Vibration propagation from a train tunnel. Measurements to validate a numerical model
Figure 1 Citytunneln will pass below Malmö City and connect Malmö central station with the Öresund bridge
To connect passenger rail service to and from the Öresund Link with the rail system north of Malmö Central Station, a tunnel is to be excavated below the central regions of the city. The tunnel, which is called the Citytunneln, will be completed as two separate full-face bored tunnels in limestone rock. Conventional cast-in-situ concrete tunnels and rampsconstructed in open cutswill connect the portals in the north and south. A new subterranean station, Triangeln, will be constructed in the section of the tunnel that is to be full-face bored. Citytunneln will pass under existing buildings that are currently used as homes, hospitals, offices, schools, etc.
A study is being carried out on behalf of Citytunnelkonsortiet in Malmö to predict vibration from the operation of Citytunneln. This is being done using a numerical finite difference model. The study has been made by the joint venture KCM, Kjessler and Mannerstråle AB, Carl Bro a/s and Maunsell Ltd with Rupert Taylor Ltd as subconsultants.
The accurate prediction of ground-borne noise and vibration from underground railways is a complex task. Whereas in airborne acoustics comparatively simple mathematical procedures may be used to process readily available source data (or readily measurable data), taking account of well-known effects such as noise barrier attenuation, transmission loss of acoustical elements, reverberation and similar concepts, the nearest analogies to these items in the prediction of ground-borne noise are many times more complex. Not only is there no simple equivalent to sound lower level, because the source impedance is an unknown (in contrast to the position with air as a medium), but the method of propagation involves several different co-existing wave types whose behaviour, except in restricted cases not found in real life, can only be approximated algebraically. Propagation through layered or anisotropic media makes matters even more complex.
A useful approach to the difficulties identified is to seek to reproduce, in the time domain, the actual dynamic behaviour of all the elements of the system using finite-difference methods. The finite difference concept is fundamentally simple although the implementation in cases other than homogeneous isotropic media is less so. This article describes a finite difference model for predicting ground-borne propagation of vibration at acoustic frequencies, and compares the predictions with measured results. The finite difference model known as FINDWAVE® and is proprietary to Rupert Taylor Ltd.
The model has been validated by applying it to a tunnel on the Copenhagen metro, and comparing the results with field measurements carried out in the tunnel and in a building above, Building 202 in the Serum Institut. The modelling has been carried out by R.M. Thornely-Taylor of Rupert Taylor Ltd. The measurements in the Copenhagen Metro (Comet) tunnel were carried out by Acoustica Carl Bro A/S, Jørgen Tornhøj Christensen.
In order to verify the predictive numerical model for the propagation of vibrations in the Malmö City Tunnel, verification measurements in the Copenhagen Metro Tunnel have been carried out. The transmission loss from the tunnel floor to the building and ground above has been measured as the reduction in vibration velocity level in the tunnel to the measured levels above.
Furthermore, the point impedance for the tunnel floor has been measured. The impedance is defined as the ratio between the applied force at the excitation point and the measured velocity at the excitation point.
The data used in the model study were as follows:
Test location: Chainage 20550 in tunnel "Betty"
Tunnel diameter: 5.45m external 4.90m internal
Tunnel depth: Top of crown 13.075m below ground level
The dynamic properties used in the model are described in the final report.
Soil properties were provided by Maunsell as their interpretation of borehole results obtained by Rambøll and Carl Bro at the Seruminstitut. Data concerning soil loss factor were provided by KM. They were evaluated from a measurements of ground vibration response at 20, 40, 57 and 80 m distance from the ground level centreline of the tunnel.
Excitation in the model was at the base of the tunnel in the vertical direction. The excitation signal produces a continuous spectrum, which can be used to compare the predicted results with the discrete frequency measurements carried out. Because of the nature of the model grid, it was necessary to excite four points on the edges of a cell representing the tunnel floor around the centreline. This will produce a slightly different driving point impedance compared with the measurements, in which the excitation took place at a point.
The model was in two parts, in order to allow for the fact that Building 202 is not aligned with the tunnel (its long axis is at approximately 45° to the tunnel). The first part of the model contained was a 50m cube, consisting of the tunnel and the soil, with the tunnel running parallel with the k axis. The second part was similar to the first, with Building 202 added, also with its long axis parallel with the k axis. The first part of the model was interfaced to the second part by interpolating the displacement for each of the lowest layer of cells in the second part, after rotating by 45°, at an interface level immediately above the tunnel crown. Each part has vibration absorbing boundaries. A detailed explanation of the interface is given in the final report.
The model was run for 0.9216 seconds (a total of 16384 times steps of 112.5 m sec) and the time history at the tunnel floor, wall and crown, at the three measurement locations, in each case in the i, j, and k directions. The i and j results for Building 202 were in directions aligned with the building axis, and were subsequently recomputed in the frame of reference of the tunnel, so that i is horizontal (lateral to the tunnel axis)_ and j is longitudinal (along the tunnel axis. Screen captures of an instant during the model run (model 1) are shown in Figure 2. and Figure3.
The model represents the material loss factor of the soil in a manner that produces a constant rate of attenuation per metre distance, at all frequencies, i.e. by setting the loss factor for a given frequency such as 80Hz, and then adjusting it inverse proportion to other frequencies.
Two examples of frequency transforms are shown in Figure 4, Figure 5 and Figure 6. The valid frequency range extends only up to 300Hz. For the measurements in and outside Building 202, the single-frequency results obtained during the measurements are also shown. The transmission loss in Figures 4 to 6 is defined as the vertical vibration velocity level on the tunnel floor minus the vertical vibration velocity in the building or in the ground.
The frequency transforms have been made without Hanning or other time-weighting. The valid upper frequency is approximately equivalent to the frequency at which a cell dimension is equal to a quarter wavelength in the medium concerned. In this case, for the upper soil layer it is approximately 725 Hz for compressional waves and approximately 320 Hz for shear waves.
Figure 7 shows the measured and modelled impedance of the tunnel floor.
Discussion of Results
It is feature of the modelled spectra that they contain significant peaks and troughs. Since the measurements were made at discrete frequencies, small frequency variation in the model spectra can lead to large differences in the discrete values.
For Citytunneln, 1/3 octave bands will be reported. For the verification measurements in the Metro they would be misleading, since the measurements were made using discrete frequencies, and therefore the comparison between measurements and model results must be done using the same bandwidth. The measurements used an infinitely narrow bandwidth (i.e. single frequency). The model should use the narrowest practicable bandwidth, which in this case is just over 1 Hz.
As the model results show, the spectrum is peaky. Given the inevitable small frequency error in the model, we do not know whether the field excitation was actually on a peak or not, and it is more helpful to show a narrow band spectrum for the model, for comparison with the discrete frequency results of the measurements. Then the influence of frequency error can be easily seen.
The vibration from operating railways usually contains most of its energy in the 50Hz to 100Hz range. The model in these circumstances of the Copenhagen Comet tunnel has predicted the transmission loss to the three locations quite well in this frequency region, particularly in the vertical direction which has the largest effect on levels of re-radiated noise inside buildings.
An overprediction of transmission loss would tend to result in an underprediction of vibration levels in buildings, and vice-versa.