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Subtitle January 2012 XX, Sweden Verification of an experimental prediction method for railway induced vibration Hans Verbraken, Geert Lombaert, Geert Degrande geert.degrande@bwk.kuleuven.be bwk.kuleuven.be/bwm Structural Mechanics,

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SLIDE 1

Subtitle January 2012 XX, Sweden

Verification of an experimental prediction method for railway induced vibration

Hans Verbraken, Geert Lombaert, Geert Degrande geert.degrande@bwk.kuleuven.be bwk.kuleuven.be/bwm Structural Mechanics, Department of Civil Engineering, KU Leuven

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SLIDE 2

Introduction

Introduction

  • n

FRA procedure Verification Conclusion Left footer January 2012 – 2/19

Railway induced vibrations and re-radiated noise in build- ings

■ Excitation mechanisms: wheel/rail roughness, rail joints,... ■ Vehicle-track interaction: dynamic axle loads. ■ Dynamic interaction between the tunnel and the soil: transfer functions. ■ Wave propagation in the soil: dynamic reciprocity theorem. ■ Dynamic soil-structure interaction. ■ Vibrations in buildings (1 to 80 Hz). ■ Re-radiated noise in buildings (16 to 250 Hz).

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SLIDE 3

Introduction

Introduction

  • n

FRA procedure Verification Conclusion Left footer January 2012 – 3/19

Prediction methods

■ Numerical predictions [Lombaert et al., JSV, 2009][François et al., CMAME, 2010]

+ Great variety in numerical models − Need for accurate parameter characterization

■ Empirical predictions

+ Soil characteristics inherently taken into account − Accurate input data is not always available

■ Hybrid predictions ■ Experimental transfer function (red) and 95% confidence region (blue) between 2 points

in the free field [Schevenels, OPTEC, 2009]

x x′ 25 50 75 100 125 150 10

−12

10

−11

10

−10

10

−9

10

−8

Frequency [Hz] Displacement [m/N] Prior model 25 50 75 100 125 150 10

−12

10

−11

10

−10

10

−9

10

−8

Frequency [Hz] Displacement [m/N] Posterior model

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SLIDE 4

The FRA procedure

Introduction FRA procedure

  • Vibration velocity

level

  • Line transfer

mobility

  • Force density

Verification Conclusion Left footer January 2012 – 4/19

FRA procedure

■ Detailed Vibration Assessment

◆ Federal Railroad Administration (FRA) and Federal Transit Administration (FTA)

[Hanson et al., FRA, 2005; Hanson et al., FTA, 2006]

High-Speed Ground Transportation Noise and Vibration Impact Assessment

Office of Railroad Development

  • U. S. Department
  • f Transportation

Federal Railroad Administration October 2005

TRANSIT NOISE AND VIBRATION IMPACT ASSESSMENT

FTA-VA-90-1003-06 May 2006 Office of Planning and Environment Federal Transit Administration

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SLIDE 5

The FRA procedure

Introduction FRA procedure

  • Vibration velocity

level

  • Line transfer

mobility

  • Force density

Verification Conclusion Left footer January 2012 – 5/19

Vibration velocity level

■ Prediction of the ground vibration velocity level in one-third octave bands [Hanson et al.,

2005, 2006] Lv = LF + TML (1)

◆ Vibration velocity level Lv = 20 log10(vRMS) [dB ref 10−8 m/s] ◆ Force density LF [dB ref N/√m ] ◆ Line transfer mobility TML [dB ref 10−8

m/s N/√m ]

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

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SLIDE 6

The FRA procedure

Introduction FRA procedure

  • Vibration velocity

level

  • Line transfer

mobility

  • Force density

Verification Conclusion Left footer January 2012 – 6/19

Line transfer mobility TML

■ Characterization of the transfer of vibration TML = 10 log10

  • hn

k=1 10

TMPk 10

  • Measurement line

Rail alignment Impact locations x y z Measurement line Rail alignment Impact locations x y z

8 16 31.5 63 125 −60 −50 −40 −30 −20 −10

TML [dB ref 10−8 (m/s)/(N/m0.5)] One−third octave band center frequency [Hz]

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SLIDE 7

The FRA procedure

Introduction FRA procedure

  • Vibration velocity

level

  • Line transfer

mobility

  • Force density

Verification Conclusion Left footer January 2012 – 7/19

Force density LF

Vibration velocity level Lv

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

Transfer mobility TML

8 16 31.5 63 125 −60 −50 −40 −30 −20 −10

TML [dB ref 10−8 (m/s)/(N/m0.5)] One−third octave band center frequency [Hz]

Force density LF = Lv − TML

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] LF [dB ref 1 N/m0.5]

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SLIDE 8

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 8/19

Moving dynamic loads

■ Analytical expressions for the vibration velocity level due to moving loads in a tunnel or

at grade [Sheng et al., 1999; Lombaert et al., 2000; Forrest and Hunt, 2006]. v(x, t) =

na

  • k=1

t

−∞

HT(xk(τ), x, t − τ)gk(τ)dτ (2)

x y z xk(τ) x x y z xk(τ) x

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SLIDE 9

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 9/19

Case

■ Bakerloo line tunnel [Gupta et al., JSV, 2009] x z 5 m 28 m 3.704 m

ρs = 1980 kg/m3 βs = 0.042 Cs = 325 m/s Cp = 1964 m/s ρs = 1980 kg/m3 βs = 0.039 Cs = 220 m/s Cp = 1571 m/s

5.5 m 60 m A B ■ Non-ballasted concrete slab track ■ Train with 28 axles (Lt = 108.33 m) ■ Unevenness FRA class 3

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SLIDE 10

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 10/19

Numerical prediction: coupled periodic FE–BE model

■ Model [Degrande et al., JSV, 2006] x y z ˜ x ˜ x + Ley ˜ x + nyLey x′ ■ (a) Transfer function, (b) time history and (c) one-third octave band spectrum of the vi-

bration velocity in the free field in point A [Gupta et al., JSV, 2009]

20 40 60 80 100 120 −60 −50 −40 −30 −20 −10

Frequency [Hz] Mobility [dB ref 10−8 m/s/N]

−20 −15 −10 −5 5 10 15 20 −2 −1 1 2 x 10

−4

Time [s] Velocity [m/s]

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

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SLIDE 11

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 11/19

Numerical prediction model

■ Transfer functions of the track-tunnel-soil system at (a) 10 Hz and (b) 40 Hz [Gupta, 2008]

(a) Animation (avi) and zoom (avi). (b) Animation (avi) and zoom (avi).

■ Response (avi) in the free field due to a carriage moving at constant speed on an uneven

rail with a single wavelength unevenness (excitation at 40 Hz).

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SLIDE 12

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 12/19

Derivation of analytical expressions for LF and TML

■ Vibration velocity

v(x, t) =

na

  • k=1

t

−∞

HT(xk(τ), x, t − τ)gk(τ)dτ (3)

■ Assumptions

◆ Fixed point loads ◆ Non-coherent and equal axle loads ◆ Frequency-averaged transfer function ◆ Equidistant point sources

■ Analytical expressions for LF and TML

Lv = 10 log10

  • g2

RMS

La

  • LF

+ 10 log10

  • La

na

  • k=1

ω2

ω1 |ˆ

hzz(xk, x, ω)|2dω ∆ω

  • TML

(4)

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SLIDE 13

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 13/19

Results

■ One-third octave band spectra of the velocity in (a) point A and (b) point B for a train

passage at a speed of 30 km/h calculated with the numerical method (grey line) and the FRA procedure (black line).

x y z A B

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

(a) (b)

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SLIDE 14

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 14/19

Assumption 1: fixed point loads

■ One-thirds octave band RMS value of the vertical velocity in (a) point A and (b) point B

due to a moving train (grey line) and due to a fixed train (black line).

x y z A B

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

(a) (b)

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SLIDE 15

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 15/19

Assumption 2: non-coherent and equal axle loads

■ One-third octave band RMS value of the vertical velocity in (a) point A and (b) point B

with coherent point loads (grey line) and non-coherent point loads (black line)

x y z A B

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

(a) (b) [Hunt, 1996; Wu and Thompson, 2001]

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SLIDE 16

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 16/19

Assumption 3: average value of the transfer function

■ One-third octave band spectra of the velocity in (a) point A and (b) point B with narrow

band (grey line) and frequency-averaged (black line) transfer function.

x y z A B

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

8 16 31.5 63 125 10 20 30 40 50 60

One−third octave band center frequency [Hz] Lv [dB ref 10−8 m/s]

(a) (b)

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SLIDE 17

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 17/19

Assumption 4: equidistant point sources

■ Transfer mobility

TML = 10 log10

  • La

na

  • k=1

ω2

ω1 |ˆ

hzz(xk, x, ω)|2dω ∆ω

  • (5)

(a) (b) (c)

La La h/2 h

TML = 10 log10

  • h

n

  • k=1

10

TMPk 10

  • (6)
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SLIDE 18

Verification

Introduction FRA procedure Verification

  • Analytical
  • Case
  • Numerical
  • Derivation
  • Results

Conclusion Left footer January 2012 – 18/19

Assumption 4: equidistant point sources

■ Transfer mobility in (a) point A and (b) point B calculated with the original axle positions

(light-grey line), with 28 equidistant axle positions (dark-grey line) and with 15 equidistant axle positions including two edge points (black line).

x y z A B

8 16 31.5 63 125 −60 −50 −40 −30 −20 −10

One−third octave band center frequency [Hz] TML [dB ref 10−8 (m/s)/(N/m0.5)]

8 16 31.5 63 125 −60 −50 −40 −30 −20 −10

One−third octave band center frequency [Hz] TML [dB ref 10−8 (m/s)/(N/m0.5)]

(a) (b)

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SLIDE 19

Conclusion

Introduction FRA procedure Verification Conclusion

  • Conclusion

Left footer January 2012 – 19/19

Conclusion

■ Separate characterization of force density and line transfer mobility leads to a good ap-

proximation of the one-third octave band vibration velocity level

■ Analytical expressions of force density and line transfer mobility are obtained based on

four assumptions

◆ Fixed train position ◆ Non-coherent axle loads ◆ Averaged transfer function ◆ Equidistant point loads

■ The results for tunnels are also validated for the case of railway traffic at grade