Vibr

ationTe

sting

& M

odal I

dent

ificat

ion

Vibration Testing & Modal Identification

Jean-Claude GolinvalReferences

• D. J. EWINS

Modal Testing : theory, practice and application

Second Edition, Research Studies Press LTD, 2000

ISBN 0 86380 218 4

• N. MAIA, J. SILVA

Theoretical and Experimental Modal Analysis

Research Studies Press LTD, 1997

ISBN 0 86380 208 7

2Two Routes to Vibration Analysis

Description of the structure in terms of its mass, stiffness and

damping properties

Theoretical Approach – Direct Problem

Experimental Approach – Inverse Problem

Structural Model Modal Model Response Model

Response Measurements

Modal Model (Identification) Structural Model

Natural frequencies, Modal damping factors,

Mode shapes

Frequency Response Functions, Impulse Response Functions

3

Définition

Développement d ’un modèle mathématique du comportement dynamique d ’une structure à partir de tests expérimentaux.

Relation fondamentale

Réponse Excitation Propriétés du système= ×

Bref historiqueAvant 1947 : mesures de la réponse seule

1947 : mesures de la réponse et de l ’excitation Kennedy, Pancu : mesures des fréquences de résonance et de l ’amortissement de structures d ’avions.

1960-70 : progrès techniques des moyens de mesures (capteurs, électronique, …) et d ’acquisition (analyseurs de spectre digitaux)

1965 : algorithme FFT (Cooley et Tukey)

Analyse et identification modale

4Single-Degree-Of-Freedom (SDOF) System

Governing equation of motion

the trial solution :

Roots :

)(tfxkxcxm =++

teXtx λ=)(

200

2

2,1 12

42

ζωωζλ −±−=−

±−= im

mkcmc

wheremm

k

00 2

candω

ζω ==

Natural frequency Viscous damping ratio

In case of no external forcing ( f(t) = 0,

leads to the requirement that : 02 =++ kcm λλ

5

Free vibration characteristic of damped SDOF system )1( <ζ

tit eeXtx2

00 1)( ζωζω −−=

Forced response to harmonic excitation: tieFtf ω=)(

)()()( ωωω FHX =

⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛ −=

+−=

020

22

21

11)()(

1)(

ωωζ

ωωωω

ωi

kcimkH

Frequency Response Function (FRF)

6

Alternative Forms of FRF

Force Structure Response

Displacement

Velocity

Acceleration

Standard FRF Inverse FRF

Displacement Receptance Admittance

Dynamic stiffness

Velocity Mobility Mechanical impedance

Acceleration Accelerance Inertance

Apparent mass

FRF = Response / Force

7

Graphical Displays of FRF Data

The FRF is a complex quantity.

))(())(( ωω HiHH ℑ+ℜ=Real Part Imaginary Part

Frequency

The most common forms of presentation are:

• Bode plot: Modulus and Phase of FRF vs. Frequency (2 graphs);

• Nyquist plot: Imaginary Part (of FRF) vs. en Real Part (of FRF) (a single graph which does not contain frequency information explicitely);

• Cartesian plots: Real Part (of FRF) vs. Frequency and Imaginary Part (of FRF) vs. Frequency (2 graphs);

8

• Bode Plot

Linear scales Log-Log scales

Receptance Mass Stiffness

)log()log(2)log())(log(/1/1)( 2

kmHkmH

−−−−

ωωωω

9

• Nyquist Plot

)(ωH

)(ωφ

• Cartesian Plots

Real Part

Imaginary Part

10

• Three-dimensional plot of SDOF system FRF

11Multi-Degree-Of-Freedom Systems

Governing equation of motion

)(tfqKqCqM =++

q = vector of displacements

f(t) = vector of forces

M = mass matrix

K = stiffness matrix

C = damping matrix

M, K and C constitute the Spatial Model.

12

Undamped MDOF SystemsFree Vibration Solution – The Modal Properties

The undamped system’s natural frequencies are the solutions of

and the corresponding mode shapes are the solutions of

02 =− MK ωdtm

( ) ( ) 02 =− rr ψMK ω

N eigensolutions

( ) ( ) ( )N

N

ψψψ 21

ωωω ≤≤≤ 21

0=+ xKxM

13

[ ][ ]r

Tr

T

k

m

=

=

ΨKΨΨMΨ

Orthogonality properties:

Let us define

( ) ( ) ( )[ ][ ] [ ]22

221

221

Nr

N

diag ωωωω =

= ψψψΨ Modal matrix

Spectral matrix

[ ]2rωandΨ constitute the Modal Model.

Modal mass matrix

Modal stiffness matrix

[ ] [ ] [ ]rrr km 12 −=ωAnd thus

14

Orthogonality properties:

in which case, the mass-normalised eigenvectors are written as

( ) ( ) ( )[ ]NxxxX 21= Mass-normalised modal matrix

Remark :Mass-normalisation is the most relevant scaling in modal testing, i.e.

[ ] [ ]1=rm

[ ][ ]2

rT

T

ω=

=

XKX

1XMX

15

Forced Response Solution – The FRF Characteristics

tiet ωFfqKqM ==+ )(

Solution of the form: tie ωQq =

Modal transformation: ( ) aXxQ ==∑=

k

N

kka

1

The equations of motion then becomes: ( ) FaXMK =− 2ω

Premultiply both sides by to obtainTX

[ ] [ ]( ) FX1XQ Tr

122 −−= ωωwhich leads to

[ ] [ ] FXa1 Tr =⎟

⎠⎞

⎜⎝⎛ − 22 ωω

16

[ ] [ ]( ) Tr X1XH

122)(−

−= ωωω

From the definition of the FRF matrix, it follows that:

( ) ( )[ ]( )

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

TN

T

rN

x

xxxH

1

2211

ωω

Hence the spectral representation of the FRF matrix:

( ) ( )∑= −

=N

k k

Tkk

122)(

ωωω

xxH

17

Any individual FRF coefficient linking coordinates r and s (called dynamic influence coefficient) is written:

( )( ) skrkkrs

N

k k

krsrs A

AH φφ

ωωω =

−= ∑

=with

122)(

Modal Constant (or Residue)

Natural Frequency

(Pole)

18

Damped Systems)(tfqKqCqM =++

Modal transformation: ( ) ( ) ηXxq == ∑=

k

N

kkt

1η

Modal matrix

Pre-multiplying the equation of motion by TX

[ ] ( ) [ ] )()()( ttkttm Tr

Tr fXηXCXη =++ η

Modal damping matrix

19

Proportional Damping

In this case, the matrix is diagonal

where [ck] = modal damping matrix.

[ ]kT c=XCX

Using the notation introduced above for the SDOF analysis, the damping coefficient of mode k is written in the form:

kkkk mc ωζ2=

The forced response to an harmonic excitation

tie ωFqKqCqM =++

is in the form:tie ωQq =

Modal transformation: ( ) aXxQ == ∑=

k

N

kka

1

20

Thus, the equation of motion takes the form:

( ) FaXCMK =+− ωω i2

Pre-multiplying par , we have:TX

[ ] [ ] [ ]( ) FX1XQ Tkkk i

122 2−

+−= ωζωωω

[ ] [ ] [ ] FXa1 Tkkk i =⎟⎠⎞

⎜⎝⎛ +− ωζωωω 222

[ ] [ ] [ ]( ) Tkkk i X1XH

122 2)(−

+−= ωζωωωωWe obtain:

and the expression of the FRF matrix is:

( ) ( )∑= +−

=N

k kkk

Tkk

i122 2

)(ωωζωω

ωxx

H

21

The receptance frequency response function (FRF) at coordinate r , resulting from a single force applied at coordinate s , is :

( )( ) ( ) ( )kskrkrs

N

k kkk

krsrs A

i

AH xx=

+−= ∑

=with

122 2

)(ωωζωω

ω

Example:

22

Particular form of proportional damping

MKC γβ +=

In this case, the eigenvalues and eigenmodes take the form:

undampeddamped

kkk

kkdk

XX =

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−=

ωγωβζ

ζωω

21

1 2 kζ

kω

0=β

0=γ

23

Viscous damping (General case)

In this case, the matrix is not diagonal anymore.

Consider the homogeneous system:

It can be shown that the eigensolutions of the appropriate equation

occur as complex conjugates.

XCXT

0qKqCqM =++

( ) 02 =++ QKCM λλ

( ) ( )Nk

kk

kk ,,1,

,*

*

=⎪⎭

⎪⎬⎫

ψψ

λλ

with: ⎟⎠⎞⎜

⎝⎛ −+−= 21 kkkk i ζζωλ

This approach is not particularly convenient for numerical application.

24

In the general case of viscous damping, it is better to recast the equations into the state-space form:

⎪⎩

⎪⎨⎧

=−=++

0qMqMFqKqCqM tie ω

or tie ω

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡0F

qq

M00K

qq

0MMC

A B+u u tie ωP=

A and B are real and symmetric matrices of dimension 2N x 2N.

25

Homogeneous equation:

Solution of the form:

Eigenvalue problem:

2N eigenvalues and eigenvectors verifying the orthogonality properties:

0uBuA =+

teλUu =

( ) 0zBA =+λ

kλ ( )kz

[ ][ ]k

Tk

T

b

a

=Β

=

ZZ

ZAZ

and which have the usual characteristic that

)2,,1(; Nkab

k

kk =−=λ

26

Forced response to harmonic excitationtie ωPuBuA =+

Development in the series of eigenmodes:

( )∑=

=N

kk

tik e

2

1zu ωη

Substituting into the state-space equation and pre-multiplying by we have:

( )Tkz

( ) )2,,1( Nkbai Tkkkkk ==+ Pzηηω

( ) )2,,1()(

Nkia kk

Tk

k =−

=λω

ηPz

Thus, the solution may be written as:

( ) ( ) tiN

k kk

Tkk e

iaω

λω∑= −

=2

1 )(Pzz

u

27

( ) ( ) ( ) ( ) Pzzzz

Q

Q

⎟⎟

⎠

⎞

−+

⎜⎜

⎝

⎛

−=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∑= )()(1 kk

TkkN

k kk

Tkk

iaiai λωλωω

However, because the eigensolutions occur in complex conjugate pairs, the solution may be rewritten as:

If we extract the response vector Q, we obtain the expression of the FRF matrix in the form:

( ) ( ) ( ) ( ))()(1 kk

TkkN

k kk

Tkk

iaia λωλω −+

−= ∑

=

zzzzH

28

A single response parameter takes the form:Modal constants

(Residues)

( ) ( ) ( ) ( )∑= −

+−

=N

k kk

kskr

kk

kskrrs iaia

H1 )()(

)(λωλω

ωzzzz

Poles (complex)

( ) ( ) ( ) ( ))()(1 kk

TkkN

k kk

Tkk

iaia λωλω −+

−= ∑

=

zzzzH

29

Complex Modes

Real mode: in-phase vibration, (standing wave)

Complex mode: out-of-phase vibration (travelling wave)

Measurement of modal complexity (complex mode shapes plotted on Argand diagrams)

30

Example:

31

a) Undamped systems

b) Damped systems (proportional damping)

c) Damped systems (general case of viscous damping)

( ) ( ) ( ) ( )∑=

−+

−=

N

k kk

kskr

kk

kskrrs iaia

H1

)()()(

λωλωω

zzzz

( )( ) ( ) ( )kskrkrs

N

k kkk

krsrs A

i

AH xx=

+−=∑

=

with1

22 2)(

ωωζωωω

( )( ) ( ) ( )kskrkrs

N

k k

krsrs A

AH xx=

−=∑

=

with1

22)(ωω

ω

The fundamental relations used for experimental modal analysis are:

Summary

32FRF Measurement TechniquesTest Planning

• Definition of the objectives

- estimation of vibration amplitude levels

- estimation of natural frequencies, mode shapes and damping factors

- model validation

• Selection of the frequency range

• Selection of measurement and excitation coordinates

• Selection of excitation devices and transducers

• Selection of the support points (to minimise external influence)

• Checking the quality of measured data

• Measured data consistency, including reciprocity

• Measurement repeatability

• Measurement reliability

33

General Layout of FRF Measurement System

Excitation mechanism

Transduction systemAnalyser

34

• Mechanical Exciters

- little flexibility or control

Excitation of the Structure

35

• Electrohydraulic Exciters

- substantial forces are generated

- possibility to apply simultaneously a static load to the dynamicvibratory load

- large displacement amplitudes

- limited frequency range

- hydraulic shakers are complex and expansive

36

• Electrodynamic Exciters

37

Characteristics of a shaker

38

Example of grounded structure

Examples of freely-supported structure

39

Exciter attachment Various mounting arrangements for exciter

(a) Ideal configuration

(b) Suspended shaker plus inertia mass

(c) et (d) Compromise configurations

(b) Unsatisfactory assembly

(c) et (d) Acceptable assembly

(e) Use of a push rod or stinger

40

Example of electrodynamic shaker used for environmental testing (qualification, fatigue)

Gearing & Watson V2664

Max. force = 26.6 kN(sinus et random) ;

Max. displ. = 50 mm(peak-peak) ;

Max. velocity = 1.52 m/s.

41

• Piezoelectric exciters

- Operational at high frequency range

- Low excitation amplitude

• Non-Contact Magnetic Excitation

- Electromagnetic force

- Measurement of the reaction force on the body of the magnet (not directly the force applied to the structure)

- Application to rotating machinery

42

• Hammer Excitation

Tip

Head

Force gauge

43

(I) Stiff tip (steel)

(II) Medium tip (vinyl)

(III) Soft tip (rubber)

Duration of the pulse

= cut-off frequency

Typical impact force pulse and spectrum

44

Use of different excitation devices

45

Consider a sinusoidal signal

tXxtXx

tXxtFf

ωωωω

ωω

sinsin

sinsin

2−=

===

Signal Response parameter

XX

XF

2ωω

Force

Displacement

Velocity

Acceleration

24

25

/10/smmsmm

μμ

1Hz15800101Hz0,5

onacceleratintdisplacemefrequency−

Example:

Not detectable

Transducers

46

Choice of the physical quantity to be measured

• Displacement : measurement of gaps, orbits of rotor inside journals, thermal dilatations, …

• Velocity : used in standards to characterise vibration intensity

• Acceleration : measurement of high frequency signals (shocks)

acceleration

Vibration amplitude

velocitydisplacement

frequency

47

Vibration Transducers

Definition

Physical quantity

(force, displacement, velocity, acceleration)

absolute or relative

TRANSDUCER

(sensitivity)

Electrical quantity

(voltage, current, charge)

Specifications• Measurement Range (e.g. 0 to 10 g,

-1 mm to +1mm)

• Frequency Range (e.g. 1hz to 1,5 kHz)

• Sensitivity (e.g. 10 pC/g, 300 mV/(cm/s))

• Transverse Sensitivity

• Environmental:

- Magnetic Sensitivity (e.g. 0,1 g/tesla)

- Acoustic Sensitivity (e.g. 0,01 pC/db)

- Temperature Transient Sensitivity

• Physical

- Weight, Dimensions

• Mounting Techniques

48

Vibration Transducer Basics

Governing equation of motion

)()()( tfyxkyxcxm =−+−+

In terms of relative displacement

)()( tRymtfzkzczm =−=++

The forced response to harmonic excitation applied to the body (with f(t)=0) is written in the form

tieYty ω=)(

titi

eZcimk

eYmtz ωω

ωωω =

+−= 2

2

)(

( )( ) ( )2

022

02

20

21 ωωζωωωω+−

=YZ

20

20

12tan

ωωωωζφ

−=

amplitude phase

)(tx)(tz

)(ty

m

k

c

49

Displacement transducer

(sismographs)

yzYZ

−=⇒⎪⎩

⎪⎨

⎧

→

∀→>

°180

1,4

0 φ

ζωω have we For

When , the amplitude peak disappears 707,0=ζ

50

Design requirement: <<<0ω

k low

m highTransducer is relatively large and heavy

Example: LVDT (Linear Variable Differential Transformer)

Frequency Range: < 500 Hz

51

Velocity Transducer (same working principles as for the displacement transducer)

Practical example

Advantages:

- reliability, robustness (no power supply needed)

- high sensitivity (e.g. 300 mV/(cm/s))

- low impedance (no conditioning amplifier needed)

Drawbacks: - heavy and large, frequency range < 2000 Hz

tU

te

∂∂=

∂∂−= λφ

Voltage Magnetic flux

52

Accelerometers

titi

eZcimk

eYmtz ωω

ωωω =

+−= 2

2

)(

( ) ( )20

220

22

20

21

1

ωωζωωωω

+−=

YZ

20

20

12tan

ωωωωζφ

−=

Amplitude

Phase

System response:

53

Measurement of the acceleration of the support

( )YkmYZ 2

20

2

0

,0 ωω

ωωω −=−→→ get we As

Acceleration of the supportAccelerometer

sensitivity

Design requirement: >>>0ω

k high

m small

Transducer is small and light

Sensitivity seismic mass÷

Optimise the choice for any

given application

( ) ( )20

220

22

20

21

1

ωωζωωωω

+−=

YZ

Amplitude:

54

Typical examples of piezoelectric accelerometers

yQ 1λ=

Sensitivity in pC/g

55

Advantages:

- high frequency range

- excellent linearity

- low sensitivity to environmental noises

- small size, lightness

Drawbacks:

- very high impedance

charge amplifier

Selection of accelerometers

56

Attachment and location of accelerometers

57

Force Transducers Impedance Heads

Combination of 2 transducers: force + acceleration

58

Laser Transducers

• Laser Doppler velocimeter (LDV)

− Characteristics for a typical device:

− Frequency range: 0-250 kHz

− Vibration velocity range: 0.01 - 20000 mm/s

− Target distance: 0.2 - 30 m

− Signal/noise: excellent (depends on target, type of scan and measurement range)

− Sensitivity: 1 - 1000 mm/s/V

• Holographic interferometry (ESPI, DSPI)

59Base excitation of mechanical systems

General case

)(2 tq

Partition of the DOF :

• n1 free displacements q1

• n2 imposed displacements q2

⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡)(22

1

2221

1211

2

1

2221

1211

tr0

qq

KKKK

qq

MMMM

Equations of motion

60

The response takes the form

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ −=

−

2

1121

11

0)(

qy

IKKI

q t

)()()( 212111 tt qMSMg +−=

(equivalent load)

= S (static condensation matrix at foundation level)

)(1111111 tgyKyM =+

Dynamic part of the response solution of

61

System submitted to global support acceleration

)(tϕu

)(2

11

2

1 tϕ⎭⎬⎫

⎩⎨⎧+

⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧=

uu

0y

qq

q

where ( ) )()( 2121111

2121

111

tt ϕuMuMguKKu

+−=−= −

Method of additional masses

⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+ )(2

1

2221

1211

2

10222221

1211

tf0

qq

KKKK

qq

MMMMM

Modification of the equations of motion

with M22 of high amplitude level 2022)( qMf ≅t

62

Effective modal masses

Governing motion equation for the unrestrained DOF

)(1111111 tgyKyM =+ )()()( 212111 tt qMSMg +−=

Normal equations

212112 )( qMSMXηηΩ +−=+ T

with

Γ = modal participation matrix of dimension n1 x n2

Concept of effective modal mass : Γ Γ T

( )∑

=

=−1

1

2n

j j

jiST mm

μeΓ

Total mass Mass associated with the support

Effective mass of mode j in direction i

63Digital Signal Processing

Different types of signals

64

Basic Theory of Fourier Analysis

A function x(t), periodic in time T, can be written in an infinite series of sinusoids:

( )( ( ))tnbtnaatx nn

n ωω sincos2

)(1

0 ++= ∑∞

=

( )Tπω 2=with

where ( )

( ) ),2,1(sin)(2

),2,1,0(cos)(2

0

0

==

==

∫

∫ndttntx

Tb

ndttntxT

a

T

n

T

n

ω

ω

Alternative forms

∑∞

−∞=

=n

tnin eXtx ω)( ∫ −=

T tnin dtetx

TX

0)(1 ωwhere

65

The Fourier Transform

A nonperiodic function x(t) which satisfies the condition

( ) ( )( ) ωωωωω dtBtAtx sin)(cos)()( += ∫∞

∞−

where ( )

( ) dtttxB

dtttxA

∫

∫∞

∞−

∞

∞−

=

=

ωπ

ω

ωπ

ω

sin)(1)(

cos)(1)(

Alternative form

ωω ω deXtx ti)()( ∫∞

∞−= ∫

∞

∞−

−= dtetxX ti ω

πω )(

21)(where

∞<∫∞

∞−dttx )(

can be represented by the integral

66

The Discrete Fourier Transform (DFT)

Let us consider a time signal which is digitised (by an A-D converter)

tΔ = time resolution

tf s Δ

= 1= sampling rate

tNT Δ= = sample length

Nf

Tf s==Δ 1

= frequency resolution

The Fourier series of the original signal (periodicity assumption) can be written:

∑∞

−∞=

=n

Ttni

n eXtxπ2

)( ∫−

=T

Ttni

n dtetxT

X0

2

)(1 π

with

1x2x

3x kx

)(tx

Nx

tNT Δ=

tk Δ t

67

If x(t) is discretised at evenly spaced time intervals , we can write at a particular time :tkti Δ=

∫−

=T

Ttni

n dtetxT

X0

2

)(1 π

Nkni

kT

tkniT

tni

k exetkxetxk πππ 222

)()(−Δ−−

=Δ=

In this case, the integral

is replaced by textN

X Nkni

k

N

kn Δ

Δ=

−

=∑

π2

1

1

The (complex) DFT is defined by

),,1(1 2

1

NnexN

X Nkni

k

N

kn ==

−

=∑

π

tΔ

68

The calculation of the DFT of a signal using the definition

requires N 2 complex multiplications (and additions)

time-consuming operation.

The basis of the FFT algorithm of radix 2 (N=2r where r is an integral number) is the partitioning of the full sequence of sample values into a number of shorter sequences:

The Fast Fourier Transform (FFT)

)1,,0(1 21

0

−==−−

=∑ Nnex

NX N

kni

k

N

kn

π

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

+−

+

−

=

−−

=∑∑ N

kni

k

N

k

Nkni

k

N

kn exex

NX

)12(2

12

12

0

22

2

12

0

1 ππ

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−

+

−

=

−−−

=∑∑ 2

2

12

12

0

22

2

2

12

021

21 N

kni

k

N

k

NniN

kni

k

N

kn exeex

NX

πππ

69

( )

( ) ( ))1

2,,0(

)()(21

2

)()(21)(

−=⎪⎭

⎪⎬

⎫

−=+

+= NnnXWnXNnX

nXWnXnX

zn

y

zn

y

where Ni

eWπ2−

=

Thus the full FFT computation scheme becomes

Example of partitioning : N=8 samples

x0 x1 x2 x3 x4 x5 x6 x7x0 x2 x4 x6 x1 x3 x5 x7x0 x4 x2 x6 x1 x5 x3 x7x0 x4 x2 x6 x1 x5 x3 x7

1x 2x 3x

4x

5x 6x 7x0x

1x 3x2x

4x

6x0x

1x4x

0x

2x

6x

3x

7x

5x7x

5x

70

Random Vibration

Consider a random process represented by an infinite set of samples

∑=∞→

=n

k

k

nx txn

t1

1)(

1 )(1lim)(μ

)()(1lim),( 1)(

11

)(11 ττ +=+ ∑

=∞→txtx

nttR k

n

k

k

nxx

∫−∞→=

2

2

22 )(1limT

TTdttx

TXRoot Mean Square Value

Definitions

Mean Value

Autocorrelation Function

71

Classes of random process

Random Process

Steady-stateNon steady-state

Ergodic(is a type of random process which requires only a single

sample, albeit of considerable length, to portray all the

statistical properties required for its definition)

Non ergodic

)(),()(

11

1

ττμμ

xxxx

xx

RttRt

=+= independent of

t1

72

For an ergodic process, it follows that the Autocorrelation function is defined as the ‘expected’ (or average) value of the product (x(t).x(t+τ))

2)0( XRxx =

( ) ( ) ( )[ ]ττ += txtxERxx .

It can been seen that:

73

Power Spectral Density (PSD)

The Auto- or Power Spectral Density (PSD), Sxx(ω), of a signal x(t)is defined as the Fourier Transform of the Autocorrelation Function i.e.

∫∞

∞−

−= ττπ

ω τω deRS ixxxx )(

21)(

The Auto-Spectral Density is a real and even function of frequency, and does in fact provide a description of the frequency composition of the original signal. For an acceleration signal for example, it has units of (m/s2)2/(rad/s).

The inverse transformation gives:

∫∞

∞−= ωωτ τω deSR i

xxxx )()(

Thus, it comes: 2)()0( XdSR xxxx == ∫∞

∞−ωω

74

A similar concept can be applied to a pair of functions such as x(t)and f(t) to produce cross correlation and cross spectral density functions.

The cross correlation function, Rxf (t) , is defined as:

∫∞

∞−

−= ττπ

ω τω deRS ixfxf )(

21)(

( ) ( ) ( )[ ]ττ += tftxERxf .

and the cross spectral density is defined as its Fourier Transform:

75

Time Domain Autocorrelation function Power Spectral Density

sinus

Narrow bandprocess

Large bandprocess

White noise

76

System Response - Time / Frequency Duality

)()()( ωωω FHX =

∫ −=t

dthftx0

)()()( τττ

)(ωH

TransfertFunction (FRF)

)(th

Impulse Response)(tf

)(ωF

In the frequency domain, we can establish the following relations:

)()()()()()(

)()()( 2

ωωωωωω

ωωω

xfxx

fffx

ffxx

SHSSHS

SHS

=

=

=

Input Output

Property of the cross power spectral densities:

)()( * ωω fxxf SS =

77

Some Features of Digital Fourier Analysis

There are a number of features of digital Fourier analysis which, if not properly treated, can give rise to erroneous results. These are generally the result of the discretisation approximation and of the need to limit the length of the time history.

Specific features are:

• aliasing,

• leakage,

• windowing,

• filtering,

• zooming,

• averaging.

78

Aliasing

Aliasing results from the discretisation of the originally continuous time history.

High-frequency signal: the existence of high frequencies is misinterpreted if the sampling rate is too slow.

Thus, a signal of frequency ω and one of (ω s - ω) (where ω s is the sampling frequency) are indistinguishable when represented as a discretised time history.

Low-frequency signal

79

Alias distortion of spectrum by DFT

(a) True spectrum of signal

(b) Indicated spectrum from DFT: frequencies higher than ωS / 2are “folded back”

Nyquist sampling theorem

2sωω ≤ ==

2maxsωω Nyquist (cut-off) frequency

Practically, one often chooses : ωω 56.2=s

80

Solution : use an anti-aliasing filter which subjects the original time signal to a low-pass, sharp cut-off filter.

True spectrum of signal

Low-pass filter

Indicated spectrum from DFT

Because the filters are not perfect, it is necessary to reject the spectral measurements in a frequency range approaching the Nyquist frequency. It is for this reason that a 1024-point transform results in a 512-line spectrum and that only the first 400 lines are given on the analyser display.

81

Leakage

Leakage results from the need to take only a finite length of time history coupled with the assumption of periodicity.

(a) Ideal signal

(b) ‘Awkward’ signal

(a) The signal is perfectly periodic in the time window T(b) The periodicity assumption is not strictly valid (discontinuity

at the end of the sample)

82

Time domain Frequency domain

83

Windowing

)()()(' twtxtx =Analysed signal Window

)()()(' ωωω WXX ⊗=Convolution product

(a) Boxcar

(b) Hanning

(c) Cosine-taper

(d) Exponential

Different types of window

84

Frequency spectra of different windows

85

Filtering

Filtering is another signal conditioning process rather like windowing, except that it is applied in the frequency domain rather that the time domain.

)()()(' ωωω WXX = )()()(' twtxtx ⊗=

86

Improving Resolution

Inadequate frequency resolution (e.g. especially for lightly-damped systems) arises because of the constraints imposed by the DFT process i.e.

• the limited number of discrete points available (N);

• the maximum frequency range to be covered (ωs/2) and/or the length of time sample available.

Solutions to this problem are:

• Increasing transform size (but it may be counterproductive to increase the fineness of the spectrum overall)

• Zero padding

• Zoom

87

Results of using zero-padding

Zero padding: It consists in adding a series of zeroes to the short sample of actual data, to increase artificially the length of the sample.

(a) DFT of data between 0and T1

(b) DFT of data padded to T2

(c) DFT of full record 0 to T2.

It reveals the presence of 2 frequency components in the signal !

88

A method consists to apply a band-pass filter to the signal and to perform a DFT between 0 and

Zoom

The objective is to obtain a finer frequency resolution by concentrating all the spectral lines (400, 800, …) on the frequency range of interest (between f1 and f2 rather than between 0 and f2).

(a) Spectrum of the signal to be analysed

(b) Band-pass filter

( )12 ωω −

89

Then because of the aliasing phenomenon, the frequency components between f1 and f2 will appear aliased in the analysis range 0 to (f2-f1) with the advantage of a finer resolution.

In this example, the resolution is 4 times finer than in the original baseband analysis.

Effective frequency translation for zoom

90

Other methods to achieve a zoom measurement are based on effectively shifting the frequency origin by multiplying the original time history by a function and then filtering the higher of the two components thus produced.

Example : suppose the signal to be analysed is:

Multiplying this by yields:

If we filter out the second component, we are left with the original signal translated down the frequency range by . In this method, it is clear that sample times are multiplied by the zoom magnification factor.

( )t1cos ω

( )tAtx ωsin)( =( )t1cos ω

( ) ( ) ( ) ( )( )ttAttAtx 111 sinsin2

cossin)(' ωωωωωω ++−==

1ω

91

Averaging

Need to enhance the quality of the measurements in terms of reliability and accuracy. A high number of samples allows to remove spurious random noise from the signals.

There are several options which can be selected when setting an analyser into average mode (exponential, linear).

Different interpretations of multi-sample averaging

(a) Sequential

(b) Overlap

92Use of Different Excitation Signals

Classes of signal used for modal testing

Periodic excitation

• stepped sine

• slow sine sweep

• periodic

• pseudo-random

• periodic random

Transient excitation

• burst sine

• burst random

• chirp

• impulse

Random excitation

• (true) random

• white noise

• narrow-band random

93

Periodic Excitation

• Stepped-sine Testing

Features :

- Step-by-step variation of the frequency of the excitation signal;

- Need to ensure steady-state conditions before the measurements are made;

- The extent of the unwanted transient response will depend on:

the proximity of a natural frequency,

the lightness of the damping,

the abruptness of the changeover.

long measurement time.

Example of rapid coarse sweep with a large frequency increment, followed by a set of small fine sweeps localised around the resonances of interest.

94

• Slow Sine Sweep Testing

(= the traditional method of FRF measurement)

Features :

- Slow and continuous variation of the excitation frequency;

- Slow sweep rate to achieve steady-state conditions;

Prescriptions of the ISO Standard:

- Linear sweep:

- Logarithmic sweep:

Distorting effect of sweep rate

22max 216 rrS ζω< Hz/min

22max 310 rrS ζω< Hz/min

95

• Periodic, Pseudo-Random and Periodic Random Excitation

Features :

- Superposition of sinusoids;

or

- Generation of a random mixture of amplitudes and phases for the various frequency components and repetition of this sequence for several successive cycles.

Optimisation of the energy in the frequency range of interest.

Advantage:

- Exact periodicity of the excitation resulting in zero leakage errors

no need to use a window.

96

Random Excitation

FRF estimates using random excitation

The principle upon which the FRF is determined using random excitation relies on the following relations:

)()()()()()(

)()()( 2

ωωωωωω

ωωω

xfxx

fffx

ffxx

SHSSHS

SHS

=

=

=rather than )()()( ωωω FHX =

where Sxx (ω) , Sff (ω), Sxf (ω) are the autospectra of the response and excitation signals and the cross spectrum between these two signals, respectively.

97

The spectrum analyser has the facility to estimate these various Power Spectral Densities (PSD). Two estimates for the FRF can becomputed using these equations. We shall denote them:

Because we have used different quantities for these two estimates, we must be prepared for the eventuality that they are not identical (as, according to theory, they should be) and to this end we shall introduce the coherence function which is defined by:

)()(

)(1 ωω

ωff

fx

SS

H =)()()(2 ω

ωωxf

xx

SSH =and

)()(

2

12

ωωγ

HH= which can be shown to be always less or equal to 1.

98

Noisy Data

)(txStructure)(tf

Noise Noise

)(tm)(tnMeasured Excitation

)(~ tx)(~ tf

Measured Response

Measured excitation and response signals:

)()()(~ tntftf += )()()(~ tmtxtx +=

Assumption: m(t) and n(t) are not correlated with x(t) and f(t).0)()()( === ωωω xmfnnm SSS

Thus, it follows:

)()()(

)()()(

~~

~~

ωωω

ωωω

mmxxxx

nnffff

SSS

SSS

+=

+=)()(~~ ωω fxxf SS =and

99

)()()(

)(1 ωωω

ωnnff

fx

SSS

H+

=

)()()()(2 ω

ωωωxf

mmxx

SSSH +=

FRF estimates:

is used when noise degrades the response signal (better indicator near antiresonance)

is used when noise degrades the force signal (better indicator near resonance)

General Case: )()()( 21 ωωω HHH ≤≤

ω

)(1 ωH

)(2 ωHAlternative:

)()()( 21 ωωω HHHV =

)(ωH

100

Coherence estimation:

( ) ( )

⎟⎠

⎞⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

++=

)()(1

)()(1

)(

)()()(

)()()(

2

2~~

ωω

ωω

ωγ

ωωω

ωωω

γ

xx

mm

ff

nn

fx

mmxx

xf

nnff

fxxf

SS

SS

SSS

SSS

)()(

2

12

ωωγ

HH=

12~~ ≤xfγ

Cases when

• There may be noise on one or other of the signals;

• The dynamic behaviour of the structure is nonlinear;

• The structure is excited by a non-recorded source of excitation (e.g. lateral coupling between the structure and the shaker).

12~~ ≤xfγ

101

Example of FRF measurement using random excitation

Coherence drop due to noise in the

response signal

Coherence drop due to noise in the excitation signal

102

Example of application

103

Transient Excitation

There are 3 types of transient excitation signals.

• Burst excitation signals

It consists of short sections of an underlying continuous signal(which may be a sine wave, a sine sweep or a random signal) followed by a period of zero input.

The duration of the burst is selected to allow the damping out of the response by the end of the measurement period (to avoid leakage errors).

104

• Rapid sine sweep or ‘chirp’

(short duration signal)

Good control of both the amplitude and the frequency content of the signal.

105

• Impulsive excitation

One practical difficulty encountered when using the hammer excitation is that of the double-hit (rebound).

- Impact from a hammer blow

- Impact controlled by a shaker attached to the structure

106

(a) Excitation pulse clearly satisfies the assumption

(b) The response history also satisfies the assumption

(c) The response history does not satisfy the assumption (for the period T selected)

(d) Exponential window

Periodicity assumption

107

(e) Results processed by exponential window

Impulsive excitation as pseudo-periodic function

108

Types of excitation

No single method is the ‘best’ and it is probably worth making use of several types in order to optimise time, effort and accuracy.

109Modal Identification Methods

1. FRF measurements

(e.g. 800 lines / spectrum)

2. Modal parameter extraction

data reduction

(e.g. 6 modes x 4 values = 24 values/spectrum)

3. Derivation of a mathematical model

(Spatial model, Modal Model)

( )∑= +−

=N

k kkk

krsrs i

AH

122 2

)(ωωζωω

ω

jkα

jkα

110

Classification of modal identification methods

Frequency Domain

(FRF’s)

Time Domain

(IRF’s or response histories)

SDOF methods MDOF methods

Single-FRF methods Multi-FRF methods

• modelling of damping effects,

• model order.

Difficulties :

111

SDOF Modal Identification Methods

SDOF means that just one resonance is considered at a time

Assumption: the modes must appear clearly separated so that theycan be analysed sequentially, one after the other.

( )( )krs

kkk

krsrs B

i

AH +

+−≅

ωωζωωω

2)( 22

( ) ( ) ( )∑∑≠== +−

++−

=+−

=N

kjj jjj

jrs

kkk

krsN

k kkk

krsrs i

A

i

A

i

AH

12222

122 222

)(ωωζωωωωζωωωωζωω

ω

!

rsH

( )333 ,, rsAζω

( )222 ,, rsAζω

( )111 ,, rsAζω

12

3

rsH

rsH

112

a) Peak-Amplitude Method (Peak-Picking)

Estimation of the damping ratio using the quality factor:

kba

kQζωω

ω2

1=−

=

rsH

rsH

kω aω bω

Half-Power points

2rsα

kω

113

b) Circle-Fit Method

Circle-fitting FRF plot in the vicinity of resonance

( ))2/tan()2/tan(2

22

bak

bak θθω

ωωζ+

−=

rωaω

bω

Effect of residual terms

aω

bω

bθaθ

rζ

detection of nonlinear behaviour

114

Concept of residual terms

To take into account the influence of modes which exist outside the frequency range of measurement and/or analysis, any FRF coefficient may be rewritten as:

rrr

jkrN

mr

m

mr

m

r

N

r rrr

jkrjk

iA

iA

ωωζωω

ωωζωωωα

2

2)(

221

1

1

122

2

2

1

1

+−⎟⎟⎠

⎞⎜⎜⎝

⎛++=

+−=

∑∑∑

∑

+==

−

=

=

High-frequency modes

Low-frequency modesModes actually identified

115

Contributions of low-, medium- and high frequency modes

( )Rrs

m

mk kkk

krsRrs

rs Ki

A

MH 1

21)(

2

1

222 ++−

+−= ∑= ωωζωωω

ω

Residual mass Residual stiffness

rsH

Rec

epta

nce

(log-

log

scal

e)

rsH

rsH

116

Examples of regenerated FRF’s

without residuals

with residuals

117

MDOF Modal Identification Methods

a) Least Squares Complex Exponential (LSCE)

Features:

• Time-domain technique

• Global estimation of natural frequency and damping of several modes simultaneously

Theoretical basis:

The general expression used for the FRF of a damped system in terms of modal parameters is:

( ) ( ) ( ) ( )

( ) ( )

)()(

)()()(

*

*

1

**

**

1

k

krsN

k k

krs

kk

kskrN

k kk

kskrrs

iA

iA

izz

izz

H

λωλω

λωρλωρω

−+

−=

−+

−=

∑

∑

=

=

118

Taking the inverse Fourier Transform, the expression for an Impulse Response Function (IRF) in terms of modal parameters writes:

( ) ( ) ( ) ⎟⎠

⎞⎜⎝

⎛=+= ∑∑==

N

k

tkrs

tkrs

N

k

tkrsrs

kkk eAeAeAth1

*

1Re2)(

* λλλ

Time sampling: tjt Δ=

( ) ⎟⎠

⎞⎜⎝

⎛=Δ ∑=

N

k

jkkrsrs ZAtjh

1Re2)(

tk

keZ Δ= λ defines the natural frequency and damping ratio of mode k (= global estimate)

( )krsA defines the displacement of point r in mode k (= local estimate)

119

The unknowns Zk may be considered as the roots of a polynomial of order 2N:

( ) ∑∏=

≠−=

=−=N

p

pp

N

kNk

k ZZZZP2

00

)( α

where the coefficients αp have to be estimated using data measured on the system.

( ) ⎟⎠

⎞⎜⎝

⎛=Δ ∑=

N

k

jkkrsrs ZAtjh

1Re2)(

For this purpose, let us consider the expression of the IRF

120

The IRF at a series of 2N equally spaced time intervals is:

( )

( )

( ) ⎟⎠

⎞⎜⎝

⎛=Δ×

⎟⎠

⎞⎜⎝

⎛=Δ×

⎟⎠

⎞⎜⎝

⎛=×

∑

∑

∑

=

=

=

N

k

NkkrsrsN

N

k

pkkrsrsp

N

kkkrsrs

ZAtNh

ZAtph

ZAh

1

22

1

1

00

Re2)2(

Re2)(

Re2)0(

α

α

α

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ ∑ ∑∑

= ==

N

k

pk

N

ppkrsrs

N

pp ZAtph

1

2

0

2

0Re2)( αα

= 0 from the definition

0)(2

0=Δ∑

=

tphrs

N

ppα

121

)2()(12

0tNhtph rsrs

N

pp Δ−=Δ∑

−

=

α

It follows that:

To estimate the coefficients αp in a least squares sense, let express this equation:

( )( ) ( )( )tNhtph rsrs

N

pp Δ+−=Δ+∑

−

=

212

0α

• for all possible time points:

• for all possible measured responses:

• for all possible excitation points: ),,1(),,1(

),,1,0()2(

i

o

t

nsnr

ntN

……

…

==

=Δ+

122

Thus the matrix:

( ) ( ) ( )( )( ) ( ) ( )

( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

( ) ( )( ) ( )( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

Δ−+Δ+Δ

Δ+−Δ+Δ

Δ−+Δ+Δ

ΔΔΔΔ−Δ

tNnhtnhtnh

tNhthth

tNnhtnhtnh

tNhththtNhthh

tnntnntnn

rsrsrs

ttt

ioioio121

121

121

22120

111111

111111

111111

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−12

1

0

Nα

αα ( )

( )( )

( )( )

( )( )

( )( )⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Δ+−

Δ+−

Δ+−

Δ+−Δ−

tnNh

tNh

tnNh

tNhtNh

tnn

rs

t

io2

2

2

122

11

11

11

Overdetermined system: A x = b

which has the solution: ( ) bAAAx TT 1−=

123

The solution:

allows to reconstruct the polynomial:

and the roots:

gives the complex eigenvalues:

If the eigenvalues may be further expressed as follows:

we obtain the values of the natural frequency ωk and the damping

ratio ζk (in the assumption of proportional damping).

1210 −= NT αααx

∑=

=N

p

pp ZZP

2

0

)( αt

kkeZ Δ= λ

),,2,1( Nkk …=λ

( )21 kkkk i ζζωλ −+−=

124

Selection of the model order

By using a different number of poles (2N) and comparing the error between the regenerated FRF’s and the original measured data, it is possible to draw a so-called ‘stabilisation diagram’:

As the number of poles increases, a number of computational modes are created in addition to the genuine physical modes which are of interest.

125

b) Least Squares Frequency Domain (LSFD)

Features:

• Frequency domain technique

• Global estimation of residues (--> complex modes)

Theoretical basis:

FRF of a damped system in the frequency band [ωmin, ωmax]:

( ) ( )Rrsk

krsm

mk k

krsRrs

rs KiA

iA

MH 1

)()(1)( *

*

2

2

1

+⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

−+−= ∑

= λωλωωω

126

( ) ( )

⎪⎭

⎪⎬⎫∗

Rrs

Rrs

krskrs

MK

AA

,

, are local estimates (which depend on the locations of the measurement point (r) and of the excitation point (s)

Receptance diagram (log-log scale) rsH R

rsK1

RrsM2

1ω

Lower residual

Upper residual

( ) ( )Rrsk

krsm

mk k

krsRrs

rs KiA

iA

MH 1

)()(1)( *

*

2

2

1

+⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

−+−= ∑

= λωλωωω

127

Calculation of the residues and( )krsA ( )∗

krsAAssumption: the poles λk have been identified (LSCE method).

( ) ( ) ( )

( ) ( ) ( )⎪⎩

⎪⎨⎧

−=

+=∗

krskrskrs

krskrskrs

ViUA

ViUAWriting:

the FRF takes the form:

( ) ( ) Rrs

kkrs

m

mkkkrsR

rsrs K

QVPUM

H 1)()(1)(2

1

2 +⎟⎟⎠

⎞⎜⎜⎝

⎛++−= ∑

=

ωωω

ω

(2 m+2) unknowns with m = m2 - m1

We define the error as: ( ) 2)(

max

min

ωωω

ωω

EXPrsrs HHE −= ∑

=

Theoretical value Measured data

128

The error is minimised by differentiating its expression with respect to each unknowns in turn,

( ) ( )

0;0

),,(;0;0 21

=∂∂=

∂∂

==∂

∂=∂

∂

rsrs

krskrs

ME

KE

mmkV

EU

E …

thus generating a set of (2 m+2) equations with (2 m+2) unknowns (m = m2-m1).

129

Extraction of the (complex) eigenmodes.

For each identified pole k (k = 1,.., m) , we have:

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )ksrnksn

kskskskss

kskks

kskksk

zzA

zzzA

zzAzzA

oo==

===

=

=

2

22

11,λ

( ) ( )kssks Az =

and thus:

( )( )

( )),,,1( srnr

zA

z oks

krskr ≠== …

A transducer should not be located where zs(k) = 0 (on a vibration node)

130

( )

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

iooo

i

i

nnsnn

rn

n

rs

s

r

HHH

H

H

H

H

H

H

1

11

1

11

ωH « roving hammer »

In practice:

131

Frequency band Partial fitting

As modes 1, 2, 3 have been identified (poles and residues), their participation in the frequency range [ωmin, ωmax] can be subtracted mathematically before identifying modes 4, 5 et 6.

FRF data associated with a bad coherence are not considered for identification

Used data

Rejected data

Remarks

rsH rsH

132

c) Ibrahim Time Domain Method (ITD, 1973-77)

Features :

• Time-domain method

• Global estimation of poles and residues

Theoretical basis:

The free response of a viscously-damped system is given by:

∑=

=n

k

tk

ket2

1)()( λzqFree response

vector

complex eigenvector z (k)(unscaled mode)

Number of modes in the response

133

Considering 2n time instants, the free response matrix is given by:

( ) ( )[ ] ( ) ( )

1 1 1 2

2 1 2 2

1 2 1 2

n

n n n

t t

n nt t

e et t

e e

λ λ

λ λ

⎡ ⎤⎢ ⎥

⎡ ⎤= ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

q q z z… …

2n n×Q

2n n×Z

2 2n n×Λ=

134

( ) ( )[ ] ( ) ( )

1 1 1 2

2 1 2 2

1 2 1 2

n

n n n

t t

n nt t

e eˆ ˆt t t t

e e

λ λ

λ λ

⎡ ⎤⎢ ⎥

⎡ ⎤+ Δ + Δ = ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

q q z z… …

Considering a second set of samples shifted by an interval of time Δtwith respect to the first set:

( ) ( )( )

( )2n 2nt t ttk j k jk

j k kk 1 k 1

t t e e eλ λλ+Δ Δ

= =+ Δ = =∑ ∑q z z

tΔQ Z Λ

tˆ

Δ =Q Z Λ

135

( ) ( )[ ] ( ) ( )

1 1 1 2

2 1 2 2

1 2 1 22 2

n

n n n

t t

n nt t

e eˆ ˆˆ ˆt t t t

e e

λ λ

λ λ

⎡ ⎤⎢ ⎥

⎡ ⎤+ Δ + Δ = ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

q q z z… …

In the same way, for a time shift of 2 Δt

with ( ) ( )2 i t

i iˆ e λ Δ=z z

2 tˆ

Δ =Q Z Λ

136

Thus, we have:

2

ou

ou

t

t

t

ˆ

ˆˆˆ

ˆ

Δ

Δ

Δ

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎡ ⎤⎡ ⎤

= =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

Q ZΛ Φ A Λ

Q Z

ZQΛ Φ A ΛQ Z

and

Eliminating Λ gives:

1i i

ˆ ˆ− =ΦΦ a a

1 ˆˆ − =ΦΦ A A

If we define:

We have:1 i t

i iˆ eλ Δ− =ΦΦ a a

Eigenvalue problem

The eigenvectors ai have their first n co-ordinates equal to the eigenmodes of the structure and λi are the complex eigenvalues.

or

or

[ ] [ ]nn aaAaaA ˆˆˆ11 …… == and

137

( ) 1i tT T

i iˆ eλ− Δ⎡ ⎤ =⎣ ⎦Φ Φ Φ Φ a a

If the number of time samples is greater than the number of modes excited, then matrices Φ are rectangular and the eigenvalue problem is transformed into:

138

Since the order of matrix may exceed the number of modes excited, it remains to separate the structural modes from the numerical modes.

For this purpose, let us write the response of the real stationsdelayed by a time-interval Δτ : i.e.

Modal Confidence Factor

( ) 1T Tˆ −⎡ ⎤⎣ ⎦Φ Φ Φ Φ

( ) ( )( )

( ) ( )2 2 2

1 1 1

k k k kn n n

t t tk k k

k k k' t e e e ' eλ τ λ τ λ λ+Δ Δ

= = == = =∑ ∑ ∑q z z z

( ) ( )t ' tτ+ Δ =q q

If zi(k) is the i th deflection of the k th mode at the real station, then the value at the assumed station delayed by Δτ is expected to be:

( ) ( )τλ Δ= kekiki zz expected,

139

The Modal Confidence Factor (MCF) is defined as:

and should be near 1.( )

( )ki

ki

',

zz expectedMCF =

140

d) Stochastic Subspace Identification Method

Features:

• Time-domain method

• Global estimation of poles and residues

Theoretical background: Stochastic model in the state-space.

Given a n-dimensional time series q[k] = q(tk), the dynamic behavior of the system may be described by the state space model

[ ] [ ] [ ][ ] [ ] [ ]1k k kk k k

+r = A r + wq = B x + v

State matrix

Output matrix

State vector

Process noise

Assumption: w and v are zero-mean Gaussian white noise.

Measurement noise

141

The covariance matrices of w and v are:

[ ][ ] [ ] [ ]( ) ( )T Tk

E k t k t tk

δ⎡ ⎤⎛ ⎞ ⎛ ⎞+ + =⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠⎣ ⎦

w Q Sw v

v S R

The output covariance matrices are defined as

Expectation

[ ] [ ]

[ ] [ ]

0T

Ti

E k k

E k i k

⎡ ⎤= ⎣ ⎦⎡ ⎤= +⎣ ⎦

Λ q q

Λ q q

Covariance matrices

142

As w[k] and v[k] are independent of the state vector r[k], the following properties can be established:

[ ] [ ]

[ ] [ ]

0

0

0 01

1

T T

T T

T

ii

E k k

E k k

−

⎡ ⎤ = +⎣ ⎦⎡ ⎤+ = = +⎣ ⎦

= +

=

r r A Σ A Q

r r G A Σ B S

Λ B Σ B R

Λ B A G

where Σ0 is the state covariance matrix and G is referred as the next state-output covariance matrix.

143

Covariance-Driven Stochastic Subspace Method

Let us construct the Hankel matrix with p block rows and q block columns of the output covariance matrix Λi

1 2

2 3 1

1 1

q

qp,q

p p p q

+

+ + +

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Λ Λ Λ

Λ Λ ΛH

Λ Λ Λ

( )q p≥

The pth-order observability and controllability matrices are respectively:

1

1

Tpp

qq

−

−

⎡ ⎤= ⎣ ⎦⎡ ⎤= ⎣ ⎦

O B B A B A

C G A G A G

144

p,q p q=H O C

1ii

−=Λ B A GThe previous relationship

leads to the following factorisation property

Let W1 and W2 be two user-defined invertible weighting matrices

of dimension p x m and q x m respectively. Performing the following singular value decomposition gives:

[ ] [ ]1 , 2 1 2 1 2 11

1 10

0 0T T

p q⎡ ⎤= =⎢ ⎥⎣ ⎦

W H W U U U SS

V V V

S1 contains 2 Nm non-zero singular values where Nm is the number of system modes.

145

It follows that the observability and controllability matrices can be recovered as

1 21 1

1 211

p

Tq

=

=

O U S

C S V

Let us write

2 11

Tpp p↑ −

−⎡ ⎤= =⎣ ⎦O B A B A B A O A

may be determined by making use of Op matrix

Matrices A and B then the eigenvalues and eigenvectors of the system may be easily computed.

146

Comparison and Correlation Techniques

• Visual inspection experimental / calculated mode-shapes.

• Visual inspection of regenerated / predicted FRFs.

• Use of mathematical tools.

Let us note:

xX an experimental (complex) eigenvector;

xA an analytical eigenvector (predicted by a model);

147Modal Scale Factor (MSF)

TX ATA A

MSF = x xx x

orTX ATX X

MSF = x xx x

depending upon which mode-shape is taken as the reference one.

148

Graphical representations

Modal Assurance Criterion (MAC)

( )10 ≤≤ MAC)()(

2

ATAX

TX

ATX

MACxxxx

xx=

149

Causes of bad correlation:

• presence of non-linearities, noisy data

• bad modal identification

• inappropriate choice of measurement coordinates

• modelling uncertainties (physical parameters, geometry)

• modelling errors (FEM)