PRINCIPAL COMPONENT ANALYSISOF THE GARTEUR SM-AG19 TESTBED DATA

Mircea Rades 1 David Ewins 2

1 Catedra Rezistenta materialelor, Universitatea Politehnica Bucuresti,Splaiul Independentei 313, 79590 Bucuresti, Romania.

2 Department of Mechanical Engineering, Imperial College of Science,Technology and Medicine, Exhibition Road, London SW7 2BX, U.K.

SUMMARY: The paper presents the principal component analysis (PCA) of the frequencyresponse functions (FRFs) measured for the GARTEUR SM-AG19 Testbed in the Centre ofVibration Engineering, at the Imperial College, London. The experimental data set consists ofFRFs for the re-tested unmodified structure, as well as for the structure modified by a massadded to the tail. It contains inconsistencies and limitations due to the single-input single-output acquisition by hammer excitation, and the non-optimal selection of responsemeasurement degrees of freedom. This prompted special analysis techniques. PCA provides areduced set of uncorrelated principal response functions (PRFs), computed as linearcombinations of the measured FRFs. The frequency dependence of the PRFs is used to definetwo new modal indicator functions able to estimate the number of active modes of vibrationfrom the restrained data set. Analysing a selected peak of each PRF allows determination ofmodal parameters by single degree of freedom identification techniques.

KEYWORDS: principal component analysis, principal response functions, componentwisemode indicator function, aggregate mode indicator function, frequency response functions,identification of modal parameters.

INTRODUCTION

Principal Component Analysis applied to measured Frequency Response Functions (FRFs)provides a simple and parsimonious description of frequency response data. Its main objectiveis to replace the measured set of FRFs by a reduced set of uncorrelated Principal ResponseFunctions (PRFs) containing (almost) as much information as there is in the original FRFs.

Algebraically, PRFs are particular linear combinations of the measured FRFs. Geometrically,the transformation from FRFs to PRFs amounts to a rotation of the coordinate axes,represented by the original FRFs, to a new coordinate system with mutually orthogonal axes.Physically, these new axes represent directions with maximum autopower.

PRINCIPAL COMPONENT ANALYSIS OF FRF’S

The Compound FRF Matrix

Suppose the measurement data set consists of N complex valued FRFs sampled at fNfrequencies. They can be arranged columnwise in a Compound FRF (CFRF) matrix,

∈A C xNNf . Each column, ia , contains the frequency-dependent elements of an individualFRF, measured at a given output/input coordinate combination. Each row contains N complexFRF values measured at the same frequency. Analysis of all FRF test data, arranged in asingle multi-frequency CFRF matrix, can smear the frequency shifts and other inaccuraciesdue to the non-simultaneous measurement of the FRFs.

Due to noise and non-linear effects, the CFRF matrix is apparently of full rank. When aneffective rank can be estimated, this matrix can be replaced by a so-called Aggregate FRF(AFRF) matrix of the same size, obtained by eliminating the linearly dependent information,i.e. the redundancy among the measured FRFs.

Principal Response Functions

The Singular Value Decomposition (SVD) of the CFRF matrix can be written

∑∑==

=σ==N

1ii

N

1i

Hiii

H AvuV UA ΣΣΣΣ (1)

where ∈U C xNNf and ∈V C NxN are the unitary matrices of the left and right singularvectors, respectively, and the superscript H denotes the conjugate transpose (Hermitian). Thesingular values iσ are arranged in non-increasing order in the real diagonal matrix ΣΣΣΣ .

The SVD decomposes the CFRF matrix into a sum of rank-one matrices Hiiii vuA σ= of the

same size as A. Each singular value is equal to the Frobenius norm (a square-root-of-sum-of-squares) of the associated iA matrix

Fii A=σ (2)and can be considered as a measure of its energy content.

Because the left and right singular vectors have unit length, the amplitude information iscontained in the singular values. The left singular vectors (LSV), iu , contain the frequencydistribution of the energy. The right singular vectors (RSV), iv , describe the spatialdistribution of the energy contained in the FRF set.

The Principal Response Functions, ip , defined as the LSVs scaled by the respective singularvalues [1], are linear combinations of the original FRFs, ja :

∑=

==σ=N

1jjjiiiii v avAup . (3)

In Eqn 3 the multiplying factors jiv are the complex valued elements of the right singular

vectors. PRFs are also the eigenvectors of the matrix HAA scaled by the square roots of therespective eigenvalues.

The matrix having the PRFs as columns isΣΣΣΣ UVAP == . (4)

If A is rank-deficient, then a reduced number of columns is retained in the matrix P.

The cross-product matrix of the uncorrelated PRFs should be diagonal, representingautopower. Indeed,

( ) 2HHHH )( ΣΣΣΣ=== VAAVAVAVPP , (5)so that the cross-power off-diagonal elements are zero. The columns of P form a set oforthogonal response functions, each one representing an amount of energy equal to the squareof the related singular value. The first PRF, corresponding to the largest singular value, is theuncorrelated response function with the largest autopower. The second PRF has the secondlargest autopower, and so on.

PRFs have peaks at the natural frequencies, as have the FRFs. The modes whose shape issimilar to the weighting RSV are enhanced, while the others are attenuated. For an adequateselection of input/output coordinate combinations, each PRF is dominated by a single mode ofvibration. Single degree of freedom (SDOF) identification techniques can be used todetermine the corresponding modal parameters [2]. For a non-optimal location of sensors andexcitation coordinates, resulting in an insufficient spatial independence of the modal vectors,and for limited spatial resolution, a PRF can have multiple peaks, especially when this isbacked by insufficient frequency resolution.

MODE INDICATOR FUNCTIONS

The Componentwise Mode Indicator Function

For each component iA of the CFRF matrix, the diagonal elements of the orthogonal

projector onto the null space of HiA exhibit minima at the natural frequencies.

The Componentwise Mode Indicator Function (CoMIF) is defined [3] by vectors of the form:( ) ( )H

iiNiiNi ff diag diag uuIAAI CoMIF −=−= + , (6)

where + denotes the pseudoinverse, and fNI is the identity matrix of order fN .

In the CoMIF plot, for each principal component, the diagonal elements of the projector ontothe plane perpendicular to the axis, having the respective LSV as the unit vector, are displayedagainst frequency. The number of curves is equal to the estimated effective rank of the CFRFmatrix, i.e. to the truncated number of its principal components. Each curve has a localminimum at a natural frequency, with the deepest trough at the natural frequency of thecorresponding dominant mode. Visual inspection of CoMIF curves reveals the number ofmodes active in a given frequency band and the dominant mode in each CoMIF curve, whichis the strongest enhanced mode in the respective PRF. The componentwise analysis allows abetter estimation of the rank of the CFRF matrix and a better understanding of thecontribution of each mode to the dynamics of the system.

The Aggregate Mode Indicator Function

For noisy data and for structures with high modal density, an overlay of the CoMIF curvesbecomes hard to interpret, so that a single-curve mode indicator function has been developed.

The Aggregate Mode Indicator Function (AMIF) has been defined [4] as:)( diag += AAAMIF . (7)

If A is rank-deficient and its effective rank is rN , then a rank-limited matrix A~ , referred toas the Aggregate FRF matrix, can be constructed retaining only the rN largest principalcomponents. The AMIF becomes:

== ∑

=

+ rN

1i

Hii diag)~~( diag uuAAAMIF . (8)

Different AMIF curves can be plotted for different values of rN , as in [5]. From Eqn 8 it isseen that the AMIF is an aggregate of vectors of the form

( )Hiiiii diag)( diag uuAAAMIF == + (9)

and the sum extends over a number of principal components equal to the estimated rank of theAFRF matrix.

The AMIF is currently implemented as the PRFMIF in the MODENT Suite of software formodal analysis by ICATS [6].

Other MIFs

The CoMIF and the AMIF are compared with two currently used mode indicator functions,the MIF and the ImMIF.

At each frequency the MIF is defined [7] as

( )

a

a aRe

1MIF N

1j

2ij

N

1jijij

i

∑

∑

=

=−= , (10)

while the somewhat complementary ImMIF is defined [8] as

( )

a

a amI

1ImMIF N

1j

2ij

N

1jijij

i

∑

∑

=

=−= . (11)

The last two locate the frequencies where the forced response is closest to the monophasecondition. The first two also locate modes with a higher degree of complexity, being based onthe information density in the FRF data set.

ANALYSIS OF THE NEW MEASUREMENT DATA

The PCA technique and the new MIFs were tested on two FRF data sets. In the following theywill be referred to as UNMOD – for the unmodified GARTEUR structure as reassembled atImperial College, and MOD1 – for the structure modified by a mass added to the tail.

Response DOFs101z 105x 105z 108z111z 112x 112z 11z12x 12z 1z 201x201y 201z 205y 206z301x 301z 302y 303x303z 5x 5z 8z

Excitation DOF: 12z

Fig. 1: The measurement locations

The experimental data-base consists of 24 complex valued FRFs, measured at 24 arbitrarilyselected locations (Fig.1) using single point hammer excitation at the right wing tip. The dataset spans a frequency range from about 0 to 100 Hz, with 0.125 Hz frequency resolution.

Measurements on the Unmodified Structure

Using the FRFs from the new unmodified data, set 1, a CFRF matrix of size 801x24 has beenconstructed. The first eleven PRFs are shown in Fig.2. An overlay of the CoMIF curves ispresented in Fig.3. As in Fig.2, the number near each local minimum indicates the index ofthe CoMIF curve (hence of the PRF) where the respective mode is dominant.

Fig. 2: First 11 PRFs for UNMOD Fig.3: Overlaid CoMIFs for UNMOD

The CoMIF plot (Fig.4) clearly reveals 10 modes of vibration between 5 and 75 Hz. Subplotscorrespond to separate CoMIFs with the index shown on the left. Each detected mode ismarked by a local minimum at the associated frequency. The deepest minimum indicates thedominant mode.

Fig. 4: CoMIF plot for UNMOD

Table 1 lists the natural frequencies and (equivalent viscous) damping ratios determined bySDOF circle fit. The index of the PRF used for modal parameter identification is given incolumn 2.

Table 1: Modal parameters for UNMOD

Mode PRFNatural

frequency,Hz

Dampingratio,

%Description

1 5 6.55 4.13 2N wing bending2 8 16.60 2.60 Fuselage rotation3 3 34.91 1.03 Antisym. wing torsion4 1 35.30 1.90 Symmetric wing torsion5 6 36.54 1.20 3N wing bending6 9 49.99 0.49 4N wing bending7 11 50.76 0.75 Inpl. wing vs. fuselage8 10 56.43 0.45 Sym. in-plane wing bending9 4 65.04 2.21 5N wing bending10 7 69.72 0.58 Tail torsion

Examples of circle fit modal analysis are illustrated in Fig.5. The almost circular shape ofPRFs in the neighbourhood of resonance indicates a good mode isolation for SDOF analysis.Modal viscous damping values are calculated as the arithmetic mean of two values, onedetermined using the two points chosen next to resonance, indicated in figures, the otherdetermined using the next close points below and above resonance.

Fig. 5: SDOF circle fit to PRFs for UNMOD

Figure 6 presents the MIF and ImMIF plots. Both fail to indicate all three close modes atabout 35 Hz. Figure 7 presents the AMIF and the complementary 1-AMIF computed for arank rN =11. They locate only two of the three modes at 35 Hz. All four MIFs from Fig.6 andFig.7 have limits, being single-curve indicators. They can be used in a first stage of theanalysis to locate not too close modes and are best suited for structures with many modeswithin the frequency band of interest. A check for close modes should be carried out in asecond stage. The CoMIF plot helps locating all active modes. Though the number of PRFsexceeds the number of active modes, there are no computational modes to be sorted out.

Fig. 6: MIF and ImMIF for UNMOD Fig.7: AMIF and 1-AMIF for UNMOD

Measurements on the Tail Modified Structure

The FRFs from the MOD1 data set have been used to build up a CFRF matrix of size 801x24.The first 14 PRFs are shown in Fig.8 and the overlaid CoMIF curves are presented in Fig.9.

Fig. 8: First 14 PRFs for MOD1 Fig.9: Overlaid CoMIFs for MOD1

The individual CoMIF curves are displayed in Fig.10 revealing eleven modes between 5 and80 Hz. The modal parameters determined by SDOF circle fit of PRFs are listed in Table 2.Column 2 shows the individual PRF used for modal identification. It can be noticed that PRFsof indices 4, 9 and 13 have not been used for modal identification, while the parameters of

Fig. 10: Individual CoMIF plots for MOD1

modes 10 and 11 were determined from PRFs 5 and 8, respectively, where dominant are themodes 1 and 9. It is believed that all these problems are due to the non-optimal test planning.

Table 2: Modal parameters for MOD1

Mode PRFNatural

frequency,Hz

Dampingratio,

%Description

1 3 6.55 2.52 2N wing bending2 11 13.95 1.85 Fuselage rotation3 2 32.40 0.97 3N wing bending4 10 35.17 0.80 Symmetric wing torsion5 1 35.54 1.58 Antisym. wing torsion6 6 38.17 0.49 Tail torsion7 14 48.78 0.44 Inpl.wing vs. fuselage8 3 49.92 1.97 4N wing bending9 7 56.46 0.25 Symmetric in-plane bending10 5 58.25 1.73 Fuselage bending11 8 78.59 0.97 5N wing bending

For comparison, the MIF and ImMIF plots are presented in Fig.11. Both fail locating mode 7at about 49 Hz and mode 4 at 35 Hz. On the contrary, mode 7 is clearly indicated in theCoMIF overlay in Fig.9 and in both the AMIF and 1-AMIF plots from Fig.12, computed foran AFRF matrix of rank 14. Mode 4 at 35 Hz can also be located in Fig.9 by a small troughoverlaid on the deeper next mode trough at 35.5 Hz.

Fig. 11: MIF and ImMIF for MOD1 Fig.12: AMIF and 1-AMIF for MOD1

CONCLUDING REMARKS

Based on the new measurements on the GARTEUR SM-AG19 Testbed, the paper shows howthe Principal Component Analysis can be used for modal parameter identification from arestricted and non-optimal data set. It is a pleading for the use of two new mode indicatorfunctions, the CoMIF and the AMIF, to determine the number of modes present in a givenfrequency range. Based on the information density of the data set, they have a different

physical background and outperform the commonly used MIFs, developed to locatefrequencies where the response is close to the monophase condition.

Basically, PCA separates the system response into incoherent components. Its aim is toreplace the measured set of FRFs by a reduced set of uncorrelated PRFs. A rank-limited set ofvirtual FRFs can be reconstructed from the truncated set of PRFs. Their analysis is notrecommended, because they represent a structure constrained by the cancellation of severalsingular values. Instead, analysis of isolated peaks in individual PRFs yields accurate resultsusing simple SDOF modal parameter identification procedures.

REFERENCES

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