EURODYN '99
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RAILWAY BRIDGES DAMPING IDENTIFICATION
USING TRAFFIC INDUCED VIBRATION
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EURODYN '99
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Plan
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Introduction
Description of parametric models
Models validation
Application to real life signals
Damping identification along decay
Conclusion
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LOGICIEL SCIENTIFIQUE
Work commissioned and supported by ERRI committee
D 214 'Railways bridges for speeds >200 km/h'
Common methods of damping measurement are :
z
z
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Introduction
logarithmic decrement or general free decay analysis
frequency response under controlled excitation (sinus or
transient)
Because of their low cost, free decay analysis methods are
prefered
z
z
due to the non stationarity of train excitation, signal processing
problems occur : spectral methods are inefficient
the time methods are well adapted to solve these problems
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Introduction
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Time methods are called parametric models
z
z
z
commonly used in speech processing and automatics since the
late 60 's
the time signal is used to identify model parameters
the parametric models proposed here are
•
•
z
AR model (Auto Regressive)
Eigen value model (Prony Pisarenko)
the logarithmic decrement method was also tested
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Parametric models
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AR model
sampled general linear system of input et and ouput xt may
be modeled by difference equation
xt + a1 xt −1 + K + a 2 m xt − 2 m = et
2m
x t = − ∑ a k x t − k + et
k =1
or in matrix form
⎡ x 2 m +1 ⎤ ⎡ x 2 m
⎥ ⎢x
⎢x
⎢ 2 m + 2 ⎥ = ⎢ 2 m +1
⎢ M ⎥ ⎢ M
⎥ ⎢
⎢
x
⎣ t ⎦ ⎣ xt −1
x1 ⎤ ⎡ − a1 ⎤ ⎡e2 m +1 ⎤
L x 2 ⎥⎥ ⎢⎢ − a 2 ⎥⎥ ⎢⎢ e2 m ⎥⎥
+
M ⎥⎢ M ⎥ ⎢ M ⎥
⎥
⎥ ⎢
⎥⎢
L x t − 2 m ⎦ ⎣ − a 2 m ⎦ ⎣ et ⎦
x 2 m −1 L
x2m
M
xt − 2
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Parametric models
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ak are called the AR coefficients
These coefficients are related to the frequencies ωk and
dampings ζk by the relationships (∆t=sampling interval)
ωk ζ k = −
ωk
( )
Ln a k
2
2 ∆t
(
a tan ℑ( a k ) / ℜ( a k )
1− ζ =
∆t
2
k
)
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Parametric models
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The solution is found through least square estimation
X = Φ xθ + E
(
θ = Φ Φx
T
x
)
−1
Φ Tx X
„
One cannot fully justify the use of the AR model unless
the input of the system consists in white noise, which is
not the case here
„
The least square estimation is a source of bias errors in
parameter estimation
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Parametric models
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The eigen value method (Prony-Pisarenko)
„
When no input is present system equation can be written
2m
as
xt + ∑ a k xt − k = 0
k =1
⎡1 ⎤ ~ ⎡1 ⎤
[X Φ x ]⎢ ⎥ = Φ x ⎢ ⎥ = {0}
⎣ − θ⎦
⎣ − θ⎦
⎡1 ⎤
~T ~ ⎡ 1 ⎤
Φ x Φ x ⎢ ⎥ = R xx ⎢ ⎥ = {0}
⎣ − θ⎦
⎣ − θ⎦
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Rxx is the covariance matrix
z
z
z
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Parametric models
parameter vector [1 -θ]T is the covariance matrix eigen vector
associated with eigen value 0
as Rxx is definite semi positive, this eigen value is the lowest one
it can be shown that in presence of noise at the output, the eigen
value remains the lowest one
The method simply consists in computing the covariance
matrix Rxx and then calculating eigen vectors. The eigen
vector whose eigen value is the lowest gives the ak
parameters
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Models validation
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Simulations have been performed to validate the different
models and estimate errors in damping estimation
z
z
AR, eigen model and logarithmic decrement have been tested
all methods are sensitive to
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•
•
•
z
size (number of samples) of data processed
sampling frequency
filtering
modal coupling
The simulation aim is to optimize the choice of signal processing
parameters
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z
Models validation
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logarithmic decrement
bias errors are small < 5%
more than 10 periods of signal are necessary to obtain less than 15 %
random error
• in the case of several modes, the filtering around each mode gives similar
results with less than 20 % random error
• results are coherent when the modes are sufficiently separated
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z
AR & Eigen models
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~10 periods of signal are sufficient
AR model is biased but the filtering reduces this effect
eigen model is unbiased (<5%)
in these conditions bias and random errors are less than 10 %
with three modes bias error may increase (15 %) when coupling between
modes is important (4% damping)
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Models validation
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LOGICIEL SCIENTIFIQUE
Conclusion
z
z
z
z
logarithmic decrement is a good estimator, if the modes are not
too close one to the other
AR & Eigen models give good results, The Eigen model gives
better results than AR model since it is unbiased
simulations gave confidence in time methods. A strategy of
mode identification is well defined and may be applied to real
life measurements
as errors in simulations are less than 15 %, greater errors must be
expected in reality
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Application to real life signals
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Train passing by were recorded on Briollay bridge on
TGV Atlantique high speed line
z
3 spans steel/concrete bridge
38 m span
Paris
Angers
53.5 m span
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Application to real life signals
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Modal analysis was performed with hammer excitation
whole bridge flexion mode @ 2.29 Hz
whole bridge torsion mode @ 3.28 Hz
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Application to real life signals
whole bridge flexion mode @ 3.66 Hz
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1stspan flexion mode @ 4.16 Hz
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Application to real life signals
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1stspan torsion mode @ 4.96 Hz
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6 train passing by are analyzed
z
z
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Application to real life signals
5 TGV
1 TER
Strategy for analyzing decay signals
z
z
z
z
measure the spectral components of the signal
separate modes to be identified in small groups
for each group, apply optimum sampling and filtering to isolate
modes
identify frequency and damping
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EURODYN '99
Application to real life signals
fe=512 Hz
fe=17 Hz
TGV2
-9.5
V
0.08
m/s2
Real
Real
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TGV2
-0.08
-10
0 s
16 s
0 s
16 s
measurement signal
30 times decimated signal
fe=512 Hz
fe=17 Hz
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EURODYN '99
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Application to real life signals
X:2.2644
X:3.5964
X:4.0959
X:4.8951
Spectra
z
very noisy
z 2 leading modes at 2.3 &
3.6 Hz
z 2 other modes at 4.1 &
4.9 Hz (not always
excited)
z first torsion mode at 3.3
Hz not excited
Hz
Hz
Hz
Hz
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Y:469.4962 um/s2
Y:68.70036 um/s2
Y:2.729211 um/s2
Y:735.5877 nm/s2
TGV2
0.001
m/s2
Mag (Log)
1E-09
0 Hz
8 Hz
TGV5
0.001
m/s2
Mag (Log)
1E-09
0 Hz
8 Hz
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Application to real life signals
identification with AR and
Eigen model
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AR model identification
0.03
m/s2
z
red points give the
synthetized curve with
model results
z a fit error is computed and
gives model confidence
Real
-0.03
6 s
16 s
Eigen model identification
0.03
m/s2
Real
-0.03
6 s
16 s
page 20
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Application to real life signals
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Damping mode 1 : 2.3 Hz
1
0.8
AR model
Eigen model
%
0.6
Log. dec.
0.4
0.2
0
TGV1d
TGV3d
TGV2d
TGV4d
TGV5d
TER2d
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Application to real life signals
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Damping mode 2 : 3.6 Hz
2
AR model
1.5
%
Eigen model
1
Log. dec.
0.5
0
TGV1d
TGV3d
TGV2d
TGV4d
TGV5d
TER2d
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Application to real life signals
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Damping mode 3 : 4.1 Hz
2
1.5
%
AR model
1
Eigen model
0.5
0
TGV1d
TGV3d
TGV2d
TGV4d
TGV5d
TER2d
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Application to real life signals
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Damping mode 4 : 4.9 Hz
2
%
1.5
AR model
Eigen model
1
0.5
0
TGV1d
TGV3d
TGV2d
TGV4d
TGV5d
TER2d
page 24
EURODYN '99
Application to real life signals
AR model
mode 1
2.3 Hz
mode 2
3.6 Hz
mode 3
4.1 Hz
mode 4
4.9 Hz
Eigen model
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logarithmic decrement
damp. (%)
0.45
scatter (%)
10
damp. (%)
0.47
scatter (%)
15
damp. (%)
0.46
scatter (%)
10
1.1
15
1.13
22
1.16
11
1.32
24
1.33
15
0.81
32
0.81
30
not
computable
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Application to real life signals
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Result synthesis
z
z
z
z
z
frequencies are estimated with a very good accuracy ~1 %
modes 1 & 2 are estimated with maximum 10-20 % dispersion
modes 3 & 4 are estimated with maximum 30 % dispersion
discrepancies are caused by measurement/estimation errors AND
bridge and excitation characteristics (non linearities, train
loading)
as modes 3 & 4 are only slightly emerging from noise, dispersion
is greater
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Damping along free decay
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Due to non linearities, damping varies along decay : the
change in damping can be estimated using the following
procedure
z
z
the damping is estimated using the Prony method in steps
of ~ 10 periods (filtering 48 dB/oct. at 1.5 X first frequency
when possible)
maximum of amplitude is measured : this amplitude is
associated with a damping value
Plot of the damping as a function of amplitude
page 27
Damping along free decay
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ζ1
ζ3
U1 U2
0.3
m/s²
U3
Real
-0.3
0 s
1 s
ζ2
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Briollay bridge
EURODYN '99
3 spans steel/concrete decks
z
z
strong dependance on amplitude
0.3 < ζ < 0.9 %
Briollay damping mode 1 : 2.3 Hz
TGV1dd
TGV3dd
1
Damping (%)
„
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TGV2dd
0.8
TGV4dd
0.6
TGV5dd
0.4
TER2dd
0.2
0
0
0.01
0.02
0.03
0.04
0.05
Acceleration (m /s²)
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PK 41 + 450
EURODYN '99
16.5 m long reinforced concrete deck
z
z
4<ζ<8%
slight increase with acceleration
PK41 + 450 damping mode 1 : 13.5 Hz
Damping (%)
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10
Train 2
8
Train 4
6
Train 8
4
Train 10
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Acceleration (m /s²)
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Valenton bridge
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46 m long steel deck
z
z
strong dependance on amplitude
good correlation between trains
Valenton damping mode 1 : 2.8 Hz
Damping (%)
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1.20
Marc1
1.00
Marc3
0.80
Marc4
0.60
TGV4
0.40
0.20
0.00
0
0.5
1
1.5
2
2.5
3
3.5
4
Velocity (m m /s)
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Tivernon bridge
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3 spans 11.7 m long steel/concrete decks
z
z
constant damping (except for 1 train)
good correlation between trains
Tivernon damping mode 1 : 9 Hz
4
Damping (%)
„
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M1
M5
3
M6
2
M7
1
0
0
0.5
1
1.5
2
2.5
velocity (m m /s)
page 32
OA 49/25
EURODYN '99
46 m long steel/concrete deck
z
z
constant damping
good correlation between trains
OA 49/25 damping mode 1 : 3.75 Hz
Damping (%)
„
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4
TGV6
3
TGV9
TGV12
2
TGV13
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Velocity (m m /s)
page 33
BIP
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2 spans 33-35 m long steel/concrete
decks
z
damping constant
BIP damping mode 1 : 3.8 Hz
2
Damping (%)
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Train9
1.5
Train11
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Velocity (m m /s)
page 34
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Damping along free decay
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Conclusion
z
two types of behaviour
•
•
z
due to the deformation amplitude, different train characteristics
and speeds lead to different results
•
z
non linearity : damping increases linearly with amplitude
constant damping
Briollay : TER 0.25 % ; TGV 0.4 -> 1%
This kind of analysis gives good confidence on measurements
and damping values
•
scattering in measurements can be explained
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Prony and AR models have proved to be very efficient
z
z
„
Conclusion
good accuracy in results
coupled modes identification
Recommendations in damping estimation
z
z
proper signal filtering is of great help
damping has to be estimated by a statistical approach
use several methods (log. dec. AR eigen method) in order to allow
assessment of errors made in the estimation procedure
• use several measurements to estimate random errors
• mean and standard deviation must be given
•
page 36
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Conclusion
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as damping may vary with vibration level
•
•
•
it should be studied throughout the decay period (when possible)
damping values must be associated with a deck vibration amplitude
trains type and speeds are of great importance in assessing the
reproductibility of results
page 37