Guideline for estimating structural damping of railway bridges Background document D5.2-S2

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Guideline for estimating structural damping of railway bridges Background document D5.2-S2

PRIORITY 6

SUSTAINABLE DEVELOPMENT GLOBAL CHANGE & ECOSYSTEMS

INTEGRATED PROJECT

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This Report is a Part of the Research Project “Sustainable Bridges” which aims to help European railways to use their bridges more efficiently by allowing higher axle loads on freight vehicles and by increasing the maximum permissible speed of passenger trains. This should be possible without causing unnecessary disruption to the carriage of goods and passengers, and without compromising the safety and economy of the working railway.

The Project has developed improved methods for computing the safe carrying capacity of bridges and better engineering solutions that can be used in upgrading bridges that are found to be in need of attention. Other re- sults will help to increase the remaining life of existing bridges by recommending strengthening, monitoring and repair systems.

A consortium, consisting of 32 partners drawn from railway bridge owners, consultants, contractors, research institutes and universities, has carried out the Project, which has a gross budget of more than 10 million Euros.

The European Commission’s 6^th Framework Programme has provided substantial funding, with the balancing funding coming from the Project partners. Skanska Sverige AB has provided the overall co-ordination of the Pro- ject, whilst Luleå Technical University has undertaken the scientific leadership.

The authors of this report have used their best endeavours to ensure that the information presented here is of the highest quality. However, no liability can be accepted by the authors for any loss caused by its use.

Figure on the front page: Railway bridge with set-up for estimating structural damping.

Project acronym: Sustainable Bridges

Project full title: Sustainable Bridges – Assessment for Future Traffic Demands and Longer Lives Contract number: TIP3-CT-2003-001653

Project start and end date: 2003-12-01 -- 2007-11-30 Duration 48 months

Document number: Deliverable D5.2-S2 Abbreviation SB-5.2-S2

Author/s: G. Feltrin and D. Gsell, Empa Date of original release: 2007-11-30

Revision date:

Project co-funded by the European Commission within the Sixth Framework Programme (2002-2006)

Dissemination Level

PU Public X

PP Restricted to other programme participants (including the Commission Services) RE Restricted to a group specified by the consortium (including the Commission Services) CO Confidential, only for members of the consortium (including the Commission Services)

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Summary

Structural damping has important effects on the dynamical behavior of a railway bridge because it is one of the main factors limiting the vibration amplitudes. Its knowledge is therefore essential for the assessment of existing bridges subjected to dynamic loads. Unfortunately, structural damping is a somewhat elusive physical quantity which, for a specific bridge, is difficult to predict a priori. The fuzziness is due to the heterogeneity of railway bridges with respect to construction material, bridge design, bearing systems and soil conditions. Fur- thermore, the current knowledge of structural damping of railway bridges is incomplete and heterogeneous because in the past different methods have been used to measure structural damping.

The objective of this guideline is to provide methods for estimating experimentally the structural damping of railway bridges. Standardizing the methods for measuring structural damping allows to set-up a homogeneous data set that permits a more systematic and accurate a priori estimation of structural damping. This guideline focuses on three experimental methods for estimating structural damping with field tests: the decay curve method, the multiple mode decay curve method and the ambient vibration method. The methods allows for a fast, cheap and reliable estimation of structural damping without any use of artificial excitation sources but just by analyzing vibrations generated by train crossings and ambient sources. With respect to the state of the art, e.g. (Institute, 1999), this guideline introduces numerically robust methods for estimating simultaneously the damping of several vibration modes using multiple input data channels. Furthermore, for each method, detailed advices are given about their practical use. This guideline was initiated by a request of WP4 that needs reliable structural damping information as input for the assessment methods developed in this work package.

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Acknowledgments

This guideline has been drafted on the basis of Contract No. TIP3-CT-2003-001653 between the European Community represented by the Commission of the European Communities and the Skanska Teknic AB contractor acting as Coordinator of the Consortium. The authors ac- knowledge the Swiss State Secretariat for Education and Research and Empa for its finan- cial support.

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1 Introduction

In a vibrating structure, damping characterises the removal of mechanical energy. Generally, the dominant part of the dissipated energy is converted into thermal energy (heat) and interaction or radiation energy (noise and soil vibration). Damping is responsible for the observed monotonic amplitude decay of a freely vibrating structure. Damping has also an important influence on the vibration amplitude of a structure subjected to time varying, external forces.

Therefore, a good knowledge of damping is fundamental for providing a reasonable accurate prediction of the dynamic response of structures subjected to dynamic loads.

A precise definition of damping is difficult to provide because of the great variety of mechanisms producing energy loss. In this document, the following definition is used:

Damping is any effect, either deliberately engendered or inherent to a structure that tends to reduce the amplitude of oscillations by dissipating mechanical energy.

For bridges, the relevant damping mechanisms are – Material damping

– Nonmaterial structural damping – Interaction or radiation damping

Material damping refers to the energy dissipation within bulk material and is associated to mechanisms on atomic, molecular or micro scale level, regardless of the precise physical process involved. Nonmaterial structural damping refers to all energy dissipation mechanisms within a structure associated to the relative motion of adjacent structural members (e.g. friction in joints, bearings, interfaces). Interaction or radiation damping refers to the energy loss in a structure generated by transferring energy to the surroundings by wave propa- gation (interaction with soil and air, radiation of sound). The overall damping of a bridge is the sum of these three damping mechanisms. Except for material damping, that can be measured in the laboratory, an experimentally precise distinction between nonmaterial structural damping and interaction or radiation damping is very difficult to achieve.

Generally, damping is generated by nonlinear processes. Therefore, damping depends on the static equilibrium state of a structure, and the amplitude and frequency of vibration. Fur- thermore, for certain materials and structural types, damping depends also on environmental parameters like temperature and humidity. The presence of water or ice might change the damping properties of a structure.

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2 Characterisation of Damping

In structural engineering, damping is typically characterized on a global level. Hence, the characterization considers the total energy loss of a structure. The structure represents a closed system that looses mechanical energy by internal energy dissipation and by energy transfer trough the system boundary (interaction or radiation damping).

2.1 Damping loss factor

The damping loss factor

ψ

is defined as the ratio of the dissipated energy per cycle E_D to the maximum potential energy E_P achieved during a cycle:

1 2

D

P

E

ψ

E

= π

.

The damping loss factor has mainly theoretical significance because the total dissipated energy E_D as well as the maximum potential energy E_P can not be measured directly. Fur- thermore, the definition applies only for stationary periodic motions. That is, the dissipated energy during a cycle is in equilibrium with the energy that the structure absorbs from the work done by external forces.

2.2 Modal damping

The state of vibration of linear or weakly nonlinear structures is commonly described as the superposition of a finite or infinite number of vibration modes. If the individual vibration modes are loosely coupled, that is, when there is no significant exchange of energy between the vibration modes, the individual modes can be considered as independent and simple vibratory systems. This allows subdividing and distributing the overall damping of a structure into individual vibration modes. A vibration mode can be modelled as a single degree of freedom system (SDOF model). That is, a system composed by a mass, a spring and a damping element. The spring and damping elements may be nonlinear. The mathematical model of the SDOF system is given by

( )

_n

( , , ) ( )

_n

( , , ) ( ) ( )

mx t

&& +

c x x

& K

x t

& +

k x x

& K

x t

=

f t ^.

( )

x t : modal displacement

( )

f t : modal force component m: modal mass

( , , )

c x xn

& K

: linear or nonlinear modal damping coefficient

( , , )

k x xn

& K

: linear or nonlinear modal stiffness coefficient.

If the exchange of energy between several vibration modes is too large to be neglected, the system has to be modelled as a multiple degree of freedom system (MDOF). In this case, the modal damping concept may still apply. However, the characterization of damping is much more complex.

For vibration modes with linear response, that is, the damping and stiffness coefficients of the SDOF model are both constant (c_n

=

c, viscous damping, and k_n

=

k, linear elastic stiffness), the characterization of damping can be associated to the concepts of damping ratio

ζ

and logarithmic decrement

δ

.

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2.2.1 Damping ratio

The damping ratio is defined as

2 2

c c

km m

ζ ⁼ ⁼ ω

^.

ω

is the natural circular frequency associated to the vibration mode. It is defined as

k

ω

= m . The associated natural frequency is defined as

1

2 2

f k

ω m

ω

π π

= = .

The damping ratio

ζ

is a non-dimensional quantity. The damping ratio is related to the loss factor by

ψ = 2 ζ

2.2.2 Logarithmic decrement

Closely related to the damping ratio is the concept of logarithmic decrement. The logarithmic decrement is defined as the natural logarithm of the ratio of two consecutive amplitude maxima of a freely vibrating ( f t

( ) ≡ 0

) SDOF system:

*

ln ( )

( )

x t x t T

δ

_{= ⎜}^⎛ ^⎞_⎟

⎝ + ⎠^,

where x t

( )

^* refers to the amplitude of vibration at time t^* and T is the period of the vibration (Figure 2.1).

0 2 4 6 8 10

-2 -1 0 1 2

a [ms-2 ]

Time [s]

x(t*) x(t*+T)

t* t+T

x(t*+mT)

t*+mT

Figure 2.1: Definition of logarithmic decrement δ .

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The relation between logarithmic decrement and damping ratio is given by

2

1 δ π ζ

= ζ

−

^and

4

² ²

ζ δ

π δ

= +

^.

For small structural damping, the latter relation can be simplified to

δ ≈ 2 πζ

and

2 ζ δ

≈ π

.

2.2.3 Nonlinear modal damping

Damping ratio and logarithmic decrement are unable to describe correctly nonlinear structural damping. In these cases, specific parameter identification techniques are used for char- acterizing nonlinear damping. However, the use of nonlinear damping models in assessment is not common practise because standard structural analysis software does not support nonlinear damping models.

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3 Experimental estimation of damping

3.1 Decay curve method

Damping can be estimated by the decay curve method. The damping is extracted from the decay of the vibration amplitude of a freely vibrating structure. The decay curve method yields a direct estimate of the logarithmic decrement

δ

.

3.1.1 Algorithm

A freely vibrating, linear vibration mode exhibits an exponentially modulated decay curve.

The input of the algorithm is

– A vector x t

( )

, t

= 0 K

t_end

= 0 K (

n_t

− Δ 1)

t containing the recorded time series. n_t is the number of samples of the time series and Δt is the time interval between two samples.

The logarithmic decrement

δ

can then be estimated by the following algorithm:

1. Extract the decay curve of an isolated vibration mode by first filtering the measured time history x t

( )

and then removing the part that is not associated to the free vibration:

( ) ( ( ))

x tm

←

filter x t .

2. Extract the local maxima x_i of the absolute amplitude x t_m( ) and the time t_i of the local maxima. This operation yields the set

{

^{( , )}t xi i

}

, where i=0KN−1^,N being the number of extracted maxima.

3. Compute the natural logarithm of the extracted maxima y_i

← ln( )

x_i and the non-

dimensional time

τ

_i

= ⋅

t_i f , where f is the natural frequency of the vibration mode. Cre- ate the set

{

^{( ,}

τ

i yi⁾

}

.

4. Compute the best fit linear regression y

=

a

τ +

b of the set

{

^{( ,}

τ

i yi⁾

}

. 5. Compute the mean value of the logarithmic decrement:

δ

ˆ= −a.

3.1.2 Remarks:

– The estimation of damping has to be based on a sufficiently large number of independent measurements. The accuracy of the damping estimation can be obtained by computing the standard deviation of the independent estimates.

– The quality of the filtering process is vital for achieving an accurate estimate of damping.

The isolation of the modal decay curve by filtering can be achieved by applying a band- pass filter or low- and high-pass filters in series. The quality of the band-pass filter can be improved by down-sampling or decimating the measured time series. When down-

sampling the recorded time series special attention has to be paid to avoid aliasing. The sampling rate has to be at least 10 times the natural frequency of the vibration mode.

– The filtering process produces a distortion of the time series at its very beginning. The part of the measured time series containing the amplitude decay shall not be affected by this distortion. This can be achieved by starting the data recording sufficiently ahead of the start of the amplitude decay curve. The magnitude of signal distortion can also be reduced by flipping the recorded data from left to right. By this operation, the flipped data series

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starts with low amplitude vibrations that reduce significantly the distortion effect. After filtering, the data series has to be flipped again from left to right.

– The structure must be freely vibrating for applying this method. That is, no significant dynamic forces have to act on the structure. In order to avoid systematic errors, the amplitudes of the decay curve have to be significantly larger than the amplitudes generated by ambient vibration sources (wind, micro tremors etc.).

– In principle, any excitation source can be used to excite the structure. The most meaning- ful estimations are obtained when the structure is excited up to the amplitude range of practical interest. In general, this can only be achieved by the crossing of a train over a bridge and it is therefore highly recommended to use this vibration source when evaluating damping.

– In general, the best results are obtained when the sensors are mounted at the position where the shape of the vibration modes achieves their maxima. The positions of the maxima can be estimated using an analytical or numerical model of the structure.

– Accelerometers or geophones (velocity sensors) are recommended for recording the decay curve. Since the accelerations can be significant, when using accelerometers, the saturation of the sensors has to be avoided. It is therefore recommended to use accelerometers with a range of at least ±50 ms^-2 (±5g). Geophones should have a range of at least ±20 mms^-1. The sensors have to be firmly mounted to avoid parasitic vibrations (see (Feltrin, 2004) for details)

– The algorithm is only applicable for well isolated vibration modes that can be extracted using an appropriate filtering process.

– Information about the excited vibration modes can be obtained by computing the power spectrum of the recorded time series.

– Structural damping is in most cases nonlinear. That is, the algorithm presented in this section does not apply for the whole decay curve. However, in these cases, the decay curve can be subdivided into segments covering different amplitude ranges. Within these segments, structural damping can be estimated with sufficient accuracy by applying the decay curve algorithm.

– A measure of the accuracy of the estimation

δ

ˆ is obtained by computing the variance

( ) ˆ

Var

δ

. The variance Var

( ) δ ˆ

is given by

2 1

( ˆ ) ( ) ˆ

( 2) ( 2) ( )

i N

i i

yy i

i N

xx i i

y y Var S

N S N

δ τ τ

=

= = −

− − −

∑ ∑

^.

ˆ

_i

y: Amplitude computed using the linear regression model y

ˆ

_i

=

a

τ

_i

+

b.

τ

: mean value of the set of non-dimensional times

τ

_i:

1 i N i i

N

τ τ

=

= ∑

= _.

3.2 Multiple mode decay curve method

The decay curve method is only valid for estimating the structural damping of a single vibration mode. In general, this requires an additional data processing step, e.g. filtering, for extracting the modal component from the recorded time series. However, in some cases, e.g.

when the natural frequencies of two modes are too close together, the different modal com-

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ponents of the recorded time series can not be isolated by a filtering process. In these cases, the structural damping has to be estimated by algorithms that are able to handle simultaneously multiple vibration modes. The multiple mode decay curve method allows to extract the natural frequencies and the damping ratios of the involved vibration modes.

3.2.1 Algorithm

– A vector x t

( )

, t

= 0 K

t_end

= 0 K (

n_t

− Δ 1)

t containing the recorded time series of the decay curve. n_t is the number of samples of the time series and Δt is the time interval between two samples.

The algorithm for computing the natural frequencies and the damping ratios is given by 1. Filter the recorded time series x t

( )

for removing unwanted low and/or high frequency

components and subtract from the filtered time series its mean value:

ˆ( ) filter( ( )) mean(filter( ( )))

x t

←

x t

−

x t .

2. Set-up the Hankel matrix H₀ and the shifted Hankel matrix H₁:

0 0

( , ) ˆ ( 1)

H

=

H i j

=

x i

+ −

j and H₁

=

H i j₁

( , ) =

x i

ˆ ( +

j

)

3. Compute the singular value decomposition of the Hankel matrix H₀: H₀

=

USV^T, where U is the left singular vector, V is the right singular vector and S is the diagonal matrix containing the singular vectors.

4. Choose a models size n and compute the system matrices A and C, and the initial state vector Z₀ of the discrete time linear system with the formulas:

1 T

n n n n

A

= Σ

U H V

Σ first rows (

_n 1/ 2_n

)

C

=

m U S

1/ 2 0

first column (

_n _n^T

)

Z

=

S V

Legend:

n: model size (dimension of the system matrix A).

m: output data channel size.

1/ 2 n Sn⁻

Σ =

: Inverse of the square root of the diagonal matrix of dimension n containing the n greatest singular values.

Un: Matrix containing the n singular left eigenvectors associated to the n greatest singular values.

Vn: Matrix containing the n singular right eigenvectors associated to the n greatest singular values.

5. Compute the eigenvalues of the system matrix A:

{ } λ

k ←eig A^{( )}.

6. Compute the natural frequencies of the vibration modes

{ }

fk ←^log(imag⁽

λ

k⁾⁾⋅ fs^{/ 2}

π

.

s

1/

f

= Δ

t is the sampling frequency of the time series x t_m

( )

.

7. Compute the damping ratios

{ } ζ

k ← −real⁽

λ ω

k^{) /} k, where

ω

_k

= 2 π

f_k.

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3.2.2 Remarks:

– The estimation of damping has to be based on a sufficiently large number of independent measurements. The accuracy of the damping estimation can be obtained by computing the standard deviation of the independent estimates.

– The dimension of the Hankel matrices affects the quality of the results. In general, the greater the dimension, the more reliable are the results.

– The structure must be freely vibrating for applying this method. That is, no significant dynamic forces have to act on the structure. In order to avoid systematic errors, the amplitudes of the decay curve have to be significantly larger than the amplitudes generated by ambient vibration sources (wind, micro-tremors etc.).

– In principle, any excitation source can be used to excite the structure. The most meaning- ful estimations are obtained when the structure is excited up to the amplitude range of practical interest. In general, this can only be achieved by the crossing of a train over a bridge and it is therefore highly recommended to use this vibration source when evaluating damping.

– Structural damping is in most cases nonlinear. In these cases, the decay curve can be subdivided into segments covering different amplitude ranges. Within these segments, structural damping can be estimated by applying the multiple mode decay curve method. A segment containing at least three full periods of the associated vibration mode is already sufficient for obtaining a reasonably accurate damping estimate. In general, for short decay curve segments, the multiple mode decay curve method is more accurate than the decay curve method.

– The best damping ratio estimates are obtained for the vibration modes with the most significant components within the decay curve.

– The accuracy of the estimation can be verified by comparing the reconstructed decay curve y t

ˆ( )

with the filtered decay curve x t

ˆ( )

. The reconstructed decay curve is computed according to the algorithm

( ) ( ), ˆ ( ) ( )

z t

+ Δ =

t Az t y t

=

C z t

starting with the initial condition z

(0) =

Z₀. A fast visual verification is obtained by displaying the power spectra of both decay curves. If the peaks of the power spectra related to the vibration modes coincide well, then the damping estimates are also accurate.

– The minimum dimension (model size) of the system matrix A is twice the number of vibration modes that significantly contribute to the decay curve. The number of vibration modes can be determined by the number of significant peaks in the power spectrum of the decay curve. For analysing the sensitivity of the damping ratio with respect to the mode size, the system matrix A should be computed for different model sizes starting from a minimum model size and incrementing the size at each step by 2. The damping ratios obtained from the models yields a measure of the estimation reliability. In general, good estimates are already obtained with the minimum model size.

3.3 Ambient vibration method

In the cases where recordings of the decay curve are not feasible, the ambient vibration method provides a mean to estimate structural damping. Ambient vibration methods for estimating mechanical systems are usually based on concepts of linear system theory. The recorded time series are used to estimate system parameters which describe the input-output

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properties of a linear, time invariant, discrete time system. Ambient vibration methods are a special class of system identification methods because they do not require any knowledge about the exciting forces (output only system identification). These methods are based on the assumption that the driving force is a broad band, stationary, stochastic process. The spe- cialized literature knows many different ambient vibration methods. Detailed information on linear system theory and related system identification methods are found in (Ljung, 1999, van Overschee and De Moor, 1996, Peeters and De Roeck, 2001, Aoki, 1990).

3.3.1 Algorithm

In this section, a simple, but numerically efficient system identification method is used for estimating natural frequencies and structural damping. The algorithm is a variant of the “Ei- gensystem Realisation Algorithm” or “Stochastic Realization Estimator” (Juang and Pappa, 1985, Aoki, 1990). The algorithm is based on covariance functions of recorded time series.

– A vector x t

( )

, t

= 0 K

t_end

= 0 K (

n_t

− Δ 1)

t containing the recorded time series. n_t is the number of samples of the time series and Δt is the time interval or time step width between two samples.

The overall algorithm is given by

8. Filter the measured time history x t

( )

for removing unwanted low and/or high frequency components and subtract from the filtered time series its mean value:

ˆ( ) filter( ( )) mean(filter( ( )))

x t

←

x t

−

x t .

9. Compute the covariance matrix R_xx of the time series x t

ˆ( )

. 10. Estimate the system matrix A of the linear system model.

11. Compute the natural frequencies f_k and modal damping ratio

ζ

_k by using the system matrix A.

The details of steps 2, 3 and 4 of the algorithm are:

Computation of the covariance matrix The covariance function is defined by

ˆ ˆ

( ) ( )

^T

( )

Rxx k

=

E x t

⎡ ⎣ ⋅

x t

− Δ

k t

⎤ ⎦

^,

where ^E

[ ]

is the expectation operator and k= L1 K ^{, where}K tΔ is the maximum time lag.

The algorithm below computes the covariance function via Fast Fourier Transform. The covariance function is computed according to the following steps:

1. Partition the time series x t

ˆ( )

into p segments with a number of samples N:

{

y tm^{( )}

}

←x t^ˆ^{( )}, for m

= L 1

p^.

2. Extend each segment y t_m

( )

with a buffer zone of at least K zeros (zero padding) for avoiding end effects. This operation yields p segments y t

ˆ ( )

_m .

3. Compute the discrete Fourier transform of each y t

ˆ ( )

_m using an FFT algorithm:

ˆ_m( ) (ˆ_m( )) Y f ←FFT y t .

4. Compute the mean value of the series Yˆ ( )_m f , m

= L 1

p^:

1

1 ˆ

( ) ( )

m p

m m

Y f Y f

p

=

= ∑

^.

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5. Compute the frequency domain covariance function by using the formula

ˆ ˆ

( ) ( ( )) ^T( ) Rxx f =conj Y f ⋅Y f .

6. Compute the time domain covariance function R_xx

( )

k by applying the inverse discrete Fourier transform to R_xx

( )

f : R_xx

( )

k

=

IFFT R

(

_xx

( ))

f .

Computation of the system matrices

12. Set-up the Hankel matrix T₀ and the shifted Hankel matrix T₁:

0 0

( , )

_xx

( )

T

=

T i j

=

R i

+

j and T₁

=

T i j₁

( , ) =

R_xx

(

i

+ +

j

1)

13. Compute the singular value decomposition of the Hankel matrix T₀: T₀

=

USV^T, where U is the left singular vector, V is the right singular vector and S is the diagonal matrix containing the singular vectors.

14. Choose a models size n and compute the system matrix A of the discrete time linear system with the formula:

1 T

n n n n

A

= Σ

U TV

Σ first rows (

_n 1/ 2_n

)

C

=

m U S

first column (

1/ 2_n _n^T

)

B

=

S V

Legend:

n: model size (dimension of the system matrix A).

m: output data channel size.

1/ 2 n Sn⁻

Σ =

: Inverse of the square root of the diagonal matrix of dimension n containing the n greatest singular values.

Un: Matrix containing the n singular left eigenvectors associated to the n greatest singular values.

Vn: Matrix containing the n singular right eigenvectors associated to the n greatest singular values.

Computation of modal parameters

15. Compute the eigenvalues of the system matrix A:

{ } λ

k ←eig A^{( )}.

16. Compute the natural frequencies of the vibration modes

{ }

fk ←^log(imag⁽

λ

k⁾⁾⋅ fs^{/ 2}

π

.

s

1/

f

= Δ

t is the sampling frequency of the time series x t_m

( )

.

17. Compute the damping ratios

{ } ζ

k ← −real⁽

λ ω

k^{) /} k, where

ω

_k

= 2 π

f_k.

3.3.2 Remarks:

– The algorithm can be applied to time series recorded with one or several sensors. When the algorithm is used with time series of several sensors, the covariance function R_xx

( )

f is a matrix.

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– Ambient vibration sources have been found to fulfil quite well the assumption of a broad band, stationary and stochastic process. When using ambient vibration measurements, the recorded time series have to be sufficiently long and in a sufficient number for assuring a representative sample. Because the driving forces are unknown, a precise figure of sufficiently long is difficult to provide. Typically, the individual time series should have a length of at least 5 minutes. The total recording time of time series should be at least 30 minutes.

If the train crossings rate is too high for allowing the recording of long time series, the number of measurements has to be increased to achieve a total recording time of at least 30 minutes. The vibrations have to be recorded between two subsequent train crossings.

– The same algorithm applies as well for forced vibration tests with a shaker. The excitation force generated by the shaker has to be a broad band, stationary, stochastic process. In particular, the excitation force does not need to be a white noise process.

– The minimum dimension (model size) of the system matrix A is twice the number of vibration modes within the analysed frequency range. The system matrix A should be computed for different model sizes starting from the minimum model size and incrementing the size at each step by 2. The damping ratios obtained from the models yields a measure of the estimation reliability. The damping ratio can be estimated by computing the mean value.

– Output only system identification algorithms produce many parasitic vibration modes. Usu- ally, the parasitic vibration modes are easily identified by exhibiting a negative or unrealis- tic large damping ratio (e.g.

ζ

_k

≥ 0.1

). Very often, these parasitic vibration modes occur in the neighborhood of a real vibration mode. In general, these parasitic modes disappear by changing the model size. Therefore, the estimation of damping should always be based on a sequence of estimations with different model sizes. The true vibration modes occur for each mode size. Vibration modes with vanishing natural frequency can also be discarded.

– For applications in ambient vibration measurements, it is recommended to use accelerometers with very high resolution (smaller than 5·10^-5 ms^-2 rms, see Table 4.1 in (Feltrin, 2004) for details) because of the very small accelerations involved. Significant errors can occur with accelerometers with insufficient resolution. The amplitude range of accelerometers with very high resolution is usually modest: typically up to 5 ms^-2. These accelerometers may saturate during train crossing. The sensors have to be firmly mounted to avoid parasitic vibrations (see (Feltrin, 2004) for details).

– The data acquisition devices should allow a sufficient amplification of the signal produced by the accelerometer.

– In principle, any stochastic identification algorithm not requiring detailed information of the excitation forces can be used for extracting natural frequencies and damping ratios.

– The dimension of the Hankel matrices affects the quality of the results. In general, the greater the dimension, the more reliable are the results.

3.4 Additional methods

The damping can be estimated by using other methods than those described in the sections 3.1, 3.2 and 3.3 of this guideline. In this section, the half power bandwidth and the phase methods are shortly described and several advices are given for their correct use. However, these methods are considered less reliable or more difficult to apply and are therefore, in general, not recommended.

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3.4.1 Half power bandwidth method

The half power bandwidth method is often cited in textbooks of structural dynamics. The damping is estimated by determining the width at the amplitude h=h_max/ 2 of the spectrum of the transfer function or resonance curve, where h_max refers to the magnitude of the peak associated to a natural frequency. The damping is estimated with the formula

2 1

WT f f

f f f

ξ = = ⁻ +

^,

where W_T

=

f₂

−

f₁ refers to the band width (see Figure 3.1a)). It is important to distinguish between the spectrum of the transfer function and the power spectrum of the transfer function. When using the power spectrum of the transfer function the band width W_T has to be estimated at the magnitude h

=

h_max

/ 2

, where h_max refers to the magnitude of the peak of the power spectrum of the transfer function. The following issues have to be considered when using the bandwidth method:

– The formula for estimating the damping is an approximation and yields accurate results only if

ξ < 0.1

.

– The bandwidth method can only be applied to single vibration modes. Natural frequencies have to be well separated in order to minimize the superposition of the vibration modes in the transfer function. The gap between the natural frequencies shall be at least 5 times the greater band width W_T.

– The damping shall be estimated using transfer functions computed with a different frequency step width

Δ

f . To be reliable, the estimated damping should not change significantly when changing the frequency step width

Δ

f .

– In case of small damping, the bandwidth method requires a fine frequency resolution of the transfer function. The frequency step width

Δ

f should be smaller than 1/5 of the bandwidth:

Δ <

f W_T

/ 5

. For the fundamental vibration mode of a bridge, the required frequency step width is typically ≈ 0.02 Hz. For achieving this frequency resolution, the total time span of the recorded time series has to be at least 50 s. On the other hand, the frequency step width

Δ

f of the spectrum of the transfer function shall not be too small to avoid local spikes in the spectrum.

Frequency [Hz]

0

Phase angle

h_max

f₁ f₂ hmax

√2

π

π2

Frequency [Hz]

f_ω b)

a)

Spectral amplitude

W_T

Δf Δϕ

Figure 3.1: a) Half power bandwidth method. b) Phase method.

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– The damping is estimated using a transfer function of the displacement, velocity or acceleration. The experimental determination of the transfer function is obtained by using shakers or impact devices. This is due to the fact that the computation of the transfer function requires the exact knowledge of the driving force. The use of shakers or impact devices is expensive and may require a restriction of the operability of a bridge during the tests.

– In principle, assuming that the source generating ambient vibration is a broad band, stochastic process, the bandwidth method can be applied to the power spectrum computed using the time series recorded with ambient vibration tests. These results have to be han- dled with care and should be based on a large number of independent measurements.

Damping estimations based on ambient vibrations records display a significantly larger scattering than estimations based on transfer functions computed using a driving force measurement. The smaller the damping the larger the number of measurements.

3.4.2 Phase method

Another method for estimating damping is the phase method. This method is not as popular as the half power bandwidth method. The phase

ϕ

of the transfer function or resonance function of a linear mass-spring-dashpot system is given by

2 2

arctan 2 f f

f f

ω ω

ϕ

= ^⎛⎜⎝

ξ

− ^⎞⎟⎠^,

where f_ω is the natural frequency. The phase curve has an angle of

π

/ 2 (90°) when the frequency is equal to the natural frequency ( f

=

f_ω,see Figure 3.1b)). Computing the slope

/

S_ϕ

=

d

ϕ

df of the phase curve at f

=

f_ω yields a simple formula for estimating the damping:

1 S f_{ϕ ω}

ξ

= .

The slope S_ϕ can be estimated by computing S_ϕ

= Δ ϕ / Δ

f at f

=

f_ω (see Figure 3.1b)) The following issues have to be considered when using the phase method:

– All items listed in section 3.4.1 regarding the half power bandwidth method apply as well for the phase method.

– Since sensors, filters and amplifiers generate phase shifts in the recorded signals, the experimentally determined phase curve may be shifted with respect to the theoretical phase curve displayed in Figure 3.1b). In this cases, the phase at f

=

f_ω is not equal to the angle

π

/ 2.

– For small structural damping, the slope of the phase curve at f

=

f_ω is very steep. There- fore, computing the slope S_ϕ using a numerical differentiation scheme may result in an in- accurate estimation of damping. The accuracy can be improved by reducing the frequency step width

Δ

f and/or using a multi-step numerical differentiation scheme.

– In general, the phase method is less sensitive to changes of the frequency step width

Δ

f as the half power band width method.

– An alternative method for estimating the damping using the phase curve is to fit the parameters f_ω and

ξ

in the equation that describes the phase. A nonlinear least square curve fitting method can be applied for estimating both parameters.

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4 Case study

4.1 Description of the bridge

The bridge used in this case study is the “Kanalbrücke Hurden”-bridge situated in Freien- bach, Switzerland. The bridge is a three-span reinforced concrete bridge with span length of 17.0, 26.4 and 17.0 m. The width of the bridge is 3.45 m. It has only one railway track. Figure 4.1a) displays a view of the bridge. The typical trains crossing the bridge are two floor passenger trains. The locomotive Re 450 has a total weight of 87 tons. The maximum axle load of the locomotive is 22 tons. A train is made of up to maximum three units containing one locomotive and three two floor carriages. The axle weight of the carriages can reach 20 tons.

The elevation of the bridge is displayed in Figure 4.2.

a) b)

Figure 4.1: a) The railway bridge Kanalbrücke Hurden.

b) Accelerometers mounted on the railway bridge Kanalbrücke Hurden and detail view of an accelerometer mounted on an aluminium plate.

17.00 26.40 17.00

8.50 8.50 7.00 6.20 6.20 7.00 8.50 8.50

S1 S2 S3 S4 S5

Figure 4.2: Elevation view of the bridge with position of the acceleration sensors.

4.2 Vibration measurements

The vibration measurements were performed at five different positions of the bridge deck.

Accelerometers were placed in the middle of each span (S1, S3 and S5) and two additional accelerometers were mounted within the middle span (S2 and S4, see Figure 4.2). Figure

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4.1b) shows the five accelerometers that have been used to measure the vertical vibrations of the deck. The accelerometers were mounted on aluminium plates that were firmly fixed with screws on the bridge deck. The recorded data was sampled with a sampling frequency of f_s

= 256

Hz. This covers the frequency range f

= K 0 128

Hz. Figure 4.3a) displays a typical time series of accelerations record at S3 during a train crossing. Figure 4.4 depicts the associated power spectra displaying significant frequency contents at approximately 6.0 and 10.8 Hz. The first peak at 6.0 Hz is associated to the first bending mode and is the most dominant vibration mode. The second peak at 10.8 Hz is associated to the second bending mode.

4.3 Method of decay curve

The method of decay curve is applied on vibrations produced by train crossings. A reliable estimation of modal damping of the first bending mode can be obtained by analysing the decay curve recorded by the sensor at S3. In the forcing phase, the acceleration records are characterized by frequency components in the range of f

= K 35 36

Hz. These vibrations are generated by irregularities of the track or by train vibrations. In this case, these vibrations can be used to estimate the end of the forcing period. This time has been estimated to 4.8 s with respect to the relative time of the record shown in Figure 4.3. Figure 4.3b) and c) shows that the velocities and displacements at position S3 are largely dominated by the first bending mode. This allows a reliable estimation of the modal damping of the first bending mode.

0 1 2 3 4 5 6 7 8

−2 0 2

a [m/s2 ]

S3 a)

0 1 2 3 4 5 6 7 8

−20 0 20

v [mm/s]

S3 b)

0 1 2 3 4 5 6 7 8

−0.4

−0.2 0 0.2 0.4

d [mm]

S3 c)

time [s]

Figure 4.3: Recorded accelerations and computed velocities and displacements of the bridge deck at S3 during train crossing.

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The solid curve displayed in Figure 4.5a) is the extracted decay curve of the velocity amplitudes. The filtered decay curve used for estimating the damping is the dashed curve. The filtering has been performed with a Chebyshev Type II band-pass filter with a pass band of

5 7

f

= K

Hz. Before filtering, the data was decimated by a factor of 2. A comparison of the two curves in Figure 4.5a) shows that in this case the first bending mode is so dominantly present in the velocities that the unfiltered curve can also be used for a damping estimation.

Figure 4.5b) displays the absolute values of the decay curve with the identified local maxima.

The logarithmic values of these local maxima are displayed in Figure 4.5c). This figure depicts a quite common amplitude dependence of structural damping. For amplitudes greater than approximately 1 mms^-1, the decay of amplitude is significantly faster than for smaller amplitudes. Assuming linear viscous damping, the logarithmic decrement for velocity amplitudes that are greater than approximately 1 mms^-1 is

δ

=0.264. This is equivalent to the damping ratio

ζ = 0.042

. For velocity amplitudes smaller than1 mms^-1, the logarithmic decrement is

δ

≈0.1 and the damping ratio is

ζ ≈ 0.016

. The continuous, straight lines of Figure 4.5c) represent the linear regression models. Very similar results are obtained by analyzing the decay curve of the displacements and the accelerations of the same record and of additional records (see Table 4.1).

Table 4.1: Logarithmic decrements of the first bending mode for different velocity ranges estimated by the decay curve method (based on recordings at point S3).

Event v_range [mm/s]

Error! Ob- jects cannot

be created from editing

field codes.

Error! Ob- jects cannot

be created from editing

field codes.

Error! Objects cannot be created from

editing field codes.

Crossing 1a 1…4 0.265 0.264 0.269

Crossing 1b 0.3…1.6 0.074 0.090 0.085 Crossing 2 0.1…0.9 0.088 0.086 0.090 Crossing 3 0.1…1.4 0.117 0.118 0.120 Crossing 4 0.5…2.6 0.267 0.278 0.272 Crossing 5 0.5…2.4 0.264 0.263 0.244 Crossing 6 0.7…3.0 0.256 0.231 0.246 Legend: vrange: Velocity range of the analyzed decay curve

δacc: Logarithmic decrement computed using accelerations δvel: Logarithmic decrement computed using velocities δdef: Logarithmic decrement computed using displacements

0 5 10 15 20 25 30

0 0.5 1 1.5 2 2.5 3 3.5x 10⁻⁴

Power spectrum (a) [m2 s−3 ]

Frequency [Hz]

S3 S4 S5

Figure 4.4: Power spectra of the accelerations S3, S4 and S5.

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0 0.5 1 1.5 2 2.5 3 3.5 4

−5 0 5

v [mms−1 ] a)

unfiltered filtered

0 0.5 1 1.5 2 2.5 3 3.5 4

0 5

|vel| [mms−1 ]

t [s]

b)

0 5 10 15 20

−2 0 2

log(|v|)

τ [−]

c)

Figure 4.5: a) Decay curve of the velocities at S4 after train crossing.

b) Peak amplitudes of the decay curve displayed in a)

c) Logarithm of the peak amplitudes of the decay curve with the linear regressions branches for estimating the damping.

4.4 Multiple mode decay curve method

The multiple mode decay curve method is applied on vibration records produced by train crossings analyzed in the previous section. The logarithmic decrements of the first two bending vibration modes are determined simultaneously. Close to the second bending vibration mode, there is a third vibration mode with a natural frequency that is approximately 1 Hz higher and which disturbs (beating effect) the estimation of the logarithmic decrement of the second bending vibration mode when applying the decay curve method. The analysis was performed using recorded accelerations. The decay curves are processed without any prior filtering of the recorded data.

Table 4.2: Logarithmic decrements of the first two bending vibration modes estimated by the multiple mode decay curve method.

δ1 δ2

Crossing S1 S2 S3 S4 S5 S1 S2 S3 S4 S5

1a 0.226 0.236 0.241 0.262 0.317 0.161 0.208 0.217 — 0.152 1b 0.126 0.114 0.113 0.111 0.097 0.079 — — 0.033 0.043 2 0.150 0.116 0.149 0.106 — — — — — 0.134 3 0.127 0.119 0.118 0.117 0.118 — — — — — 4 — — 0.299 0.260 — 0.136 0.141 0.118 — 0.198 5 0.252 0.215 0.219 0.236 0.282 — — — — — 6 — 0.300 0.268 0.246 — 0.126 0.123 0.106 — 0.179

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Table 4.2 summarises the damping estimations of the first two bending vibration modes obtained with the multiple mode decay curve method. The first bending mode can be estimated at each sensor position. This is not the case for the second bending mode that is more likely to be detected at sensor positions S1 and S5 than at other positions. In addition, the train crossings 3 and 5 do not excite sufficiently the second bending mode to provide reliable damping estimations (different trains have different impacts on the vibration modes). For a specific train crossing, the damping estimates can vary between the sensor positions. How- ever, a variation of up to 20 % with respect to the average value is the rule independently of the applied method.

Figure 4.6 displays the results of the analysis of the data segment 1a of the accelerations at position S5 after train crossing 1 (the same data segment of Figure 4.5). The outcomes of the multiple mode decay curve method allow to reconstruct the analyzed decay curve by computation. Comparing the computed decay curve with the recorded provides a fast and reliable mean to verify the accuracy of the damping estimation. Figure 4.6b) compares the recorded and computed decay curve time histories and Figure 4.6b) compares the power spectrum of the decay curves. Both dominant peaks in Figure 4.6b) are very well reproduced by the computed decay curve demonstrating the high accuracy of the multiple mode decay curve method. In this case, the chosen model size is 12.

Table 4.3 displays a comparison of the estimated damping ratios and of the two bending modes. The estimation obtained from the records of the sensor at position S4 were used for the first bending mode. For the second bending mode, the records of the sensor at position S5 were preferred since the amplitude of the mode shapes is significantly greater at this position. The logarithmic decrements of the first bending mode agree well. The deviations are within the estimation accuracy. The damping ratios of the second bending mode could not be

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.2 0 0.2

a [ms−2 ]

t [s]

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.2 0 0.2

a [ms−2 ]

t [s]

b)

0 5 10 15 20 25 30

0 5x 10⁻⁶

P(a) [m2 s−3 ]

f [Hz]

c) recorded computed

recorded computed

Figure 4.6: a) Decay curve of the accelerations at S5 after train crossing 1 (The analyzed segment is represented with a dotted line).

b) Comparison of the recorded and computed time history of the decay curve.

c) Comparison of the recorded and computed power spectra of the decay curve.

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estimated at each train crossing, since the magnitude of the modal response to train crossing was too small to provide reliable estimates (crossings 3 and 5). The estimates agree well in the cases with small amplitudes of the third vibration mode (crossing 4 and 6) and less well in the other cases (values in parenthesis). Figure 4.7a) displays the filtered decay curve of the accelerations of the second bending mode at S5. The beating of the decay curve (Figure 4.7a) and b)) is a direct effect of the residual components of the third vibration mode. These components manifest themselves in the power spectrum of the decay curve by small peaks in the neighbourhood of the major peak that is associated to the second vibration mode (Figure 4.7c)). These residuals reduce the accuracy of the damping estimate.

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.1 0 0.1

a [ms−2 ] ^a)

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.05 0.1

|a| [ms−2 ]

t [s]

b)

0 5 10 15 20 25 30

0 2 4x 10⁻⁶

P(a) [m2 s−3 ]

f [Hz]

c)

Figure 4.7: a) Filtered decay curve of the accelerations at S5 after train crossing.

b) Peak amplitudes of the decay curve displayed in a) c) Power spectrum of the decay curve displayed in a)

Table 4.3: Comparison of the logarithmic decrements of the first two bending vibration modes esti- mated by the decay curve (DCM) and the multiple mode decay curve method (MDCM).

δ1 δ2

Event DCM MDCM DCM MDCM

Crossing 1a 0.265 0.262 (0.110) 0.152

Crossing 1b 0.074 0.111 0.036 0.043

Crossing 2 0.088 0.106 (0.093) 0.134

Crossing 3 0.117 0.117 — —

Crossing 4 0.267 0.260 0.166 0.198

Crossing 5 0.264 0.236 — —

Crossing 6 0.256 0.246 0.182 0.179

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4.5 Ambient vibration method

Figure 4.8 displays the acceleration time series at S3 and the associated power spectra of the vertical bridge deck accelerations at S3, S4 and S5 generated by ambient excitation sources. The spectra reveal at least three vibration modes. The peaks at 6.15 Hz and 11.0 Hz are associated to the natural frequencies of the first two bending modes of the railway bridge. These natural frequencies are greater than those measured during train crossing (see Figure 4.4, 6.0 Hz and 10.8 Hz). The frequency shift is mainly due to the train mass and, but to a much lower degree, to the larger vibration amplitudes occurring during train crossing. The root mean square of the ambient acceleration amplitudes at the position S3 is 1.14⋅10^-4 ms^-2. That is, the ambient vibration amplitudes are approximately 4 orders of magnitude smaller than the vibration amplitudes during a train crossing. The peak at 4.63 Hz is associated to the fundamental vibration mode of the adjacent road bridge. This peak demonstrates that, in this case, the traffic of the road bridge is one of the dominant ambient vibration sources. Furthermore, it demonstrates that not every peak of the power spectrum computed from ambient vibration measurements is associated to a vibration mode of the bridge.

Discriminating real vibration modes from parasitic ones is one of the most challenging as- pects of ambient vibration measurements techniques and therefore a source of errors. In this case, the measurements recorded during train crossing allow to discriminate the vibration modes of the railway bridge from those of the adjacent road bridge.

The time series were filtered with a Chebyshev Type II low-pass filter with a pass band of

0 20

f

= K

Hz. Afterwards, the mean value of the time series was removed. The covariance function was computed with segments of 4048 samples (16 seconds). The maximum time lag of the covariance function was 2048 samples (8 seconds). The size of the Hankel matrices was chosen to 128x128. The system parameters were computed starting from a model size of 8 up to a model size of 32.

Table 4.4 displays the estimated natural frequencies and damping ratios. The values f₁, f₂ and ζ1, ζ2 have been obtained with the accelerations at S3 and the values f3, f4 and ζ3, ζ4 with

0 20 40 60 80 100 120

−1

−0.5 0 0.5

1x 10⁻³

a [ms−2 ]

Time [s]

0 5 10 15 20 25 30

0 0.5

1x 10⁻⁸

Power spectrum (a) [m2 s−3 ]

Frequency [Hz]

S3 S4 S5

Figure 4.8: Time series of deck accelerations at S3, and power spectra at S3, S4 and S5 due to am- bient excitation sources.

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the accelerations at S5. At this position, the vibration modes at 11 and 12 Hz have an important component (Figure 4.8). The results in Table 4.4 show that the influence of the model size on the natural frequencies is small. This is an important indication that the identified modes are reliable. The influence of the model size on the damping ratios is generally greater. A standard deviation of 20% or 30% of the mean value is quite common. The damping ratio of the first bending mode (

ζ

₂

≈ 0.019

) is compatible with that for small vibration amplitudes, velocity amplitudes smaller than1 mms^-1, computed using the decay curve method (

ζ

₂

≈ 0.016

,

δ

≈0.1, see Table 4.1). The damping ratio of the parasitic vibration mode generated by the coupling with the road bridge is definitely too small for concrete bridges (

ζ

₁

≈ 0.003

). This is an additional indication that something might be wrong with that vibration mode.

Table 4.4: Estimated natural frequencies and damping ratios using the system identification algorithm.

Frequency [Hz] Damping ratio [-]

Model

size f1 f2 f3 f4 ζ1 ζ2 ζ3 ζ4

8 4.67 6.22 11.07 12.14 0.0030 0.0195 0.0119 0.045 10 4.66 6.21 11.06 12.15 0.0039 0.0187 0.0128 0.041 12 4.66 6.22 11.04 12.15 0.0037 0.0182 0.0181 0.028 14 4.66 6.21 11.03 12.20 0.0038 0.0181 0.0153 0.030 16 4.67 6.22 11.01 12.23 0.0056 0.0194 0.0142 0.030 18 4.66 6.21 11.02 12.23 0.0043 0.0160 0.0142 0.030 20 4.65 6.23 11.02 12.22 0.0032 0.0189 0.0140 0.031 22 4.66 6.23 11.02 12.22 0.0020 0.0196 0.0148 0.030 24 4.65 6.23 11.02 12.51 0.0022 0.0196 0.0187 0.034 26 4.64 6.23 11.02 12.22 0.0023 0.0205 0.0140 0.032 28 4.64 6.23 11.01 12.31 0.0013 0.0206 0.0104 0.034 30 4.66 6.23 10.99 12.29 0.0019 0.0197 0.0125 0.032 32 4.67 6.22 10.98 12.31 0.0030 0.0195 0.0133 0.030 m 4.66 6.22 11.02 12.24 0.0031 0.0191 0.0142 0.0328 s 0.01 0.01 0.03 0.10 0.0012 0.0012 0.0023 0.0049 Legend: m: Mean value

s: Standard deviation

Guideline for estimating structural damping of railway bridges Background document D5.2-S2