Study on the P2 Resonance Frequency of Vehicle Track System

doi:10.3901/JME.2019.08.118

Abstract

Abstract: The P2 resonance of vehicle and track coupling system is transformed to the natural frequencies of system with unspring mass and track. The mode shape and characteristic equation are discussed. The influences of length of track model, unspring mass, track mass, contact stiffness between wheel and rail, flexural rigidity of the rail, stiffness of foundation and vehicle speed on the natural frequencies especially the P2 resonance frequency are analyzed thoroughly based on the characteristic equation. When the length of track model is longer than 25 m, its influence on P2 resonance and the second natural frequency can be neglected. The increasing contact stiffness will increase the frequency of P2 resonance slightly, but has almost no influence on the second natural frequency. Both the mass of track and unspring mass have significant effects on P2 resonance. The frequency of P2 resonance is much lower than that of the track system without unspring mass. If the track mass is neglected, the P2 resonance of vehicle/track system is actually a vibration system of single degree of freedom. The frequency of P2 resonance grows evidently with the increase of track stiffness. The vibrational mass of track and the flexural rigidity of the rail have slight influence on P2 resonance, while the effect of speed on P2 resonance is not obvious. For application, one method of calculating track stiffness and the frequency of P2 resonance is proposed based on the free vibration of track system, and the effectiveness of this method is verified by experiment.

Key words: Laplace transformation, P2 resonance, track, track stiffness, unspring mass, vehicle

CLC Number:

U270
U211

GUAN Qinghua, ZHOU Yeming, LI Wei, WEN Zefeng, JIN Xuesong. Study on the P2 Resonance Frequency of Vehicle Track System[J]. Journal of Mechanical Engineering, 2019, 55(8): 118-127.

References

[1] JENKINS H H,STEPHENSON J E,CLAYTON G A,et al. The effect of track and vehicle parameters on wheel/rail vertical dynamic forces[J]. Railway Engineering Journal,1974,3(1):2-16.
[2] RADFORD R W. Wheel/rail vertical forces in high-speed railway operation[J]. Journal of Engineering for Industry,1977,99(4):849-858.
[3] 许实儒,徐维杰,仲延禧. 钢轨接头处轮轨冲击力的模拟分析[J]. 铁道学报,1989(S1):99-109. XU Shiru,XU Weijie,ZHONG Yanxi. A simulation analysis of wheel/rail impact force at the rail joint[J]. Journal of China Railway Society,1989(S1):99-109.
[4] GRASSIE S L,KALOUSEK J. Rail corrugation:characteristics,causes and treatments[J]. Proceedings of the Institution of Mechanical Engineers,Part F:Journal of Rail and Rapid Transit,1993,207(1):57-68.
[5] KNOTHE K L,GRASSIE S L. Modelling of railway track and vehicle/track interaction at high frequencies[J]. Vehicle System Dynamics,1993,22(3-4):209-262.
[6] NIELSEN J C O,LUNDÉN R,JOHANSSON A,et al. Train-track interaction and mechanisms of irregular wear on wheel and rail surfaces[J]. Vehicle System Dynamics,2003,40(1-3):3-54.
[7] POPP K,KRUSE H,KAISER I. Vehicle-track dynamics in the mid-frequency range[J]. Vehicle System Dynamics,1999,31(5-6):423-464.
[8] JIN X,WU L,FANG J,et al. An investigation into the mechanism of the polygonal wear of metro train wheels and its effect on the dynamic behaviour of a wheel/rail system[J]. Vehicle System Dynamics,2012,50(12):1817-1834.
[9] DIANA G, CHELI F, BRUNI S,et al. Dynamic interaction between rail vehicles and track for high speed train[J]. Vehicle System Dynamics,1995,24(1):15-30.
[10] TAO G,WANG L,WEN Z,et al. Experimental investigation into the mechanism of the polygonal wear of electric locomotive wheels[J]. Vehicle System Dynamics,2018,56(6):883-899.
[11] PATIL S P. Response of infinite railroad track to vibrating mass[J]. Journal of Engineering Mechanics,1988,114(4):688-703.
[12] GRASSIE S L,GREGORY R W,HARRISON D,et al. The dynamic response of railway track to high frequency vertical excitation[J]. Journal of Mechanical Engineering Science,1982,24(2):77-90.
[13] CLARK R A,DEAN P A,ELKINS J A,et al. An investigation into the dynamic effects of railway vehicles running on corrugated rails[J]. Journal of Mechanical Engineering Science,1982,24(2):65-76.
[14] ONO K,YAMADA M. Analysis of railway track vibration[J]. Journal of Sound and Vibration,1989,130(2):269-297.
[15] MÜLLER S,KRZYZYNSKI T,ILIAS H. Comparison of semi-analytical methods of analysing periodic structures under a moving load[J]. Vehicle System Dynamics,1995,24(s1):325-339.
[16] 李成辉,万复光. 高速铁路轨道位移波分析[J]. 铁道学报,1993,15(2):76-79. LI Chenghui,WAN Fuguang. An analysis of the track deformation wave of high speed railway[J]. Journal of China Railway Society,1993,15(2):76-79.
[17] ZHAI W M,SUN X,A detailed model for investigating vertical interactions between railway vehicle and track[J]. Vehicle System Dynamics,1994,23(s1):603-615.
[18] NIELSEN J C O,IGELAND A. Vertical dynamic interaction between train and track influence of wheel and track imperfections[J]. Journal of Sound and Vibration,1995,187(5):825-839.
[19] 练松良,刘富. 轨道刚度变化对轮轨冲击荷载的动力影响[J]. 同济大学学报,2002,30(4):427-430. LIAN Songliang,LIU Fu. Effect of track stiffness uneven on wheel/rail impact load[J]. Journal of Tongji University,2002,30(4):427-430.
[20] LEI X,NODA N A. Analyses of dynamic response of vehicle and track coupling system with random irregularity of track vertical profile[J]. Journal of Sound and Vibration,2002,258(1):147-165.
[21] 娄平,曾庆元. 车辆-轨道-桥梁系统竖向运动方程的建立[J]. 铁道学报,2004,26(5):71-79. LOU Ping,ZENG Qingyuan. Formulation of equations of vertical motion for vehicle-track-bridge system[J]. Journal of China Railway Society,2004,26(5):71-79.
[22] WU T X,THOMPSON D J. A double Timoshenko beam model for vertical vibration analysis of railway track at high frequencies[J]. Journal of Sound and Vibration,1999,224(2):329-348.
[23] SHENG X,XIAO X,ZHANG S. The time domain moving Green function of a railway track and its application to wheel-rail interactions[J]. Journal of Sound and Vibration,2016,377:133-154.
[24] 孙文静,周劲松,宫岛. 基于格林函数法的车辆-轨道垂向耦合系统随机振动分析[J]. 中国铁道科学,2015,36(1):61-67. SUN Wenjing,ZHOU Jinsong,GONG Dao. Analysis on random vibration of vehicle-track vertical coupling system with Green function method[J]. China Railway Science,2015,36(1):61-67.
[25] 孙宇,翟婉明. 基于格林函数法的车辆-轨道垂向耦合动力学分析[J]. 工程力学,2017,34(3):219-226. SUN Yu,ZHAI Wanming. Analysis of the vehicle-track vertical coupled dynamics based on the Green's function method[J]. Engineering Mechanics,2017,34(3):219-226.
[26] 李霞,李伟,温泽峰,等. 普通短轨枕轨道结构钢轨波磨初步研究[J]. 机械工程学报,2013,49(2):109-115. LI Xia,LI Wei,WEN Zefeng,et al. Preliminary study on the rail corrugation of the fixed-dual short sleepers track[J]. Journal of Mechanical Engineering,2013,49(2):109-115.
[27] 李霞,李伟,申莹莹,等. 基于轨道振动理论的梯形轨枕轨道钢轨波磨研究[J]. 机械工程学报,2016,52(22):121-128. LI Xia,LI Wei,SHEN Yingying,et al. Study on the rail corrugation of the ladder-type sleepers track based on the track vibration theory[J]. Journal of Mechanical Engineering,2016,52(22):121-128.
[28] 赵国堂. 铁路轨道刚度的确定方法[J]. 中国铁道科学,2005,26(1):1-6. ZHAO Guotang. Method for determining the rigidity of railway track[J]. China Railway Science,2005,26(1):1-6.
[29] 童大埙. 铁路轨道[M]. 北京:中国铁道出版社,1995. TONG Daxun. Railroad track[M]. Beijing:China Railway Publishing House,1995.
[30] 翟婉明. 车辆-轨道耦合动力学[M]. 北京:科学出版社,2015. ZHAI Wanming. Vehicle-track coupled dynamics[M]. Beijing:Science Press,2015.