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
Introduction Railway induced vibrations and re-radiated noise in build- ings Introduction • n FRA procedure Verification Conclusion ■ 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). Left footer January 2012 – 2/19
Introduction Prediction methods ■ Numerical predictions [Lombaert et al., JSV, 2009][François et al., CMAME, 2010] Introduction • n + Great variety in numerical models − Need for accurate parameter characterization FRA procedure ■ Empirical predictions Verification + Soil characteristics inherently taken into account Conclusion − 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] Displacement [m/N] −8 Displacement [m/N] −8 10 10 Prior model Posterior model −9 −9 10 10 x ′ −10 −10 10 10 −11 −11 x 10 10 −12 −12 10 10 0 25 50 75 100 125 150 0 25 50 75 100 125 150 Frequency [Hz] Frequency [Hz] Left footer January 2012 – 3/19
The FRA procedure FRA procedure ■ Detailed Vibration Assessment Introduction ◆ Federal Railroad Administration (FRA) and Federal Transit Administration (FTA) FRA procedure [Hanson et al., FRA, 2005; Hanson et al., FTA, 2006] • Vibration velocity level • Line transfer mobility • Force density High-Speed Ground Transportation Noise and Vibration Impact Assessment Verification U. S. Department Conclusion TRANSIT NOISE AND VIBRATION of Transportation Federal Railroad IMPACT ASSESSMENT Administration October 2005 FTA-VA-90-1003-06 May 2006 Office of Railroad Development Office of Planning and Environment Federal Transit Administration Left footer January 2012 – 4/19
The FRA procedure Vibration velocity level ■ Prediction of the ground vibration velocity level in one-third octave bands [Hanson et al., Introduction 2005, 2006] FRA procedure • Vibration velocity level L v = L F + TM L (1) • Line transfer mobility ◆ Vibration velocity level L v = 20 log 10 ( v RMS ) [dB ref 10 − 8 m / s] • Force density ◆ Force density L F [dB ref N / √ m ] Verification m / s ◆ Line transfer mobility TM L [dB ref 10 − 8 Conclusion N / √ m ] 60 50 L v [dB ref 10 −8 m/s] 40 30 20 10 0 8 16 31.5 63 125 One−third octave band center frequency [Hz] Left footer January 2012 – 5/19
The FRA procedure Line transfer mobility TM L � � Introduction TMP k h � n ■ Characterization of the transfer of vibration TM L = 10 log 10 k =1 10 10 FRA procedure • Vibration velocity level Rail alignment Impact locations • Line transfer z Impact locations Measurement line mobility Measurement line z y y • Force density x x Verification Conclusion Rail alignment TM L [dB ref 10 −8 (m/s)/(N/m 0.5 )] 0 −10 −20 −30 −40 −50 −60 8 16 31.5 63 125 One−third octave band center frequency [Hz] Left footer January 2012 – 6/19
The FRA procedure Force density L F Introduction 60 FRA procedure 50 L v [dB ref 10 −8 m/s] • Vibration velocity 40 level Vibration velocity level L v 30 • Line transfer mobility 20 10 • Force density 0 8 16 31.5 63 125 Verification One−third octave band center frequency [Hz] Conclusion TM L [dB ref 10 −8 (m/s)/(N/m 0.5 )] 0 −10 −20 Transfer mobility TM L −30 −40 −50 −60 8 16 31.5 63 125 One−third octave band center frequency [Hz] 60 50 L F [dB ref 1 N/m 0.5 ] 40 Force density L F = L v − TM L 30 20 10 0 8 16 31.5 63 125 One−third octave band center frequency [Hz] Left footer January 2012 – 7/19
Verification Moving dynamic loads ■ Analytical expressions for the vibration velocity level due to moving loads in a tunnel or Introduction at grade [Sheng et al., 1999; Lombaert et al., 2000; Forrest and Hunt, 2006]. FRA procedure Verification � t n a � • Analytical H T ( x k ( τ ) , x , t − τ ) g k ( τ ) dτ v ( x , t ) = (2) • Case −∞ k =1 • Numerical • Derivation • Results Conclusion z y x k ( τ ) x x k ( τ ) z y x x x Left footer January 2012 – 8/19
Verification Case ■ Bakerloo line tunnel [Gupta et al., JSV, 2009] Introduction FRA procedure 5.5 m 60 m Verification z • Analytical A B x • Case ρ s = 1980 kg / m 3 C s = 325 m / s 5 m C p = 1964 m / s β s = 0 . 042 • Numerical • Derivation ρ s = 1980 kg / m 3 C s = 220 m / s • Results C p = 1571 m / s β s = 0 . 039 Conclusion 28 m 3.704 m ■ Non-ballasted concrete slab track ■ Train with 28 axles ( L t = 108 . 33 m ) ■ Unevenness FRA class 3 Left footer January 2012 – 9/19
Verification Numerical prediction: coupled periodic FE–BE model ■ Model [Degrande et al., JSV, 2006] Introduction ˜ x + n y L e y FRA procedure Verification • Analytical x + L e y ˜ ˜ • Case x z • Numerical y x • Derivation • Results x ′ Conclusion ■ (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] −4 x 10 0 60 2 Mobility [dB ref 10 −8 m/s/N] −10 50 L v [dB ref 10 −8 m/s] Velocity [m/s] 1 −20 40 0 −30 30 −40 20 −1 −50 10 −2 −60 0 −20 −15 −10 −5 0 5 10 15 20 0 20 40 60 80 100 120 8 16 31.5 63 125 Time [s] Frequency [Hz] One−third octave band center frequency [Hz] Left footer January 2012 – 10/19
Verification Numerical prediction model ■ Transfer functions of the track-tunnel-soil system at (a) 10 Hz and (b) 40 Hz [Gupta, 2008] Introduction FRA procedure Verification • Analytical • Case • Numerical • Derivation • Results Conclusion (a) (b) Animation (avi) and zoom (avi). 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). Left footer January 2012 – 11/19
Verification Derivation of analytical expressions for L F and TM L ■ Vibration velocity Introduction FRA procedure � t n a � Verification H T ( x k ( τ ) , x , t − τ ) g k ( τ ) dτ v ( x , t ) = (3) • Analytical −∞ k =1 • Case ■ Assumptions • Numerical ◆ Fixed point loads • Derivation ◆ Non-coherent and equal axle loads • Results ◆ Frequency-averaged transfer function Conclusion ◆ Equidistant point sources ■ Analytical expressions for L F and TM L � ω 2 � � � � n a ω 1 | ˆ h zz ( x k , x , ω ) | 2 dω g 2 � RMS L v = 10 log 10 + 10 log 10 L a (4) L a ∆ ω k =1 � �� � � �� � L F TM L Left footer January 2012 – 12/19
Verification Results ■ One-third octave band spectra of the velocity in (a) point A and (b) point B for a train Introduction passage at a speed of 30 km/h calculated with the numerical method (grey line) and the FRA procedure FRA procedure (black line). Verification • Analytical • Case • Numerical • Derivation • Results 60 60 Conclusion z y 50 50 L v [dB ref 10 −8 m/s] L v [dB ref 10 −8 m/s] x 40 40 A 30 30 20 20 B 10 10 0 0 8 16 31.5 63 125 8 16 31.5 63 125 One−third octave band center frequency [Hz] One−third octave band center frequency [Hz] (a) (b) Left footer January 2012 – 13/19
Verification Assumption 1: fixed point loads ■ One-thirds octave band RMS value of the vertical velocity in (a) point A and (b) point B Introduction due to a moving train (grey line) and due to a fixed train (black line). FRA procedure Verification • Analytical • Case • Numerical • Derivation 60 60 • Results z y 50 50 L v [dB ref 10 −8 m/s] L v [dB ref 10 −8 m/s] Conclusion x 40 40 A 30 30 20 20 B 10 10 0 0 8 16 31.5 63 125 8 16 31.5 63 125 One−third octave band center frequency [Hz] One−third octave band center frequency [Hz] (a) (b) Left footer January 2012 – 14/19
Verification 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 Introduction with coherent point loads (grey line) and non-coherent point loads (black line) FRA procedure Verification • Analytical • Case • Numerical • Derivation 60 60 • Results z y 50 50 L v [dB ref 10 −8 m/s] L v [dB ref 10 −8 m/s] Conclusion x 40 40 A 30 30 20 20 B 10 10 0 0 8 16 31.5 63 125 8 16 31.5 63 125 One−third octave band center frequency [Hz] One−third octave band center frequency [Hz] (a) (b) [Hunt, 1996; Wu and Thompson, 2001] Left footer January 2012 – 15/19
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