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Some investigations on modal identification methods of ambient vibration structures

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Date: 2009/12/05

Some investigations on modal identification methods of ambient vibration structures

Le Thai Hoa

Wind Engineering Research Center

Tokyo Polytechnic University

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 loads &

Micro tremor

Indirect & direct identifications

Experimental

Modal Analysis

Output-only

Identification

Ambient

Vibration Tests

Operational

Modal Analysis

Random/Stochastic

Shaker (Harmonic)

Hummer (Impulse)

Sine sweep (Harmonic)

Base servo (White noise, Seismic loads)

FRF identification

Transfer Functions

Experimental

Modal Analysis

Input-output

Identification

Forced

Vibration Tests

Operational

Modal Analysis

Output-only

Identification

Deterministic/Stochastic

Removing harmonic & input effects

Modal identification methods

Ambient vibration – Output-only system identification

Time domain

Frequency domain

Time-frequency plane

Wavelet Transform (WT)

Ibrahim Time Domain (ITD)

Frequency Domain Decomposition (FDD)

Eigensystem Realization Algorithm (ERA)

Hilbert-Huang Transform (HHT)

Enhanced Frequency Domain Decomposition (EFDD)

Random Decrement Technique (RDT)

Stochastic Subspace Identification (SSI)

Applicable in conditions and combined

[Time-scale Plane]

Commercial and industrial uses

Academic uses, under development

Commercial Software for OMA

ARTeMIS Extractor 2009 Family

The State-of-the-Art software for Operational Modal Analysis

ODS: Operational Deflection Shapes

FDD: Frequency Domain Decomposition EFDD: Enhanced Frequency Domain Decomposition

SSI: Stochastic Subspace Identification

UPC: Unweighted Principal Component PC: Principal Component CVA: Canonical Variate Algorithm

Uses of FDD, RDT and SSI

For MDOF Systems

Power

Spectral

Density

Matrix

SYY(n)

Modal

Parameters

FDD

Response time series

Y(t)

(POD, SVD…)

FDD

EFDD

(POD, SVD…)

ITD

RD

Functions

DYY(t)

RDT

RDT

MRDT

CovarianceMatrix

RYY(t)

Direct method

SSI-COV

(POD, SVD…)

SSI

[Time-scale Plane]

SSI-DATA

Direct method

Data Matrix HY(t)

(POD, SVD…)

Comparison FDD, RDT and SSI

FDD

- 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

Current trends in modal identification:

Combination between identification methods

Refined techniques of identification methods

Comparisons between identification methods

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

Time Domain

Multi-mode RDT

Modal Parameters

Random Decrement Function RDF

RDF-ITD & ERA

RDT to refine modal identification

RDF-SSI-Covariance

RDF-SSI-Data

Output

Response

Time series

Y(t)

Frequency Domain

RDT

Power

Spectral

Matrix

RDF-BF

RDF-FDD

Wavelet Transform (WT)

Hilbert-Huang Transform

Possibilities of RDT

combined with other

modal identification

methods

Time-Frequency Plane

Time-frequency Domain

Frequency Domain Decomposition (FDD)

Random Decrement

Technique (RDT)

Frequency Domain Decomposition

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

FDD 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

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

Frequency Domain Decomposition

- Output spectral matrix estimated from output data

Output

response

PSD matrix

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

Frequencies & Damping Ratios

Identification

Where: Spectral eigenvalues (Singular values) &

Spectral eigenvectors (Singular vectors)

Mode shapes

Identification

- ith modal shape identified at selected frequency

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

0

0

&

to

Triggering condition Xo

Xo

RD function (Free decay)

Random Decrement Techniques

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

(Auto-RD signature)

Conditional correlation functions

(Cross-RD signature)

N : Number of averaged time segments

X0 : Triggering condition (crossing level)

k : Length of segment

Combined FDD-RDT diagram

FDD

POD, SVD, QR…

Natural Frequencies

1st Spectral Eigenvalue

Response

Data Matrix

Y(t)

Cross Power Spectral Matrix SYY(n)

Free Decay Fun. & Damping Ratios

1st Spectral Eigenvector

Mode Shapes

FDD-RDT

POD, SVD, QR…

Natural Frequencies

RDT

1st Spectral Eigenvalue

RD

Fun.

DYY(t)

Data Matrix

Y(t)

Cross Power Spectral Matrix SYY(n)

Free Decay Fun. & Damping Ratios

1st Spectral Eigenvector

Mode Shapes

Damping only

FDD

BPF

RDT

Natural Frequencies

Response Series at Filtered Frequencies

Free Decay Fun. & Damping Ratios

…

at fi

- Stochastic Subspace
- Identification(SSI)
- Covariance-driven SSI
- Data-driven SSI

SSI

- 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 representation

:

,

- Continuous stochastic state-space model

state-space model

Second-order equations

First order equations

A: state matrix; C: output matrix

X(t): state vector; Y(t): response vector

- Discrete stochastic state-space model

wk: process noise (disturbances, modeling, input)

vk : sensor noise

vk

wk

wk , vk : zero mean white noises

with covariance matrix

C

yk

A

Stochastic system

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

shifted t

Block Toeplitz matrix

past

future

s: number of block rows

N-2s: number of block columns

s: number of block rows

SSI-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

Hankel

k: number of singular values

k: system order

Toeplitz

- Observability matrix & system matrices

&

- Modal parameters estimation

Mode shapes:

Poles:

Frequencies:

Damping:

Flow chart of SSI algorithm

Data Matrix

[Y(t)]

SSI-COV

Covariance

Block Teoplitz Matrix

RP [], RF[],

Data past/ future

Data Rearrangement

Parameter s

Block Hankel Matrix

HP[], HF[]

Data

Data order reduction

Orthogonal Projection

Ps

SSI-DATA

Hankel matrix

Parameter k

POD

Observability Matrix

Os

System Matrices

A, C

Toeplitz matrix

POD

Modal Parameters

Stabilization Diagram

Numerical example

Fullscale ambient measurement

5 minutes record

Z

Floor 5

Floor5

X

Y

Floor 4

Floor4

Floor 3

Floor3

Floor 2

Floor2

Five-storey steel frame

Floor 1

Floor1

Ground

Ground

(X)

Output displacement

Random decrement functions

Floor 5

Parameters

level crossing: segment: 50s

no. of sample: 30000

no. of samples in segment: 5000

Floor 4

Floor 3

Floor 2

Spectral eigenvalues

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

FDD

Eigenvalue1: 99.9%

Eigenvalue2: 0.07%

Eigenvalue3: 0.01%

Eigenvalue4: 0%

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

FDD-RDT

Eigenvalue1: 100%

Eigenvalue2: 0%

Eigenvalue3: 0%

Eigenvalue4: 0%

Spectral eigenvectors

FDD

99.9%

0.01%

0.07%

0%

Spectral eigenvectors

FDD-RDT

100%

0%

0%

0%

Mode shapes estimation

Mode 1

Mode 2

Mode 3

FDD

Mode 4

Mode 5

MAC

Mode shapes comparison

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Identified auto PSD functions

FDD

MAC=95%

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

MAC=98%

Identified free decay functions

FDD

Mode 1

Mode 2

Mode 3

Mode 4

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

Mode 5

Unclear with modes 2 & 5

Identified free decay functions

FDD

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Better

FDD - Band-pass filtering

Floor 5

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

Damping ratio via FDD-BPF

Free decay functions

Floor 5

Mode 2

Mode 1

Mode 3

Mode 4

Uncertainty in damping ratios estimated from free decay functions at modes 4 & 5

Mode 5

FDD - Band-pass filtering

Floor 1

X1(t)

f1=1.73Hz

f2=5.34Hz

f3=8.82Hz

f4=13.67Hz

f5=18.02Hz

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

Damping ratio via FDD-BPF

Free decay functions

Floor 1

Mode 2

Mode 1

Mode 3

Mode 4

Mode 5

Damping ratio via FDD-BPF

Selected free decay functions for damping estimation

Mode 2

Mode 1

Mode 3

Mode 4

Mode 5

Numerical example

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

Parameters formulated

Data parameters

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

Hankel matrix parameters

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

System order parameters

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

Projection functions

s=50

s=100

s=150

Data after orthogonal projection look like time-shifted sine functions

Effects of s on energy contribution

system orders

(k)

(s)

(k)

(s)

k=10 90-96% Energy

k=15 92-97% Energy

k=20 93-98% Energy

(k)

(s)

Frequency diagram

13.67Hz

mode 4

mode 5

18.044Hz

1.74Hz

5.34Hz

8.82Hz

mode 2

mode 1

mode 3

s=50

k=5:5:60

PSD of response time series

Frequency diagram

mode 4

mode 5

mode 2

mode 1

mode 3

s=20:10:120

k=60

Frequency diagram

mode 5

mode 4

mode 3

mode 1

mode 2

s=20:10:120

k=60

PSD of response time series

Damping diagram

mode 1

0.18%

mode 3

mode 2

0.46%

0.22%

s=50

k=5:5:60

Damping diagram

mode 1

0.18%

mode 2

mode 3

0.22%

0.47%

s=20:10:120

k=60