755. A survey of Hopf bifurcation analysis in nonlinear railway wheelset dynamics

Transcription

1 755. A survey of Hopf bifuration analysis in nonlinear railway wheelset dynamis Hamid M. Sedighi, Kourosh H. Shirazi Department of Mehanial Engineering, Shahid Chamran University, Ahvaz, Iran hmsedighi@gmail.om, k.shirazi@su.a.ir (Reeived 7 Deember 0; aepted 4 February 0)0; aepted 4 Deember 0) Abstrat. This artile attempts to analyze the Hopf bifuration behavior of a railway wheelset in the presene of dead-zone and yaw damper nonlinearities. A model that is more preise than Yang and Ahmadian is investigated. Using Bogoliubov-Mitropolsky averaging method and ritial speed, the amplitude of the limit yle in the presene of the mentioned nonlinearities is taken into onsideration. To solve these nonlinear equations analytially, the integration interval has been divided into three sub-domains. Two-dimensional bifuration diagrams are provided to illustrate the mehanism of formation of Hopf bifuration. These diagrams an be used for design of stable wheelset systems. Keywords: Hopf bifuration, hunting, rail wheelset, disontinuous nonlinearity, analytial approah. Introdution High-speed railway vehiles are assuming an ever-inreasing importane in today s transportation infrastrutures. As the veloity of rail vehile inreases, the vehile beomes less stable and ultimately exhibits rigorous osillations, namely Hunting. It has been only last two deades that analyses have been made inorporating some of the more important nonlinearities that lead to the ourrene of the hunting, suh as learanes between omponents, the wheel flange ontating fores, dry frition in suspension omponents et. Yang and Ahmadian [] reported that De Pater [] used Krylov and Bogoliubov [3] method to examine limit yle behavior of a two-axle bogie with ylindrial wheels. Law and Brand [4] used the same method to analyze the dynamis of a single wheelset having urved wheel profiles and flange ontat, where they modeled the effets of this flange fore by a linear rail spring with a dead-band equal to the flange learane. The nonlinear studies led to more advaned researh that used bifuration and haos theory in dynami systems. Huilgol [5] first investigated the Hopf bifuration in a wheelset, in the presene of the nonlinear ontat fore between the wheel and the rail. Later, Lohe and Huilgol [6] found an asymmetri osillation in their numerial simulation. Subsequently, a group of sientists led by True Hans [7] further studied the bifurations in two rail bogie models, where nonlinear reep fore and dead-band wheel/rail ontat fore are onsidered. They found periodi, bi-periodi and haoti behavior in this model and stated that subritial Hopf bifuration is popular in rail vehiles. The relationship between the damping and the ritial hunting speed of a truk has been studied by Wikens [8, 9]. Chung and Shim [0] studied the Hopf bifuration in a rail bogie. They found that introduing hystereti nonlinearity leads to superritial bifuration. Pombo and Ambrósio [] analyzed the implementation of a multi joint wheel rail ontat model to railway dynamis in small radius urved traks. Yang and Ahmadian [] analyzed Hopf bifuration in a wheelset in the presene of nonlinear yaw damper in a primary suspension system. The flange fore is modeled as a linear spring besides a nonlinear fourth order damping inluding a dead-zone due to the wheel/rail learane. Sedighi et al. [3-8] have modeled the dead-zone nonlinearity and other disontinuities using ontinuities based funtion to failitate the severe omputational issues that are enountered in the analytial investigations of nonlinear problems. This investigation emphasizes the influenes of suspension nonlinearities and the wheel/rail interfae nonlinearities on Hopf bifuration. Frequeny of the limit yle is found 344 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

2 analytially. Also using the Averaging method a relation between the limit yle amplitude and parameters of the system is introdued. These relations lead to several analytial riteria for predition of possibility of hunting behavior. Two-dimensional bifuration diagrams are depited to study the Hopf bifuration in the system. Governing equations of motion of a single-axle wheelset The wheelset is onsidered to be a 4 degrees of freedom system and is illustrated in Fig.. Fig.. Free-body diagram for wheelset The equations of motion are organized using Newton's seond law of motion for x, y, z and ϕ as follows: m ɺɺ w y= Fly+ Fry N r sin( βr ϕ) + Nl sin( βl+ ϕ) + Fs, y FT () Iwzψɺɺ ( Iwx Iwy) ϕθ ɺ ɺ = a( Frx Flx ) aψ Nr sin( βr ϕ) Nl sin( βl ϕ) M + M + aψ F F + M bf () ( ), ( ) ɺ ɺ sin( ) sin( ) ( ) zr zl ry ly s z d Iwxϕɺɺ Iwy Iwz θψ = Nr βr ϕ Rr Nl βl + ϕ Rl + M yr + M yl ψ + R l Flx ψ F ly + a Nr os( βr ϕ) + Nl os( βl + ϕ) + R r Frx ψ F ry + a Fry + F lz (3) m ɺɺ wz= Frz+ Flz + Nr os( βr ϕ) + Nl os( βl+ ϕ) mwg (4) where: F = f ξ, F = f ξ f ξ, M = f ξ f ξ (5) xp 33 x yp y sp zp y sp and ξ is the Kalker's reepages, for roll, pith and yaw rates of the left and the right ontat planes. Fig. provides that the suspension fores in the lateral diretion, F s, y and the suspension moments in the vertial diretion, M s, z, ating on the wheelset: F = K y C yɺ, M = K b ψ (6) s, y y y s, z x Fig.. The wheelset model VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

3 The nonlinear longitudinal yaw damping fore desribed as follows: F d and the flange ontat fore F T is 3 4 CVψ + CVψ + C3Vψ + C4Vψ Vψ > 0 Kr ( y δ) y > δ Fd =, FT = 0 δ y δ 3 4 CVψ CVψ C3Vψ C4Vψ Vψ 0 + < Kr ( y + δ) y < δ (7) where Vψ = bψ ɺ and the oeffiients C to C 4 are obtained from experimental tests on the atual dampers by []. The onstant K r is the wheel/rail ontat lateral stiffness, and δ is the flange learane. Assuming ϕ= λy a, Rɺ = λϕɺ, Rl = Rr = R0, the vertial wheelset displaement equation of motion an be also negleted. Thus equations of motion are redued as follows: y f λ f y V m ɺɺ w y + m w gλ + R0 y V ψ ψ λ + ɺ + ɺ + = Fs, y FT a V a V a R 0 (8) λ V af33 f λ a f33 Iwzψɺɺ + ( Iwy I wx) yɺ + λy R0 y V ψ + ɺ + ψɺ a R0 R0 V a V f y mgλaψ + ψ ɺ + θɺ λ = M s, z bfd V a (9) Comparison between equation (8) with results of Yang and Ahmadian indiated that the λ V term ( Iwy I wx) yɺ has been omitted and the Eulerian aeleration term was deletion in a R0 their paper. As illustrated in the result setion, this term hanges the ritial speed of the system. Analyti behavior of the nonlinear model Based on the ahieved results for V and ω in the numerial simulations, the nonlinear behavior of the system an be analyzed using the Averaging method, Expanding A( V ) about V and ombining equations (0) and () gives: Xɺ = A V X+ ε F X, µ, ε, F X, µ, ε = BX+ F X = 0, f,0, f ( ) ( ) ( ) ( ) { } 0 4 In whih the parameters f, f 4 are nonlinear funtions in terms of dead-zone and nonlinear T A V orresponding to ± iω are yaw dampers. The eigenvetors of ( ) 0 A V and ( ) ξ = α ± iβ and η = p + iq, respetively. Using the Averaging method [3], we an obtain the approximate solution of equation (0) as follows: X= a( α osϕ β sinϕ) = a α + β os ( ϕ+ γ), () where γ= tg ( β α) and the time dependent variables a and ϕ are defined as: da dϕ = εh( a), = ω+ εg ( a) () at dt 0 T (0) 346 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

4 Using symboli alulations, the funtions H ( a ) and G ( ) π γ H( a) = f i ( p i osϕ+ q i sinϕ) dϕ π γ i= π γ G ( a) = f i ( q i osϕ p i sinϕ) dϕ πa γ i= a an be expressed as: The steady state solution (limit yle) ourred if H( a ) = 0. Equation (3) an be used to examine the amplitude and the phase of the limit yle. To solve for the amplitude of the stationary limit yle, we assume: π γ H( a) = f i ( p i osϕ+ q i sinϕ) dϕ= 0 π (4) γ i= whih has the nontrivial solution a. To solve equation (4) for a, it is divided to three subdomain intervals as shown in Fig. 3: aros( S) γ π γ+ aros( S) π γ H( a) =... dϕ... dϕ... dϕ π + + γ π γ aros( S) π aros( S) (3) (5) where the term S in the integral domain is limit yle, we obtained a from (4) and substituted in (3): π γ δ a α + β. To solve for the frequeny of the ε Ω = ω + f i ( q i osϕ p i sinϕ) dϕ πa (6) γ i= therefore the long-term behavior of the system an be obtained by substituting solution equation (4) for a into equation () that is given as follows: X = a α + β os( Ω t + θ + γ) (7) where ϑ= tg ( α β), the theoretial as well as the numerial simulation limit yle are indiated in Fig. 4. This figure onfirms the soundness and effetiveness of the introdued EFs. Fig. 3. Disretization of lateral ontat fore for analytial approah VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

5 Results In order to analyze the influene of the system parameters on the hunting behavior, we solve equations (8) and (9) numerially. By varying forward speed V and plot stable response of the lateral displaement vs. speed, bifuration diagrams generate as shown in Figs. 5-. In Fig. 5 the diret numerial solution of equation of motion is depited. As this figure indiates the V from numerial solution is 60 km/h. Fig. 4. Comparison of theoretial (symbols) and numerial simulation (ontinues line) limit yles Fig. 5. Bifuration diagram for set of main parameters The effet of the rail stiffness ( K r ) on the ritial speed is onsidered. Fig. 6 shows that the rail stiffness has no remarkable effet on hunting speed, however it redues hunting amplitude. Also, Fig. 7 indiates that smaller flange learane redues the amplitude of limit yle with no signifiant effets on ritial speed. 348 Fig. 6. The effet of rail lateral stiffness on ritial speed VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

6 Through Fig. 8 to Fig., the effet of variation of the parameters C y, K y, K x and λ on ritial speed are indiated. Inreasing lateral stiffness as well as lateral damping and yaw stiffness, raises the ritial speed V and lowers the hunting amplitude. Comparing Fig. 8 to Fig. 0 it is observed that the hunting speed shows more sensitivity to hange of the yaw stiffness relative to the other parameters suh as the lateral damping. As Fig. indiates inreasing wheelset oniity, dereases the ritial speed V and inreases the hunting amplitude, while the hunting speed and amplitude shows more sensitivity to hange of the wheelset oniity. Fig. 7. The effet of flange learane on ritial speed Fig. 8. The effet of lateral damping on ritial speed Fig. 9. The effet of lateral stiffness on ritial speed VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

7 Fig. 0. The effet of yaw spring stiffness on ritial speed Fig.. The effet of wheelset oniity on ritial speed Conlusions In this paper Hopf bifuration in a railway wheelset was studied through a nonlinear model. Novel proedure for modeling of disontinuous nonlinearities has been employed to predit analytial response of nonlinear vibration in the time domain. It appears from the present work that the method an signifiantly alleviate the analytial investigation of the nonlinear problems. The authors believe that the introdued proedure has speial potential to be applied to other strong nonlinearities suh as preload, dead-zone and saturation disontinuous. Additionally, the effets of suspension parameters suh as lateral damping and stiffness, yaw stiffness and wheelset oniity on ritial speed were also investigated. The results of the investigation demonstrate that:. Inreasing gauge learane redues the amplitude of hunting. Gauge learane does not have a signifiant effet on the ritial speed.. Yaw stiffness has a major effet on hunting veloity and an be an important design parameter, while inreasing lateral damping has less effet on inrease of the ritial speed. 3. Inreasing rail lateral stiffness does not signifiantly affet the ritial speed, but redues hunting amplitude. Referenes [] Yang S., Chen E. The Hopf bifuration in a railway bogie with hystereti nonlinearity. Journal of the China Railway Soiety, Vol. 5, 993, p VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

8 [] Pater D. The approximate determination of the hunting movement of a railway vehile by aid of the method of Krylov and Bogoliubov. Applied Sientifi Researh, Vol. 9, 960, p [3] Bogoliubov N. N., Mitropolsky Y. Asymptoti Method in the Theory of Nonlinear Osillations. Delhi, India: Hindustan Publishing Corp, 96, p [4] Law E. H., Brand R. S. Analysis of the nonlinear dynamis of a railway vehile wheelset. Dynami Systems, Measurement and Control, Vol. 95, 973, p [5] Huilgol R. R. Hopf-Friedrihs bifuration and the hunting of a railway axle. PMM Journal of Applied Mathematis and Mehanis, Vol., 978, p [6] Lohe M. A., Huilgol R. R. Flange fore effets on the motion of train wheelset. Vehile System Dynamis, Vol., 98, p [7] True H., Kaas-Petersen C. A bifuration analysis of nonlinear osillations in railway vehiles. The Dynamis of Vehiles on Road and on Traks, 8th IAVSD Symp., 984, p [8] Wikens A. H. The hunting stability of railway vehile wheelsets and bogies having profiled wheels. International Journal of Solids and Strutures, Vol., 965, p [9] Wikens A. H. Stati and dynami instabilities of bogie railway vehiles with linkage steered wheelsets. Vehile System Dynamis, Vol. 6, 996, p. -6. [0] Chung W. J., Shim J. K. Influene fators on ritial speed hysteresis in railway vehiles. Japan Soiety Mehanial Engineering International Journal, Vol. 46, 003, p [] Pombo J. C., Ambrósio J. C. Appliation of a wheel-rail ontat model to railway dynamis in small radius urved traks. Multibody System Dynamis, Vol. 9, 008, p [] Yang S., Ahmadian M. The Hopf bifuration in a rail wheelset with nonlinear damping. Pro. of Transportation Division, International Mehanial Engineering Congress and Exposition, Atlanta, 996, p [3] Sedighi H. M., Shirazi K. H., Zare J. Novel equivalent funtion for deadzone nonlinearity: applied to analytial solution of beam vibration using He s parameter expanding method. Latin Amerian Journal of Solids and Strutures, 0, in press. [4] Sedighi H. M., Shirazi K. H., Reza A., Zare J. Aurate modeling of preload disontinuity in the analytial approah of the nonlinear free vibration of beams. Proeedings of the Institution of Mehanial Engineers, Part C: Journal of Mehanial Engineering Siene, 0, doi: 0.77/ [5] Sedighi H. M., Reza A., Zare J. Dynami analysis of preload nonlinearity in nonlinear beam vibration. Journal of Vibroengineering, Vol. 3, 0, p [6] Sedighi H. M., Reza A., Zare J. Study on the frequeny amplitude relation of beam vibration. International Journal of the Physial Sienes, Vol. 6, 0, p , doi: /IJPS.556. [7] Sedighi H. M., Shirazi K. H. A new approah to analytial solution of antilever beam vibration with nonlinear boundary ondition. ASME Journal of Computational and Nonlinear Dynamis, 0, doi: 0.5/ [8] Sedighi H. M., Shirazi K. H., Noghrehabadi A. R., Yildirim A. Asymptoti investigation of bukled beam nonlinear vibration. Iranian Journal of Siene and Tehnology, Transation B. Engineering, 0, in press. VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN