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Vibration from Underground Trains * Pipe-in-Pipe Software Hugh Hunt Cambridge University Engineering Department

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“Common sense will carry one a long way but no ordinary mortal is endowed with an inborn instinct for vibrations”. “Vibrations are too rapid for our sense of sight … common sense applied to these phenomena is too common to be other than a source of danger”. Professor C E Inglis, FRS, James Forrest Lecture, 1944

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The basis for “common sense” Stiffness Frequency = Mass Damping ? Continuous media ? The mkc model

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Vibration from Railways Source: Talbot … where is the “mass-on-spring”? … how do I use my “intuition”?

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Floating Slab Track Source: Tiflex Ltd Vibration isolation of rail track in tunnels

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Undersleeper pads Source: Edilon

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Novel designs for vibration control Source: Pandrol Ltd

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Floating Slab Track 2 Insertion Gain for Floating Slab Track Insertion gain (dB) Isolation frequency 6.3Hz Source: GERB

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Benefit, typically 10 to 20 dB Amplification at new resonance typically 10dB frequency m k response What is Insertion Gain? T1T1 before “insertion” m k k additional T2T2 after “insertion” frequency Insertion Gain 0 dB T1T1 T2T2

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The Pipe-in-Pipe model (a) (b) a h x y z 1 R 2 R e r e z e r z Tunnel: thin-walled cylinder radius R wall thickness t infinite length Soil: thick-walled cylinder inner radius R outer radius infinite length Coupling Discrete Fourier transform around Continuous Fourier transform along z

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isolation dB ref mm (rms) vertical vibration virtual free surface

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isolation rms insertion gain (dB) Insertion Gain Acknowledgements: Dr James Forrest Dr Mohammed Hussein

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This is a demonstration of the Pipe-in-Pipe software. It is freeware and can be obtained by following the PiP link at I am hugely indebted to Dr Mohammed Hussein from the University of Nottingham for his enormous contribution to the development of PiP software. When downloading the software you will need to enter a username and password. These are available on request. This is a requirement of Mathworks (ie MATLAB) to permit use of the MCR-installer which is a suite of DLLs needed to run PiP as a pre-compiled MATLAB executable. Please be patient when installing and running PiP for the first time – it may take a few minutes to get started. Subsequent uses are quick.

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This shows how different – by as much as 5dB - the vibration levels can be when the material properties are changed by only 15%. This begins to put into question the possibility of prediction accuracy any better than ± 10dB

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This shows how an increase in the bending stiffness of the slab leads, in this case, to an increase in vibration

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This shows how an increase in the mass and bending stiffness of the slab plus a reduction in the natural frequency of the slab from 40Hz to 25Hz leads to a reduction in vibration

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It is convenient to plot Case2, Case 3 and Case 4 as “Insertion Gains” measured relative to Case 1. These show clearly where the changes made lead to reduction in vibration (ie IG < 0dB) 0dB

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Here the measurement point is shifted from being 20m directly above the tunnel to being 20m above and 5m to one side. Note that the response is as much as 5dB different at certain frequencies. This again puts into question the possibility of obtaining prediction of vibration within an accuracy of better than ±10dB

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The PiP software also allows the user to plot contour plots such as these which illustrate where the vibration is greatest at different frequencies. This can help guide the location and distribution of piled foundations. The PiP software can also include the effect of rigid bedrock and the latest version of PiP (soon to be released) includes a free surface and layered soil

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Validation – PiP vs. coupled FEM-BEM Source: Mohammed Hussein / Shashank Gupta / Lars Rikse The PiP software is in excellent agreement with other models

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Conclusions The mass-on-spring model is not good for predicting Insertion Gain The PiP model is a very fast and convenient tool for computing vibration from railway tunnels The use of PiP puts into question the possibility of obtaining better than ±10dB prediction accuracy from any model – no matter how detailed. PiP is freeware and is easy to use.

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