Generalization of the Fourier transformbased meth

Authors: Xiaozhen Sheng

Publish Date: 2015/03/10

Volume: 23, Issue: 1, Pages: 12-29

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Abstract

A Fourier transformbased method has been developed for calculating the response of a railway track as an infinitely long uniform periodic structure subject to moving or stationary harmonic loads The track may become a nonuniform periodic structure by for example rail dampers which are installed between sleepers to control rolling noise and roughness growth The period of the structure may become greater than the sleeper spacing For those new situations the current version of the method cannot be directly applied it must be generalized and this is the aim of this paper Generalization is performed by applying periodic conditions to each type of support and summarizing contributions from all types of support Responses of the rail sleeper and damper are all formulated as an inverse Fourier transform from wavenumber domain to spatial domain The generalized method is applied to investigate dynamics of a typical track with rail dampers of particular design It is found that the rail dampers can significantly suppress the pinned–pinned vibration of the original track widen the stop bands and increase vibration decay rate along the rail However it is also found that a new pinned–pinned mode is created by the dampers and between about 450 and 1300 Hz dampers vibrate stronger than the rail making noise radiation from the dampers a potential issue These concerns must be fully investigated in the future The formulae presented in this paper provide a powerful tool to do thatA major concern for the railway industry is the growth of rail roughness the formation of rail corrugation and the generation of wheel/rail noise It is now well known that these unwanted phenomena are generated from dynamical interactions between moving wheels and rails Due to operations of highspeed trains and roughness of short wavelengths on the wheel/rail rolling surfaces wheel/rail interactions are of high frequencies up to several thousand Hertz and involve complex vibrational wave propagations and resonances in the track structure For a track with sleepers a major track type used worldwide it is normally modeled to be a periodic structure consisting of an infinitely long uniform main structure ie the rails attached at a given spacing ie the sleeper spacing by an infinite number of supports ie the railpad/sleeper/ballastOne of the important aspects in dealing with dynamics of a periodic structure such as a railway track is to calculate the free vibration characteristics ie propagation constants and associated modes These free vibration characteristics are utilized in a number of methods dealing with forced vibration of the periodic structure and also helpful to develop understandings of mechanisms involved In 1 Mead gives an extensive review on the subject with focus mainly on work contributed from Southampton before 1996 Two methods are often used to calculate propagation constants the receptance or Green’s function method and the transfer matrix method The first method is normally applied in an analytical manner and therefore restricted to simple main structures and supports 2 With the finite element method FEM involved the transfer matrix method can account for more complex structures but leads to heavy computations because of the necessary longitudinal discretization In addition to that the eigenvalue equation may become illconditioned due to the presence of evanescent waves with high vibration decay rates as noted by Gry and Gontier 3 4 when dealing with vibrations of a rail on periodic supports at acoustic frequencies One of the measures to overcome these shortcomings is given in these two references in which displacement variation of the rail in the longitudinal direction is synthesized using some sort of modesThe periodic structures dealt with by Gry and Gontier are uniform as defined by some researchers ie no attachment is presented between the sleepers Brown and Byrne propose a method to deal with the socalled nonuniform periodic structure 5 The basis of the method presented in that paper substructuring using wave shape coordinate reduction is to divide the nonuniform periodic structure into a number of uniform substructures along the periodic axis and to reduce the number of coordinates in each substructureAnother important aspect in dealing with track dynamics is to calculate the response of a railway track as a periodic structure subject to a moving or stationary harmonic load Results from such calculations can not only further reveal the dynamical characteristics of the track but can also provide a basis either in the timedomain or in the frequencydomain for dealing with wheel/rail interactions Different approaches have been developed to analyze the response of a track as a uniform periodic structure to fast moving harmonic loads of high frequencyResponses of a discretely supported rail to moving loads can be modeled in the timedomain eg 6 7 by solving differential equations as an initialvalue problem Timedomain approaches require the track to be truncated into a finite length To minimize the effect of wave reflections from the truncations and to be able to account for high frequency vibration the track model must be sufficiently long In fact when dealing with interactions between a highspeed train and a railway track the entire train should be taken into account This is not only because intervehicle couplings in a highspeed train are much stronger 7 therefore having a significant effect on dynamics of both the train and the track but also due to the longdistance propagating vibration waves induced in the track by the high speed Therefore the track model must be much longer than the train up to 450 m and the rail must be modeled using either the FEM 6 or the modal superposition method 7 This would generate a large number of differential equations of timevarying coefficients It is timeconsuming to solve these equations due not only to the large number of equations but also to the very small timesteps required for high frequencies For a periodic excitation extra time is also required to allow the steadystate solution to achieveComputational efficiency and accuracy can be significantly increased using methods based on the periodic structure theory Vibration of an infinite and periodically supported by springs beam subject to a moving harmonic load has been investigated in 8 In this study the author employs the Euler beam theory which is only valid for frequencies below 250 Hz on a single segment and combines the periodic conditions to produce the steadystate response of the periodically supported beam The periodic conditions are also used in 9 to investigate the steadystate responses of different periodic structures including a railway track to a moving load Nordborg 10 also used a periodically supported Euler beam to represent a railway track subject to a moving load However the varying stiffness of the track is calculated using a quasistatic approach This quasistatic approach has also been used by other researchers eg 11Another method which has been used to deal with forced vibrations of and wheel interactions with a track as an infinitely long periodic structure is the Green function method 12 13 14 which is based on the Duhamel integration and working in the timedomain The Green function of the track is defined as the response of the track at a location due to a unit impulsive force a Dirac delta function applied at the same or another location To account for multiple and moving loads Green functions for a large number of different locations to simulate a wheel travels over a sleeper bay are required Green functions are normally computed as the inverse Fourier transform of the corresponding frequency response function which is the response of the track to a unit harmonic load at different frequencies Therefore it is essential to be able to calculate the response of a periodic railway track subject to a harmonic load of high frequency

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