identification methods for structural systems · 2014-11-25 · modal id methods as already...

$: Identification Methods for Structural Systems · 2014-11-25 · Modal ID Methods As already mentioned in Lecture 5 (slide 5), the FRF can be written in partial fraction expansion$
Identification Methods for Structural Systems

Prof. Dr. Eleni Chatzi

Lecture 9 - 23 April, 2013

Institute of Structural Engineering Identification Methods for Structural Systems 1

Identification Methods

The work is done either in

the frequency domain - Modal ID methodsusing the Frequency Response Function (FRF) information

or in the

time domain - Time Domain ID methodswhere an analytical or numerical (FE) model of the system orinformation from the Impulse Response Function (IRF) are used.

Reminder: The IRF is the inverse Fourier Transform of the FRF.


Identification Methods

Applications

Troubleshooting

Finite Element model updating

Structural Modification

Reduction of the Order of mathematical models

Response Prediction

Structural Damage Detection

Active Vibration Control


Modal ID Methods

Modal Id Methods

aim at determining the inherent dynamic characteristics of a systemin forms of natural frequencies, damping factors and mode shapes,and using them to formulate a mathematical model for its dynamicbehavior.The formulated mathematical model is referred to as modal modeland the information for the characteristics are known as its modaldata.

Modal analysis is based upon the fact that the vibration response ofa linear time-invariant (LTI) dynamic system can be expressed as thelinear combination of a set of simple harmonic motions called thenatural modes of vibration.This concept is akin to the use of a Fourier combination of sine andcosine waves to represent a complicated waveform.

source: Jimin He, Zhi-Fang Fu, “Modal Analysis”, Butterworth-Heinemann


Modal Analysis Techniques

1 Classification based on the number of input and outputlocations:

SISO - SIMO - MIMO - MISO

2 Based on the type of identified properties:

Direct Methods Indirect MethodsMechanical Modal

k,m,c (ωi , ζi , etc)


Modal ID Methods

Basic Concepts

Assume an N-degree of freedom system with damping:

Mx + Cx + Kx = F

x is the N × 1 state space vector of the system and M,C,K are thesystem mass, damping and stiffness matrices.

In steady state conditions (i.e. once the free response has dieddown), we showed that one can establish the relationship betweenthe complex amplitude of the response X and the input F = F0e

iωt

as:

X = H(iω) · F

where H is the FRF matrix


Modal ID Methods

As already mentioned in Lecture 5 (slide 5), the FRF can be written inpartial fraction expansion. (i.e poles/residues form):

H(ω) =N∑

k=1

Ak

iω − λk+

A∗k

iω − λ∗k

where Ak is the modal constant matrix with components kAij related tocomponents φik , φjk of eigenvector k through the formula:

kAij = qkφikφjk and qk is a modal participation factor

In addition, index k corresponds to the k-th mode and the poles of thedenominator are simply the roots of the characteristic polynomial:

λk , λ∗k = −ωkζk ± i

(ωk

√1− ζ2

k

)


Modal ID Methods

The Frequency Response Function

We already discussed that the component Hij of the FRF matrix, H,corresponds to a particular output response at point i due to aninput force at point j .

Based on this, Modal testing is an experimental technique used toderive the modal model of a LTI vibratory system.Combinations of excitation and response at different locations leadto a complete set of frequency response functions (FRFs) which canbe collectively represented by an FRF matrix of the system.This matrix is usually symmetric, reflecting the structural reciprocityof the system i.e. Hij = Hji .


Basic Concepts

The Impulse Response Function hij(ω), is the equivalent of Hij(ω) inthe time domain, hence it can be obtained using IFT and theproperty

F−1

{1

a + iω

}= e−at , t > 0

or hij(t) = (kAijeλk t +k A

∗ije

λ∗k t)

with λk = −ωrζk + iωk

√1− ζ2

k

Ultimately this can also be compactly written as:

hij(t) =2N∑k=1

kAijeλk t

(with a slightly different notation for kAij , λk)


Basic Concepts

Modal ID Fundamentals

Returnign to the definition of the FRF, since the matrices of residualsAk , are of rank one, meaning that they can be decomposed as:

Ak = ΦkΦTk =

Φ1k...

ΦNk

[ Φ1k · · · ΦNk

]Φk : eigenvector of mode k

then it follows that essentially, the FRF of a linear mdof system withN dofs is the sum of N sdof FRFs (modal superposition) and thetransfer function matrix is completely characterized by the modalparameters i.e the roots of the character polynomial λm and themode shape vectors Φm, m = 1, . . . , N


Basic Concepts

Mode Shapes and Operating Deflection Shapes

At or near the natural frequency of a mode, the overall vibrationshape will be dominated by the mode shape of the resonance.

Applying a harmonic force at one of the degrees of freedom, say r, ata frequency iωdn (damped frequency of the nth mode) results in adisplacement vector, X (ωdn), which is approximately proportional tothe mode shape vector of mode n, i.e Φn.

In reality there is always a small contribution of the other modes.

Within this context of properly exciting the structure, modal testingwas also known as resonance testing, as it initiated using variousshakers properly tuned in order to put the structure into resonance.


Modal Testing

The practice of modal testing involves measuring the FRFs or impulseresponses of a structure. The FRF measurement can simply be done byasserting a measured excitation at one location of the structure in theabsence of other excitations and measure vibration responses at one ormore location(s). The excitation can be of a selected frequency band,stepped sinusoid, transient, random or white noise. It is usually measuredby a force transducer at the driving point while the response is measured byaccelerometers or other probes.

Given input and output time histories the FRF can be calculated throughthe output/input Fourier Transform ratio:

Y (ω) = H(ω)X (ω)⇒

H(ω) =Y (ω)

X (ω)where X (ω) = F {x(t)} ,Y (ω) = F {y(t)}

Important Note: When the Input corresponds to an impulse i.e.

x(t) = δ(t) , the Fourier transform is F {x(t)} = 1⇒ H(ω) = Y (ω)


Modal Testing

For a structural system what is usually measured as input is the force(F ) inputted to the system. However the measured output can bedisplacement (U), velocity (U) or acceleration (U).

Therefore, based on the available measurements we distinguishbetween the following cases:

H(ω) =U(ω)

F (ω)Receptance FRF

H(ω) =U(ω)

F (ω)Mobility FRF

H(ω) =U(ω)

F (ω)Accelerance FRF

Also, when the response and excitation coordinates coincide (i = j),the FRF component Hij is referred to as a point FRF. For i 6= j wehave a transfer FRF.


Modal Testing

As it turns out, using fundamental properties from signal processing,the previous formulation for the estimation of the FRF is equivalentto the following ratio:

H(ω) =Syx(ω)

Sxx(ω)where

Sxx stands for Power Spectral Density andSyx stands for Cross Spectral Density

This calculation proves to yield more smoothed results in thepresence of noise and is therefore preferable.

In MATLAB the FRF is estimated via the buil-in tfestimate function.


Modal Testing

Definitions -Power Spectral Density

The PSD is the FT of autocorrelation function (ACF) of a signal. Itrefers to the amount of power per unit (density) of frequency(spectral) as a function of the frequency.

Sxx(ω) =

∫ ∞−∞

Rxx(t)e−iωtdt = F (Rxx)

This is a positive, real valued function of frequency, often called thespectrum of a signal.

Note: For the PSD to exist, the Random Process (signal) has to beat least Wide Sense Stationary (WSS).


The PSD

PSD Properties

1 PSD of a real valued process is symmetric S(−x) = S(x)

2 Continuous and differentiable on [−π, π]

3 Has a derivative equal to 0 at x = 0

4 Describes the mean signal power:

Var(Xt) = Rxx(0) =1

π

∫ π0 S(ω)dω (df = dω

2π )


The PSD

Examples:Consider the following random signals with different bandwidths

(source:http://cnx.org)


The PSD

ACF

The width of the ACF tells us howslowly a signal is fluctuating orhow band-limited it is.

In the upper plot, the ACF decaysquickly to zero as t increases,whereas, for a slowly fluctuatingsignal (lower plot) the ACF decaysmuch more slowly.


The PSD

PSD

Here you see the correspondingPSDs

In the upper plot energy isscattered on a broader band offrequencies

In the lower plot, it isconcentrated on a narrow band


The PSD

Important FT and PSD Properties:

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

Normalized Frequency (×π rad/sample)

Pow

er/fr

eque

ncy

(dB

/rad/

sam

ple) Welch Power Spectral Density Estimate

0 2000 4000 6000 8000 10000-4

-2

0

2

4x 10

5

time

WN

MATAB code % Generate 10000 point White Noise WN=wgn(1,10000, 100); plot(WN) xlabel('time') ylabel('WN') figure % Generate the Spectrum pwelch(WN) ylim([0 110])

PSD for White Noise signal

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

time

Impu

lse

0 20 40 60 80 1000

0.5

1

1.5

2

point

FT a

mpl

itude

MATAB code % Generate Impulse imp=zeros(1,100); imp(1)=1; hold on plot(imp) xlabel('time') ylabel('Impulse') % Generate the Fourier Transform figure plot(abs(fft(imp))) xlabel('point') ylabel('FT amplitude')

FT for Impulse


The PSD

Important FT and PSD Properties:

MATAB code % Generate Sinusoid, freq=8*pi rad/s -> 8*pi/(2*pi)=4Hz fs=100; %sampling freq. t=[0:1/fs:10]; s=sin(8*pi*t); plot(t,s) xlabel('time t') ylabel('sin(t)') figure % Generate the Spectrum pwelch(s,[],[],1024,fs)

PSD for Sinusoid

MATAB code % Generate Sinusoid, freq=8*pi rad/s -> 8*pi/(2*pi)=4Hz fs=100; %sampling freq. t=[0:1/fs:10]; s=sin(8*pi*t); plot(t,s) xlabel('time t') ylabel('sin(t)') figure

FT for Sinusoid

0 2 4 6 8 10-1

-0.5

0

0.5

1

time t

sin(

t)

0 10 20 30 40 50-80

-60

-40

-20

0X: 4.004Y: -0.9063

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB

/Hz)

Welch Power Spectral Density Estimate

0 2 4 6 8 10-1

-0.5

0

0.5

1

time tsi

n(t)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Hz

FT a

mpl

itude

%Plot the FFT of the signal L=length(s); NFFT= 2^nextpow2(L); % Next power of 2 from length of x; %FFT points need to be larger than the length of the signal Y = fft(s,NFFT)/L; f = fs/2*linspace(0,1,NFFT/2+1); %Horizontal axis frequency equivalent in Hz figure(2) plot(f,2*abs(Y(1:NFFT/2+1)));hold on %Plot only half xlabel('Hz') ylabel('FT amplitude')


Modal ID Methods

Peak Picking Method

At the vicinity of a resonance, the FRF is dominated by thecontribution of that vibration mode and the contributions of othervibration modes are negligible. If this assumption holds, then theFRF from an MDoF system or a real structure can be treated as theFRF from an SDoF system momentarily. The SDoF mathematicalmodel can then be used with curve fitting in order to derive themodal parameters.


Modal ID Methods

Peak Picking method - Half Power method (Lecture 3)

ω

2R 1R

2Q

( )11H iω

1ω 2ω nω

Q

( )11 : Point FRFH iω

ω

Identified Quantities (per peak)

Natural Frequency ωn = ωpeak & Damping ζn =ω2 − ω1

2ωn

Modal constant (related to the mode shape) can be proven to beequal to: An = 2Qζnω

2n


Modal ID Methods

Peak Picking method - Half Power method

The latter is derived from the SDoF system TF, (see Lecture 3) written formode n:

H(ω) =An

ω2n − ω2 + 2iζnωωn

hence, for ω = ωn

Q = |H(ωn)| =An

2ζnω2n

⇒ An = 2Qζnω2n


Peak Picking method

Mode Shape Extraction

As already stated, the modal constant components are (per modek): kAij = qkφijφjk . Therefore when looking at the FRF pointcomponent Hii , for excitation at i and response at i, we have that foreach mode k = 1, · · · ,N:

kAii = qkφikφik ⇒ φik is defined

(qk is a scaling factor and does not affect the estimate of the mode)Then the j-th component of mode k can be obtained either from:

the point FRF: Hjj ⇒ φjk =√

kAjj or

the transfer FRF: Hij ⇒ φjk =kAij

φik


Peak Picking method

Disadvantages

This method relies on the peak FRF value, which is very difficult tomeasure accurately

Therefore, it is not capable of producing accurate modal data

Damping is estimated from half power points only. No other FRF datapoints are used. The half power points have to be interpolated, as itis unlikely that they are two of the measured data points

Requires knowledge (recorded data) of the input signal (excitation)

Cannot handle noise efficiently

The assumption of SDoF behavior in the vicinity of resonance is notaccurate, in practice the nearby modes will most likely also contribute.


The FDD method

The Frequency Domain Decomposition Method (FDD)

This method which is very similar to the peak picking method is used forthe identification of the modal characteristics of the structure, given onlyambient measurements.

Assume the standard input X (t) - output Y (t) relationship, based on theFourier Transform:

Y (ω) = H(ω)X (ω)

When assuming wide sense stationary processes one can equivalently writethe following using PSDs:

Gyy (ω) = H∗(ω)Gxx(ω)HT (ω) (1)

where Gxx is the (r × r) power spectral density (PSD) matrix of the input,

r is the number of inputs, Gyy is the (m × m) PSD matrix of the

responses, m is the number of responses, H(ω) is the (m × r) frequency

response function (FRF) matrix and the overbar and superscript T denote

the complex conjugate and transpose, respectively.Institute of Structural Engineering Identification Methods for Structural Systems 27

The FDD method

The method is applicable for cases where the Power Spectral Densityof the input is constant, i.e. Gxx(ω) = C . This applies for the caseof broadband random input excitations like white noise (such asambient excitations). In this way FDD is developed for output onlymodal identification where only measurements of the response areavailable.

Proof

Remember that the PSD Gxx(ω) is the FT of the autocorrelationfunction Rxx(t). The autocorrelation function essentially indicatesthe correlation between values of the process at different points intime, as a function of the two times t1, t2 or of the time difference(lag - τ).


The FDD method

By definition:

A continuous time random process w(t) where t ∈ < is a whitenoise process if and only if its mean function and autocorrelationfunction satisfy the following:

µw (t) = E {w (t)} = 0Rww (t1, t2) = E {w (t1)w (t2)} = Cδ (t1 − t2)

i.e. it is a zero mean process for all time and has infinite power atzero time shift since its autocorrelation function is the Dirac deltafunction. The above autocorrelation function implies the followingpower spectral density, since the FT of Dirac Delta is unity:

Gww (iω) = C


The FDD method

As already mentioned the FRF can be written in partial fraction expansion.(i.e poles/residues form)

H(iω) =N∑

m=1

Am

iω − λm+

A∗m

iω − λ∗mwhere Am = Φmγ

Tm

Φm : modal vector

γm : modal participation vector

Suppose the input y(t) is white noise, then the PSD is a constant matrix⇒ Gxx(iω) = C . Hence,

(??)⇒ Gyy (iw) =N∑

m=1

N∑k=1

[Am

iω − λm+

A∗m

iω − λ∗m

]C

[Ak

iω − λk+

A∗k

iω − λ∗k

]Hwhere H denotes the conjugate transpose (Hermitian transpose) matrix (∗T ).


The FDD method

After some math manipulations, this can be further simplified to:

Gyy (ω) =N∑

m=1

Am

iω − λm+

A∗miω − λ∗m

+Bm

−iω − λm+

B∗m−iω − λ∗m

where Am is the mth residue matrix of the output PSD and it is anN × N matrix itself:

Am = RmC (N∑

K=1

R∗TK−λm − λK

+RTK

−λm − λK) (2)

Therefore the contribution of only the mth mode in expression (2) is:

Am =RmCR

∗Tm

−2Re(λm)


The FDD method

When the damping is light, the residue becomes proportional to themode shape vector: (dominating term)

Dm ∝ AmCA∗Tm = Φmγ

TmCγmΦT

m = dmΦmΦTm

In fact, at a certain frequency ω, only a limited number of modeswill contribute significantly. Let this set be denoted by Sub(ω). This,the response spectral density can be written:

Gyy (ω) =∑

m∈Sub(ω)

dmΦmΦTm

iω + λm+

d∗mΦ∗mΦ∗Tmiω − λ∗m

Source: Brincker R., Zhang L., Palle A.,“Modal Identification from Ambient

Responses using Frequency Domain Decomposition”


Identification Algorithm FDD

Steps

1 From a set of given output measurements y(t), ∈ <m×1 obtainthe PSD Gyy (ω) ∈ <m×m. This can be done using Matlab’sfunction cpsd(x , x)

Note: cpsd stands for cross spectral density. It is in generalwritten for two separate signals x(t), y(t) and it is equal to thepower spectral density when x = y .

2 Gyy (ω) is “decomposed” by taking the Singular valueDecomposition (SVD) of the Gyy (ω) matrix


Identification Algorithm FDD

Basics: Singular value Decomposition - SVD

Suppose A is an m × n matrix whose entries ∈ C (complex) or <then:

A = UΣVH

whereU is a m ×m unitary matrix: UHU = UUH = Im

Σ is a m × n diagonal matrix with Σii ≥ 0

VH is a n × n unitary matrix

Σii are called the singular values of A

(H denotes conjugate transpose)


Identification Algorithm

Properties:

AHA = (VΣHUH)(UΣVH) = V(ΣHΣ)VH

AAH = UΣVHVΣHUH = U(ΣΣH)UH

Hence, the columns of V (right singular vectors) are eigenvectors ofAHASo we can write at each discrete frequency ωi :

Gyy (ωi ) = UiΣiUHi

where Ui = [Ui1, . . . ,Uim]

Note that here, V = U because Gyy is a normal matrix(i.e. it issquare and GH

yyGyy = GyyGHyy



3 Near a peak corresponding to the k-th mode in the spectrum,that mode (or maybe a closely spaced mode) will be dominating.Assume the case of no closely spaced modes then the firstsingular vector is an estimate of that mode shape:φ = Ui1 since only the k-th term dominates in Equation 1.

The corresponding singular value defines the PSD of thecorresponding single degree of freedom (sdof) system. As anote, this (partial) PSD is identified by isolating that peak andcomparing the mode shape estimate φ with the singular vectorsobtained for frequency lines around the peak. Based on a modalcriterion (MAC) we can assess the similarity of that vector tothe ones around the peak and set an appropriate threshold,according to which the virtual “SDoF” area is obtained.



How to properly isolate the SDoF vincinity?

Isolate a peak

Frequency (Hz)

Sing

ular

Val

ues

Using the Modal AssuranceCriterion (MAC). This essentiallycompares the relationship between 2mode shapes by comparing theirlinearity.

Assume two modal vectors φ, ψ. One expression for the MAC criterion is:

MAC =|ψTφ|2

(ψTψ)(φTφ)

for MAC = 1 ⇒ the vectors are consistent (similar)

MAC = 0 ⇒ not consistent

As long as the SVDs around that peak singular value give high MAC values

⇒ that singular vector belongs to the SDoF PSD.Institute of Structural Engineering Identification Methods for Structural Systems 37


Frequency

Sing

ular

Val

ues

Time

Acce

lera

tion Isolate a peak – Equivalent sdof PSD Equivalent sdof time history

Tn

4 After we define the limits of the sdof, we can either apply the 1/2power method to find ζ or we can take the signal back to thefrequency domain. There, we define fd (damped frequency) as 1/Td

from the zero crossings (or peak distance) and ζ from the logarithmicdecrement method.

5 In case of closely spaced modes, the first singular vector will always bea good estimate of the strongest mode.In the case though that these two closely spaced modes areorthogonal, the first two singular vectors can give unbiased estimatesof the corresponding mode shapes.



Variations of the method

Enhanced Frequency Domain Decomposition (EFDD)This method uses the transformation in time domain for thederivation of ωn, ζn

Curve Fit frequency Domain Decomposition (CFDD)This method does not go to time domain but uses a SDoFcurve fitter in the frequency domain in order to obtain, ωn, ζn


Stabilization Chart

Consideration

Selection of model order - Stabilization Chart

Estimated poles corresponding to physically relevant system modestend to appear, for all estimation orders (i.e. assumed frequencypeaks), at nearly identical locations

While the so-called mathematical poles, i.e. poles resulting from themathematical solution of the normal equations but meaningless withrespect to the physical interpretation, tend to vary in location.

The occurrence of this last class of poles is mainly due to thepresence of noise in the measurements.


Stabilization ChartStabilization Chart

or 0M ≈⋅ with ∑ =− ⋅⋅−= ioNN

k kkTkk1

1 SRSTM . The size of the square matrix M is n+1, and thus much smaller than the original normal equation (29). To remove the parameter redundancy of transfer function model (20) (and to avoid the trivial solution with all coefficient equal to zero), a constraint has to be imposed on the coefficients of the transfer functions. This can be done, for instance, by imposing that one of the coefficients is equal to a non-zero constant value. Assume, for instance, that the last coefficient of D is constrained to 1 (i.e. coefficient n+1). In that case, the Least Squares (LS) estimate of D is given by

+⋅−

=−

1

)}1,:1({)]:1,:1([ˆ

1

LS

nnnn MM (34)

Once LSˆ is known, (32) can be used to derive all LSˆ coefficients. This approach is more time

efficient that solving (29) directly (approximately 22io NN times faster). The mode shape vectors are

derived using (14) and (15).

3.1.5 Stabilization chart

In modal analysis, a stabilization chart is an important tool that is often used to assist the user in separating physical poles from mathematical ones. A stabilization chart is obtained by repeating the analysis for increasing model order n. For each model order, the poles are calculated from the estimated denominator coefficients. The stable poles (i.e. the poles with a negative real part) are then presented graphically in ascending model order in a so-called “stabilization chart” (see Figure 6). Estimated poles corresponding to physically relevant system modes tend to appear for each estimation order at nearly identical locations, while the so-called mathematical poles, i.e. poles resulting from the mathematical solution of the normal equations but meaningless with respect to the physical interpretation, tend to jump around. These mathematical poles are mainly due to the presence of noise on the measurements.

(a) (b)

Figure 6. Stabilization chart obtained with (a) a time-domain estimator (LSCE) and (b) the frequency-domain least-squares estimator.

The LSCE estimator (Least Squares Complex Exponential) is probably the most frequently used technique in industry. The LSCE estimator is a time-domain technique that makes use of impulse response functions (16) to derive the modal parameters. In Figure 6 the stabilization chart of the LSCE estimator is compared with the proposed frequency-domain least squares estimator. It turns out that in many applications, the frequency-domain estimator is able to generate quite clear stabilization compared to the LSCE approach.

Frequency vs Model Order


Frequency vs Model Order


identification methods for structural systems · 2014-11-25 · modal id methods as already...

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