Regeneration of frequency response functions from poles and zeros: a discussion with implications for cepstrum-based operational modal analysis

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1 Regeneation of fequency esponse functions fom poles and zeos: a discussion with implications fo cepstum-based opeational modal analysis Wade A. SMITH 1 ; Robet B. RANDALL 1, Univesity of New South Wales, Austalia ABSTRACT Opeational modal analysis (OMA) sees to detemine a stuctue s dynamic chaacteistics fom esponse-only measuements, which compise both excitation and tansmission path effects. The cepstum has been used successfully in a numbe of applications to sepaate these souce and path effects, afte which the poles and zeos of the tansfe function can be obtained via a cuve-fitting pocess. The contibutions fom the individual poles and zeos can then be added (in log magnitude) to egeneate the fequency esponse function (FRF). This pape discusses a numbe of obsevations elating to this FRF egeneation pocess, as well as a numbe of boade points explaining FRFs fom a pole-zeo pespective. Among the topics coveed in the discussion ae: the equied distibution of poles and zeos fo the successful egeneation of FRFs; node points and wea modes in a pole-zeo model; the diffeences in pole-zeo distibution between eceptance, mobility and inetance FRF foms; and, how to deal with the vey low fequency egion when egeneating FRFs. It is hoped that the discussion will assist in the application of cepstum-based OMA methods and will lead to impoved undestanding of the FRF egeneation pocess and of fequency esponse functions moe boadly. Keywods: Modal analysis, Vibating sufaces and stuctues, I-INCE Classification of Subjects Numbe(s): 75.6 (See 1. INTRODUCTION Cepstum-based opeational modal analysis (OMA) was developed in the 199s (1-5), and has ecently eceived futhe attention (6-8). OMA sees to detemine the dynamic chaacteistics of a stuctue fom esponse-only measuements, which compise excitation and tansmission path effects, which must be sepaated to obtain the latte. In cetain cases, mostly involving a single input, the cepstum can be used to conduct the sepaation, povided the two effects dominate distinct egions of the cepstum. This elaxes the common assumption in OMA techniques that the input must be fequentially white. Once the sepaation is complete, the fequency esponse functions (FRFs) of the system can be egeneated using the poles and zeos (esonances and anti-esonances) identified by cuve-fitting the cepstum of the coesponding tansmission path. Though this pocess, the system is implicitly descibed using a pole-zeo model, as opposed to the moe common pole-esidue model, the two being identical only fo complete models including all poles and zeos. Yet in pactice thee will almost inevitably be some tuncation, in which only poles and zeos in a limited fequency band ae used to descibe the system as a whole. The FRFs egeneated fom such a model have coectly located poles and zeos in the consideed fequency band, but ae subject to the effects of out-of-band poles and zeos, which manifest as a distotion of the geneal slope of the egeneated FRFs. A numbe of techniques have been poposed to coect fo this magnitude distotion, as outlined in Section.4 and addessed by the authos in (7). Though this pocess of efining techniques to egeneate FRFs fom poles and zeos, the authos have identified a numbe of issues that waant 1 wade.smith@unsw.edu.au b.andall@unsw.edu.au Inte-noise 14 Page 1 of 1

2 Page of 1 Inte-noise 14 futhe discussion and explanation, and this is the basis of the pesent pape. It is hoped the discussion will assist in the application of cepstum-based OMA methods and will lead to impoved undestanding of the FRF egeneation pocess.. BACKGROUND ON CEPSTRUM-BASED OMA.1 The cepstum defined Vaious definitions fo the cepstum exist (9); hee we shall use the so-called eal cepstum xˆ and complex cepstum xˆ, defined fo some time signal x(t) as: xˆ whee X f xt c 1 1 log X f and ˆ log X f x c (1) is the (complex-valued) Fouie tansfom of x(t). Table 1 includes a numbe of tems often employed when using the cepstum. Note that the eal cepstum includes no phase infomation, so the oiginal time signal is not ecoveable afte lifteing in the cepstum domain. Howeve, the pocess fom the amplitude spectum to the eal cepstum is evesible, and so the amplitude spectum can be obtained afte cepstal editing (1), as exploited in this pape. The complex cepstum is also eal-valued, but includes phase infomation fom the spectum so is fully invetible; thus, x(t) can be ecoveed afte lifteing the cepstum. Table 1 Cepstum teminology Fequency domain Fequency Spectum Filteing Low-pass filte High-pass filte Cepstum domain Quefency Cepstum Lifteing Shot-pass lifte Long-pass lifte. Sepaation of souce and path effects While OMA possesses a numbe of advantages ove EMA (whee excitation foces ae measued), its eliance on esponse-only measuements equies diffeent techniques in ode to extact modal popeties. Response measuements, of couse, typically compise both excitation and tansmission path effects, and these need to be sepaated befoe the stuctual popeties can be detemined. Fo a linea time-invaiant (LTI) system subjected to a single input x(t), the system esponse y(t) is the convolution of the input and the impulse esponse function h(t): t d y( t) h( t) x( t) h( ) x () In the fequency domain, expessing the elationship in tems of the complex specta (o Fouie tansfoms of the time ecods), the convolution becomes multiplicative: Y( f ) H( f ) X ( f ) (3) in which H(f) is the fequency esponse function (FRF). Fo the eal cepstum the espective moduli can be used. Taing the log and the invese Fouie tansfom, we obtain the cepsta, in which souce and path effects ae additive: yˆ hˆ xˆ and yˆ hˆ xˆ c (4) Thus in cicumstances in which the input occupies defined egions of the cepstum, the souce effects can easily be sepaated (via appopiate lifteing) fom those of the tansmission path, which usually occupy a boade quefency ange. Examples of such inputs include fequentially smooth and flat foces, which ae concentated in the low quefency egion, such as a hamme blow, which does not have to be white (4). Such an input can be emoved fom the esponse cepstum via long-pass lifteing, leaving only the path cepstum, fom which the system s FRF can be ecoveed. It should be noted that this is a much less estictive equiement than the common white noise input assumption made by most othe OMA techniques. c c Page of 1 Inte-noise 14

3 Inte-noise 14 Page 3 of 1.3 Repesentation of tansmission path effects Employing a pole-zeo (as opposed to pole-esidue) model, the tansfe function (fom which the FRF is obtained by evaluating along the imaginay axis) can be epesented as: Nz * B s z n n s z 1 n N p * s p s p H( s) (5) 1 whee N p is the numbe of modal fequencies (o poles) consideed in the model, p ae the system s (complex) poles, N z is the numbe of (complex) zeos (anti-esonances), z n, consideed in the model, and B is a scaling facto. As can be seen, the poles and zeos occu in complex conjugate pais, although in some cases thee ae also eal poles and zeos, as discussed in late sections. Any eal stuctue will have an infinite numbe of modes, and so by limiting N p (and N z ) to a finite value (by esticting the fequency ange of inteest, say), the model becomes a tuncated epesentation of the complete system. The numeato and denominato in Eq. (5) can be visualised as being the poduct of distances fom a given point in the s-plane to the zeos and poles, espectively. Thus poles and zeos outside the fequency band of inteest manifest as a distotion of the magnitude of H(s) in a fequency-dependent manne; the in-band poles and zeos will be epesented accuately, but the esidues will be distoted. Tansfe functions with a geate imbalance between numbes of poles and zeos will be moe susceptible to this magnitude distotion. In the context of a chain -lie stuctue, such as a beam, the geate the distance between souce and esponse, the geate the imbalance between the numbes of poles and zeos (5). Fo a sampled sequence, the tansfe function of Eq. (5) can be expessed in the z-plane as: Nzi 1 Nzo 1 a z 1 b z 1 1 N pi 1 N po 1 c z 1 d z Bz H( z) (6) 1 1 whee a and c epesent the zeos and poles inside the unit cicle, and 1/b and 1/d the zeos and poles outside the unit cicle, espectively (whee a, b, c, d 1). Based on a moe geneal expession by Oppenheim and Schafe (11), Gao and Randall () showed that fo a stable minimum phase system (applies to many simple mechanical systems), the complex cepstum coesponding to H(z) can be expessed in tems of the zeos and poles inside the unit cicle: log B n N pi / n N / ˆ A zi n c Aa hc ( n) cos( cn) cos( an) n (7) 1 n 1 n n in which n is the quefency index and whee, fo example, a pai of poles, c, has been eplaced by n ( / n) Ac cos( cn), in which Ac c and c c epesent, fo each pole, the damping and the damped natual fequency, espectively, and liewise fo the zeos a. (Implicit in the above is that the Fouie tansfoms of Eq. (1) have been eplaced by z-tansfoms in the calculation of the cepstum.) Thus, accoding to Eq. (7), by cuve-fitting a lifteed fom of the esponse cepstum (to emove souce effects), one can obtain the tansfe function s poles and zeos. Fom these poles and zeos, the FRF can be egeneated to within a scaling facto, since B is not ecoveed fom the cuve-fitting pocess. Howeve, the obtained FRF will still be subject to the peviously-mentioned distotion associated with tuncation. In geneal, theefoe, to obtain the system s tue FRFs, the FRF egeneation pocess needs to account fo both oveall and elative scaling factos..4 Magnitude equalisation of egeneated FRFs A numbe of techniques have been poposed to geneate a magnitude equalisation cuve to coect fo the effects of tuncation, including the phantom zeo appoach of Randall et al. (5). Hanson et al. (1, 13) applied cepstum OMA to cetain multiple-input-multiple-output (MIMO) applications. Thei technique to detemine the magnitude equalisation cuve was to compae the OMA-egeneated FRF with one obtained fom an updated finite element model. The poles and zeos found via the OMA Inte-noise 14 Page 3 of 1

4 Page 4 of 1 Inte-noise 14 pocess wee used to update the FE model, fom which an FRF was detemined and compaed with the OMA-egeneated FRF. The (log magnitude) diffeence between the two FRFs was then used to develop the equalisation cuve, which was in tun used to coect the distoted slope of the OMA-egeneated FRF. One difficulty that aises in basing the magnitude equalisation cuve on the diffeence between two FRFs is that any misalignment of the poles and zeos in the FRFs esults in a seies of pea/notches that need to be smoothed o filteed out. Hanson et al. (1, 13) achieved this by tansfoming the diffeence cuve into the cepstum domain, shot-pass lifteed the esulting cepstum since the pea/notches would typically occupy the highe quefency egion and then tansfoming bac to the fequency domain. The esulting equalisation cuve, howeve, had a noticeable ipple thoughout the fequency ange, pesumably caused by end effects duing the Fouie tansfom(s). The pesent authos poposed a simple altenative appoach to obtain a smoothed equalisation cuve, based on polynomial cuve fitting of the (log magnitude) diffeence between the egeneated and efeence FRFs (7). The efeence FRF could be found by a finite element model, as used by Hanson et al., o simply the FRF measued though conventional expeimental modal analysis (EMA) techniques, but not unde opeational conditions. The method has a numbe of advantages ove the afoementioned phantom zeos and cepstal filteing appoaches. When executed with a basic constained optimisation algoithm, fo example, it allows the use to specify a paticula point, o points, though which the scaled FRF cuve must pass. This would be especially useful fo the coect scaling of the vey low fequency (VLF) egion of the FRF, which was not executed well with the lifteing appoach (7). This is paticulaly impotant because the behaviou of stuctues in the VLF egion is usually well undestood theoetically but is often mased in EMA-based measuements. Fig. 1 shows the poposed OMA pocess summay as outlined by the authos in (7). 3. INITIAL EXPERIMENTAL RESULTS Figue 1 Summay of poposed OMA pocess (7) 3.1 Expeimental setup The technique outlined in the pevious sections was applied to a steel beam test ig, as outlined in (13) and shown in Fig.. The beam ( x 5 x 1 mm) was suppoted on soft spings to epesent fee-fee bounday conditions, with the highest igid-body mode at 5 Hz well sepaated fom the fist bending mode of 95 Hz. The beam was excited with a fequentially smooth input though a shae mounted at one end of the Page 4 of 1 Inte-noise 14

5 Inte-noise 14 Page 5 of 1 beam (point 1 in Fig. ) and on its longitudinal centeline. This ensued fo simplicity that only the bending modes wee excited. The excitation was bust andom noise with a cycle of 5ms on / 5ms off. The noise, with fequency content anging fom 48 Hz, was subjected to coloued filteing to appoximate pin noise. This was done to intoduce distotion of the esponse specta an intentional violation of the typical OMA white noise assumption while maintaining a fequentially smooth input, to allow fo cepstum-based souce-path sepaation. Such distotion means that both the oveall and elative scaling of each mode is lost in the esponse specta. The acceleation esponse was measued at 11 equi-spaced points along the longitudinal centeline of the beam, as shown in Fig.. Each esponse was measued fo a duation of ten minutes and was sampled at 496 Hz. The long measuement ecod allowed fo sufficient aveaging to obtain fo each esponse a smooth spectal density. The applied foce was also measued to allow fo the detemination of efeence FRFs though conventional EMA techniques. Point 11 Point 1 Figue Repesentation of steel beam test setup (adapted fom (1)) 3. Results and discussion Shown in Fig. 3 (top) ae the fequency esponse functions (expessed as acceleance) egeneated using the cepstum OMA appoach befoe caying out any scaling. These wee fist epoted by the authos in (7). Included in the plots fo compaison ae the FRFs obtained though conventional EMA techniques. The FRFs elate to points 1 (diving point) and 11, which wee chosen to epesent the two extemes, one having (nealy) equally balanced poles and zeos, and the othe having only poles. The effect of this (im)balance between numbes of poles and zeos is clealy seen in the geneal slope of the egeneated FRFs as compaed with thei measued countepats, with the egeneated FRF of point 11 exhibiting significant distotion (up to 4 db diffeence fom the EMA value). Note that the poles and zeos above about 16 Hz ae not epesented in the egeneated FRFs because they wee filteed out pio to conducting the cepstum cuve-fitting (based on Eq. (7)). Also filteed fom the esponse specta pio to the cepstum cuve-fitting wee the low-fequency igid-body modes of the beam oscillating on the soft sping suppots, which wee not of inteest in this study and could have distoted the esponse cepsta. Shown in Fig. 3 (bottom) ae the scaled vesions of the egeneated FRFs, again plotted alongside the EMA esults. The egeneated FRFs have been scaled by the polynomial-based equalisation cuve, as outlined peviously. It can be seen that now the egeneated and measued FRFs ae almost indistinguishable fom one anothe. Fig. 4 shows the equalisation cuves used to scale the egeneated FRFs fom Fig. 3. Also plotted ae the log magnitude diffeence cuves between the measued and egeneated FRFs, in which the seies of pea notches is clealy seen. To obtain the equalisation cuves, polynomials wee fitted to the diffeence cuves with the only constaint being that the cuves pass though the theoetical zeo-fequency acceleance value (shown as a small dot in Fig. 4). This pocess was implemented as a constained linea least-squaes optimisation poblem using MATLAB s in-built lsqlin function. Although this equalisation pocess ultimately poduced vey good esults, the authos wee concened about the FRF behaviou in the VLF egion. In paticula, it was thought that outside the fequency ange containing poles and zeos, the equalisation cuve should be monotonic, as fom obsevation it seems that the effects of (high fequency) out-of-band modes have a monotonic natue. Yet the equalisation cuve fo the tansfe point in Fig. 4 is fa fom monotonic, equiing a tenth ode polynomial to popely epesent the abupt change in cuvatue in the VLF egion. This pompted an investigation of FRF behaviou in this fequency ange. Inte-noise 14 Page 5 of 1

6 Magnitude (db) Magnitude (db) Log FRF (db) Log FRF (db) Log FRF (db) Log FRF (db) Page 6 of 1 Inte-noise Figue 3 Dive point (left) and tansfe (ight) FRF compaison. Top: effect of tuncation: measued ( ), egeneated fom OMA without scaling ( ). Bottom: effect of polynomial-based equalisation cuve: measued ( ), egeneated fom OMA and scaled ( ) (7) Figue 4 Dive point (left) and tansfe (ight) equalisation cuves obtained fom polynomial cuve-fitting, as used in Figue 3: equalisation cuves ( ), aw diffeence cuves between measued and OMA-egeneated FRFs showing seies of pea/notches ( ) (7) 4. INVESTIGATING FRF BEHAVIOUR IN THE VERY LOW FREQUENCY REGION 4.1 Fixed bounday system To investigate the behavio of the egeneated FRFs in the VLF egion, a simple undamped DOF model was established, as shown in Fig. 5. Figue 5 Undamped DOF model Page 6 of 1 Inte-noise 14

7 Inte-noise 14 Page 7 of 1 Assuming steady state conditions, we can expess the output of the system as a function of the excitation foce and FRF tems: X( ) H ( ) F( ) (8) in which X X T and F T easily obtained as: X 1, F F 1,, and H is the eceptance FRF matix, which can be 1 m H ( ) 4 (9) a a1 a m1 tot in which a m1m, a1 m1 m1 m, a 1 and tot 1. The oots of the denominato and numeato in Eq. (9) epesent, fo the given FRF, the poles and zeos, espectively. This equation illustates o confoms to a numbe of well-nown points fo the FRFs of chain-lie stuctues: that thee geneally exists (almost) as many zeos as poles in diving point FRFs (H 11 and H ) but as the distance between the input-output points inceases the numbe of zeos deceases, until at the exteme tansfe case thee ae no zeos (H 1 and H 1 ). This can be physically undestood as exteme tansfe points altenating in elative phase elationships in successive esonances, while diving points maintain the same phase elationships in successive esonances and thus equie a zeo between each pai of poles to evese the phase jump fom the last esonance. Anothe featue of Eq. (9) is that in geneal thee will be no DC (zeo fequency) poles o zeos, but non-dc poles and zeos will occu symmetically about the zeo-fequency axis. (In the geneal case involving damping, it is well-nown that the poles and zeos will eithe be puely eal o appea in complex conjugate pais; this is assumed but not poven hee.) This fom of the FRF epesents the eceptance case, but mobility ( X / F) and especially acceleance ( X / F) ae often used, in which case we need to multiply H by j and (j), espectively, to give the coect FRF fom: H ( ) jh ( ) and H ( ) H ( ) (1) m In binging exta tems into the numeato, DC zeos ae ceated one in the mobility case and two in the acceleance case. 4. Fee bounday system Now let us tae the fee suppot case, which is the one most often used in laboatoy modal testing. Setting 1 = in Fig. 5, the eceptance FRFs matix becomes: in which a m a 1 m a a m H ( ) (11) m and the constant tem (a ) in the denominato of Eq. (9) vanishes to ceate two DC poles, which is not supising since with any fee bounday stuctue we should expect igid body modes. Following the same convesion to mobility and acceleance as in Eq. (1) again yields one DC zeo in the mobility case and two in the acceleance case. 4.3 DC poles and zeos summaised These DC pole-zeo chaacteistics ae summaized in Table. It can be seen that the only cases fo which the DC poles and zeos balance ae the fixed bounday eceptance FRF and the fee bounday acceleance FRF. Table DC pole-zeo chaacteistics FRF fom Fixed bounday Fee bounday Receptance no pole, no zeo two poles, no zeo Mobility no pole, one zeo two poles, one zeo Acceleance no pole, two zeos two poles, two zeos Inte-noise 14 Page 7 of 1

8 Page 8 of 1 Inte-noise 14 Now we can elate these DC pole-zeo chaacteistics bac to the well-nown classical (log-log) FRF foms, as given fo example in (14, 15) and shown in Fig. 6 fo ou DOF model. It can be seen that the FRF is asymptotic to lines of constant mass o stiffness (staight lines on a log-log scale) in the VLF and vey high fequency (VHF) egions, and in paticula we note that the asymptotic slope of a given FRF in the VLF egion is detemined by the balance of DC poles and zeos, taing an intege value (on a log-log scale) fom in the fee bounday eceptance case (two poles, no zeo) to + in the fixed bounday acceleance case (no pole, two zeos). Pehaps a moe familia epesentation of these gadients is in tems of decibels (shown in the figues), whee a slope of 1 is db/decade unsupisingly, the oll-off gadient fo a fist ode filte. Figue 6 Dive point fequency esponse functions fo fixed (left) and fee (ight) bounday conditions: eceptance (top), mobility (middle) and acceleance (bottom); coesponding mass and stiffness lines epesenting asymptotic VLF ( ) and VHF ( ) chaacteistics Yet despite this explanation fo the behaviou of FRFs in the VLF egion, it does not explain the VLF zeo-gadient chaacteistics of the balanced DC pole-zeo cases (fixed bounday eceptance and fee bounday acceleance), which of couse is what pompted the initial investigation. 4.4 New esults including negative fequency poles and zeos It has since been ealised that to achieve these chaacteistics with a pole-zeo model, the negative fequency poles and zeos (o, moe geneally, the complex conjugate pais) must be included in the egeneation pocess. To neglect them is in fact to tuncate the model unnecessaily, and although ealie papes (, 1) had focused pecisely on this point in egad to high fequency modes, the same tuncation effect fom the negative fequency egion had not been appeciated. Indeed, this was so even in the pesent authos ecent wo (7), as discussed peviously and shown in Fig. 3, whee the (unscaled) egeneated tansfe FRF (top-ight) does not follow the mass line in the VLF egion, and consequently the coesponding equalisation cuve (Fig. 4) equies an abupt change in cuvatue to achieve a sufficient coection. In the context of a polynomial-based equalisation cuve (7), this means a much highe ode polynomial would be used than what was tuly equied. Page 8 of 1 Inte-noise 14

9 Log FRF (db) Magnitude (db) Inte-noise 14 Page 9 of 1 Fig. 7 illustates this point by using the same data as in Fig. 3, but now including the egeneated FRF (tansfe case only) using both positive and negative fequency poles and zeos. It is seen hee that the new FRF confoms to the expected VLF behaviou, flattening out to the mass line nea zeo fequency. Meanwhile, the coesponding FRF diffeence cuve on which the equalisation cuve is based now shows the basic monotonic tend that had initially been expected Figue 7 Tansfe FRFs (left) and coesponding aw diffeence cuves between measued and OMA-egeneated FRFs (ight): measued ( ), egeneated fom OMA without negative fequency poles and zeos ( ), egeneated fom OMA with negative fequency poles and zeos ( ) 5. IMPLICATIONS FOR REGENERATING FRFS FROM POLES AND ZEROS It is clea fom the peceding section that a cetain distibution of poles and zeos is equied to give the best egeneation of FRFs. That section concentated on FRF chaacteistics in the VLF egion, whee DC and negative fequency poles and zeos wee shown to dictate VLF behaviou. But it is pehaps woth mentioning a couple of futhe points that will hopefully guide the analyst in the FRF egeneation pocess. A ey point hee is that all FRF infomation should be contained in the poles and zeos, except fo an oveall scaling facto and the effects of out-of-band modes, and so any FRF phenomenon should be descibable in tems of its poles and zeos. 5.1 Wea modes and node points One phenomenon that is pehaps not so clea fom a pole-zeo pespective is that of a wea mode. Unlie in a pole-esidue model, whee modal contibutions ae scaled explicitly, a pole-zeo model elies on the scaling effect of neighbouing poles and zeos, the individual contibutions of which ae simply added (in log magnitude) to give the complete FRF. Thus it becomes clea that to epesent a wea mode with a pole-zeo model, the placement of a zeo in close poximity to the pole would geneally be equied (this may not be so if the pole is in a egion of elative modal spasity). Following on fom wea modes, we must also be able to descibe node points with a pole-zeo model. Given that poles ae global system popeties, they do not simply disappea in the event of node points. Thus the only way to negate the effects of a pole is though cancellation with a zeo, i.e. a zeo must occu at exactly the same fequency as the pole to be negated. This was explained by Motteshead and Lallement (16), and although pole-zeo cancellation is well-nown in the contol field, its use in descibing vibation nodes does not seem to have been widely appeciated. The wea pole and node point concepts can be easily lined, at least in chain-lie stuctues, by consideing a zeo to pass by a given pole, educing the stength of the pole until the point of intesection, wheeupon a node occus. 5. Pactical implications The pactical implications of the above points ae paticulaly appaent when consideing zeos. Unlie poles, the zeos in pactical measuements ae often buied in noise and ae not always so clea. In the application of cepstum OMA, this maes it difficult to choose how many zeos to use when establishing the pole-zeo model. This issue was consideed ecently by Dacemann et al. (8), whee diffeent numbes of zeos wee used to egeneate FRFs fo stuctual health monitoing puposes. In such cases, a thoough undestanding of expected pole-zeo distibutions can help the analyst to egeneate bette FRFs. Fo example, the nowledge that in a pole-zeo model, a wea mode can only be epesented by placing a zeo in close poximity allows the analyst to achieve such a featue. Inte-noise 14 Page 9 of 1

10 Page 1 of 1 Inte-noise CONCLUSIONS This pape has discussed a numbe of issues suounding the egeneation of fequency esponse functions fom poles and zeos. This is paticulaly elevant to cepstum-based opeational modal analysis, which equies the use of a pole-zeo model to epesent the system. Emphasis is given to the behaviou of FRFs in the vey low fequency egion, and in paticula to the specific distibution of poles and zeos equied to egeneate FRFs coectly. It is pointed out that negative fequency poles and zeos should be included in the egeneation pocess, along with the coect balance of DC poles and zeos. It is hoped the discussion will assist in the application of cepstum-based OMA methods and will lead to impoved undestanding of the FRF egeneation pocess and of fequency esponse functions moe boadly. ACKNOWLEDGEMENTS The authos would lie to than David Hanson fo the use of his expeimental data. This eseach was suppoted by the Austalian Reseach Council and SpectaQuest, though Linage Poject [LP11738]. REFERENCES 1. Gao Y, Randall RB, editos. Zeos Vs Residues. 17th Intenational Semina on Modal Analysis; 199; K. U. Leuven, Belgium. Katholiee Univesiteit Leuven.. Gao Y, Randall RB. Detemination Of Fequency Response Functions Fom Response Measuements I. Extaction Of Poles And Zeos Fom Response Cepsta. Mechanical Systems and Signal Pocessing. 1996;1(3): Gao Y, Randall RB. Detemination Of Fequency Response Functions Fom Response Measuements II. Regeneation Of Fequency Response Functions Fom Poles And Zeos. Mechanical Systems and Signal Pocessing. 1996;1(3): Randall RB, Gao Y. Extaction Of Modal Paametes Fom The Response Powe Cepstum. Jounal of Sound and Vibation. 1994;176(): Randall RB, Gao Y, Sestiei A. Phantom zeos in cuve-fitted fequency esponse functions. Mechanical Systems and Signal Pocessing. 1994;8(6): Dacemann U, Smith WA, Randall RB. Damage identification based on esponse-only measuements using cepstum analysis and atificial neual netwos. Stuctual Health Monitoing. 14;13(4): Randall RB, Smith WA, Cepstum-based opeational modal analysis: egeneation of fequency esponse functions. Poceedings of 7th Austalasian Congess on Applied Mechanics (ACAM 7); Decembe; Adelaide, Austalia. 8. Dacemann U, Smith WA, Li J, Randall RB. On the use of the cepstum and atificial neual netwos to identify stuctual mass changes fom esponse-only measuements. ISMA; Septembe; Leuven, Belgium Childes DG, Sinne DP, Kemeait RC. The cepstum: A guide to pocessing. Poceedings of the IEEE. 1977;65(1): Randall RB. Vibation-based Condition Monitoing: Industial, Aeospace and Automotive Applications. West Sussex, United Kingdom: John Wiley & Sons, Ltd; Oppenheim AV, Schafe RW. Discete-Time Signal Pocessing. Thid ed. Oppenheim AV, edito. Uppe Saddle Rive, NJ: Peason Highe Education; Hanson D, Randall RB, Antoni J, Thompson DJ, Wates TP, Fod RAJ. Cyclostationaity and the cepstum fo opeational modal analysis of MIMO systems Pat I: Modal paamete identification. Mechanical Systems and Signal Pocessing. 7;1(6): Hanson D, Randall RB, Antoni J, Wates TP, Thompson DJ, Fod RAJ. Cyclostationaity and the cepstum fo opeational modal analysis of MIMO systems Pat II: Obtaining scaled mode shapes though finite element model updating. Mechanical Systems and Signal Pocessing. 7;1(6): Ewins DJ. Modal Testing: Theoy, Pactice and Application. Second ed. Baldoc, England: Reseach Studies Pess Ltd;. 15. Salte JP. Steady-State Vibation. Hampshie, UK: Kenneth Mason; p. 16. Motteshead JE, Lallement G. Vibation Nodes, and the Cancellation of Poles and Zeos by Unit-Ran Modifications to Stuctues. Jounal of Sound and Vibation. 1999;(5): Page 1 of 1 Inte-noise 14