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The dynamics of slab tracks and floating slab tracks are analyzed by multibeam models for the track and by integration in the wave-number domain for the soil, which is modeled as a layered half-space. Frequency-dependent compliances and force transfers are calculated for a great variety of track and soil parameters. The distribution of the load and the displacements along the track is investigated as well as the wave propagation perpendicular to the track and the ground vibration amplitudes. The floating slab track has a dominating plate-mat resonance and a strong high-frequency reduction. A track-soil resonance can also be recognized for an unisolated slab track in the case of layered soils. Generally, there is a strong damping of the track by the soil. The reduction effect of the slab mat is mainly owing to the elimination of this strong damping. The continuous soil yields slightly different rules for the displacements and force densities than those of a Winkler support. The total force transfer from the rail to the soil is the best criterion to judge the effectiveness of a floating slab track in reducing the ground vibration at some distance from the railway line. The total force transfer is easier to calculate than the double Fourier integrals of the ground vibration amplitudes, namely in the far field, and it has the best correlation with the reduction of the ground vibration.
Train-induced ground vibrations are all generated by the vehicle, by static or dynamic vehicle loads. The most important and most accepted excitation are the dynamic wheel loads from the passage over track irregularities. Dynamic wheel loads will be compared from parallel axle-box and ground vibration measurements at more than seven sites. Some low-frequency excitation of ground vibrations, typically between 10 and 30 Hz, cannot be found in the axle-box measurements. Therefore, other vehicle modes, such as rigid bogie modes, flexible carriage modes, rigid and flexible wheelset modes, have been analysed for additional excitation forces. These vehicle dynamics analyses give an explanation for higher axle-box results at high frequencies, but not for the excitation of the higher low-frequency ground-vibration component. Finally, the effect of the moving static train loads will be analysed. For a regular track and soil, the moving static train loads yield the quasi-static response which exists only in the low-frequency nearfield of the track. If the support stiffness is randomly varying along the track, the pulses on the track generate an additional low-frequency component which is called the irregular pulse responses.
This component will be demonstrated by numerical analysis where all axle pulses are superposed in frequency domain.
Irregularities of the track are a main cause of train-induced ground vibration, and track maintenance is of great importance. Although geometric irregularities at the wheel-rail contact are widely used, other types of irregularities, such as stiffness irregularities, irregularities from different track positions and irregularities in the wave propagation, were analysed in the present study. The track behaviour was investigated by a multi-beam-on-soil model. This track model is coupled with a vehicle model to calculate the vehicle–track interaction. The track model was also used for the track filtering, which transfers a track support error to the equivalent rail irregularity or, conversely, the sharp axle pulse on the rail to a smoother pulse on the soil. In the case in which this filtering varies randomly along the track, the pulses of the moving static load induce a certain ground Vibration component (“the scatter of axle pulses”). This effect was calculated by the superposition of axle pulses in the frequency domain and by a stochastic simulation. Simultaneous vehicle, track and soil measurements at a certain site were used to evaluate the different excitation and ground Vibration components. The agreement between calculations and axle-box and soil measurements is good. The ground vibrations calculated from rail irregularities and corresponding dynamic loads, however, clearly underestimate the measured ground vibration amplitudes. Only the static load that is moving over a varying track support stiffness can produce the important mid-frequency ground Vibration component by the scatter of axle pulses.
In this article, the passage of different trains over different bridges will be studied for resonant excitation. The intensity of the resonance will be estimated in frequency domain by using three separated spectra. At first, the excitation spectrum of the modal forces is built by the mode shape and the passage time of the train over the bridge. The second spectrum is the frequency response function of the bridge which include the modal frequency, damping and mass. The third part is the spectrum of the axle sequence of the train. The influences of train speed, bridge length, bridge support, track irregularities, and train type on the resonance amplitudes will be analysed for each of these spectra separately for getting a better insight. A variety of axle-sequence spectra and corresponding rules will be presented for different vehicles and trains. As examples, the passage of a slow freight train over a long-span bridge, a normal passenger train over a medium-span bridge, and a high-speed train over a short bridge will be analysed. Corresponding measurements show the amplification, but also the cancellation of the subsequent axle responses. Namely in one of the measurement examples, the first mode of the bridge was amplified and the second mode was cancelled at a low speed of the train and vice versa at a higher speed.
The layered soil is calculated in the frequency wavenumber domain and the solutions for fixed or moving point or track loads follow as wavenumber integrals. The resulting point load solutions can be approximated by simple formula. Measurements yield the specific soil parameters for the theoretical or approximate solutions, but they can also directly provide the point-load solution (the transfer function of that site). A prediction method for the train-induced ground vibration has been developed, based on one of these site-specific transfer functions. The ground vibrations strongly depend on the regular and irregular inhomogeneity of the soil. The regular layering of the soil yields a cut-on and a resonance phenomenon, while the irregular inhomogeneity seems to be important for high-speed trains. The attenuations with the distance of the ground vibration, due to point-like excitations such as vibrator, hammer, or train-track excitations, were investigated and compared. All theoretical results were compared with measurements at conventional and high-speed railway lines, validating the approximate prediction method.
Construction work, such as pile driving and soil compaction, or road and railway traffic excite nearby buildings, and the perceptible or audible vibration can be a nuisance for nearby inhabitants. A simplified building model has been created for these situations, which includes the effects of soil-structure interaction, the low-frequency amplification along the height of the building as well as the high-frequency reduction and the floor resonances. The model consists of one wall for all supporting structures (walls and columns) and one floor for each storey. The effect of different floor resonance frequencies is included in a stochastic procedure. The soil is modelled by a spring and a viscous damper, and the free-field amplitudes of the soil are applied under this soil element.
The model can be calculated by transfer matrices or in a continuous wave-type version where an analytical solution can be evaluated numerically. The building response in the high-frequency (acoustic) region is calculated as mean values over wider frequency bands. The approach to an infinite building model can be found for these high frequencies and the corresponding soil-structure transfer can be described by the ratio of impedances at foundation level.
The rules for choosing the parameters to obtain realistic results are derived from complex calculations for example, for the stiffness and damping of building foundations and many measurements as for the damping of floor resonances. The influences on the floor resonance from the soil (damping) and the supporting structure (detuning) are important. Some more effects will be discussed by the simplified and detailed models and by measurements to establish a good understanding of ground-induced building vibrations.
Train-induced ground vibration can be excited by wheel and track irregularities and by two kinds of irregularities of the soil, by geometric irregularities or by the spatially varying soil stiffness. For both types of irregularities, the effective track irregularity on top of the track is calculated in wavenumber domain and with wavenumber integrals. For a general multi-beam track model, the wavenumber integrals are solved numerically. The irregularities of the soil are filtered by the track when transferred from the bottom to the top of the track. The high-wavenumber irregularities are strongly reduced due to the bending stiffness of the track and the compliance of the support. In addition, soft track elements reduce directly the stiffness variation of the support. Therefore, the mitigation effect of elastic track elements for these excitation components seems to be important. For under-sleeper pads and slab tracks, calculation and measurements are presented including additional excitation components and the dynamic vehicle–track interaction, and the relevance of the excitation mechanisms is discussed based on the dynamic forces which are acting on the ground. Due to the restricted amplitudes, the parametric excitation by the stiffness variation seems to be less important than the geometric irregularities. The calculations yield the correct trends of the measurements and many details of the measured ballast, slab, and under-sleeper-pad tracks.
Train-induced ground vibrations are generated by static and dynamic axle loads which can be calculated by vehicle-track-soil models and the vehicle and track irregularities. A fast prediction method has been developed which uses approximate transfer functions of layered soils. In the present contribution, this prediction method is used for the inverse calculation of the axle-load spectra from the measured ground vibration. The layered soils of some measuring sites show very differing ground vibration spectra in the amplitude range of 0.0001–1.0 mm/s as a consequence of the soft layer and stiff half-space, differing layer frequencies, as well as the far- and near-field measuring points. The back-calculation, however, yields axle-load spectra within a single order of magnitude around 1 kN. Axle-box measurements confirm the amplitude level of the axle loads. This standard axle-load spectrum can be used for a basic prediction at a new site. The separation of train and site-specific components allows a better evaluation of railway vibrations, for example, of different trains and different tracks. By eliminating the effects of differing soil characteristics, an important mid-frequency component has been found which lies between 8 and 32 Hz depending on the train speed. The origin of this dominant mid-frequency component is discussed using advanced prediction methods like moving constant loads, scattered axle impulses and axle-sequence spectra.
The reduction of train-induced ground vibration by elastic elements such as rail pads and sleeper pads has been analyzed by a combined finite-element boundary-element method. The dynamic compliance of the track, the transfer function of the total force on the ground and the ground vibration ratios have been calculated for a variety of isolated and un-isolated track systems. It has been found that the soil force transfer, which describes the excitation force of the soil, is an appropriate quantity to predict the reduction of the ground vibration and the effectiveness of isolated tracks. All force transfer functions of isolated tracks display a vehicletrack resonance where the wheelset on the compliant track is excited by wheel and track irregularities. At higher frequencies, considerable reductions of the amplitudes are observed as the benefit of the resilient element. The influence of the stiffness of the rail or sleeper pads, the ballast and the soil, and the mass of the sleeper and the wheelset on the resonance frequency and the reduction has been investigated. Sleeper pads are advantageous due to the higher mass that is elastically supported compared to the rail-pad track system. The combination of elastic rail and sleeper pads has been found to be disadvantageous, as the second resonance occurs in the frequency range of intended reduction.