D.J. Thompson

A theoretical model for ground vibration from trains generated by vertical track irregularities

A model is developed for predicting ground vibrations due to vertical track irregularities. This model incorporates vehicles, a track and a layered ground, and uses the moving axle loads and the vertical rail irregularities as its inputs. Outputs include the dynamic wheel–rail forces and the displacement power spectra of the track and the ground surface. Results from this model are presented for a single-axle vehicle model and a British Mark 3 passenger coach running on different tracks (a ‘lighter ballasted track’, a ‘heavier ballasted track’ and a slab track) at different speeds (25, 60 and 83 m/s). Based on these results, the effects of track structure, vehicle speed and frequency range on the observed vibration levels are identified. The different roles of the moving axle loads and the roughness-induced dynamic loads are indicated, at different frequencies and for train speeds below and above the lowest ground wave speed.

A comparison of a theoretical model for quasi-statically and dynamically induced environmental vibration from trains with measurements

This paper presents comparisons between a theoretical ground vibration model and measured data at three sites. The model, which is briefly outlined here, encompasses both the quasi-static and dynamic mechanisms of excitation. The vertical dynamics of a number of vehicles travelling at a constant speed on an infinite track are coupled to a semi-analytical model for a three-dimensional layered ground. This model is also used to demonstrate the roles of the two components of vibration at different frequencies and for train speeds below and above the lowest ground wave speed. It is found that, in most practical cases, the dynamic component gives rise to the higher level of vibration.

The Quantification of Structure-Borne Transmission Paths by Inverse Methods - Part 1: Improved Singular Value Rejection Methods

Structure-borne sound from installed machinery is often transmitted into a receiver structure via many connection points and several co-ordinate directions at each of them. In order to quantify the contributions from the various connection points, the operational forces at the interfaces, or an equivalent set of forces at some other locations, should be determined. These forces may be combined with measured transfer functions to determine their contributions to the sound at the receiver locations. Inverse methods are becoming widely used, in which a matrix of measured accelerances is inverted at each frequency and used with operational acceleration data to find the forces. Due to poor conditioning of this matrix, however, the results can often be unreliable. In this paper, using both simulations and measurements, an assessment is made of the success and failure of various strategies for dealing with the problems of ill conditioning, in particular over-determination and singular value rejection. In each case the test structure is a rectangular plate, and a wide frequency range is covered to include regions of both low and high modal overlap. Critical for the rejection of singular values is a suitable threshold. It is established that previously used thresholds, based on estimates of error in either accelerances or operational responses, cannot be used universally. An alternative approach is developed in which the accelerance matrix is perturbed by a different amount for each sample of the operational responses. Based on this approach a more robust strategy is proposed which takes account simultaneously of the effect of errors in both the accelerances and operational responses.

Modelling ground vibration from railways using wavenumber finite- and boundary-element methods

A mathematical model is presented for ground vibration induced by trains, which uses wavenumber finite- and boundary-element methods. The track, tunnel and ground are assumed homogeneous and infinitely long in the track direction (x-direction). The models are formulated in terms of the wavenumber in the x-direction and discretization in the yz-plane. The effect of load motion in the x-direction is included. Compared with a conventional, three-dimensional finite- or boundary-element model, this is computationally faster and requires far less memory, even though calculations must be performed for a series of discrete wavenumbers. Thus it becomes practicable to carry out investigative study of train-induced ground vibration. The boundary-element implementation uses a variable transformation to solve the well-known problem of strongly singular integrals in the formulation. A ‘boundary truncation element’ greatly improves accuracy where the infinite surface of the ground is truncated in the boundary-element discretization. Predictions of vibration response on the ground surface due to a unit force applied at the track are performed for two railway tunnels. The results show a substantial difference in the environmental vibration that could be expected from the alternative designs. The effect of a moving load is demonstrated in a surface vibration example in which vibration propagates from an embankment into layered ground.

A theoretical study on the influence of the track on train-induced ground vibration

An investigation is presented on the nature of train-induced ground vibration propagation. It is based on a theoretical model for the track and a layered ground. Results are given of the responses of the ground and track to a moving harmonic or quasi-static load on the rails. The dispersion characteristics of the propagating modes of vibration in the track and the ground are presented and the excitation of vibration in the ground via the track is discussed in relation to these propagating wavenumbers. An important feature of the coupled system is the coincidence of a propagating wavenumber in the track and the ground that gives rise to the main peak in the vibration spectrum in the frequency range of interest. It has been observed, in some cases, that when the train speed reaches a value close to the speed of propagating waves in the ground, the response to the quasi-static axle loads of the train reaches a peak. The relationship between this critical speed and the wave speeds in the track and ground is considered in order to investigate the effectiveness of controlling this peak response load speed by increasing the bending stiffness of the track/embankment structure or by reducing its mass. It is found that such treatments may, or may not, have a significant effect depending on the ground stiffness and layering. For the multiple quasi-static moving axle loads of a train the loading has strong, closely spaced harmonic components. The effect on the vibration spectrum of the superposition of vibration from multiple axles is shown to lead to the reinforcement or suppression of some frequencies as a function of axle spacing and speed. This is demonstrated with calculated results.

The quantification of structure-borne transmission paths by inverse methods. Part 2: Use of regularization techniques

The inversion of an ill-conditioned matrix of measured data lies at the heart of procedures for the quantification of structure-borne sources and transmission paths. In an earlier paper the use of over-determination, singular value decomposition and the rejection of small singular values was discussed. In the present paper alternative techniques for regularizing the matrix inversion are considered. Such techniques have been used in the field of digital image processing and more recently in relation to nearfield acoustic holography. The application to structure-borne sound transmission involves matrices, which vary much more with frequency and from one element to another. In this study Tikhonov regularization is used with the ordinary cross-validation method for selecting the regularization parameter. An iterative inversion technique is also studied. Here a form of cross-validation is developed allowing an optimum value of the iteration parameter to be selected. Simulations are carried out using a rectangular plate structure to assess the relative merits of these techniques. Experiments are also performed to validate the results. Both techniques are found to give considerably improved results compared to singular value rejection.

Theoretical Investigation of Wheel/Rail Non-Linear Interaction due to Roughness Excitation

A study is presented of the non-linear dynamic interaction between a wheel and rail, excited by rough-ness on the wheel and rail contact surfaces. A moving irregularity model is used to represent the wheel/ rail interaction process in the time-domain. A low order multiple degree-of-freedom system is developed to approximate the infinite track for numerical simulations. The effects of the non-linear contact on the wheel/rail dynamic interaction are investigated through calculations, analysis and comparisons with the results from a linear contact model. The difference between the non-linear and linear interactions is found to be small if the roughness level is not extremely severe and a typical static contact preload exists. The difference increases for low preloads or for high roughness amplitudes. For example, if the wheel and rail surfaces are in good condition (r.m.s. amplitudes of roughness below 15 mm), the linear model can be used without significant error for all static loads down to 25 kN (equivalent to an unloaded container wagon). When the track is corrugated with an r.m.s. amplitude of 25 mm, good agreement between linear and non-linear models is obtained for static loads of 50 kN and above, typical of passenger stock or loaded freight vehicles, but differences of up to 4 dB in one-third octave force levels are found at 25 kN. Differences between the linear and non-linear models are found to occur when the r.m.s. roughness amplitude is more than 0.35 times the static deflection of the contact zone; significant loss of contact occurs at amplitudes about 1.5 times greater than this.

On the impact noise generation due to a wheel passing over rail joints

Abstract Impacts occur when a railway wheel encounters discontinuities such as rail joints. A model is presented in which the wheel/rail impacts due to rail joints are simulated in the time domain. The impact forces are transformed into the frequency domain and converted into the form of an equivalent roughness input. Using Track–Wheel Interaction Noise Software (TWINS) and the equivalent roughness input, the impact noise radiation is predicted for different rail joints and at various train speeds. It is found that the impact noise radiation due to rail joints is related to the train speed, the joint geometry and the static wheel load. The overall impact noise level from a single joint increases with the speed V at a rate of roughly 20 log 10 V .

Track Dynamic Behaviour at High Frequencies. Part 1: Theoretical Models and Laboratory Measurements

SUMMARY Three alternative theoretical models of the dynamic behaviour of railway track in the frequency range 50–6000 Hz are described. Using these models various physical phenomena of importance to rolling noise generation are pointed out, and the advantages and short-comings of the three models are compared. Finally, laboratory measurements are described in which the vertical and lateral dynamic stiffnesses of rail fastener systems has been measured under preload in the range 100–1000 Hz.

Identification, modelling and reduction potential of railway noise sources: a critical survey

Environmental requirements for railway operations will become tighter in the future. In particular, annoyance due to railway noise has to be taken carefully into account in the expansion of freight traffic as well as in new high speed line projects. Reduction of noise at source can be more attractive than the use of noise barriers but this requires a thorough understanding of the source mechanisms. This paper presents a critical survey of the identification and modelling of railway noise sources and summarizes the current knowledge of the physical source phenomena (mainly rolling and aerodynamic sources) as well as the potential for noise reduction. Future research perspectives are also given. These concern, in particular, improvements to source modelling, especially for aerodynamic noise, investigation of other sources and development of more advanced models for predicting railway noise in the environment. These should include a better description of the sources, obtained from modelling.