15) 11 12 𝑠 cos 𝑠 1 0 1 (𝑏𝑎2/4) 𝜂= . Equation (12)governsonlytheslowmotion𝑤 of system (𝜔2 +3𝑏𝑎2/4) (5). Note that the fast excitation affects the slow behavior of 2 system (12) by adding (𝐸22𝑟𝑓) /2 to the linear stiffness. By The instantaneous frequency ](𝑎, 𝜃) can be approximated by numerical simulations, the probability densities 𝑝(𝑎) 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 ] (𝑎,) 𝜃 =𝑏0 (𝑎) +𝑏2 (𝑎) cos 2𝜃 +4 𝑏 (𝑎) cos 4𝜃 +6 𝑏 (𝑎) cos 6𝜃, fast excitation parameter 𝐸22 ⋅𝑟𝑓, the bigger the difference (16) between the two amplitudes. where 4. Effect of Fast Parametric Excitation 3𝑏𝑎2 𝜂2 𝑏 (𝑎) = √(𝜔2 + )(1− ) Inthelastsection,anapproximateequationgoverningonly 0 4 16 the slow motion of system (5) is obtained by perturbation. In thefollowing,wewilldiscusstheeffectofthefastharmonic 3𝑏𝑎2 𝜂 3𝜂3 𝑏 (𝑎) = √(𝜔2 + )( + ) excitation on this slow system in greater detail. 2 4 2 64

2 2 (17) 4.1. Effect on Resonant Frequency. Let 𝜔 =[1+(𝐸22𝑟𝑓) /2]. 3𝑏𝑎2 𝜂2 𝑏 (𝑎) = √(𝜔2 + )(− ) In order to study the effect of the fast parametric excitation on 4 4 16 the resonant frequency of system (12), we will first consider the free response of system (12)governedby 2 𝜂3 2 3𝑏𝑎 2 2 3 𝑏6 (𝑎) = √(𝜔 + )( ). 𝐷1𝑤+𝜔 𝑤+𝑏𝑤 =0. (13) 4 64 4 Shock and Vibration

1.8 Substitute (20)into(12)andtreat(20) as generalized van 𝑤, 𝐷 𝑤 𝐴, Γ 1.7 der Pol transformation from 1 to ;thefollowing equations for 𝐴 and Γ canbeobtained: 1.6 3 𝑑𝐴 1.5 =𝐹1 (𝐴, Θ,𝑠 𝑟 𝑇1)+𝐺1 (𝐴, Θ) 𝜉(𝑇1), 2 𝑑𝑇1 1 (21) 1.4 𝑑Γ 𝜔 =𝐹 (𝐴, Θ, 𝑟 𝑇 )+𝐺 (𝐴, Θ) 𝜉(𝑇 ), 1.3 2 𝑠 1 2 1 𝑑𝑇1 1.2 where 1.1 𝐹1 (𝐴, Θ,𝑠 𝑟 𝑇1) 1 1 =− (2𝜁𝐴𝑉 (𝐴, Θ) Θ 0.9 𝜔2 +𝑏𝐴2 sin 0 0.5 1 1.5 2 a 2 +[−𝐸11 +𝐸12𝑟𝑠 ]𝐴cos Θ cos (𝑟𝑠𝑇1)) Figure 3: Average resonant frequency for different values of the fast ×𝑉(𝐴, Θ) sin Θ, excitation parameter 𝐸22 ⋅𝑟𝑓. Line 1: no fast excitation; line 2: 𝐸22 ⋅𝑟𝑓 = 0.3; line 3: 𝐸22 ⋅𝑟𝑓= 0.9. 𝐹2 (𝐴,Θ,𝑟𝑠𝑇1) 1 =− (2𝜁𝐴𝑉 (𝐴, Θ) Θ 𝜔2𝐴+𝑏𝐴3 sin By integrating (16)withrespectto𝜃 from 0 to 2𝜋,anaverage 2 frequency +[−𝐸11 +𝐸12𝑟𝑠 ]𝐴cos Θ cos (𝑟𝑠𝑇1)) ×𝑉(𝐴, Θ) Θ 1 2𝜋 cos 𝜔 (𝑎) = ∫ ] (𝑎,) 𝜃 𝑑𝜃=𝑏(𝑎) (18) 2𝜋 0 ℎ 0 𝐺 (𝐴, Θ) =− 0 𝑉 (𝐴, Θ) Θ 1 𝜔2 +𝑏𝐴2 sin of the oscillator is obtained. The approximate relation ℎ 𝐺 (𝐴, Θ) =− 0 𝑉 (𝐴, Θ) Θ, 2 𝜔2𝐴+𝑏𝐴3 cos 𝜃 𝑇 ≈ 𝑎 𝑇 +𝛾 ( 1) 𝜔 ( ) 1 (19) (22) ℎ will be used in the averaging process that follows. Note that and 0 has been specified in (6). the resonant frequency ](𝑎, 𝜃) depends on both the amplitude 𝑎 and phase 𝛾.InFigure3, the average resonant frequency 4.2.1. The Case with ℎ0 =0. Firstly, we consider the pure 𝜔(𝑎) isplottedagainsttheamplitude𝑎 for different values of parametric harmonic excitation case. Neglect the diffusion 𝐸22 ⋅𝑟𝑓. As the fast excitation parameter 𝐸22 ⋅𝑟𝑓 increases, the terms and (21)canberewrittenas average resonant frequency of the system also increases. 𝑑𝐴 1 =− 2 2 𝑑𝑇1 𝜔 +𝑏𝐴 4.2. Effect on Resonant Response. Wenow proceed to examine the effect of fast parametric excitation on the resonant ×(2𝜁𝐴𝑉(𝐴, Θ) sin Θ response of system (12). It is reasonable to assume that neither 2 light damping nor a weak random excitation will destabilize +[−𝐸11 +𝐸12𝑟𝑠 ]𝐴cos Θ cos (𝑟𝑠𝑇1)) system (12). In this case the response of system (12)canbe regarded as a random spread of the periodic solutions of ×𝑉(𝐴, Θ) sin Θ, (23) system (13). As a consequence, 𝑑Γ 1 =− 2 3 𝑑𝑇1 𝜔 𝐴+𝑏𝐴 𝑤(𝑇1)=𝐴cos Θ(𝑇1), ×(2𝜁𝐴𝑉(𝐴, Θ) sin Θ 𝐷1𝑤(𝑇1)=−𝐴𝑉(𝐴, Θ) sin Θ(𝑇1), (20) 2 +[−𝐸11 +𝐸12𝑟 ]𝐴 Θ (𝑟𝑠𝑇1)) Θ(𝑇1)=Φ(𝑇1)+Γ(𝑇1), 𝑠 cos cos ×𝑉(𝐴, Θ) cos Θ. where 𝐴, Θ, Φ, Γ,and𝑉 are all random processes. The instantaneous and average frequencies of system (12)areof The nonlinear system (23) is subjected to harmonic para- thesameformsgivenby(15)and(18). metric excitations and there is the possibility of parametric Shock and Vibration 5 resonance. Since large response of the pantograph-catenary 1.8 system may cause malfunctions in power collection, we will 1.6 emphasize the primary parametric resonance case. Assume that in primary parametric resonance there exists 1.4 1.2 𝑟𝑠 =2+𝜀𝜂, (24) 1.0 𝜔 (𝐴) 3 a 0.8 2 where 𝜔(𝐴) is the average frequency of system (12)and𝜀𝜂 ≪ 1 𝑇 0.6 is the small detuning parameter. Multiply (24)by 1 and 1 utilize the approximate relation (19)toobtain 0.4 0.2 𝑟𝑠𝑇1 =2Θ+𝜀𝜂𝜔𝑇1 −2Γ. (25) 0.0 Δ=𝜀𝜂𝜔𝑇 −2Γ 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Introduce a new variable 1 so that (25)can r be rewritten as s Figure 4: Frequency response for primary parametric resonance. 𝑟 𝑇 =2Θ+Δ. 𝑠 1 (26) Dashed lines are unstable. Line 1: no fast excitation; line 2: 𝐸22 ⋅𝑟𝑓= 0.3; line 3: 𝐸22 ⋅𝑟𝑓= 0.6. Substitute (26)into(23)andaverage(23)withrespectto the rapidly varying process Θ from 0 to 2𝜋 to generate the following averaged differential equations: 1.0 Δa 𝑑𝐴 2 2 0.9 = (−𝐴 [512𝜁𝜔 + 320𝜁𝑏𝐴 Δa 𝑑𝑇 1 0.8 ΔΩ 2 −64(2𝑏0 −𝑏4)(−𝐸11 +𝐸12𝑟𝑠 ) sin Δ]) 0.7 Δa, ΔΩ −1 × (512 (𝜔2 +𝑏𝐴2)) 0.6 ΔΩ 2 0.5 𝑑Δ (2𝑏0 +2𝑏2 +𝑏4)(−𝐸11 +𝐸12𝑟 ) cos Δ =𝑟 −2𝑏 + 𝑠 . 𝑠 0 2 2 0.4 𝑑𝑇1 4(𝜔 +𝑏𝐴 ) (27) −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

E22 · rf Equation (27)involvesonlyslowlyvaryingprocesses𝐴 and Δ 𝑑𝐴/𝑑𝑇 =𝑑Δ/𝑑𝑇=0 .Byletting 1 1 ,(27) gives the frequency Figure 5: The amplitude Δ𝑎 at 𝑟𝑠 = 2.0 andthewidthoftheresonant response relation: region ΔΩ as functions of the fast excitation parameter 𝐸22 ⋅𝑟𝑓.

512𝜁𝜔2𝐴 + 320𝜁𝑏𝐴3 2 [ ] ℎ =0̸ 2 4.2.2. The Case with 0 . In practical railway engineering, 64𝐴 (2𝑏0 −𝑏4)(−𝐸11 +𝐸12𝑟𝑠 ) random disturbances are always present and the term 𝜉(𝑇1) (28) 2 2 2 cannot be neglected. Suppose that the stochastic excitation 4(𝑟𝑠 −2𝑏0)(𝜔 +𝑏𝐴 ) 𝜉(𝑇 ) 2𝐷 +[ ] =1. 1 is a weak Gaussian white noise with intensity .Then, 2 (2𝑏0 +2𝑏2 +𝑏4)(−𝐸11 +𝐸12𝑟𝑠 ) (21) can be modeled as Stratonovich stochastic differential equations and transformed into the following Itoequationsˆ

Numerical results are obtained for 𝜁 = 0.1, 𝑏 = 0.5, 𝐸11 = 1.2, by adding Wong-Zakai correction terms [18]: 𝐸12 = 0.1,and𝐸21 = 0.5 and shown in Figures 4 and 5.Figure4 displays the frequency response under primary 𝑑𝐴=𝑚1 (𝐴, Θ,𝑠 𝑟 𝑇1)𝑑𝑇1 +𝜎1 (𝐴, Θ) 𝑑𝐵(𝑇1), parametric resonance for different values of the fast excitation (29) 𝑑Γ=𝑚(𝐴, Θ, 𝑟 𝑇 )𝑑𝑇 +𝜎 (𝐴, Θ) 𝑑𝐵(𝑇), parameter 𝐸22 ⋅𝑟𝑓.Itisobservedthatfastparametric 2 𝑠 1 1 2 1 excitation shifts the resonant peaks to the right, which means that a higher frequency of the slow excitation is required where 𝑑𝐵(𝑇1) is the unit Wiener process and to produce the resonant response. The dependence of the Δ𝑎 𝑟 = 2.0 amplitude at 𝑠 and the width of the resonant region 𝜕𝐺1 𝜕𝐺1 ΔΩ 𝐸 ⋅𝑟 𝑚 (𝐴, Θ, 𝑟 𝑇 )=𝐹 (𝐴, Θ, 𝑟 𝑇 )+𝐷(𝐺 +𝐺 ), as a function of the fast excitation parameter 22 𝑓 1 𝑠 1 1 𝑠 1 1 𝜕𝐴 2 𝜕Θ is shown in Figure 5.Theamplitudeandthewidthofthe 𝜕𝐺 𝜕𝐺 resonant region are reduced appreciably by fast parametric 𝑚 (𝐴, Θ, 𝑟 𝑇 )=𝐹 (𝐴, Θ, 𝑟 𝑇 )+𝐷(𝐺 2 +𝐺 2 ), excitation. 2 𝑠 1 2 𝑠 1 1 𝜕𝐴 2 𝜕Θ 6 Shock and Vibration

𝑏 (𝐴, Θ) =𝜎2 (𝐴, Θ) =2𝐷𝐺2, 1 1 1 3 (30) 2 2 2 1 𝑏2 (𝐴, Θ) =𝜎2 (𝐴, Θ) =2𝐷𝐺2, 2 and 𝐹𝑖 and 𝐺𝑖 (𝑖 = 1, 2) are given in (22). In primary parametric resonance, utilize (26)andaveragetherapidly ) a

Θ 2𝜋 (

varying process from 0 to to generate the averaged Itoˆ p stochastic differential equations: 1

𝑑𝐴=𝑚1 (𝐴, Δ) 𝑑𝑇1 + 𝜎1 (𝐴) 𝑑𝐵(𝑇1), (31) 𝑑Δ=𝑚2 (𝐴, Δ) 𝑑𝑇1 + 𝜎2 (𝐴) 𝑑𝐵(𝑇1), where the averaged drift and diffusion coefficients are 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2 2 a 𝑚1 (𝐴, Δ) = (−𝐴 [512𝜁𝜔 + 320𝜁𝑏𝐴

2 Figure 6: Stationary probability density 𝑝(𝑎) for different values of −64 (2𝑏0 −𝑏4)(−𝐸11 +𝐸12𝑟𝑠 ) sin Δ]) the fast excitation parameter 𝐸22 ⋅𝑟𝑓. Line 1: no fast excitation; line 𝐸 ⋅𝑟 𝐸 ⋅𝑟 2 2 −1 2: 22 𝑓 = 0.4; line 3: 22 𝑓 = 0.8. Dotted lines are from direct × (512 (𝜔 +𝑏𝐴 )) numerical simulations of the nonlinear model (5).

ℎ2𝐷(4𝑏2𝐴2 + 12𝑏𝜔2𝐴2 + 32𝜔4) + 0 , 64𝐴(𝜔2 +𝑏𝐴2)3 0.8

𝑚2 (𝐴, Δ) =𝑟𝑠 −2𝑏0

2 (2𝑏0 +2𝑏2 +𝑏4)(−𝐸11 +𝐸12𝑟 ) cos Δ + 𝑠 , 4(𝜔2 +𝑏𝐴2) 0.6 E[a], 𝜎∗10 E[a] ℎ2𝐷 (16𝜔2 + 10𝑏𝐴2) 2 0 𝑏1 (𝐴) = 𝜎 (𝐴) = , 1 16(𝜔2 +𝑏𝐴2)2 𝜎∗10 0.4 ℎ2𝐷 (16𝜔2 + 14𝑏𝐴2) 2 0 𝑏2 (𝐴) = 𝜎 (𝐴) = . 2 16𝐴2(𝜔2 +𝑏𝐴2)2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 E · r (32) 22 f Figure 7: Mean and variance of the amplitude of the slow system The Fokker-Planck-Kolmogorov (FPK) equation associ- (12). ated with the Itoequations(ˆ 31)is 𝜕𝑝 𝜕 𝜕 =− (𝑚 𝑝) − (𝑚 𝑝) The nonlinear three-dimensional parabolic problem as given 𝜕𝑇 𝜕𝑎 1 𝜕Δ 2 1 in (33)–(35) does not admit an easy solution, analytically (33) 1 𝜕2 1 𝜕2 or numerically. Fortunately, in practical applications we are + (𝑏 𝑝) + (𝑏 𝑝) , more interested in the stationary solution of the FPK equation 2 𝜕𝑎2 1 2 𝜕Δ2 2 (33). In this case, (33) can be simplified by letting 𝜕𝑝/𝜕𝜏 = 0 𝑝(𝑎, Δ) where 𝑝=𝑝(𝑎,Δ,𝑇1) is the probability density of amplitude . Then, the joint stationary probability density is 𝑎 and phase Δ.Theinitialconditionfor(33)is obtained readily by using the finite difference method. The stationary probability density of the amplitude 𝑝(𝑎) can be 𝑝(𝑎, Δ) 𝑝=𝛿(𝑎−𝑎0)𝛿(Δ−Δ0), 𝑇1 =0, (34) obtained from by 2𝜋 and the boundary conditions for (33)are 𝑝 (𝑎) = ∫ 𝑝 (𝑎, Δ) 𝑑Δ. (36) 0 𝑝=finite,𝑎=0, Numerical results for 𝑝(𝑎) and 𝑝(𝑎, Δ) of system (12)in 𝜕𝑝 parametric resonance are obtained for 𝜁=0.1, 𝑏=0.5, 𝑝, 󳨀→ 0 , 𝑎 󳨀→ ∞, 𝐸 =1.2𝐸 = 0.1 𝐸 =0.5𝑟 = 2.0 𝐷 = 1.0 𝜕𝑎 11 , 12 , 21 , 𝑠 , , ℎ0 = 0.1,andshowninFigures6–8.Figure6 shows the 𝑝(𝑎,Δ+2𝑛𝜋,𝑇1 |𝑎0,Δ0,𝑇10)=𝑝(𝑎,Δ,𝑇1 |𝑎0,Δ0,𝑇10). stationary probability density 𝑝(𝑎) for different values of the (35) fast excitation parameter 𝐸22 ⋅𝑟𝑓. Fast parametric excitation Shock and Vibration 7

1.4 3.5 1.2 3 1 2.5 0.8 2 0.6 1.5 p(a, 𝛿) p(a, 𝛿) 1 0.4 0.5 0.2 0 0 6 6 5 2 5 4 1.5 4 3 1.61.6 3 1 2 0.808 1.21.2 𝛿 2 1 0 0.404 1 0.505 𝛿 0 a 0 0 a

(a) (b)

Figure 8: Joint stationary probability density 𝑝(𝑎, Δ). (a): No fast excitation; (b) 𝐸22 ⋅𝑟𝑓 = 0.8. shifts the probability density curve to the left, changing both Conflict of Interests the peak height and shape. Even when the fast excitation is small, the response of the slow system (12)maychange The authors declare that there is no conflict of interests dramatically. This observation is reinforced in Figure 7,in regarding the publication of this paper. which the mean and variance of the amplitude 𝑎 of system (12) change significantly upon adding fast parametric excitation. Acknowledgments This reflects the increased stiffness of the slow system12 ( ) under fast excitation. Finally, direct numerical simulations of The authors gratefully acknowledge the financial sup- the nonlinear model (5) are performed to generate 𝑝(𝑎).As blackport provided by the Natural Science Foundation of showninFigure6, data from direct numerical simulations China (nos. 10932009 and 11372271), 973 Program (no. closely match those generated by (36), thus validating the 2011CB711105), National Key Technology Support Program analysis presented. The joint probability density 𝑝(𝑎, Δ) of the (no.2009BAG12A01)andNaturalScienceFoundationof slow system (12)isplottedinFigure8. Zhejiang Province (no. LY12A02004). Opinions, findings, and conclusions expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors. 5. Conclusions In the present paper, the effect of fast parametric excitation References on a stochastically excited pantograph-catenary system has been investigated. A nonlinear model of the pantograph- [1] G. Galeotti and P. Toni, “Nonlinear modeling of a railway catenary system has been adopted, wherein the stiffness of pantograph for high speed running,” Transactions on Modelling the nonlinear spring has a time-varying component char- and Simulation,vol.5,pp.421–436,1993. acterizedbybothlowandhighfrequencies.Theoverall [2]P.H.Poznic,J.Jerrelind,andL.Drugge,“Experimental parametrically induced motion of the system is separated into evaluation of nonlinear dynamics and coupled motions in a two parts: a dominant low-frequency vibration which is the pantograph,” in Proceedings of the ASME International Design main motion and a small high-frequency vibration which Engineering Technical Conferences and Computers and Informa- affects the low-frequency motion by altering the stiffness. tion in Engineering Conference (DETC ’09), pp. 619–626, San Using perturbation, an approximate equation governing only Diego, Calif, USA, September 2009. the low-frequency motion has been derived. An averaging [3] L. Drugge, T. Larsson, A. Berghuvud, and A. Stensson, “The method for harmonic functions has been applied to obtain nonlinear behavior of a pantograph current collector suspen- the primary resonant response of the low-frequency motion. sion,” in Proceedings of the ASME Design Engineering Technical Analytical results show that the effect of fast parametric Conferences,pp.1–7,LasVegas,Nev,USA,1999. excitation is not negligible. The addition of even a small [4]S.Levy,J.A.Bain,andE.J.Leclerc,“Railwayoverheadcontact amount of high-frequency parametric excitation may dra- systems, catenary-pantograph dynamics for power collection at matically increase the resonant frequency and change the high speeds,” Journal of Engineering for Industry,vol.90,no.4, primary resonant response of a system. From a theoretical pp. 692–699, 1968. viewpoint, an investigation of a Duffing oscillator subjected [5] J.-W. Kim, H.-C. Chae, B.-S. Park, S.-Y. Lee, C.-S. Han, and J.- to both stochastic and parametric forces has been conducted H.Jang,“Statesensitivityanalysisofthepantographsystemfora to study the surprising effect of high-frequency input. Practi- high-speed rail vehicle considering span length and static uplift cally speaking, many structures outside railway engineering force,” Journal of Sound and Vibration,vol.303,no.3–5,pp.405– can be modeled as a stochastically driven nonlinear system 427, 2007. excited by both slow and fast parametric excitations. Hence, [6]G.Poetsch,J.Evans,R.Meisingeretal.,“Pantograph/catenary the results of this investigation could be useful in other dynamics and control,” Vehicle System Dynamics,vol.28,no.2- applications. 3, pp. 159–195, 1997. 8 Shock and Vibration

[7] T. X. Wu and M. J. Brennan, “Basic analytical study of pantograph-catenary system dynamics,” Vehicle System Dynam- ics, vol. 30, no. 6, pp. 443–456, 1998. [8] T. X. Wu and M. J. Brennan, “Dynamic stiffness of a railway overhead wire system and its effect on pantograph-catenary system dynamics,” Journal of Sound and Vibration,vol.219,no. 3, pp. 483–502, 1999. [9] D. Tcherniak and J. J. Thomsen, “Slow effects of fast harmonic excitation for elastic structures,” Nonlinear Dynamics,vol.17,no. 3, pp. 227–246, 1998. [10]R.BourkhaandM.Belhaq,“Effectoffastharmonicexcitation on a self-excited motion in van der Pol oscillator,” Chaos, Solitons & Fractals,vol.34,no.2,pp.621–627,2007. [11] J. J. Thomsen, “Some general effects of strong high-frequency excitation: stiffening, biasing and smoothening,” Journal of Sound and Vibration,vol.253,no.4,pp.807–831,2002. [12] J. J. Thomsen, “Using fast vibrations to quench friction-induced oscillations,” Journal of Sound and Vibration,vol.228,no.5,pp. 1079–1102, 1999. [13] J. S. Jensen, “Non-linear dynamics of the follower-loaded double pendulum with added support-excitation,” Journal of Sound and Vibration,vol.215,no.1,pp.125–142,1998. [14] J. S. Jensen, “Quasi-static equilibria of a buckled beam with added high-frequency excitation,” DCAMM Report, Technical University of Denmark, Copenhagen, Denmark, 1998. [15] J. S. Jensen, “Pipes conveying fluid pulsating with high frequency,” DCAMM Report no. 563, Technical University of Denmark, Copenhagen, Denmark, 1998. [16] E. P. Popov and I. P. Paltov, Approximate Methods for Analyzing Nonlinear Automatic Systems, Fizmatgiz, Moscow, Russia, 1960. [17] Z. Xu and Y. K. Cheung, “Averaging method using generalized harmonic functions for strongly non-linear oscillators,” Journal of Sound and Vibration,vol.174,no.4,pp.563–576,1994. [18] Z. L. Huang and W. Q. Zhu, “Averaging method for quasi- integrable Hamiltonian systems,” Journal of Sound and Vibra- tion,vol.284,no.1-2,pp.325–341,2005. [19] Z. L. Huang, W. Q. Zhu, and Y. Suzuki, “Stochastic averaging of strongly non-linear oscillators under combined harmonic and white-noise excitations,” Journal of Sound and Vibration,vol. 238, no. 2, pp. 233–256, 2000. [20] G. O. Cai and Y. K. Lin, “Nonlinearly damped systems under simultaneous broad-band and harmonic excitations,” Nonlinear Dynamics,vol.6,no.2,pp.163–177,1994. [21] R. Haiwu, X. Wei, M. Guang, and F. Tong, “Response of a Duffing oscillator to combined deterministic harmonic and random excitation,” Journal of Sound and Vibration,vol.242, no. 2, pp. 362–368, 2001.