iris chapter05

8/15/2019 Iris Chapter05

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The IRIS Damage Assessment Methodology 5-x

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Natural frequency f [Hz]

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o d e l o r d e r n

Authors:

Michael Döhler

Falk Hille

Laurent MevelWerner Rücker

5Estimation of Modal Parameters and

their Uncertainty Bounds fromSubspace-Based System Identification

Motivation

Operational Modal Analysis of existing structures is performed using output only vi-

bration measurements. To evaluate the quality of the resulting modal parameters (natural

frequencies, damping ratios, mode shapes) from system identification it is essential to

know their statistical uncertainty content.

Main Results

An innovative and efficient algorithm to estimate the modal parameters and their un-

certainties from subspace-based system identification has been developed. Applications

to output only measurements have been demonstrated on the S101 Bridge case.


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5 Estimation of Modal Parameters and their Uncertainty Bounds from Subspace-Based SystemIdentification

5-1 Introduction

Subspace-based system identification methods have proven to be efficient for theidentification of linear-time-invariant systems (LTI), fitting a linear model to input/output

or output only measurements taken from a system. An overview of subspace methods

can be found in [Benveniste and Fuchs, 1985; Benveniste and Mevel, 2007; Döhler and

Mevel, 2012; Van Overschee and De Moor, 1996].

During the last decade, subspace methods found a special interest in mechanical, civil

and aeronautical engineering for modal analysis, namely the identification of vibration

modes of structures from the eigenvalues (natural frequencies and damping ratios) and

observed eigenvectors (mode shapes) of an LTI system. For Operational Modal Analysis,

the identification of a structure under operation conditions, it is often impractical to ex-

cite the structure artificially, so that vibration measurements are taken under unmeasuredambient excitation. Therefore, identifying an LTI system from output-only measurements

is a basic service in vibration analysis, see e. g. [Basseville et al., 2001; Peeters and de

Roeck,1999].

For any system identification method, the estimated modal parameters are afflicted

with statistical uncertainty for many reasons, e. g. finite number of data samples, unde-

fined measurement noises, non-stationary excitations, model order reduction etc. Then

the system identification algorithms do not yield the exact system matrices and identi-

fication results are subject to variance errors. For many system identification methods,

the estimated parameters are asymptotically normal distributed, e. g. for estimates from

prediction error methods [Ljung, 1999], maximum likelihood methods [Pintelon et al.,2007], or the here considered subspace methods [Benveniste et al., 2000; Chiuso and

Picci, 2004]. A detailed formulation of the covariance computation for the modal parame-

ters from covariance-driven stochastic subspace identification is given in [Reynders et al.,

2008], where covariance estimates are based on the propagation of first-order perturba-

tions from the data to the modal parameters. These methods are very attractive for modal

analysis, as covariance estimates are obtained in one shot: From the same data set that

is used to estimate the modal parameters, all the covariance information is obtained by

cutting the available data into blocks, which is then propagated to the modal parameters,

without the need of computing the modal parameters on the blocks.

The variance information on the modal parameters is essential for many applications.

From statistical theory it is known that uncertainty bounds of estimated natural frequen-

cies are much smaller than those of damping ratios [Gersch, 1974]. To evaluate the quality

especially of the estimated damping ratios from system identification, it is essential to

know their uncertainty bounds. Also, a comparison of modal parameters estimated from

different data sets is not meaningful unless one knows the uncertainty bounds of the

parameters to evaluate if a significant change happened or not.

In this chapter, the identification of the modal parameters from output-only measure-

ments and the computation of their uncertainty bounds is described. They are demon-

strated on the system identification of S101 Bridge.


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Stochastic Subspace Identification 5-2

5-2 Stochastic Subspace Identification

Stochastic Subspace Identification methods are state-of-the-art methods for modalparameter estimation. They provide unbiased and consistent estimates, even under non-

stationary excitation [Benveniste and Fuchs, 1985; Benveniste and Mevel, 2007]. In this

section, an overview of the identification algorithm is given.

5-2-1 Models and Parameters

The behaviour of a mechanical system is assumed to be described by a stationary

linear dynamical system

( ) ( ) ( ) ( )Mz t Cz t Kz t v t + + = E.5-1

where t denotes continuous time, M, C , K d d ×∈ are the mass, damping and stiffness ma-

trices, high-dimensional vector d z ∈ collects the displacements of the degrees of free-

dom of the structure and the external force v is unmeasured and considered as noise. The

eigenstructure of [E.5-1] with the modes i µ and mode shapes (observed eigenvectors)r

i ψ ∈ is a solution of

2det( ) 0i i M C K µ µ + + = ,2( ) 0i i i M C K µ µ φ + + = , i i Lψ φ = ,

where matrix r d L ×

∈ maps the r sensor locations to the d degrees of freedom of the struc-ture. Observing model [E.5-1] at the r sensor locations (e. g. by acceleration, velocity or

displacement measurements) and sampling it at some rate 1/τ yields the discrete model

in state-space form

1k k k

k k k

x Fx v

y Hx w + = +

= + E.5-2

where n nF ×∈ is the state transition matrix, r nH ×∈ is the observation matrix, nk x ∈ are

the states of the system and r k y ∈ the output measurements at the discrete times t k τ = ,

where n is the system order. The vectors v k and w

k are the unmeasured input and output

disturbances. The eigenstructure of system is given by

det( ) 0i F I λ − = , ( ) 0i i F I λ φ − = , i i H ϕ φ = . E.5-3

The eigenstructure of the continuous system [E.5-1] is related to the eigenstructure of

the discrete system [E.5-2] by

i

i eτµ λ = , i i ψ ϕ = . E.5-4

From the eigenvalues , the natural frequencies f i and damping ratios i ρ of the sys-

tem are directly recovered from


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2 2

2

i i

i

a bf

πτ

+= ,

2 2

i i

i i

b

a b ρ

−=

+ where

Im( )arctan

Re( )i

i

i

a λ

λ = , lni i b λ = . E.5-5

5-2-2 Covariance-Driven Stochastic Subspace Identification

To obtain the eigenstructure of system [E.5-3] and the modal parameters from vibra-

tion measurements 1,...,( )k k N p q y = + + the stochastic subspace identification algorithm is used

[Benveniste and Fuchs, 1985; Peeters and de Roeck, 1999] . In the first step, the block

Hankel matrix H is filled with the output correlations ( )i k i k R y y

1 2

2 3 1

1 2

( )

q

q

i

p p p q

R R R

R R R

Hank R

R R R

+

+ + +

= =

H

, E.5-6

that can be estimated from

ˆ T r r i k i k R y y ×= ∈∑ . E.5-7

In the case of measurements at many coordinates, a subset of0

r r ≤ sensors can be

chosen as reference sensors in the output vector r k y ∈ , corresponding to the reference

output vectors 0(ref) r k y ∈ . Then, the output correlations

0(ref)

1

1ˆN

r r T

i k i k

k

R y y N

×+

=

= ∈∑

are estimated instead of E.5-7 to increase computation time and to improve the quality

of the results [Peeters and de Roeck, 1999]. The parameters p and q in the block Hankel

matrix E.5-6 are chosen such that { }0min , pr qr n≥ , with in general 1 p q+ = .Matrix H possesses the factorization property H = O C into observability matrix O and

stochastic controllability matrix C, where O is obtained from H by a singular value decom-

position (SVD) and truncation at the desired model order

1

1 0

0

( ) T U U V D

= D

H ,def

1/2 ( 1)

1 1

p r n

p

H

HF U

HF

+ ×

= D = ∈

O

. E.5-8

From the observability matrix O the matrices H in the first block row and F from a least

squares solution of


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Stochastic Subspace Identification 5-2

F =O O with

1 p

H

HF

HF −

=

O

,

HF

HF

HF

E.5-9

are obtained. The eigenstructure ( i λ , i ϕ ) of the system E.5-3 and the modal parameters are

finally obtained from E.5-3 to E.5-5.

5-2-3 Stabilization Diagram

In Operational Modal Analysis, the eigenstructure of mechanical, civil and aeronauti-

cal structures is identified from output-only data under ambient excitation. The model

order n of the discrete-time system E.5-2 is generally unknown (and is in fact infinity forthe mechanical system in most practical cases).

On the other side, the input and output noise processes ( )k k v and ( )k k w in model E.5-2

are hardly stationary Gaussian white noise sequences in practice. It was shown that the

identified system parameters converge to the true values when the number of data sam-

ples goes to infinity also under non-stationary excitation [Benveniste and Fuchs, 1985;

Benveniste and Mevel, 2007]. Moreover, coloured noise can be considered. Then, the

estimated eigenstructure corresponds to a combined system of a higher model order,

where the modes corresponding to the noise dynamics are also contained.

Thus, the selection of the model order in E.5-8 is a major practical issue. In order to

retrieve a desired number of modes, an even larger model order must be assumed whileperforming identification. A number of spurious modes appears in the identified model

due to this over-specification, as well as due to coloured noise that appears in practice.

Techniques from statistics to estimate the best model order may lead to a model with the

best prediction capacity, but one is rather interested in a model containing only the physi-

cal modes of the investigated structure, while rejecting the spurious modes.

Based on the observation that physical modes remain quite constant when estimated

at different over-specified model orders, while spurious modes vary, they can be distin-

guished using so-called stabilization diagrams. The system is identified truncating in E.5-8

at multiple model orders [Döhler and Mevel, 2012], and frequencies from this multi-order

system identification are plotted against the model order. From the modes common to

many models and using further stabilization criteria, such as threshold on damping val-

ues, low variation between modes and mode shapes of successive orders etc., the final

estimated model is obtained. Like this, stabilization diagrams provide a GUI where the

user is assisted in selecting the identified modes of an investigated structure.


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5-3 Covariance Estimates of Modal

Parameters The statistical uncertainty of the estimated modal parameters is necessary to assess

the quality of the estimates from the data. The uncertainty bounds of damping ratios are

particularly interesting, as the estimates of damping ratios are in general afflicted with a

high uncertainty according to statistical theory, and are thus difficult to obtain.

When estimated from a finite number of data samples, not the “true” parameters of the

system are obtained, but estimates that are naturally subject to variance errors depending

on the data and the estimation method, as the input of the system E.5-2 is unmeasured

noise. A variance analysis of the system matrices obtained from Stochastic Subspace Iden-

tification is made e. g. in [Chiusi and Picci, 2004] and expressions for their computationin the context of structural vibration analysis are given in [Döhler et al., 2013; Döhler

and Mevel, 2013; Reynders et al., 2008]. These computations are based on the propaga-

tion of first-order perturbations from the data to the identified parameters by a sensitivity

analysis. Like this, the uncertainties of the modal parameters at a chosen system order can

be computed from the uncertainty of the block Hankel matrix H, whose covariance ΣH

can be estimated by cutting the sensor data into blocks on which instances of the Hankel

matrix are computed. In the following, the underlying theory and the computation are

explained in detail.

5-3-1 Uncertainty Propagation

First, the principle of the uncertainty propagation is stated. Let θ be some parameter

vector and ˆN θ its estimate based on N data samples, whose expected valueˆ

N N θ θ = E tends

to*

θ as N goes to infinity. Define the estimated covariance ( )ˆ ˆ ˆcov( ) ( )( )T N N N N N θ θ θ θ θ = − −E and let ˆN θ fulfill the Central Limit Theorem

*ˆ( ) (0, )d N N θ θ − → Σ . E.5-10

for N → ∞ , where Σ is the asymptotic covariance and designates the normal distribu-

tion. As the number of data samples N is usually large, the distribution of N̂ θ is approxi-

mated to be normal with 1ˆcov( )N N θ ≈ Σ. Property E.5-10 is fulfilled directly for estimates

e. g. from subspace methods, maximum likelihood or prediction error methods.

Now, let ( )f θ be a vector-valued function of the parameter. Suppose that its first de-

rivative, the sensitivity matrix*

( )f f θ = exists in *θ and that it is full row rank. Using the

Taylor approximation

2* * *ˆ ˆ ˆ( ) ( ) ( ) ( )N f N N f f Oθ θ θ θ θ θ = + − + − ,

it follows

*

ˆ( ( ) ( )) (0, )d T

N f f N f f θ θ − → Σ ,


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Covariance Estimates of Modal Parameters 5-3

which is known as the delta method , and the covariance of ˆ( )N f θ can be approximated by

ˆ ˆcov( ( )) cov( ) T N f N f f θ θ ≈ . E.5-11

Note that*

( )f f θ = in the derivation above. A consistent estimate of the sensitivity is

then obtained from ˆ( )f N θ . Thus, starting from the covarianceˆcov( )N θ of an estimate

ˆN θ ,

that can for example be obtained from a sample covariance, the covariance of a function

of this estimate is obtained in E.5-11.

For subspace-based system identification, this initial covariance estimate is the covari-

ance of the vectorized Hankel matrix ΣH

cov(vec( ))= H that can be obtained as a sample

covariance by cutting the available data into blocks and computing instances of H on

each block, where vec is the column-stacking vectorization operator. As the system ma-

trices and subsequently the modal parameters are functions of H, it is the objective to

compute the sensitivities of these parameters with respect to vec(H) to finally obtain theircovariance from E.5-11.

In order to obtain the desired sensitivity matrices, first-order perturbations are used.

Let the vector θ be close to the true value*

θ and let the scalar ε be the perturbation

magnitude, writing*

(0)θ θ = (meaning no perturbation) and*

( )θ θ ε = . Then, a first-order

perturbation is defined as

2*

( )Oθ

θ ε θ θ θ ε ε

∂D = = − + =

∂,

and a first-order perturbation of ( )f θ yields

( )

( ) f f f

f θ θ

θ ε ε θ ε ε

∂ ∂D = = = D

∂ ∂ .

Thus, the desired sensitivity matrices can be obtained by applying first-order pertur-

bations. The following definitions are needed in the following sections. Let I a the identity

matrix of size a×a and 0a,b

the zero matrix of size a×b. Define the selection matrices

1 ,[ 0 ] pr pr r S I = , 2 ,[0 ] pr r pr S I = ,

such that 1S =O O , 2S =O O . Also, some properties of the vectorization operator are im-

portant. For ,a b ∈ define the permutation

, ,

, , ,

1 1

a ba b b a

a b k l l k

k l

E E = =

= ⊗∑∑ ,

where ,,a b

k l E is a matrix of size a×b that is equal to 1 at position (k ,l ) and zero elsewhere,

and ⊗ denotes the Kronecker product. Then, for any matrix ,a b X ∈ it holds

,vec( ) vec( )T

a b X X =

For any compatible matrices X , Y and Z the vectorization operator yields the relation

vec( ) ( )vec( )T XYZ Z X Y = ⊗


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5-3-2 Covariance of the Hankel Matrix

The covariance ΣH

cov(vec( ))= H of the vectorized Hankel matrix is the starting point

on the right hand side in E.5-11 before propagating it to the covariance of estimatesthat are a function of this matrix. For its calculation, the available data is separated into

nb blocks of the same length N

b for simplicity, with

b bn N N ⋅ = . Each block may be long

enough to assume statistical independence between the blocks. The correlations

( ) (ref )

1 ( 1)

1ˆb

b

jN j T

i k i k

k j N b

R y y N

+= + −

= ∑ E.5-12

are computed each of the blocks and the corresponding Hankel matrix

( )ˆ( ) j j i Hank R=H E.5-13

is filled. Then, 11

b

b

n

j n j == ∑H H and the covariance of the Hankel matrix in follows from the

covariance of the sample mean as

( )( )1

1vec( ) vec( ) vec( ) vec( )

( 1)

bn T

j j

j b bn n =Σ = − −

− ∑H H H H H .

As this matrix of dimension 0 0( 1) ( 1) p qrr p qrr + × + can get very large and for practicalapplications the number of available data blocks is usually smaller,

0( 1)bn p qrr < + , it is pro-

posed to store instead the matrix [Döhler and Mevel, 2013]

1 2 ... bnT t t t = H where ( )

1vec( ) vec( )

( 1) j j

b b

t n n

= −−

H H . E.5-14

Then, the relationship

T T T Σ =H H H E.5-15

holds and will be used for the computations.

5-3-3 Sensitivities of the System Matrices

The sensitivities of the system matrices A and C are obtained in two steps: First, a per-

turbation DH of the Hankel matrix is propagated to a perturbation DO of the observability

matrix, and second, a perturbation DO is propagated to perturbations D A and DC in the

system matrices.

In order to obtain DO, the sensitivities of the singular values and vectors in E.5-8 are

necessary. They have been derived in [Pintelon et al., 2007].


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Lemma 1 [Pintelon et al., 2007]. Let i σ , ui and v i be the i -th singular value, left and

right singular vector of the matrix 0( 1) p r qr + ×∈H and DH be a first-order perturbation on

H. Then it holds

( ) vec( )T i i i v uσ D = ⊗ DH , vec( )i

i i

i

uB C

v

D = D

D H

where

0

1( 1)

1

i

i

p r

i T

qr

I B

I

σ

σ

+ −=

−

H

H,

0 0

( 1)

( 1) ,

( )1

( ( ))

T T

i p r i i

i T T

i qr i i p r qr i

v I u uC

u I v v σ

+

+

⊗ −=

⊗ − .

Using this result, the sensitivity of the observability matrix is derived.

Lemma 2 [Reynders et al., 2008]. Let Bi and C

i be given in Lemma 1 for i = 1, …, n. Then,

,vec( ) vec( )D = D

O HO H E.5-16

where 0( 1) ( 1), p rn p rqr + × +∈

O H with

0

1 1 1 1

1/2 1/2, 1 1 3 1 ( 1) ( 1)

( )1

( ) ( 0 )2( )

T

N p r p r qr

T

n n n n

v u B C

I U S I v u B C

+

− + + ×

+

⊗

= ⊗ Σ + Σ ⊗ ⊗

O H , 3 ( 1) ,n nk n k k S E − +∑

Proof. From the definition of O follows

1/2 1/2 1/2 1/211 1 1 1 1 1 1 1 12U U U U −D = DΣ + D DΣ = Σ DΣ + D ΣO ,

1/2 1/211 1 1 1 ( 1) 12

vec( ) ( )vec( ) ( )vec( )n p r I U I U −

+D = ⊗ Σ DΣ + Σ ⊗ DO .

The sensitivities of Σ1 and U

1 are obtained from Lemma 1 by stacking as

1 1 1( )

= vec( )

( )

T

T

n n n

v u

v u

σ

σ

D ⊗

D D ⊗

H ,

1 1

1

1

vec = vec( )

n n

B C U

V B C

+

+

D

D D

H

and the assertion follows.

The results of [Reynders et al., 2008] on the sensitivity of the system matrices are col-

lected in the following lemma.


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Lemma 3 [Reynders et al., 2008]. Let the system matrix A be obtained from O in E.5-9

and C from the first block row of O. Then, a perturbation on O is propagated to A and C by

,vec( ) vec( ) A AD = DO O , ,vec( ) vec( )C C D = D

O O E.5-17

where2 ( 1)

,

n p rn

A

× +∈O

, ( 1),rn p rn

C

× +∈O

, with

1, 2 1 2 2 ( 1) ,( ) ( ) (( ) ( ) )T T T T T

A n p r nI S A S S A S+ + −

+= ⊗ − ⊗ + − ⊗O O O O O O O ,

, ,0C n r r pr I I = ⊗ O

Proof. Using the product rule for a perturbation on 1( )T T A + −= =O O O O O O and Kro-

necker algebra leads to the assertion.

5-3-4 Sensitivities of the Modal Parameters

The natural frequencies and the damping ratios are functions of the eigenvalues of

system matrix A and the mode shapes depend on both A and C . In [Reynders et al., 2008],

the sensitivity derivations for the eigenvalues and eigenvectors of a matrix and subse-

quently for the modal parameters are stated, based on derivations in [Golub and van

Loan, 1996; Pintelon et al., 2007]. They are summarized in the following.

Lemma 4. Let i λ , i φ and i χ be the i -th eigenvalue, left eigenvector and right eigenvec-

tor of A with

i i i Aφ λ φ = ,* *

i i i A χ λ χ = ,

where * denotes the complex conjugate transpose. Then,

, vec( )i i A Aλ λ D = D , , vec( )i i A Aφ φ D = D ,

where21

,i

n

Aλ

×∈ ,2

,i

n n

Aφ

×∈ , with

*, *1

( )i

T

A i i

i i

λ φ χ χ φ

= ⊗ ,*

, *( )

i

T i i A i n i n

i i

I A I φ φ χ

λ φ χ φ

+

= − ⊗ −

.

Lemma 5. Let i λ and i φ be the i -th eigenvalue and left eigenvector of A and

ln( ) / ( ) /i i i i b a i λ λ τ τ = = + the eigenvalue of the corresponding continuous-time state

transition matrix as in E.5-5. Let furthermore the natural frequency f i and the damping

ratio i ρ be given in E.5-5. Then,

, vec( )i i f Af AD = D , , vec( )i i A A ρ ρ D = D , , , vec( )vec( )i i A C AC

ϕ ϕ D D = D

, E.5-18


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where ,i f A ,21

,i

n

A ρ

×∈ ,2( )

, ,i

r n rn

A C ϕ

× +∈ , with

, , ,

, , ,

Re( )

Im( )i i i i

i i i i

f A f A

A A

λ λ

ρ ρ λ λ

=

, , , ,i i T

A C A i r C I ϕ φ φ

= ⊗ , and

, 2 2

2 2 2,

1/ (2 ) 0 Re( ) Im( ) Re( ) Im( )1

Im( ) Re( )| | | | 0 100/ | | Im( ) Re( )Im( )

i i

i i

f i i i i

i i i i i i i i

λ

ρ λ

π λ λ λ λ

λ λ τ λ λ λ λ λ λ

×

= ∈ − −

.

5-3-5 Summary of the Computation

From Equations E.5-16, E.5-17 and E.5-18 it follows that perturbations on a frequency

f i , damping ratio i ρ and mode shape i ϕ can be written as

,

, ,,

vec( )i

i

Ai

A

Ai

f λ

ρ ρ

D = D

D O O H

H

,,

, , ,,

vec( )i

A

i A C

C

ϕ ϕ

D = D

O

O H

O

H

.

From this expression and E.5-11, the covariance of a frequency and damping ratio of

a mode i yields

, ,

, , , ,, ,

cov i i i i

T

f A f Ai T T A A

A Ai

f

ρ ρ ρ

= Σ

O O H H O H O

, E.5-19

and the covariance of the real and imaginary parts of a mode shape yields

, , , ,, ,

, ,, , , ,, ,

Re( ) Re( )Re( )cov

Im( ) Im( )Im( )i i

i i

T T

A C A C A Ai T

A C A C C C i

ϕ ϕ

ϕ ϕ

ϕ

ϕ

= Σ

O O

O H H O H

O O

. E.5-20

From these covariances, the standard deviations of the respective modal parameters

are obtained from the square roots of the diagonal entries. The following procedure sum-

marizes an efficient computation of the covariance estimates for the modal parameters at

some model order n, which is equivalent to computing the products E.5-19 to E.5-20. It

takes into account the relation T T T Σ =H H H

in E.5-15 for a covariance estimate of the vector-

ized Hankel matrix, reducing the size of the involved matrices significantly:

1) Compute matrix T H from n

b blocks of the available data in equations E.5-12, E.5-13,

E.5-14

2) Compute products , , A AT T = O O H H for frequency and damping covariance, and

, ,C C T T = O O H H for mode shape covariance

3) For each mode i compute the covariance of the frequency and damping estimate

from


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Bridge S101 [VCE, 2009] F.5-1

covi i i i

i T

f f

i

f T T ρ ρ

ρ

=

where

,

,

i

i i

i

f A

f A

A

T T ρ ρ

=

,

and the covariance of the mode shape estimate from

Re( )

covIm( ) i i

i T

i i

T T ϕ ϕ ϕ

ϕ

=

where

, ,

, ,

Re( )

Im( )

i A C A

i

i A C C

T T

T

ϕ

ϕ

ϕ

=

.

For a detailed discussion of this computation and its numerical optimization the inter-

ested reader is referred to E.5-14.

5-4 Modal Analysis of S101 Bridge

5-4-1 S101 Bridge

Within the IRIS project an extensive measurement campaign of the prestressed con-

crete road bridge S101 [Döhler et al., 2011a; 2011b; VCE, 2009] was taken, which was

planned and organized by the Austrian company VCE and accomplished by VCE and the

University of Tokyo [VCE, 2009].

For vibration measurements a BRIMOS® measurement system was used, consisting in15 sensor locations on the bridge deck, where 14 sensors were placed on one side of the

deck and one sensor on the other side of the deck. In each location the bridge deck’s ver-

tical, longitudinal and transversal directions were measured. Altogether, 45 acceleration

sensors were applied. All values were recorded permanently with a sampling frequency

of 500 Hz. During the three days measurement campaign 714 data files with 165 000 data

points were produced. In this study only the first dataset is used.


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Modal Analysis of S101 Bridge 5-4

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100

4.104.04 4.05 4.06 4.07 4.08 4.09

Damping ratio ρ [%]




Stabilization diagrams containing the natural frequencies of the first five

modes with their confidence intervals (horizontal bars)

Zoom on first mode: estimated natural frequencies (left) and corresponding

damping ratios (right) at different model orders

F.5-2

F.5-3

5-4-2 Confidence Intervals of Modal Parameters

Restricting the system identification to the first five modes in the frequency range

[0–18 Hz], the data was downsampled from sampling rate 500 Hz by factor 5. System

identification and the covariance computation was done with the covariance-driven

subspace identification detailed in chapter 5-2 with parameters p + 1 = q = 35 at model or-

ders n = 1,…,100. All r = 45 sensors were used and r 0 = 3 reference sensors were chosen. The covariance computation on all identified modes at the different model orders was

accomplished with the strategy explained in chapter 5-3. For the covariance estimate on

the Hankel matrix, the data was cut into nb = 100 blocks. In F.5-2 (left), the stabilization

diagram of the natural frequencies vs. the model order are presented, where a confidence

interval (± one standard deviation) of each frequency is plotted as a horizontal bar. In


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mode f [Hz] [%]

1 4.036 0.12 0.78 15

2 6.281 0.08 0.56 20

3 9.677 0.18 1.3 14

4 13.27 0.13 1.5 13

5 15.72 0.37 1.3 17

Mode 1 Mode 2

Mode 5

Mode 3 Mode 4

Overview of the estimated first five modes with natural frequencies f , their

variation coefficient / 100f f f σ σ = ⋅ , the damping ratios ρ and their variation

coefficient / 100 ρ ρ σ σ ρ = ⋅

T.5-1

First five mode shapes in vertical direction on one side of the deck (line) and the

sensor on the other side of the deck (red point) with their 2 ϕ σ ± uncertainty bounds

F.5-4

F.5-2 (left) it can be seen that the (true) structural modes seem to have much lower un-

certainty bounds than spurious modes. With this observation, a threshold of 1 % was put

on the variation coefficient of the frequencies (standard deviation divided by frequency),which leads to a much clearer diagram in F.5-2 (right). Such a cleaned diagram would also


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References 5

be easier to evaluate for an automated modal parameter extraction. In F.5-3, the frequen-

cy and damping ratio of first mode identified at the different model orders is magnified for

a better visibility of the confidence bounds. Using the information from these confidence

bounds, values for the frequency and the damping ratio of a mode can be chosen that areoptimal for different model orders, as the true model order is unknown.

From the stabilization diagrams, the modal parameters are chosen. In T.5-1, the sys-

tem identification results with their variation coefficients are summarized. As expected,

the variation coefficients of the frequency estimates are very low (lower than 0.5 %), while

the estimates of the damping ratios show much higher variation coefficients (up to 20 %

in this case). In F.5-4, the real parts of the obtained mode shapes in the vertical direction at

the 14 sensors of one side of the bridge deck are displayed with their uncertainty bounds

(±2 standard deviations). From the 15th sensor on the other side of the bridge deck, whose

contribution is marked as the red point in F.5-4, information about the kind of the mode

is obtained: modes 1, 3 and 5 are vertical bending modes and modes 2 and 4 are torsionalmodes.

5-5 Conclusion

In this chapter, a method was presented to identify modal parameters – frequencies,

damping ratios and mode shapes – and their uncertainty bounds from output-only vibra-

tion measurements. The knowledge of the uncertainty bounds is essential to evaluate any

identified parameter and as it is known that especially estimated damping values showlarge uncertainties in system identification.

In operational modal analysis, where the system order is unknown and the input can-

not be measured, output-only subspace-based identification methods are very conveni-

ent. It was shown how the uncertainty bounds on the identified modal parameters can

improve system identification in this case by giving additional information to choose

modes in a stabilization diagram from identification results at different model orders. An

efficient implementation of the method was suggested and successfully applied to the

identification of modal parameters from output-only vibration data of S101 Bridge.

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iris chapter05

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