The fictitious force method for efficient calculation of vibration from a tunnel embedded in a multi-layered half-space

This paper presents an extension of the Pipe-in-Pipe (PiP) model for calculating vibrations from underground railways that allows for the incorporation of a multi-layered half-space geometry. The model is based on the assumption that the tunnel displacement is not influenced by the existence of a free surface or ground layers. The displacement at the tunnel–soil interface is calculated using a model of a tunnel embedded in a full space with soil properties corresponding to the soil in contact with the tunnel. Next, a full space model is used to determine the equivalent loads that produce the same displacements at the tunnel–soil interface. The soil displacements are calculated by multiplying these equivalent loads by Green?s functions for a layered half-space. The results and the computation time of the proposed model are compared with those of an alternative coupled finite element–boundary element model that accounts for a tunnel embedded in a multi-layered half-space. While the overall response of the multi-layered half-space is well predicted, spatial shifts in the interference patterns are observed that result from the superposition of direct waves and waves reflected on the free surface and layer interfaces. The proposed model is much faster and can be run on a personal computer with much less use of memory. Therefore, it is a promising design tool to predict vibration from underground tunnels and to assess the performance of vibration countermeasures in an early design stage.     

The effect of a twin tunnel on the propagation of ground-borne vibration from an underground railway

Accurate predictions of ground-borne vibration levels in the vicinity of an underground railway are greatly sought after in modern urban centres. Yet the complexity involved in simulating the underground environment means that it is necessary to make simplifying assumptions about this system. One such commonly made assumption is to ignore the effects of neighbouring tunnels, despite the fact that many underground railway lines consist of twin-bored tunnels, one for the outbound direction and one for the inbound direction.This paper presents a unique model for two tunnels embedded in a homogeneous, elastic fullspace. Each of these tunnels is subject to both known, dynamic train forces and dynamic cavity forces. The net forces acting on the tunnels are written as the sum of those tractions acting on the invert of a single tunnel, and those tractions that represent the motion induced by the neighbouring tunnel. By apportioning the tractions in this way, the vibration response of a two-tunnel system is written as a linear combination of displacement fields produced by a single-tunnel system. Using Fourier decomposition, forces are partitioned into symmetric and antisymmetric modenumber components to minimise computation times.The significance of the interactions between two tunnels is quantified by calculating the insertion gains, in both the vertical and horizontal directions, that result from the existence of a second tunnel. The insertion-gain results are shown to be localised and highly dependent on frequency, tunnel orientation and tunnel thickness. At some locations, the magnitude of these insertion gains is greater than 20 dB. This demonstrates that a high degree of inaccuracy exists in any surface vibration prediction model that includes only one of the two tunnels. This novel two-tunnel solution represents a significant contribution to the existing body of research into vibration from underground railways, as it shows that the second tunnel has a significant influence on the accuracy of vibration predictions for underground railways.     

Predicting surface vibration from underground railways through inhomogeneous soil

Noise and vibration from underground railways is a major source of disturbance to inhabitants near subways. To help designers meet noise and vibration limits, numerical models are used to understand vibration propagation from these underground railways. However, the models commonly assume the ground is homogeneous and neglect to include local variability in the soil properties. Such simplifying assumptions add a level of uncertainty to the predictions which is not well understood. The goal of the current paper is to quantify the effect of soil inhomogeneity on surface vibration. The thin-layer method (TLM) is suggested as an efficient and accurate means of simulating vibration from underground railways in arbitrarily layered half-spaces. Stochastic variability of the soil's elastic modulus is introduced using a K–L expansion; the modulus is assumed to have a log-normal distribution and a modified exponential covariance kernel. The effect of horizontal soil variability is investigated by comparing the stochastic results for soils varied only in the vertical direction to soils with 2D variability. Results suggest that local soil inhomogeneity can significantly affect surface velocity predictions; 90 percent confidence intervals showing 8 dB averages and peak values up to 12 dB are computed. This is a significant source of uncertainty and should be considered when using predictions from models assuming homogeneous soil properties. Furthermore, the effect of horizontal variability of the elastic modulus on the confidence interval appears to be negligible. This suggests that only vertical variation needs to be taken into account when modelling ground vibration from underground railways.     

Modelling of continuous and discontinuous floating slab tracks in a tunnel using a periodic approach

This paper presents a periodic approach to couple a track and a tunnel-soil system of different periodicity. The periodicity of the track and the tunnel-soil system is exploited using the Floquet transform to efficiently formulate the problem in the frequency-wavenumber domain as well as to limit the discretization effort to a reference cell. The track and the tunnel-soil system are modelled as two separate systems of different periodicity and are coupled in the frequency-wavenumber domain. A coupled periodic finite element-boundary element method is used to model the tunnel-soil system, while a periodic finite element model or an analytical approach is used to model the track.A general analytical formulation to compute the response of three-dimensional periodic media that are excited by moving loads is discussed. It is shown that the response due to moving loads on the track can be calculated from the transfer function of the track-tunnel-soil system and the axle loads.A methodology for computing the transfer functions of the coupled track-tunnel-soil system as well as the computation of dynamic forces accounting for the interaction between the moving vehicle and the periodic track are described. The model accounts for quasi-static forces as well as dynamic forces due to parametric excitation and unevenness excitation.The methodology has been used to assess the vibration isolation efficiency of continuous and discontinuous floating slab tracks. It is concluded that both continuous and discontinuous floating slab tracks have a similar efficiency in the frequency range well above the isolation frequency of the slabs, which is usually higher than the slab passage frequency. In case of discontinuous slab tracks, the parametric excitation is found to be important, which results in a poorer performance of the track at low frequencies.     

A Study of the Suitable Measurement Location and Metrics for Assessing the Vibration Source Strength Based on the Field-Testing Data of Nanchang Underground Railway

Underground railway vibration source strength is one of the key values used for environmental impact assessment and the evaluation of mitigation measure’s performance. However, currently there is no international standard of measuring the underground railway vibration source strength for such purposes. The available local standards and industrial guidelines do not agree on measurement locations as well as the metrics for presenting the source strength. This has caused many confusions. This paper aims to study the suitable measurement location and metrics using the data from a large scale field-testing carried out at the Nanchang underground railway (Metro Line 1, China) in 2017. 200 passing trains were recorded during the test at two different sections of the railway line, one with the spring floating slab installed and the other without. Three locations were chosen at each section, including one in the middle of the track and two on the tunnel wall at different heights. Based on the results of statistical analysis, the maximum of z-weighted vertical vibration level (VLzmax) obtained at a lower measurement location on the tunnel wall is the best for representing the underground railway vibration source strength, which is 76.66 dB obtained from this study.     

Potential low frequency ground vibration (<6.3 Hz) impacts from underground LRT operations

Vibration sensitive research activities at the laboratories of the University of Washington (UW) Physics and Astronomy Building (PAB) were a critical issue for the design of the Sound Transit Link Light Rail LRT system in Seattle, Washington. A study was conducted to measure and predict low frequency ground vibration generated by the LRT operations. The University's concern was an on-going research experiment in gravity, which had sensitivity to vibration below 6.3 Hz. The experiment was located on an independent concrete slab in an area cut-out from the building foundation with no connection to the building structure. Another concern was the planned future construction of a Life Sciences Center with vibration sensitive test equipment. This paper presents the results of a study to estimate the ground displacement at these buildings using empirical measured data of a similar deep tunnel transit system and finite difference modelling analysis.     

Fully three-dimensional analysis of high-speed train-track-soil-structure dynamic interaction

In this paper, a general and fully three dimensional multi-body-finite element-boundary element model, formulated in the time domain to predict vibrations due to train passage at the vehicle, the track and the free field, is presented. The vehicle is modelled as a multi-body system and, therefore, the quasi-static and the dynamic excitation mechanisms due to train passage can be considered. The track is modelled using finite elements. The soil is considered as a homogeneous half-space by the boundary element method. This methodology could be used to take into account local soil discontinuities, underground constructions such as underpasses, and coupling with nearby structures that break the uniformity of the geometry along the track line. The nonlinear behaviour of the structures could be also considered. In the present paper, in order to test the model, vibrations induced by high-speed train passage are evaluated for a ballasted track. The quasi-static and dynamic load components are studied and the influence of the suspended mass on the vertical loads is analyzed. The numerical model is validated by comparison with experimental records from two HST lines. Finally, the dynamic behaviour of a transition zone between a ballast track and a slab track is analyzed and the obtained results from the proposed model are compared with those obtained from a model with invariant geometry with respect to the track direction.     

Voids at the tunnel-soil interface for calculation of ground vibration from underground railways

Voids at the tunnel-soil interface are not normally considered when predicting ground vibration from underground railways. The soil is generally assumed to be continuously bonded to the outer surface of the tunnel to simplify the modelling process. Evidence of voids around underground railways motivated the study presented herein to quantify the level of uncertainty in ground vibration predictions associated with neglecting to include such voids at the tunnel-soil interface. A semi-analytical method is developed which derives discrete transfers for the coupled tunnel-soil model based on the continuous Pipe-in-Pipe method. The void is simulated by uncoupling the appropriate nodes at the interface to prevent force transfer between the systems. The results from this investigation show that relatively small voids () can significantly affect the rms velocity predictions in the near-field and moderately affect predictions in the far-field. Sensitivity of the predictions to void length and void sector angle are both deemed to be significant. The findings from this study suggest that the uncertainty associated with assuming a perfect bond at the tunnel-soil interface in an area with known voidage can reasonably reach and thus should be considered in the design process.     

Modelling the length distribution of etched tracks taking into account fragmentation due to partial annealing

Fission track length measurements on apatite minerals are of great interest in thermo-chronology studies. RBS analyses on Durango apatites irradiated by Krypton ions have previously shown the progressive fragmentation of tracks by annealing. This study made for different irradiation energies corresponding to those of uranium fission fragments has been completed by TEM and AFM on etched tracks. In all cases, segmentation figures have been observed, that clearly necessitates to introduce fragmentation of latent tracks to model the length distribution of uranium etched fission tracks. This presentation concerns the case of a homogeneous population of tracks with partial amorphization. The next step will consist in considering multiple sub-populations with different segmentation rates. This algorithm is a new aspect to be included in our model, which already takes into account diffusion and crystallography in the track etching process.     

Canard轨迹原理

为了精确的求出Canard轨迹的描述方程,提出了一个分段表达的非线性模型,能直接求出一个非线性二阶自治方程的相轨迹方程的解析解,能定量求出Canard轨迹的描述方程和参数条件,可以用两类方程简单描述所有的Canard轨迹. 关键词： 非线性系统 Canard轨迹 弛张振荡     

Prediction of vibrations induced by underground railway traffic in Beijing

This paper examines the problem of subway induced vibrations on line 4 of Beijing metro, which is currently under construction and is planned to pass in close proximity of the Physics Laboratory of Beijing University. The laboratory has a lot of equipment that is very sensitive to traffic induced vibrations and future operation of metro line 4 is a matter of concern. Hence, it is important to study the influence of subway induced vibrations inside the laboratory and to propose a viable solution to mitigate the vibrations. In this paper, the tunnel north of Chengfulu station is modelled using a coupled periodic FE-BE model and the free-field response due to moving trains is predicted. In addition, vibration measurements have been performed on the site of the Physics Laboratory to estimate the existing vibration levels due to road traffic. The predicted and measured vibrations are superimposed to assess the vibrations due to the combined effect of road and railway traffic in the vicinity of the Physics Laboratory. Apart from the numerical investigations, vibration measurements have also been performed on a similar site at line 1 of Beijing metro to substantiate the estimated results on metro line 4. Finally, it is studied how the vibrations can be controlled using a floating slab track, which is widely used as an effective measure of vibration isolation in tunnels. The efficiency of a 7.9 Hz floating slab track as a vibration countermeasure is assessed in this paper. This study demonstrates the applicability of the numerical model for the relevant assessment of subway induced vibrations and its use to study the performance of different track structures in the tunnel.     

热超声倒装键合中换能杆末端多维振动的相关性研究

换能杆末端的振动情况对于超声键合机理的研究有着重要意义；换能杆末端的实际振动模式直接影响着芯片键合的质量。利用高精度非接触式激光多普勒测振仪采集了换能杆末端俯仰、水平、轴向振动速度，用相干函数方法分析了3个方向的相互耦合关系。分别对换能杆末端有未安装劈刀的情况做了比较，得出了在换能系统工作频率附近，换能杆末端轴向振速与水平、俯仰振速的相关程度。这对实际键合中，换能系统末端的劈刀安装长度的选择以及换能杆工作频率的选择有指导意义。     

Studies on the nuclear tracks in CR-39 plastics

A review was given for our recent studies on the latent tracks in CR-39 nuclear track detector. The radial size of track core has been determined through UV spectral measurements combined to the model of track overlapping and by AFM observations of slightly etched detectors. The track core radius was found to be about a few nano-meters and almost proportional to the cubic root of stopping power. As a control study, the etching properties of CR-39, irradiated by low-LET radiation, has been examined. The observed depth dependence and dose-rate dependence of the bulk etch rate of the irradiated CR-39 were explained that the damage formation process was governed by the reaction between the radiation induced radicals and the oxygen supplied from the air. This indicated that latent tracks in CR-39 are produced through local radiation induced oxidation process along the ion paths. Studies on vibration spectra, near-IR, FT-IR and Raman spectra, of CR-39 have also been carried out to estimate the chemical structure of the latent tracks. The creation of OH group in irradiated CR-39 has been confirmed.     

Ground-borne noise and vibration from underground rail systems

Ground-borne noise is one of the main causes of environmental impact from urban rail transit systems. The vibration resulting from track-train interaction is transmitted through the tunnel 3tructure and the surrounding ground to adjacent buildings. The resulting vibrations of the walls and floors of these buildings cause secondary radiation of noise. This paper presents a method for estimating A-weighted sound levels as well as noise and vibration spectra due to ground-transmitted vibration in buildings near subways.     

Reply to Comments on Chapter 12 of “Railway Noise and Vibration: Mechanisms, Modelling and Means of Control”, by D. Thompson (with contributions from C. Jones and P.-E. Gautier), Elsevier, 2009

In this authors’ reply it is stressed that the book “Railway Noise and Vibration: Mechanisms, Modelling and Means of Control, Elsevier (2009)” is a textbook which does not set out to provide a historical review. Moreover, the topic of the increase in vibration above a critical speed is considered by the authors to be a comparatively rare phenomenon within the topic of ground vibration from surface railways, the latter itself only being dealt with in a single chapter of the book.     

Impact of sound and vibration of the North-South high-speed railway connection through the city of Antwerp Belgium

In the European High-Speed Train Network the infrastructure of the North-South connection in Antwerp needs significant modifications. For the section between Berchem and Antwerp Central Station the existing track on the high level embankment will be incorporated into concrete structures providing a three level track access to the station. For the section between Antwerp Central Station and Dam two drilled tunnels are planned providing the station with pass-through facilities instead of being an “end” station as at present. The paper focuses on the methods of practical research and the resulting measures regarding the impact of sound and vibration on the environment.An essential part of this study is the impact of the planned construction of a double railway tunnel underneath the city of Antwerp. At certain locations, the distance between the foundations of the houses and the top of the tunnel is only 4 m. The study considers the projected vibration levels on the rail, the tunnel invert, building foundations and upper floors of the buildings. Also the ground-borne noise is evaluated. The study identifies the measures necessary at the rail mounting level. As a result, a floating slab has been proposed and the effects on the environment are estimated.     

滚球混沌运动演示实验

以滚球在3个凹形槽上的受力体现作用的非线性,导轨由电机推动作简谐振动,滚球在导轨上作受迫(滚动)振动,可以把导轨作简谐振动的频率或振幅作为调整参量.当受迫振动的滚球振幅小时,滚球只在1个凹形槽中作简谐振动;当振幅渐渐增大时,可呈现周期倍增和渐至混沌状态.通过观察安装在同一振动滑板上的2套导轨上的滚球的运动,则可以演示混沌运动对初值的敏感性.     

Vibration interaction in a multiple flywheel system

This paper investigates vibration interaction in a multiple flywheel system. Flywheels can be used for kinetic energy storage in a satellite Integrated Power and Attitude Control System (IPACS). One hitherto unstudied problem with IPACS is vibration interaction between multiple unbalanced wheels. This paper uses a linear state-space dynamics model to study the impact of vibration interaction. Specifically, imbalance-induced vibration inputs in one flywheel rotor are used to cause a resonant whirling vibration in another rotor. Extra-synchronous resonant vibrations are shown to exist, but with damping modeled the effect is minimal. Vibration is most severe when both rotors are spinning in the same direction.     

Effect of track stiffness on vibration levels in railway tunnels

The paper discusses the difficulties in comparing and drawing conclusions from vibration measurements made in different railway tunnels. A number of new measurements are presented, which have been made and analyzed on a consistent basis. The track stiffness has been determined for each of the measurements. Graphs are presented which indicate the level of rail and tunnel floor vibration likely to occur for different track stiffnesses. The variance in the measurement results also indicates the potential error in using the data to predict vibration levels in new situations.