Some investigations on modal identification methods of ambient vibration structures Le Thai Hoa Wind...

Some investigations on modal identification methods of

ambient vibration structures

Le Thai HoaWind Engineering Research Center

Tokyo Polytechnic University

Date: 2009/12/05

Contents

1. Frequency-domain modal identification of ambient vibration structures using combined Frequency Domain Decomposition and Random Decrement Technique

2. Time-domain modal identification of ambient vibration structures using Stochastic Subspace Identification

3. Time-frequency-domain modal identification of ambient vibration structures using Wavelet Transform

Introduction

Modal identification of ambient vibration structures has become a recent issue in structural health monitoring, assessment of engineering structures and structural control Modal parameters identification: natural frequencies, damping and mode shapes Some concepts on modal analysis Experimental/Operational Modal Analysis(EMA/OMA) Input-output/Output-only Modal Identification Deterministic/ Stochastic System Identification Ambient/ Forced/ Base Excitation Tests Time-domain/ Frequency-domain/ Time-scale plane–based modal identification methods Nonparametric/ Parametric identification methods SDOF and MDOF system identifications ….

Vibration tests/modal identification

Ambient Vibration Tests

ForcedVibration Tests

Random/StochasticRandom/Stochastic

Deterministic/Deterministic/StochasticStochastic

ExperimentalModal Analysis

OperationalModal Analysis

Output-only Identification

Ambient loads &Micro tremor

Shaker (Harmonic)Hummer (Impulse)Sine sweep (Harmonic) Base servo (White noise, Seismic loads)

ExperimentalModal Analysis

OperationalModal Analysis

Input-output Identification

Output-only Identification

FRF identificationTransfer Functions

Removing harmonic & input effects

Indirect & direct identifications

Modal identification methods

Time domainTime domain Frequency domainFrequency domain Time-frequency planeTime-frequency plane

Ibrahim Time Domain (ITD)

Random Decrement Technique (RDT)

Stochastic Subspace Identification (SSI)

Frequency Domain Decomposition (FDD)

Wavelet Transform (WT)

Hilbert-Huang Transform (HHT)

[[Time-scale PlaneTime-scale Plane]]

Enhanced Frequency Domain Decomposition (EFDD)

Eigensystem Realization Algorithm (ERA)

Ambient vibration – Output-only system identificationAmbient vibration – Output-only system identification

Applicable in conditions and combined

Commercial and industrial uses

Academic uses, under development

Commercial Frequency domain Time domain

Package ODS FDDPeak Picking(No damping)

EFDD(Damping)

SSI(UPC)

SSI(PC)

SSI(CVA)

ARTeMIS Light

ARTeMIS Handy

ARTeMIS Pro

ARTeMIS Extractor 2009 FamilyThe State-of-the-Art software for Operational Modal Analysis

ODS: Operational Deflection ShapesFDD: Frequency Domain Decomposition EFDD: Enhanced Frequency Domain DecompositionSSI: Stochastic Subspace Identification UPC: Unweighted Principal Component PC: Principal Component CVA: Canonical Variate Algorithm

Commercial Software for OMA

http://www.svibs.com/index.aspx

Uses of FDD, RDT and SSI

[[Time-scale PlaneTime-scale Plane]]

For MDOF SystemsFor MDOF Systems

Response time series

Y(t)

Power SpectralDensity MatrixSYY(n)

Modal Parameters

FDD

(POD, SVD…)

EFDD

(POD, SVD…)

RDFunctions

DYY(t)

RDTITD

MRDT

CovarianceMatrixRYY(t)

Direct method SSI-COV

(POD, SVD…)

Direct method Data Matrix HY(t)

SSI-DATA

(POD, SVD…)

FD

DR

DT

SS

I

Comparison FDD, RDT and SSIComparison FDD, RDT and SSI

Advantages: Dealing with cross spectral matrix, good for natural frequencies and mode shapes estimation Disadvantage: based on strict assumptions, leakage due to Fourier transform, damping ratios, effects of inputs and harmonics; closed frequencies

FDD

RDT Advantages: Dealing with data correlation, removing noise and initial, good for damping estimation, SDOF systems Disadvantage: MDOF systems, short data record, natural frequencies and mode shapes combined with other methods

SSI Advantages: Dealing with data directly, no leakage and less random errors, direct estimation of frequencies, damping Disadvantage: Stabilization diagram, many parameters

Current trends in modal identification:

Combination between identification methods

Refined techniques of identification methods

Comparisons between identification methods

RDT to refine modal identification

OutputResponse

Time seriesY(t)

RDTRandom

Decrement Function

RDF

RDF-ITD & ERA

RDF-SSI-Covariance

Modal Parameters

RDF-BF PowerSpectralMatrix RDF-FDD

RDF-SSI-Data

Wavelet Transform (WT)

Hilbert-Huang Transform

Time Domain

Time-Frequency Plane

Frequency Domain

Time-frequency DomainPossibilities of RDT combined with othermodal identification methods

Multi-mode RDT

Frequency Domain Decomposition (FDD)

Random DecrementTechnique (RDT)

oFDD for output-only identification based on strict points (1) Input uncorrelated white noises Input PSD matrix is diagonal and constant (2) Effective matrix decomposition of output PSD matrix Fast decay after 1st eigenvector or singular vectors for approximation of output PSD matrix (3) Light damping and full-separated frequencies

Frequency Domain DecompositionFrequency Domain Decomposition

o Relation between inputs excitation X(t) and output response Y(t) can be expressed via the complex FRF function matrix:

o Also FRF matrix written as normal pole/residue fraction form, we can obtain the output complex PSD matrix:

o Output spectral matrix estimated from output data

o Output spectral matrix is decomposed (SVD, POD…)

Where: Spectral eigenvalues (Singular values) & Spectral eigenvectors (Singular vectors)

o ith modal shape identified at selected frequency

)()()()()()()(1

Tyi

N

iyiyi

TyyyyyS

rxll

k

ry

ry

ry

y

]:1[

]:1[

]:1[

2

1

cutllll

l

l

fxlxlyyyyyy

yyyyyy

yyyyyy

yy

SSS

SSS

SSS

S

)(...)()(

............

)(...)()(

)(...)()(

)(

21

22212

12111

)()()()( 111 TyyyyyS

)(),( yy

)(1 iyi

i

Frequencies & Frequencies & Damping Damping RatiosRatiosIdentificationIdentification

Mode shapesMode shapesIdentificationIdentification

Frequency Domain DecompositionFrequency Domain Decomposition

Output Output responseresponse

PSD matrixPSD matrix

Random Decrement Techniques

Triggering condition Xo

Xo

RD function (Free decay)

RDT extracts free decay data from ambient response of structures (as averaging and eliminating initial condition)

&

to

0 0

Random Decrement Techniques

RD functions (RD signatures) are formed by averaging N segments of X(t) with conditional value Xo

(Auto-RD signature)

(Cross-RD signature)

N : Number of averaged time segmentsX0 : Triggering condition (crossing level)k : Length of segment

Conditional correlation functions

Combined FDD-RDT diagram

Response Data Matrix

Y(t)

Cross Power Spectral Matrix

SYY(n)

1st Spectral Eigenvalue

1st Spectral Eigenvector

Natural Frequencies

Free Decay Fun. & Damping Ratios

Mode Shapes

POD, SVD, QR…

Data Matrix

Y(t)

Cross Power Spectral Matrix

SYY(n)

1st Spectral Eigenvalue

1st Spectral Eigenvector

Natural Frequencies

Free Decay Fun. & Damping Ratios

Mode Shapes

RDFun.

DYY(t)

POD, SVD, QR…RDT

FDD-RDT

FDD

Damping only

Natural Frequencies

…FDD Response Series at

Filtered FrequenciesFree Decay Fun. &

Damping Ratios

RDTBPFat fi

Stochastic Subspace Identification(SSI)

o Covariance-driven SSI

o Data-driven SSI

SSISSI SSI is parametric modal identification in the time domain. Some main characteristics are follows: Dealing directly with raw response time series Data order and deterministic input signal, noise are reduced by orthogonal projection and synthesis from decomposition SSI has firstly introduced by Van Overschee and De Moor (1996). Then, developed by several authors as Hermans and Van de Auweraer(1999); Peeters (2000); Reynder and Roeck (2008); and other.

SSI has some major benefits as follows: Unbiased estimation – no leakage Leakage due to Fourier transform; leakage results in unpredictable overestimation of damping No problem with deterministic inputs(harmonics, impulse) Less random errors: Noise removing by orthogonal projection

State-space representationState-space representation Continuous stochastic state-space model

wk: process noise (disturbances, modeling, input)vk : sensor noise

:

Discrete stochastic state-space model

state-space model

Second-order equations First order equationsA: state matrix; C: output matrix X(t): state vector; Y(t): response vector

yk

vk

C

Stochastic system

A

wk

(

,

wk , vk : zero mean white noises with covariance matrix

Data reorganizing Response time series as discrete data matrix N: number of samples M: number of measured points

Reorganizing data matrix either in block Toeplitz matrix or block Hankel matrix as past (reference) and future blocks

Block Hankel matrix Block Toeplitz matrix

s: number of block rowsN-2s: number of block columns

shifted t

past

future

s: number of block rows

SSI-COV and SSI-DATASSI-COV and SSI-DATA Projecting future block Hankel matrix on past one (as reference): conditional covariance

Data order reduction via decomposing, approximating projection matrix Ps using first k values & vectors

Observability matrix & system matrices

k: number of singular valuesk: system order

&

Modal parameters estimationMode shapes:

Poles:

Frequencies: Damping:

Hankel

Toeplitz

Flow chart of SSI algorithmData Matrix

[Y(t)]

Data Rearrangement

Block Teoplitz MatrixRP [], RF[],

Block Hankel MatrixHP[], HF[]

SSI-COV

SSI-DATAOrthogonal Projection

Ps

Observability MatrixOs

System MatricesA, C

Modal ParametersStabilization Diagram

Covaria

nce

DataData order reduction

POD

POD

Hankel matrix

Data past/ future

Toeplitz matrix

Parameter k

Parameter s

Modal identification of ambient vibration structures using combined Frequency Domain Decomposition and Random Decrement Technique

Numerical example

Fullscale ambient measurement

Five-storey steel frame

GroundGround

Floor1Floor1

Floor2Floor2

Floor3Floor3

Floor4Floor4

Floor5Floor5

X

Y

Z

0 50 100 150 200 250 300-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3

Time (s)

Dis

p.

(m

)

Floor5

0 50 100 150 200 250 300-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3

Time (s)

Dis

p.

(m

)

Floor4

0 50 100 150 200 250 300-1

-0.5

0

0.5

1x 10

-3

Time (s)

Dis

p.

(m

)

Floor3

0 50 100 150 200 250 300-1

-0.5

0

0.5

1x 10

-3

Time (s)

Dis

p.

(m

)

Floor2

0 50 100 150 200 250 300-5

0

5x 10

-4

Time (s)D

isp

. (m

)

Floor1

0 50 100 150 200 250 300-1

0

1x 10

-4

Time (s)

Dis

p. (m

)

Ground

Output displacement (X)

Floor 5

Floor 4

Floor 3

Floor 2

Floor 1

Ground

5 minutes record

Random decrement functions

0 5 10 15 20 25 30 35 40 45 50-8

-6

-4

-2

0

2

4

6

8x 10

-4 Floor5

Time (s)

Dis

p.(m

) Floor 5

0 5 10 15 20 25 30 35 40 45 50-8

-6

-4

-2

0

2

4

6

8x 10

-4 Floor4

Time (s)D

isp.(m

) Floor 4

0 5 10 15 20 25 30 35 40 45 50-5

0

5x 10

-4 Floor3

Time (s)

Dis

p.(m

) Floor 3

0 5 10 15 20 25 30 35 40 45 50-4

-3

-2

-1

0

1

2

3

4x 10

-4 Floor2

Time (s)

Dis

p.(m

) Floor 2

level crossing: segment: 50sno. of sample: 30000no. of samples in segment: 5000

Parameters

Spectral eigenvalues

0 5 10 15 20 25 30

10-10

10-5

100

Frequency (Hz)

No

rma

lize

d e

ige

nva

lue

s

Eigenvalue1Eigenvalue2Eigenvalue3Eigenvalue4Eigenvalue5Eigenvalue6

13.69Hz11.45Hz

8.84Hz10.16Hz

19.75Hz

18.12Hz

1.73Hz

5.35Hz

Mode 1 Mode 2Mode 3

Mode 4Mode 5

0 5 10 15 20 25 3010

-40

10-30

10-20

10-10

100

Frequency (Hz)

Nor

mal

ized

PS

D

Singular value 1Singular value 2Singular value 3Singular value 4Singular value 5Singular value 6

1.73Hz 5.34Hz

8.82Hz10.16Hz

13.67Hz 18.02Hz

Mode 1 Mode 2Mode 3 Mode 4Mode 5

FDD

FDD-RDT

Eigenvalue1: 99.9% Eigenvalue2: 0.07%Eigenvalue3: 0.01%Eigenvalue4: 0%

Eigenvalue1: 100% Eigenvalue2: 0%Eigenvalue3: 0%Eigenvalue4: 0%

Natural frequencies (Hz) FDD FDD-RDTmode 1 1.73 1.73mode 2 5.35 5.34mode 3 8.84 8.82mode 4 13.69 13.67mode 5 18.12 18.02

Spectral eigenvectors

99.9%

0.07%

0.01%

0%

FDD

FDD-RDT

Spectral eigenvectors

100%

0%

0%

0%

Mode shapes estimation

0 0.25 0.5 0.75 1Ground

Floor1

Floor2

Floor3

Floor4

Floor5Mode 1

FEMIdentified

-1 -0.5 0 0.5 1Ground

Floor1

Floor2

Floor3

Floor4

Floor5Mode 2

FEMIdentified

-1 -0.5 0 0.5 1Ground

Floor1

Floor2

Floor3

Floor4

Floor5Mode 3

FEMIdentified

-1 -0.5 0 0.5 1Ground

Floor1

Floor2

Floor3

Floor4

Floor5Mode 4

FEMIdentified

-1 -0.5 0 0.5 1Ground

Floor1

Floor2

Floor3

Floor4

Floor5Mode 5

FEMIdentified

Mode 1Mode 1

Mode 4Mode 4 Mode 5Mode 5

}}{{

||),(

2

ATAE

TE

ATE

AEMAC

Mode 2Mode 2 Mode 3Mode 3FDD

MACMAC

Mode shapes comparisonMode 1Mode 1



0 0.2 0.4 0.6 0.8 1Ground

Floor 1

Floor 2

Floor 3

Floor 4

Floor 5

Normalized amplitude

Mode 1

FEMFDDFDD-RDT

-1 -0.5 0 0.5 1Ground

Floor 1

Floor 2

Floor 3

Floor 4

Floor 5


Mode 2

FEMFDDFDD-RDT

-1 -0.5 0 0.5 1Ground

Floor 1

Floor 2

Floor 3

Floor 4

Floor 5


Mode 3

FEMFDDFDD-RDT

-1 -0.5 0 0.5 1Ground

Floor 1

Floor 2

Floor 3

Floor 4

Floor 5


Mode 4

FEMFDDFDD-RDT

-1 -0.5 0 0.5 1Ground

Floor 1

Floor 2

Floor 3

Floor 4

floor 5


Mode 5

FEMFDDFDD-RDT

Identified auto PSD functions

MAC=98%

MAC=95%

FDD

Mode 1Mode 2

Mode 3Mode 4

Mode 5

Identified free decay functions

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

Nor

mal

ized

am

plitu

de

Free decay function of mode 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

Nor

mal

ized

am

plitu

de


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

Nor

mal

ized

am

plitu

de


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

Nor

mal

ized

am

plitu

de


Mod

e 1

Mod

e 1

Mod

e 4

Mod

e 4

Mod

e 2

Mod

e 2

Mod

e 3

Mod

e 3

FDD

Mod

e 5

Mod

e 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)N

orm

aliz

ed a

mpl

itude


Uncertainty in damping ratios estimation from free decay functions of modes 3 & 4

Unclear with modes 2 & 5

Identified free decay functionsM

ode

1M

ode

1

Mod

e 4

Mod

e 4

Mod

e 2

Mod

e 2

Mod

e 3

Mod

e 3

Mod

e 5

Mod

e 5

Better

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

No

rma

lize

d a

mp

litu

de

Free decay function for mode 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

No

rma

lize

d a

mp

litu

de

Free decay function for mode 2 FDD

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

No

rma

lize

d a

mp

litu

de


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)N

orm

aliz

ed

am

plit

ud

e


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

No

rma

lize

d a

mp

litu

de


FDD - Band-pass filtering

-2

0

2x 10

-3 Bandpass Filtering

-2

0

2x 10

-3

-2

0

2x 10

-4

Am

p.(

m)

-5

0

5x 10

-5

-5

0

5x 10

-5

0 50 100 150 200 250 300-2

0

2x 10

-5

Time (s)

Filtered at mode 1

Filtered at mode 2

Filtered at mode 3

Filtered at mode 4

Filtered at mode 5

Original output

X5(t)

f1=1.73Hz

f2=5.34Hz

f3=8.82Hz

f4=13.67Hz

f5=18.02Hz

Response time series at Floor 5 has been filtered on spectral bandwidth around each modal frequency

0 2 4 6 8 10 12 14 16 18 2010

-30

10-25

10-20

10-15

10-10

10-5

Frequency (Hz)

PSD

Bandpass Filtering

Filtered at mode 1

Filtered at mode 2 Filtered

at mode 3Filtered at mode 5

Filtered at mode 4

Floor 5

Damping ratio via FDD-BPF

0 5 10 15 20 25 30 35 40 45 50-5

0

5x 10

-4

Time (s)

Am

p. (

m)

RD function

Filtered at mode 1

Mod

e 1

Mod

e 1

0 5 10 15-4

-2

0

2

4x 10

-5

Time (s)

Am

p.(m

)

RD function

Filtered at mode 2

Mod

e 2

Mod

e 2

0 1 2 3 4 5 6 7 8 9 10-4

-2

0

2

4x 10

-6

Time (s)

Am

p.(m

)

RD function

Filtered at mode 3

Mod

e 3

Mod

e 3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3x 10

-6

Time (s)A

mp.

(m)

RD function

Filtered at mode 4

Mod

e 4

Mod

e 4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3x 10

-6

Time (s)

Am

p.(m

)

RD function

Filtered at mode 5

Mod

e 5

Mod

e 5 Uncertainty in damping

ratios estimated from free decay functions at modes 4 & 5

Free decay functionsFloor 5

FDD - Band-pass filtering

Response time series at Floor 1 has been filtered on spectral bandwidth around each modal frequency

Floor 1

-5

0

5x 10

-4 Bandpass Filtering

-5

0

5x 10

-4

-2

0

2x 10

-5

-1

0

1x 10

-5

0 50 100 150 200 250 300-1

0

1x 10

-5

Time (s)

-5

0

5x 10

-5

Am

p.(

m)

Original output

Filtered at mode 1

Filtered at mode 2

Filtered at mode 3

Filtered at mode 4

Filtered at mode 5

f1=1.73Hz

f2=5.34Hz

f3=8.82Hz

f4=13.67Hz

f5=18.02Hz

X1(t)

0 2 4 6 8 10 12 14 16 18 2010

-30

10-20

10-10

100

Frequency (Hz)

PSD

Bandpass Filtering

Filtered at mode 3

Filtered at mode 2

Filtered at mode 5

Filtered at mode 1

Filtered at mode 4

Damping ratio via FDD-BPFM

ode

1M

ode

1

Mod

e 2

Mod

e 2

Mod

e 3

Mod

e 3

Mod

e 4

Mod

e 4

Mod

e 5

Mod

e 5

Free decay functions

0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5x 10

-4

Time (s)

Am

p.(m

)

RD function

Filtered at mode 1

0 5 10 15-3

-2

-1

0

1

2

3x 10

-5

Time (s)

Am

p.(m

)

RD function

Filtered at mode 2

0 1 2 3 4 5 6 7 8 9 10-5

0

5x 10

-6

Time (s)

Am

p.(m

)

RD function

Filtered at mode 3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1

0

1

2x 10

-6

Time (s)A

mp.

(m)

RD function

Filtered at mode 4

0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1x 10

-6

Time (s)

Am

p.(m

)

RD function

Filtered at mode 5

Floor 1

Damping ratio via FDD-BPFSelected free decay functions for damping estimation

Mod

e 1

Mod

e 1

Mod

e 2

Mod

e 2

Mod

e 3

Mod

e 3

Mod

e 4

Mod

e 4

Mod

e 5

Mod

e 5

0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1x 10

-6

Time (s)

Am

p.(m

)

RD function

Filtered at mode 5

0 5 10 15-3

-2

-1

0

1

2

3x 10

-5

Time (s)

Am

p.(m

)

RD function

Filtered at mode 2

0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5x 10

-4

Time (s)

Am

p.(m

)

RD function

Filtered at mode 1

0 1 2 3 4 5 6 7 8 9 10-4

-2

0

2

4x 10

-6

Time (s)

Am

p.(m

)

RD function

Filtered at mode 3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1

0

1

2x 10

-6

Time (s)A

mp.

(m)

RD function

Filtered at mode 4

Time-domain modal identification of ambient vibration structures using Stochastic Subspace Identification

Numerical example

Parameters formulated

Number of measured points: M=6 Number of data samples: N=30000 Dimension of data matrix: MxN=6x30000

Number of block row: s=20:10:120 (11 cases) Number of block columns: N-2s Dimension of Hankel matrix: 2sMx(N-2s)

Data parameters

Hankel matrix parameters

System order parameters

Number of system order: k=5:5:60 (12 cases) (Number of singular values used)

Projection functions

0 50 100 150 200 250 300-4

-2

0

2

4

6

8

10x 10

-4

Time (ms)

Am

plit

ude

After orthogonal projection

s=50, column=10

0 50 100 150 200 250 300-6

-4

-2

0

2

4

6

8x 10

-4

Time (ms)

Am

plit

ud

e


s=50, column=50

0 50 100 150 200 250 300-4

-2

0

2

4

6x 10

-4

Time (ms)

Am

plit

ude


s=50, column=200

0 50 100 150 200 250 300-5

0

5x 10

-4

Time (ms)

Am

plit

ude


s=50, column=500

0 100 200 300 400 500 600-6

-4

-2

0

2

4

6

8x 10

-4

Time (ms)

Am

plit

ude


s=100, column=10

0 100 200 300 400 500 600-6

-4

-2

0

2

4

6

8x 10

-4

Time (ms)

Am

plit

ude


s=100, column=50

0 100 200 300 400 500 600 700 800 900-6

-4

-2

0

2

4

6x 10

-4

Time (ms)

Am

plit

ud

e


s=100, column=10

0 100 200 300 400 500 600 700 800 900-6

-4

-2

0

2

4

6x 10

-4

Time (ms)

Am

plit

ud

e


s=150, column=50

0 100 200 300 400 500 600 700 800 900-5

0

5x 10

-4

Time (ms)

Am

plit

ud

e


s=150, column=200

0 100 200 300 400 500 600 700 800 900-4

-2

0

2

4

6x 10

-4

Time (ms)

Am

plit

ud

e


s=150, column=500

0 100 200 300 400 500 600-5

0

5x 10

-4

Time (ms)

Am

plit

ud

e


s=100, column=200

0 100 200 300 400 500 600-5

0

5x 10

-4

Time (ms)

Am

plit

ud

e


s=100, column=500

s=50s=50 s=100s=100 s=150s=150

Data after orthogonal projection look like time-shifted sine functions

Effects of s on energy contribution

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

Number of singular values

Singular values

s=20s=40s=60s=80s=100s=120

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Number of singular values

Pe

rce

nta

ge

(%

)

Energy contribution

s=20s=40s=60s=80s=100s=120

1 2 3 4 5 6 7 8 9 1030

40

50

60

70

80

90

100

Number of singular values)

Pe

rce

nta

ge

(%

)

Energy cummulation

s=20s=40s=60s=80s=100s=120

20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

Number of block row s

Singular velues

Singular value 1Singular value 2Singular value 3Singular value 4Singular value 5

20 40 60 80 100 1200

10

20

30

40

50


Pe

rce

nta

ge

(%

)

Energy contribution

Singular value 1Singular value 2Singular value 3Singular value 4Singular value 5

20 40 60 80 100 12030

40

50

60

70

80

90

100


Pe

rce

nta

ge

(%

)Energy cummulation

Fisrt singular valueFirst 2 singular valuesFirst 3 singular valuesFirst 4 singular valuesFirst 5 singular values

system orders (k)

(k)

(k)

(s)

(s)

(s)

k=10 90-96% Energyk=15 92-97% Energyk=20 93-98% Energy

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60N

um

be

r o

f p

ole

s

Natural frequency (Hz)

Frequency diagram

Frequency diagram

0 2 4 6 8 10 12 14 16 18 2010

-15

10-10

10-5


PS

D

X5

mod

e 1

mod

e 2

mod

e 3

mod

e 4

mod

e 5

1.74

Hz

5.34

Hz

PSD of response time series

8.82

Hz

13.6

7Hz

18.0

44H

z

s=50k=5:5:60Natural frequencies (Hz)

FDD SSImode 1 1.73 1.74mode 2 5.35 5.34mode 3 8.84 8.82mode 4 13.69 13.67mode 5 18.12 18.04

Frequency diagram

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60S

yste

m o

rder

k


mod

e 1

mod

e 2

mod

e 3

mod

e 4

mod

e 5

s=20:10:120k=60

Frequency diagram

0 2 4 6 8 10 12 14 16 18 2020

30

40

50

60

70

80

90

100

110

120N

um

be

r o

f b

lock r

ow

s s


0 2 4 6 8 10 12 14 16 18 2010

-15

10-10

10-5


PS

D

X5

PSD of response time series

mode 1

mod

e 2

mod

e 3

mod

e 4

mod

e 5

s=20:10:120k=60

Damping diagram

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

Num

ber

of p

oles

Damping ratio (%)

s=50k=5:5:60

mode 1

0.18%

mode 2

0.22% mod

e 3

0.46%

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

70

80

Sys

tem

ord

er k

Damping ratio (%)

Damping diagram

s=20:10:120k=60

mode 1

0.18%mod

e 2

0.22% mod

e 3

0.47%

Some investigations on modal identification methods of ambient vibration structures Le Thai Hoa Wind...

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