Determining the Accuracy f Mdal Parameter Estimatin Methds by Michael Lee Ph.D., P.E. & Mar Richardsn Ph.D. Structural Measurement Systems Milpitas, CA Abstract The mst cmmn type f mdal testing system tday uses an FFT analyzer t measure a set f Frequency Respnse Functins (FRFs) frm a structure, and then uses a parameter estimatin (curve fitting) methd t determine the structure s mdal prperties frm the FRF measurements. The curve fitting methd typically fits an analytical mdel t the FRF data, (r its equivalent Impulse Respnse data) and estimates f the unnwn mdal parameters f the mdel are determined by this prcess. These parameter estimates are then assumed t be the crrect mdal parameters f the structure. In this paper, a number f standard test cases f synthesized FRFs are presented fr testing mdal parameter estimatin methds. Twelve different FRFs are presented, that are synthesized using the parameters fr three mdes. Frequency spacings between the mdes and mdal damping values are varied t mae up the different cases, which range frm light mdal cupling (r mdal density) t very heavy cupling. Randm nise is als added t the synthesized FRFs t simulate nisy measurements, giving a ttal f twenty-fur different test cases. The advantage f this apprach t curve fitter testing is, f curse, that the right answers (the mdal parameters used t synthesize the FRFs) are nwn, and can therefre be used as the basis fr determining the accuracy f the fitter. Tw different curve fitting methds, an SDOF (single mde-at-a-time) and an MDOF (multiple mdes-at-a-time) ratinal fractin plynmial fitter, were tried n the test case FRFs, and the results are presented. In publishing these standard test cases the authrs hpe t encurage the adptin f a suite f published test cases by the mdal testing cmmunity which culd then be used t qualify the accuracy f cmmercially available mdal testing sftware. Intrductin Mdal parameters are defined as the eigenvalues and eigenvectrs f the linear dynamic mdel fr a vibratry structure. This linear mdel can be written in terms f FRFs as: { ( ω) } [ H ( ω) ]{ F( ω) } X = (1) Page 1 where: { X (ω)} = n-vectr f Furier transfrmed displacement respnses { (ω)} F = n-vectr f Furier transfrmed frce inputs matrix f FRFs [ H (ω)] = ( n by n) n = number f test degrees-f-freedm (DOFs) n the structure ω = the frequency variable The FRF matrix can be written in terms f mdal parameters as: mdes = 1 [ H ( ω )] = [ R ] 2 j( jω p ) [ R ] 2 j( jω p ) where: [ ] R = ( n by n) p () σ + matrix f residues fr mde = jω = cmplex ple lcatin fr mde () σ = mdal damping fr mde () ω = mdal frequency fr mde () mdes = the number f mdes in the mdel - dentes the cmplex cnjugate Curve fitting, then amunts t matching the analytical expressin (2), r an abbreviated, r equivalent frm f (2), t experimental FRF data ver a chsen frequency range. During the prcess, sme, r all f the mdal parameters in the mdel are determined. It is straightfrward t shw that the mde shape can be btained frm a rw r clumn f the residue matrix [ R ] fr each mde (), since the residues are related t the mde shape by the frmula: [ ] A { u }{ u } t (2) R = =1,, mdes (3) where: { u } = the mde shape fr mde (), an n-vectr A = a scaling cnstant fr mde () t - dentes the transpse f the mde shape
Therefre, at least ne rw r clumn f FRF measurements are typically made, (frm the matrix f pssible measurements), and these measurements are curve fit t btain the mdal ple lcatins (frequency and damping), and a rw r clumn f mdal residues fr each mde in the mdel. Each mde is represented in an FRF by tw cmplex parameters, a cmplex ple lcatin and a cmplex residue, r a ttal f fur numbers. Types f Curve Fitters During the past 20 years, numerus different curve fitting algrithms have been develped fr fitting FRFs. They can all be gruped int fur classes: Lcal SDOF Lcal MDOF Glbal PlyReference Lcal SDOF Fitters: These curve fitters perate n ne measurement at a time, and estimate the parameters f ne mde at a time. Sme curve fitters nly estimate ne f the fur unnwns per mde. Fr instance, mdal frequency can be apprximated by simply using the frequency f a resnance pea, if ne exists in the FRF data. Lcal SDOF fitters will usually give satisfactry results n data that cntains lightly cupled mdes, i.e. lw mdal density. This is illustrated in the test cases later in this paper. Lcal MDOF Fitters: These fitters als perate n ne measurement at a time, but they can simultaneusly estimate the parameters f multiple mdes at a time. If a set f FRFs cntains mdes which are heavily cupled (resulting frm the cmbined effect f heavy damping and small mdal frequency separatin), then an MDOF fitter is usually required t adequately identify the mdal parameters. These fitters typically apply expressin (2) t the data in a least squared errr sense. That is, a set f parameters fr tw r mre mdes is fund which minimizes the squared difference between the FRF data and the mdel, with mdes > 1. Glbal Fitters: Expressin (2) maes it clear that all f the FRFs f a structure cntain the same denminatr, hence the same mdal ple lcatins. Only the numeratrs, r residues, are different frm measurement t measurement. Glbal fitters tae advantage f this fact and use all, r a large number f, the FRFs t estimate the ples first, and then estimate the residues during a secnd pass thrugh the data. This prcess yields ne glbal estimate f frequency and damping fr each mde, and usually prvides better mde shape estimates, especially near ndal pints where a mde s residues are small and nt well defined. PlyReference Fitters: This class f fitters extends the idea f a glbal fitter t include multiple references, r multiple rws r clumns, f the FRF matrix. Equatin (3) shws that every rw r clumn f the residue matrix cntains the mde shape f each mde. PlyReference fitters tae advantage f n 2 Page 2 this fact and btain additinal estimates f the mde shape by curve fitting multiple rws r clumns f data frm the FRF matrix. These multiple estimates are then cmbined in a manner which favrs the references where each mde is mre strngly represented, (i.e. its mdal participatin is greater), t yield a better estimate f each mde shape. Repeated rts, (i.e. tw r mre mdes at apprximately the same frequency but with different mde shapes), can als be fund frm multiple rws r clumns f FRF data. A single rw r clumn is nt sufficient fr this. The test cases presented here are nly useful fr testing the accuracy f Lcal SDOF and Lcal MDOF curve fitters. Additinal cases are needed t test Glbal and PlyReference curve fitters. Surces f Errr When any ind f parameter estimatin prcedure is applied t a set f experimentally determined data, a number f errrs can ccur. In particular, when curve fitting a set f FRF measurements, the fllwing prblems must be dealt with: Insufficient frequency reslutin Measurement distrtin Measurement nise Determining the mdel size, r number f mdes Insufficient frequency reslutin: This may r may nt be a prblem depending upn the type f curve fitter used. Fr instance, an SDOF circle fitter [1] estimates the residue f a single mde by fitting the equatin f a circle t FRF data in Nyquist (real versus imaginary) frmat. T use this methd, sufficient frequency reslutin is required in the vicinity f the FRF resnance peas t apprximate circles. In general, SDOF curve fitters are applied t FRF data in the vicinity f each f the resnance peas. Cnsequently, there must he a sufficient number f data pints in the vicinity f each resnance pea t btain accurate results. Hw many is enugh? The answer depends n the type f curve fitter used, but a general rule f thumb is that five t ten data pints between the half pwer pints (71% f the FRF magnitude at the pea), is sufficient. MDOF fitters are less sensitive t frequency reslutin since they apply the wavefrm generated by expressin (2) t the FRF data ver a range f frequencies. Therefre, they are mre sensitive t the shape f the FRF data, and can estimate the mdal parameters with far mre accuracy than the FRF frequency reslutin itself, prvided that the shape f the data clsely matches the shape f expressin (2). Measurement distrtin: Frm a curve fitting standpint, the mst detrimental cntaminant f FRF measurements is distrtin. Distrtin is caused either by nn-linear behavir f the structure, r by windwing effects in the analyzer.
Expressin (2) creates a wavefrm fr the linear dynamics f a structure. Therefre, nly the linear dynamics f the structure can be represented in the FRFs, if they are t be accurately matched t expressin (2) ver a range f frequencies. Many structures behave nn-linearly, hwever. SO sme means must be emplyed during the measurement prcess t filter ut the nn-linear part f the structural mtin and reserve nly the linear part. A cmmn methd fr ding this is t use randm excitatin and frequency dmain signal averaging t average away the nnlinear prtin f the mtin in the crss and autpwer spectrum averages that are used t frm the FRFs. The ther cmmn cause f distrtin in FRF measurements is due t truncatin f the time dmain signals by the finite sampling time perid f the FRF analyzer. This windwing effect is called leaage. Leaage distrts the FRF wavefrm, especially in the vicinity f the resnance peas, where the data is mst critical fr curve fitting. Leaage can be eliminated by using peridic signals, r it can be minimized by using specially shaped time dmain windws. In the test cases presented here, n attempt is made t simulate distrtin. Nevertheless, t successfully apply curve fitting, every effrt shuld be made during the measurement prcess t eliminate, r at least minimize, distrtin. Measurement nise: Numerus surces f nise can cntaminate FRF measurements, thus maing it mre difficult t estimate mdal parameters. The different types f nise and hw they are dealt with will nt be discussed here, but suffice it t say that every effrt shuld be made during the measurement prcess t reduce nise t a minimum. In the test cases presented here, Gaussian randm nise is added t the synthesized measurements t simulate measurement nise. Even thugh least squared errr curve fitters, lie the nes used here, are designed t estimate parameters in the presence f nise, it will be shwn that nise des reduce the accuracy f the parameter estimates. Mdel size, r number f mdes: The mst critical step in curve fitting is picing the mdel size, r equivalently, determining hw many mdes are represented in the FRF data. The prblems already mentined (frequency reslutin, measurement distrtin, and measurement nise), tgether with mdal density, and repeated rts (mdes at the same frequency with different mde shapes) all have a direct effect n determining the crrect mdel size. The mdel size, in turn, directly affects the accuracy f the parameter estimates btained by curve fitting. Mst cmmercially available curve fitters require that the peratr chse the mdel size. Singular value decmpsitin (SVD) and errr-based iterative methds have been develped recently which can assist the peratr in chsing the mdel size, but nise, distrtin, high mdal density, and repeated Page 3 rts can still mae if difficult t chse the crrect mdel size. In this paper, we cncentrate n just tw f the causes f errr in curve fitting, mdal cupling (r density) and nise. Curve Fitting Test Cases Twelve different FRFs were synthesized using expressin (2) and parameters fr three mdes. The residues f the mdes remained fixed at the fllwing values: Mde...residue... N. Magnitude Phase 1 100 0 2 100 180 3 100 0 Mdal damping was made the same fr each mde in each case, but varied frm case t case. case 1: frequencies = 50, 100, 150 Hz damping, = 0.5 Hz case 2: frequencies = 50, 100, 150 Hz damping = 1 Hz case 3: frequencies = 50, 100, 150 Hz damping = 5 Hz case 4: frequencies = 50, 100, 150 Hz damping = 10 Hz --------------------------------------------------- case 5: frequencies = 95, 100, 105 Hz damping = 0.5 Hz case 6: frequencies = 95, 100, 105 Hz damping = 1 Hz case 7: frequencies = 95, 100, 105 Hz damping = 5 Hz case 8: frequencies = 95, 100, 105 Hz damping = 10 Hz ---------------------------------------------------- case 9: frequencies = 99, 100, 101 Hz damping = 0.5 Hz case 10: frequencies = 99, 100, 101 Hz damping = 1 Hz case 11: frequencies = 99, 100, 101 Hz damping = 5 Hz case 12: frequencies = 99, 100, 101 Hz damping = 10 Hz The FRFs are synthesized ver the frequency range (0 Hz t 200 Hz) using 3201 spectral lines. This gives a frequency reslutin f f = 0.0625 Hz. As a percentage f critical damping the mdal damping varies frm apprximately 0.3% t 20%, a realistic range f damping fr the majrity f structures. The FRFs fr test cases 1 thrugh 12 are shwn in Figure 1. Test cases 13 thrugh 24 are generated by adding 2.5% randm nise t each f the cases 1 thrugh 12. The blc f
randm nise that is added t the 12 FRFs shwn in Figure 2. The FRFs fr test cases 13 thrugh 24 are shwn in Figure 3. Curve Fitting Results SDOF Fitter: First, an SDOF ratinal fractin plynmial fitter [2] was applied t the twelve test cases, t identify the parameters f the center (100 Hz) mde. The SDOF fitter was restricted t a frequency band f data in the vicinity f the 100 Hz mde, t minimize the influence f the ther tw (higher and lwer frequency) mdes. The fllwing curve fitting bands were used fr SDOF fitting: SDOF Curve Fitting Frequency Bands cases (1 t 4) & (13 t 16): 75 Hz t 125 Hz cases (5 t 8) & (17 t 20): 97.5 Hz t 102.5 Hz cases (9 & 10) & (21 t 22): 99.5 Hz t 100 5 Hz cases (11 & 12) &. (23 t 24): 94 Hz t 106 Hz The SDOF curve fitting results are given in Figure 4 (n nise), and Figure 5 (nise added). The errrs are the magnitude f the differences between the crrect values fr each f the mdal parameters and the SDOF fitter estimates. MDOF Fitter: An MDOF ratinal fractin plynmial fitter [2] was als applied t the 24 test case FRFs, t simultaneusly estimate the parameters if all three mdes. The MDOF fitter was als restricted t frequency bands f FRF data in the vicinity f the three resnance peas, which, generally speaing, gives better results. The fllwing curve fitting bands were used fr MDOF fitting: (13 t 18) allw slightly mre errr than the nn-nisy cases (1 t 6), as expected. Examining the FRFs in Figure 1. it is clear that the mdal peas f all three mdes we clearly discernible in the first 6 cases, but nt fr the remaining cases. This indicates a rule f thumb fr applying SDOF fitters, namely, nly use an SDOF fitter where a resnance pea is clearly evident. The MDOF fitter yielded accurate results fr mre cases than the SDOF fitter, as expected. As shwn in Figures 6 & 7, the MDOF fitter yielded accurate results fr cases (1 t 10), (13 t 18), and case 21. 0ne nticeable attribute f the MDOF; fitter is that fr the nn-nisy cases (Figure 6), when the mdal density became t great, (cases 11 & 12), all fur f the mdal parameters simultaneusly incurred large errrs. The synthesized FRF test cases used here had sufficient frequency reslutin, ( f = 0.0625 Hz), and yet the MDOF fitter culd nt accurately identity the parameters fr cases 11 & 12. These tw are bth cases f very high mdal density, r they culd als be classified as repeated rt cases. Typically, a Glbal r a PlyReference fitter is needed t successfully handle such cases MDOF Curve Fitting Frequency Bands cases (1 t 4) & (13 t 16): 25 Hz t 175 Hz cases (5 t 8) & (17 t 20): 75.5 Hz t 124.5 Hz cases (9, 10 & 12) & (21, 22 & 24): 92 Hz t 108 Hz cases 11 & 23: 70.5 Hz t 129.5 Hz Even thugh the parameters f all three mdes were estimated, nly the errrs f the parameters f the center (100 Hz) mde are given in Figure 6 (n nise) and Figure 7 (nise added). Cnclusins Bth the SMS StarMdal SDOF and MDOF plynmial curve fitters were applied t 12 different synthesized FRFs. with and withut additive nise. The curve fitting estimates f the mdal parameters f these FRFs were then cmpared with the nwn answers. The magnitudes f the differences between the estimates and the crrect answers are given in Figures 4 t 7. The SDOF results in figures 4 & 5 mae it clear that the.sdof fitter btained sufficiently accurate results fr cases (1 t 6) and (13 t 18). (Thse cases fr which all fur mdal parameters had small errrs were cnsidered sufficiently accurate). The curve fitter estimates f the additive nise cases Figure 2. 2.5% Randm Nise Blc Page 4
References [1] Richardsn, M. Mdal Analysis Using Digital Test Systems, Seminar n Understanding Digital Cntrl and Analysis in Vibratin Test Systems, Shc and Vibratin Infrmatin Center Publicatin, Naval Research Labratry, Washingtn D.C., May 1975. [2] Frmenti, D & Richardsn, M. Parameter Estimatin frm Frequency Respnse Measurements Using Ratinal Fractin Plynmials, Prceedings f the 1st Internatinal Mdal Analysis Cnference, Orland, Flrida, Nv. 8-10, 1982. [3] Brwn, D. et.al. Parameter Estimatin Techniques fr Mdal Analysis, SAE Paper N 790221, 1979. [4] Vld, H. & Leuridan, J. A Generalized Frequency Dmain Matrix Estimatin Methd fr Structural Parameter Identificatin 7th Internatinal Seminar n Mdal Analysis, Leuven University, Belgium, 1982. [5] Zhang, L. et.al. A Plyreference Frequency Dmain Methd fr Mdal Parameter Identificatin ASME Design Engineering Cnference, Cincinnati, Ohi, Sept. 1985. Page 5
Figure 1. FRFs fr Test Cases 1 12 Figure 3. FRFs fr Test Cases 11 24 Page 6
Figure 4. SDOF Curve Fitting Errrs: Cases 1-12 Figure 5. SDOF Curve Fitting Errrs: Cases 13-24 Page 7
Figure 6. MDOF Curve Fitting Errrs: Cases 1-12 Figure 7. MDOF Curve Fitting Errrs: Cases 13-24 Page 8